Recent zbMATH articles in MSC 58https://www.zbmath.org/atom/cc/582022-05-16T20:40:13.078697ZWerkzeugBook review of: É. Ghys, A singular mathematical promenadehttps://www.zbmath.org/1483.000112022-05-16T20:40:13.078697Z"Kontorovich, Alex"https://www.zbmath.org/authors/?q=ai:kontorovich.alex-vReview of [Zbl 1472.00001].Book review of: É. Ghys, A singular mathematical promenadehttps://www.zbmath.org/1483.000132022-05-16T20:40:13.078697Z"Mégy, Damien"https://www.zbmath.org/authors/?q=ai:megy.damienReview of [Zbl 1472.00001].Book review of: É. Ghys, A singular mathematical promenadehttps://www.zbmath.org/1483.000242022-05-16T20:40:13.078697Z"Tabachnikov, Sergei"https://www.zbmath.org/authors/?q=ai:tabachnikov.serge-lReview of [Zbl 1472.00001].Smooth manifold of one-dimensional lattices and shifted latticeshttps://www.zbmath.org/1483.111362022-05-16T20:40:13.078697Z"Smirnova, Elena Nikolaevna"https://www.zbmath.org/authors/?q=ai:smirnova.elena-nikolaevna"Pikhtil'kova, Ol'ga Aleksandrovna"https://www.zbmath.org/authors/?q=ai:pikhtilkova.olga-aleksandrovna"Dobrovol'skiĭ, Nikolaĭ Nikolaevich"https://www.zbmath.org/authors/?q=ai:dobrovolskii.n-n"Rebrova, Irina Yur'evna"https://www.zbmath.org/authors/?q=ai:rebrova.irina-yurevna"Rodionov, Aleksandr Valer'evich"https://www.zbmath.org/authors/?q=ai:rodionov.aleksandr-valerevich"Dobrovol'skiĭ, Nikolaĭ Mikhaĭlovich"https://www.zbmath.org/authors/?q=ai:dobrovolskii.n-mSummary: In the previous work, the authors laid the foundations of the theory of smooth varieties of number-theoretic lattices. The simplest case of one-dimensional lattices is considered.
This article considers the case of one-dimensional shifted lattices. First of all, we consider the construction of a metric space of shifted lattices by mapping one-dimensional shifted lattices to the space of two-dimensional lattices.
In this paper, we define a homeomorphic mapping of the space of one-dimensional shifted lattices to an infinite two-dimensional cylinder. Thus, it is established that the space of one-dimensional shifted lattices \(CPR_2\) is locally a Euclidean space of dimension 2.
Since the metric on these spaces is not Euclidean, but is ``logarithmic'' , unexpected results are obtained in the one-dimensional case about derivatives of basic functions, such as the determinant of the lattice, the hyperbolic lattice parameter, the norm at least, the Zeta function and lattice hyperbolic Zeta function of lattices.
The paper considers the relationship of these functions with the issues of studying the error of approximate integration over parallelepipedal grids as the determinant of the lattice, the hyperbolic lattice parameter, the norm at least, the Zeta function and lattice hyperbolic Zeta function of lattices.
Note that the geometry of metric spaces of multidimensional lattices and shifted multidimensional lattices is much more complex than the geometry of an ordinary Euclidean space. This can be seen from the paradox of nonadditivity of the length of a segment in the space of shifted one-dimensional lattices. From the presence of this paradox, it follows that there is an open problem of describing geodesic lines in the spaces of multidimensional lattices and multidimensional shifted lattices, as well as in finding a formula for the length of the arcs of lines in these spaces. Naturally, it would be interesting not only to describe these objects, but also to obtain a number-theoretic interpretation of these concepts.
A further direction of research may be the study of the analytical continuation of the hyperbolic zeta function on the spaces of lattices and multidimensional lattices. As is known, an analytical continuation of the hyperbolic zeta function of lattices is constructed for an arbitrary Cartesian lattice. Even the question of the continuity of these analytic continuations in the left half-plane on the lattice space has not been studied. All these, in our opinion, are relevant areas for further research.Hodge theory of the Turaev cobracket and the Kashiwara-Vergne problemhttps://www.zbmath.org/1483.140152022-05-16T20:40:13.078697Z"Hain, Richard"https://www.zbmath.org/authors/?q=ai:hain.richard-mDenote the set of free homotopy classes of maps \(S^1 \to X\) in a topological space \(X\) by \(\lambda(X)\) and the free \(\mathbb{Q}\)-module it generates by \(\mathbb{Q}\lambda(X)\). When \(X\) is an oriented surface with a nowhere vanishing vector field \(\xi\), there is a map
\[
\delta_\xi : \mathbb{Q}\lambda(X) \to \mathbb{Q}\lambda(X)\otimes \mathbb{Q}\lambda(X),
\]
called the \textit{Turaev cobracket}, that gives \(\mathbb{Q}\lambda(X)\) the structure of a Lie coalgebra. The cobracket was first defined by [\textit{V. G. Turaev}, Mat. Sb., Nov. Ser. 106(148), 566--588 (1978; Zbl 0384.57004)] on \(\mathbb{Q}\lambda(M)/\mathbb{Q}\) (with no framing) and lifted to \(\mathbb{Q}\lambda(M)\) for framed surfaces in [\textit{V. G. Turaev}, Ann. Sci. Éc. Norm. Supér. (4) 24, No. 6, 635--704 (1991; Zbl 0758.57011)] and [\textit{A. Alekseev} et al., ``The Goldman-Turaev Lie bialgebra and the Kashiwara-Vergne problem in higher genera'', Preprint, \url{arXiv:1804.09566}]. The cobracket \(\delta_\xi\) and the Goldman bracket [\textit{R. Hain}, ``Hodge theory of the Goldman bracket'', Preprint, \url{arXiv:1710.06053}]
\[\{\ ,\ \} : \mathbb{Q}\lambda(X)\otimes \mathbb{Q}\lambda(X) \to \mathbb{Q}\lambda(X)
\] endow \(\mathbb{Q}\lambda(X)\) with the structure of an involutive Lie bialgebra [\textit{V. G. Turaev}, Ann. Sci. Éc. Norm. Supér. (4) 24, No. 6, 635--704 (1991; Zbl 0758.57011); \textit{M. Chas}, Topology 43, No. 3, 543--568 (2004; Zbl 1050.57014); \textit{N. Kawazumi} and \textit{Y. Kuno}, Ann. Inst. Fourier 65, No. 6, 2711--2762 (2015; Zbl 1370.57009)].
The value of the cobracket on a loop \(a \in \lambda(X)\) is obtained by representing it by an immersed circle \(\alpha : S^1 \to X\) with transverse self intersections and trivial winding number relative to \(\xi\). Each double point \(P\) of \(\alpha\) divides it into two loops based at \(P\), which we denote by \(\alpha'_P\) and \(\alpha_P''\). Let \(\epsilon_P = \pm 1\) be the intersection number of the initial arcs of \(\alpha_P'\) and \(\alpha_P''\). The cobracket of \(a\) is then defined by
\[
\delta_\xi(a) = \sum_P \epsilon_P(a'_P \otimes a''_P - a''_P \otimes a'_P),\tag{1}
\]
where \(a_P'\) and \(a_P''\) are the classes of \(\alpha_P'\) and \(\alpha_P''\), respectively.
The powers of the augmentation ideal \(I\) of \(\mathbb{Q}\pi_1(X,x)\) define the \(I\)-adic topology on it and induce a topology on \(\mathbb{Q}\lambda(X)\). \textit{N. Kawazumi} and \textit{Y. Kuno} [Ann. Inst. Fourier 65, No. 6, 2711--2762 (2015; Zbl 1370.57009)] showed that \(\delta_\xi\) is continuous in the \(I\)-adic topology and thus induces a map
\[
\delta_\xi : \mathbb{Q}\lambda(X)^\wedge \to \mathbb{Q}\lambda(X)^\wedge\widehat{\otimes}\mathbb{Q}\lambda(X)^\wedge
\]
on \(I\)-adic completions. This and the completed Goldman bracket give \(\mathbb{Q}\lambda(X)^\wedge\) the structure of an involutive completed Lie bialgebra [loc. cit.].
Now suppose that \(X\) is a smooth affine curve over \(\mathbb C\) or, equivalently, the complement of a non-empty finite set \(D\) in a compact Riemann surface \(\overline{X}\). In this case \(\mathbb Q\lambda(X)^\wedge\) has a canonical pro-mixed Hodge structure [\textit{R. M. Hain}, \(K\)-Theory 1, No. 3, 271--324 (1987; Zbl 0637.55006)]. In particular, it has a \textit{weight filtration}
\[
\cdots \subseteq W_{-2} \mathbb Q\lambda(X)^\wedge \subseteq W_{-1} \mathbb Q\lambda(X)^\wedge \subseteq W_0 \mathbb Q\lambda(X)^\wedge = \mathbb Q\lambda(X)^\wedge
\]
and its complexification \(\mathbb C\lambda(X)^\wedge\) has a \textit{Hodge filtration}
\[
\cdots \supset F^{-2} \mathbb C\lambda(X)^\wedge \supset F^{-1} \mathbb C\lambda(X)^\wedge \supset F^{0} \mathbb C\lambda(X)^\wedge \supset F^1\mathbb C\lambda(X)^\wedge = 0.
\]
The Hodge filtration depends on the algebraic structure on \(X\), while the weight filtration is topologically determined and so does not depend on the complex structure. The weight filtration on \(\mathbb Q\lambda(X)^\wedge\) is the image of the weight filtration of \(\mathbb Q\pi_1(X,x)^\wedge\), which is determined uniquely by the conditions that \(W_{-1} \mathbb Q\pi_1(X,x)^\wedge = I\), \(W_{-2} \mathbb Q\pi_1(X,x)^\wedge=\ker\{I\to H_1(\overline{X})\}\), and by the condition that \(W_{-m-2}\mathbb Q\pi_1(X,x)^\wedge\) is the ideal generated by \(W_{-1}W_{-m-1}\) and \(W_{-2}W_{-m}\). This pro-mixed Hodge structure contains subtle geometric and arithmetic information about \(X\). The first main result of the paper is that the Turaev cobracket is compatible with this structure.
Theorem 1.
If \(\xi\) is a nowhere vanishing holomorphic vector field on \(X\) that is meromorphic on \(\overline{X}\), then
\[
\delta_\xi : \mathbb Q\lambda(X)^\wedge\otimes\mathbb Q(-1) \to\mathbb Q\lambda(X)^\wedge\widehat{\otimes}\mathbb Q\lambda(X)^\wedge
\]
is a morphism of pro-mixed Hodge structures, so that \(\mathbb Q\lambda(X)^\wedge\otimes\mathbb Q(1)\) is a complete Lie coalgebra in the category of pro-mixed Hodge structures.
We call such a framing \(\xi\) an \textit{algebraic framing}. The main result of [\textit{R. Hain}, Geom. Topol. 24, No. 4, 1841--1906 (2020; Zbl 1470.14017)] asserts that
\[
\{\ ,\ \} : \mathbb Q\lambda(X)^\wedge\otimes\mathbb Q\lambda(X)^\wedge \to \mathbb Q\lambda(X)^\wedge \otimes \mathbb Q(1)
\]
is a morphism of mixed Hodge structure (MHS), so that \(\mathbb Q\lambda(X)^\wedge\otimes\mathbb Q(-1)\) is a complete Lie algebra in the category of pro-mixed Hodge structures.
Corollary 1. If \(\xi\) is a quasi-algebraic framing of \(X\), then \(\big(\mathbb Q\lambda(X)^\wedge,\{\ ,\ \},\delta_\xi\big)\) is a ``twisted'' completed Lie bialgebra in the category of pro-mixed Hodge structures.
By ``twisted'' one means that one has to twist both the bracket and cobracket by \(\mathbb Q(\pm 1)\) to make them morphisms of MHS. There is no one twist of \(\mathbb Q\lambda(X)\) that makes them simultaneously morphisms of MHS. Let \(\vec{\mathsf v}\) be a non-zero tangent vector of \(\overline{X}\) at a point of \(D\). Standard results in Hodge theory (see [\textit{R. Hain}, Geom. Topol. 24, No. 4, 1841--1906 (2020; Zbl 1470.14017)]) imply:
Corollary 2. Hodge theory determines torsors of compatible isomorphisms
\[
\big(\mathbb Q\lambda(X)^\wedge,\{\ ,\ \},\delta_\xi\big) \overset{\simeq}\longrightarrow \Big(\prod_{m\ge 0}\operatorname{Gr}^W_{-m}\mathbb Q\lambda(X)^\wedge,\operatorname{Gr}^W_\bullet\{\ ,\ \},\operatorname{Gr}^W_\bullet\delta_\xi\Big)\tag{2}
\]
of the Goldman-Turaev Lie bialgebra with the associated weight graded Lie bialgebra and of the completed Hopf algebras
\[
\mathbb Q\pi_1(X,\vec{\mathsf v})^\wedge \overset{\simeq}\longrightarrow\prod_{m\ge 0}\operatorname{Gr}^W_{-m}\mathbb Q\pi_1(X,\vec{\mathsf v})^\wedge\tag{3}
\]
under which the logarithm of the boundary circle lies in \(\operatorname{Gr}^W_{-2} \mathbb Q\pi_1(X,\vec{\mathsf v})^\wedge\). These isomorphisms are torsors under the prounipotent radical \(U^{\mathrm{MT}}_{X,\vec{\mathsf v}}\) of the Mumford-Tate group of the MHS on \(\mathbb Q\pi_1(X,\vec{\mathsf v})^\wedge\).
Let \(\overline{S}\) be a closed oriented surface of genus \(g\) and \(P=\{x_0,\dots,x_n\}\) a finite subset. Set \(S=\overline{S}-P\).
Assume that \(S\) is hyperbolic; that is, \(2g-1+n>0\). Suppose that \(\xi_o\) is a framing of \(S\). Denote the index (or local degree) of \(\xi_o\) at \(x_j\) by \(d_j\). Let \({\mathbf{d}} = (d_0,\dots,d_n) \in \mathbb Z^{n+1}\) be the vector of local degrees of \(\xi_o\). The Poincaré-Hopf Theorem implies that \(\sum d_j = 2-2g\). Also denote the category of mixed Tate motives unramified over \(\mathbb Z\) by \({\mathsf{MTM}}(\mathbb Z)\). Denote the prounipotent radical of its tannakian fundamental group \(\pi_1({\mathsf{MTM}},\omega^B)\) (with respect to the Betti realization \(\omega^B\)) by \(\mathcal K\). Denote the relative completion of the mapping class group of \((\overline{S},P,\vec{\mathsf v}_o)\) by \(\mathcal G_{g,n+u}\) and its prounipotent radical by \(\mathcal U_{g,n+u}\). (See [\textit{R. Hain}, J. Am. Math. Soc. 10, No. 3, 597--651 (1997; Zbl 0915.57001)] for definitions.) These act on \(\mathbb Q\pi_1(S,\vec{\mathsf v}_o)^\wedge\). Denote the image of \(\mathcal U_{g,n+u}\) in \(\operatorname{Aut}\mathbb Q\pi_1(S,\vec{\mathsf v}_o)^\wedge\) by \(\overline{\mathbb U}_{g,n+u}\). The vector field \(\xi_o\) determines a homomorphism \(\overline{\mathbb U}_{g,n+u} \to H_1(\overline{S})\) that depends only on the vector \({\mathbf{d}}\) of local degrees of \(\xi\). Denote its kernel by \(\overline{\mathbb U}_{g,n+u}^{\mathbf{d}}\). \textit{Y. Ihara} and \textit{H. Nakamura} [J. Reine Angew. Math. 487, 125--151 (1997; Zbl 0910.14010)] construct canonical smoothings of each maximally degenerate stable curve \(X_0\) of type \((g,n+1)\) over \(\mathbb Z[[q_1,\dots,q_N]]\) for all \(n \ge 0\), where \(N= \dim \mathbb M_{g,n+1}\). Associated to each tangent vector \(\vec{\mathsf v} = \pm \partial/\partial q_j\) of \(\overline{\mathbb M}_{g,n+1}\) at the point corresponding to \(X_0\), there is a limit pro-MHS on \(\mathbb Q\lambda(X)^\wedge\), that we denote by \(\mathbb Q\lambda(X_{\vec{\mathsf v}})^\wedge\).
Hypothesis. The limit MHS on \(\mathbb Q\lambda(X_{\vec{\mathsf v}})^\wedge\) is the Hodge realization of a pro-object of \({\mathsf{MTM}}(\mathbb Z)\). Equivalently, the Mumford-Tate group of the MHS on \(\mathbb Q\lambda(X_{\vec{\mathsf v}})^\wedge\) is isomorphic to \(\pi_1({\mathsf{MTM}},\omega^B)\).
Theorem 4. If \(2g+n>1\) (i.e., \(S\) is hyperbolic), then the group \(\widehat{\mathbb U}_{g,n+u}^{\mathbf{d}}\) does not depend on the choice of a quasi-algebraic structure \(\phi : (\overline{S},P,\vec{\mathsf v}_o,\xi_o) \to (\overline{X},D,\vec{\mathsf v},\xi)\). The group \(\overline{\mathbb U}_{g,n+u}^{\mathbf{d}}\) is normal in \(\widehat{\mathbb U}_{g,n+u}^{\mathbf{d}}\). If we assume Hypothesis 1, there is a canonical surjective group homomorphism \(\mathcal K \to \widehat{\mathbb U}_{g,n+u}^{\mathbf{d}}/\overline{\mathbb U}_{g,n+u}^{\mathbf{d}}\), where \(\mathcal K\) denotes the prounipotent radical of \(\pi_1({\mathsf{MTM}})\).
From [\textit{R. Hain}, J. Am. Math. Soc. 10, No. 3, 597--651 (1997; Zbl 0915.57001)] that the completion of \(\Gamma_{g,m+\vec{r}}\) relative to \(\rho : \Gamma_{g,m+\vec{r}} \to {\mathrm{Sp}}(H_{\mathbb Q})\) is an affine \(\mathbb Q\)-group \(\mathcal G_{g,m+\vec{r}}\) that is an extension
\[
1 \to \mathcal U_{g,m+\vec{r}} \to \mathcal G_{g,m+\vec{r}} \to {\mathrm{Sp}}(H) \to 1
\]
of affine \(\mathbb Q\)-groups, where \(\mathcal U_{g,m+\vec{r}}\) is prounipotent. There is a Zariski dense homomorphism \(\tilde{\rho} : \Gamma_{g,m+\vec{r}} \to\mathcal G_{g,m+\vec{r}}(\mathbb Q)\) whose composition with the homomorphism \(\mathcal G_{g,m+\vec{r}}(\mathbb Q) \to {\mathrm{Sp}}(H_{\mathbb Q})\) is \(\rho\). When \(g=0\), \({\mathrm{Sp}}(H)\) is trivial and \(\mathcal G_{0,m+\vec{r}}\) is the unipotent completion \(\Gamma_{0,m+\vec{r}}^{\mathrm{un}}\). The action of the mapping class group \(\Gamma_{g,n+u}\) on \(\mathbb Q\pi_1(S,\vec{\mathsf v}_o)\) induces an action on \(\mathbb Q\lambda(S)\) which preserves the Goldman bracket. The stabilizer of \(\xi_o\) preserves the Turaev cobracket. The universal mapping property of relative completion implies that \(\mathcal G_{g,n+u}\) acts on \(\mathbb Q\pi_1(S,\vec{\mathsf v}_o)^\wedge\) and \(\mathbb Q\lambda(S)^\wedge\). Since the image of the mapping class group in \(\mathcal G_{g,n+u}\) is Zariski dense, this action preserves the Goldman bracket.
A quasi-complex structure
\[
\phi : (\overline{S},P,\vec{\mathsf v}_o,\xi_o)\to (\overline{X},D,\vec{\mathsf v},\xi)
\]
on \((\overline{S},P,\vec{\mathsf v}_o,\xi_o)\) determines an isomorphism \(\Gamma_{g,n+u} \cong \pi_1(\mathbb M_{g,n+u},\phi_o)\). The corresponding MHS on the relative completion \(\mathcal G_{g,n+u}\) corresponds to an action of \(\pi_1(\mathsf{MHS})\) on \(\mathcal G_{g,n+u}\). The quasi-complex structure \(\phi\) determines a semi-direct product
\[
\pi_1(\mathsf{MHS}) \ltimes \mathcal G_{g,n+u}.
\]
Since the natural homomorphism \(\mathcal G_{g,n+u} \to \operatorname{Aut} \mathbb Q\pi_1(X,\vec{\mathsf v}_o)^\wedge\) is a morphism of MHS [\textit{R. Hain}, J. Am. Math. Soc. 10, No. 3, 597--651 (1997; Zbl 0915.57001)], the monodromy homomorphism extends to a homomorphism
\[
\pi_1(\mathsf{MHS}) \ltimes\mathcal G_{g,n+u} \to \operatorname{Aut}\mathbb Q\pi_1(X,\vec{\mathsf v}_o)^\wedge.
\]
Denote its image by \(\widehat{\mathcal G}_{g,n+u}\) and the image of \(\mathcal G_{g,n+u}\) by \(\overline{\mathcal G}_{g,n+u}\). It is normal in \(\widehat{\mathcal G}_{g,n+u}\). The group \(\widehat{\mathcal G}_{g,n+u}\) is an extension
\[
1 \to \widehat{\mathbb U}_{g,n+u} \to \widehat{\mathcal G}_{g,n+u} \to {\mathrm{GSp}}(H) \to 1,
\]
where \({\mathrm{GSp}}\) denotes the general symplectic group and \(\widehat{\mathbb U}_{g,n+u}\) is prounipotent. One can argue as in [\textit{R. Hain} and \textit{M. Matsumoto}, J. Inst. Math. Jussieu 4, No. 3, 363--403 (2005; Zbl 1094.14013)] that, if \(g\ge 3\), then then \(\mathcal U^{\mathrm{MT}}_{X,\vec{\mathsf v}} \to \widehat{\mathbb U}_{g,n+u}\) is an isomorphism if and only if \(\pi_1(\mathsf{MHS}) \to{\mathrm{GSp}}(H)\) is surjective; the Griffiths invariant \(\nu(\overline{X}) \in \operatorname{Ext}^1_{\mathsf{MHS}}(\mathbb Q,PH^3(\operatorname{Jac}\overline{X}(2)))\) of the Ceresa cycle in \(\operatorname{Jac}\overline{X}\) is non-zero; and if the points \(\kappa_j := (2g-2)x_j - K_{\overline{X}} \in (\operatorname{Jac} \overline{X})\otimes \mathbb Q\), \(0\le j \le n\), are linearly independent over \(\mathbb Q\). This holds for general \((\overline{X},D,\vec{\mathsf v})\).
Proposition. For each complex structure \(\phi : (\overline{S},P,\vec{\mathsf v}_o) \to (\overline{X},D,\vec{\mathsf v})\), the coordinate ring \(\mathcal O(\widehat{\mathcal G}_{g,n+u}/\overline{\mathcal G}_{g,n+u})\) has a canonical MHS. These form an admissible variation of MHS over \(\mathbb M_{g,n+u}\) with trivial monodromy. Consequently, the MHS on \(\mathcal O(\widehat{\mathbb U}_{g,n+u}/\overline{\mathbb U}_{g,n+u})\) does not depend on the complex structure \(\phi\).
Reviewer: Mohammad Reza Rahmati (León)A counterexample to a conjecture of Nollet and Xavierhttps://www.zbmath.org/1483.141072022-05-16T20:40:13.078697Z"Braun, Francisco"https://www.zbmath.org/authors/?q=ai:braun.francisco"Dias, Luis Renato Gonçalves"https://www.zbmath.org/authors/?q=ai:dias.luis-renato-g"Venato-Santos, Jean"https://www.zbmath.org/authors/?q=ai:venato-santos.jeanIn studying the Jacobian conjecture, the reviewer and \textit{F. Xavier} conjectured that a local diffeomorphism \(F: \mathbb R^n \to \mathbb R^n\) must be injective if the pre-image of every affine hyperplane \(\pi\) is connected or empty [Discrete Contin. Dyn. Syst. 8, 17--28 (2002; Zbl 1008.58009)]. This conjecture, which implies the Jacobian conjecture, holds for \(n=2\) and has been proved for \(n \geq 3\) under the additional hypothesis that \(H^{n-1} (F^{-1} (\pi))=0\) by \textit{E. Cabral Balreira} [Comment. Math. Helv. 85, No. 1, 73--93 (2010; Zbl 1194.58034)]. Here the authors provide a very readable counterexample with a concrete example, namely the map \(F: \mathbb R^3 \to \mathbb R^3\) given by \(F(x,y,z) = (z^5 + e^x \cos y, z^3 + e^x \sin y, z)\). It is easy to see that \(F\) is a local diffeomorphism (its Jacobian determinant is \(e^{2x} \neq 0\)) which is not injective (it is periodic in the \(y\) variable). The hard part is showing that the pre-image of every plane is connected. For this the authors write an explicit equation of a plane \(\pi\) and show that in each \(F^{-1} (\pi)\) is connected in several different cases. Their example has the feature that \(H^1 (F^{-1} (\pi))\) is infinitely generated for most \(\pi\), so in view of Balreira's result one might ask if the conjecture holds assuming \(H^{n-1} (F^{-1} (\pi))\) is finitely generated.
Reviewer: Scott Nollet (Fort Worth)From torus bundles to particle-hole equivariantizationhttps://www.zbmath.org/1483.180202022-05-16T20:40:13.078697Z"Cui, Shawn X."https://www.zbmath.org/authors/?q=ai:cui.shawn-xingshan"Gustafson, Paul"https://www.zbmath.org/authors/?q=ai:gustafson.paul-p|gustafson.paul"Qiu, Yang"https://www.zbmath.org/authors/?q=ai:qiu.yang"Zhang, Qing"https://www.zbmath.org/authors/?q=ai:zhang.qing.4|zhang.qing.2|zhang.qing|zhang.qing.3|zhang.qing.1Quantum topology emerged from the discovery of the Jones polynomials [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)] and the formulation of topological quantum field theory (TQFT) [\textit{M. Atiyah}, Publ. Math., Inst. Hautes Étud. Sci. 68, 175--186 (1988; Zbl 0692.53053); \textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010)] in the 1980s, revealing deep connections between the algebraic/quantum arena of tensor categories and the topoogical/classical arena of \(3\)-dimensional manifolds. Precisely speaking, quantum invariants of \(3\)-dimensional manifolds and \((2+1)\)-dimensional TQFTs are to be constructed from modular tensor categories, two fundamental families in \((2+1)\)-dimensions being the Reshetikhin-Turaev [Zbl 0725.57007] and Turaev-Viro [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)] TQFTs, both of which are based on certain tensor categories.
The following two works lie at the backdrop of this paper.
\begin{itemize}
\item Inspired by \(M\)-theory in physics, the authors in [\textit{G. Y. Cho} et al., J. High Energy Phys. 2020, No. 11, Paper No. 115, 58 p. (2020; Zbl 1456.81341)] proposed another relation between tensor categories and \(3\)-dimensional manifolds in the opposite direction, outlining a program to construct modular tensor categories from certain classes of closed oriented \(3\)-dimensional manifolds. A central structure under study is an \(\mathrm{SL}(2,\mathbb{C})\) flat connection corresponding to a conjugacy class of morphisms from the fundamental group to \(\mathrm{SL}(2,\mathbb{C})\). The manifolds are required to have finitely many non-Abelian \(\mathrm{SL}(2,\mathbb{C})\) flat connections, each of which must be gauge equivalent to an \(\mathrm{SL}(2,\mathbb{R})\) or \(\mathrm{SU}(2)\) flat connection. Classical invariants such as the Chern-Simons invariant and twisted Reidemeister torsion also play a significant role in the construction.
\item In [\textit{S. X. Cui}, ``From three dimensional manifolds to modular tensor categories'', Preprint, \url{arXiv:2101.01674}], the authors mathematically explored the program in greater detail, systematically studying two infinite families of \(3\)-dimensional manifolds, namely Seifert fibered spaces with three singular fibers and torus bundles over the circle whose monodromy matrix \ has odd trace. It was shown that the first family being related to the Temperley-Lieb-Jones category [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] realizes modular tensor categories, while the second family is related to the quantum group category of type \(B\).
\end{itemize}
Although the efforts in the above two works suggest a far-reaching connection between \(3\)-dimensional manifolds and premodular tensor categories, there remain many questions to be resolved.
\begin{itemize}
\item[1.] The program currently only provides an algorithm in computing the modular \(S\)- and \(T\)-matrices, leaving other data such as \(F\)-symbols and \(R\)-symbols untouched.
\item[2.] Even for the modular data, the computation for the \(S\)-matrix essentially follows a try-and-error procedure.
\item[3.] There are a number of subtleties in choosing the correct set of characters as simple objects, determining the proper unit object, etc.
\end{itemize}
The principal objective in this paper is to apply the program to torus bundles over the circle with SOL geometry [Zbl 0561.57001]. The examples of Seifert fibered spaces in the second work covered six of the eight geometries, the ones left being the hyperbolic and SOL. Since the program concerns closed manifolds whose Chern-Simons invariants are all real, hyperbolic manifolds are surely excluded. A torus bundle over the circle is uniquely determined by the isotopy class of gluing diffeomorphism, called the monodromy matrix, which is an element of \(\mathrm{SL}(2,\mathbb{Z})\). Torus bundles whose monodromy is Anosov are of SOL geometry.
For a finite Abelian group \(G\) and a quadratic form \(q:G\rightarrow\mathbb{C}\), \(\mathcal{C}(G,q)\) denotes the pointed premodular category whose isomorphism classes of simple objects are \(G\) and whose topological twist is given by \(q\). There is a \(\mathbb{Z}_{2}\)-action, called the particle-hole symmetry, on \(\mathcal{C}(G,q)\) defined by sending each simple object to its dual or its inverse viewed as a group element. \(\mathcal{C}(G,q)^{\mathbb{Z}_{2}}\) denotes the \(\mathbb{Z}_{2}\)-equivariantization of \(\mathcal{C}(G,q)\) with respect to the particle-hole symmetry. The principal result is the following theorem.
Theorem 1. For each torus bundle over the circle \(M_{A}\) with monodromy matrix \(A\), \(N:=\left\vert Tr(A)+2\right\vert \), there is an associated finite Abelian group \(G_{A}\) isomorphic to \(\mathbb{Z}_{r}\times\mathbb{Z}_{N/r}\) for some integer \(r\geq1\) (Lemma 1) and a quadratic form \(q_{A}(x):=\exp(\frac{2\pi i\widetilde{q}(x)}{N})\), \(\widetilde{q}:G\rightarrow\mathbb{Z}_{N}\) (Lemma 3) such that the modular data realized by \(M_{A}\) coincide with those of \(\mathcal{C}(G_{A},q_{A})^{\mathbb{Z}_{2}}\).
A synopsis of the paper goes as follows.
\begin{itemize}
\item[\S 2] reviews some basic facts about premodular categories, recalling the program of constructing premodular categories from \(3\)-dimensional manifolds.
\item[\S 3] is devoted to computing the modular data of the equivariantization of a pointed premodular category under the particle-hole symmetry.
\item[\S 4] states and proves the main theorem concerning the construction of premodular categories from torus bundles.
\end{itemize}
Reviewer: Hirokazu Nishimura (Tsukuba)Nonlinear conditions for ultradifferentiabilityhttps://www.zbmath.org/1483.260232022-05-16T20:40:13.078697Z"Nenning, David Nicolas"https://www.zbmath.org/authors/?q=ai:nenning.david-nicolas"Rainer, Armin"https://www.zbmath.org/authors/?q=ai:rainer.armin"Schindl, Gerhard"https://www.zbmath.org/authors/?q=ai:schindl.gerhardSummary: A remarkable theorem of Joris states that a function \(f\) is \(C^{\infty}\) if two relatively prime powers of \(f\) are \(C^{\infty}\). Recently, Thilliez showed that an analogous theorem holds in Denjoy-Carleman classes of Roumieu type. We prove that a division property, equivalent to Joris's result, is valid in a wide variety of ultradifferentiable classes. Generally speaking, it holds in all dimensions for non-quasianalytic classes. In the quasianalytic case we have general validity in dimension one, but we also get validity in all dimensions for certain quasianalytic classes.Semiconcavity and sensitivity analysis in mean-field optimal control and applicationshttps://www.zbmath.org/1483.301062022-05-16T20:40:13.078697Z"Bonnet, Benoît"https://www.zbmath.org/authors/?q=ai:bonnet.benoit"Frankowska, Hélène"https://www.zbmath.org/authors/?q=ai:frankowska.heleneSummary: In this article, we investigate some of the fine properties of the value function associated with an optimal control problem in the Wasserstein space of probability measures. Building on new interpolation and linearisation formulas for non-local flows, we prove semiconcavity estimates for the value function, and establish several variants of the so-called sensitivity relations which provide connections between its superdifferential and the adjoint curves stemming from the maximum principle. We subsequently make use of these results to study the propagation of regularity for the value function along optimal trajectories, as well as to investigate sufficient optimality conditions and optimal feedbacks for mean-field optimal control problems.On the monotonicity of the best constant of Morrey's inequality in convex domainshttps://www.zbmath.org/1483.350072022-05-16T20:40:13.078697Z"Fărcăşeanu, Maria"https://www.zbmath.org/authors/?q=ai:farcaseanu.maria"Mihăilescu, Mihai"https://www.zbmath.org/authors/?q=ai:mihailescu.mihaiIn this interesting paper, the authors obtain some monotonicity properties of the best constant from Morrey's inequality in convex and bounded domains from the Euclidean space \(\mathbb{R}^{D}\) (\(D\ge1\)). Using these monotonicity properties, they give a new variational characterization of the best constant from Morrey's inequality on convex and bounded domains for which the maximum of the distance function to the boundary is small. The authors also show that this variational characterization does not hold true on convex and bounded domains for which the maximum of the distance function to the boundary is larger than one.
Reviewer: Meng Qu (Wuhu)Decay rates for Kelvin-Voigt damped wave equations. II: The geometric control conditionhttps://www.zbmath.org/1483.350262022-05-16T20:40:13.078697Z"Burq, Nicolas"https://www.zbmath.org/authors/?q=ai:burq.nicolas"Sun, Chenmin"https://www.zbmath.org/authors/?q=ai:sun.chenminAuthors' abstract: ``We study in this article decay rates for Kelvin-Voigt damped wave equations under a geometric control condition. When the damping coefficient is sufficiently smooth (C1 vanishing nicely, see the following equation: \(|\nabla a| \leq C a^{1/2}\)) we show that exponential decay follows from geometric control conditions (see [\textit{N. Burq} and \textit{H. Christianson}, Commun. Math. Phys. 336, No. 1, 101--130 (2015; Zbl 1320.35062); \textit{L. Tebou}, C. R., Math., Acad. Sci. Paris 350, No. 11--12, 603--608 (2012; Zbl 1255.35039)] for similar results under stronger assumptions on the damping function).''
For Part I, see [\textit{N. Burq}, SIAM J. Control Optim. 58, No. 4, 1893--1905 (2020; Zbl 1452.35030)].
Reviewer: Kaïs Ammari (Monastir)Finite time blow-up for the heat flow of \(H\)-surface with constant mean curvaturehttps://www.zbmath.org/1483.350482022-05-16T20:40:13.078697Z"Li, Haixia"https://www.zbmath.org/authors/?q=ai:li.haixiaSummary: We consider an initial boundary value problem for the heat flow of the equation of surfaces with constant mean curvature which was investigated in [\textit{T. Huang} et al., Manuscr. Math. 134, No. 1--2, 259--271 (2011; Zbl 1210.53012)], where global well-posedness and finite time blow-up of regular solutions were obtained. Their results are complemented in this paper in the sense that some new conditions on the initial data are provided for the solutions to develop finite time singularity.Chiti-type reverse Hölder inequality and torsional rigidity under integral Ricci curvature conditionhttps://www.zbmath.org/1483.350512022-05-16T20:40:13.078697Z"Chen, Hang"https://www.zbmath.org/authors/?q=ai:chen.hang|chen.hang.1Summary: In this paper, we prove a reverse Hölder inequality for the eigenfunction of the Dirichlet problem on domains of a compact Riemannian manifold with the integral Ricci curvature condition. We also prove an isoperimetric inequality for the torsional rigidity of such domains. These results extend some recent work of \textit{N. Gamara} et al. [Open Math. 13, 557--570 (2015; Zbl 06632233)] and \textit{D. Colladay} et al. [J. Geom. Anal. 28, No. 4, 3906--3927 (2018; Zbl 1410.58016)] from the pointwise lower Ricci curvature bound to the integral Ricci curvature condition. We also extend the results from Laplacian to \(p\)-Laplacian.Li-Yau multipLier set and optimal Li-Yau gradient estimate on hyperboLic spaceshttps://www.zbmath.org/1483.350592022-05-16T20:40:13.078697Z"Yu, Chengjie"https://www.zbmath.org/authors/?q=ai:yu.chengjie"Zhao, Feifei"https://www.zbmath.org/authors/?q=ai:zhao.feifeiThis paper is motivated by finding sharp Li-Yau-type gradient estimate for positive solution of heat equations on complete Riemannian manifolds with negative Ricci curvature lower bound. To reach this aim, the authors first introduce the notion of Li-Yau multiplier set and show that it can be computed by heat kernel of the manifold, then, an optimal Li-Yau-type gradient estimate is obtained on hyperbolic spaces by using recurrence relations of heat kernels on hyperbolic spaces. Lastly, as an application of the previous results, a sharp and interesting Harnack inequalities on hyperbolic spaces is shown.
Reviewer: Vincenzo Vespri (Firenze)Asymptotic behavior of solutions of the Dirichlet problem for the Poisson equation on model Riemannian manifoldshttps://www.zbmath.org/1483.350782022-05-16T20:40:13.078697Z"Losev, Alexander Georgievoch"https://www.zbmath.org/authors/?q=ai:losev.alexander-georgievoch"Mazepa, Elena Alexeevna"https://www.zbmath.org/authors/?q=ai:mazepa.elena-alexeevnaSummary: The paper is devoted to estimating the speed of approximation of solutions of the Dirichlet problem for the Poisson equation on non-compact model Riemannian manifolds to their boundary data at ``infinity''. Quantitative characteristics that estimate the speed of the approximation are found in terms of the metric of the manifold and the smoothness of the inhomogeneity in the Poisson equation.Maximal regularity of parabolic transmission problemshttps://www.zbmath.org/1483.351062022-05-16T20:40:13.078697Z"Amann, Herbert"https://www.zbmath.org/authors/?q=ai:amann.herbertSummary: Linear reaction-diffusion equations with inhomogeneous boundary and transmission conditions are shown to possess the property of maximal \(L_{\mathrm{p}}\) regularity. The new feature is the fact that the transmission interface is allowed to intersect the boundary of the domain transversally.Estimates for the first eigenvalues of bi-drifted Laplacian on smooth metric measure spacehttps://www.zbmath.org/1483.351402022-05-16T20:40:13.078697Z"Araújo Filho, Marcio Costa"https://www.zbmath.org/authors/?q=ai:araujo-filho.marcio-costaSummary: In this paper we obtain lower bounds for the first eigenvalue to some kinds of the eigenvalue problems for Bi-drifted Laplacian operator on compact manifolds (also called a smooth metric measure space) with boundary and \(m\)-Bakry-Emery Ricci curvature or Bakry-Emery Ricci curvature bounded below. We also address the eigenvalue problem with Wentzell-type boundary condition for drifted Laplacian on smooth metric measure space.Qualitative properties of stationary solutions of the NLS on the hyperbolic space without and with external potentialshttps://www.zbmath.org/1483.352262022-05-16T20:40:13.078697Z"Selvitella, Alessandro"https://www.zbmath.org/authors/?q=ai:selvitella.alessandro|selvitella.alessandro-mariaSummary: In this paper, we prove some qualitative properties of stationary solutions of the NLS on the Hyperbolic space. First, we prove a variational characterization of the ground state and give a complete characterization of the spectrum of the linearized operator around the ground state. Then we prove some rigidity theorems and necessary conditions for the existence of solutions in weighted spaces. Finally, we add a slowly varying potential to the homogeneous equation and prove the existence of non-trivial solutions concentrating on the critical points of a reduced functional. The results are the natural counterparts of the corresponding theorems on the Euclidean space. We produce also the natural virial identity on the Hyperbolic space for the complete evolution, which however requires the introduction of a weighted energy, which is not conserved and so does not lead directly to finite time blow-up as in the Euclidean case.Dynamical properties in an SVEIR epidemic model with age-dependent vaccination, latency, infection, and relapsehttps://www.zbmath.org/1483.352912022-05-16T20:40:13.078697Z"Sun, Dandan"https://www.zbmath.org/authors/?q=ai:sun.dandan"Li, Yingke"https://www.zbmath.org/authors/?q=ai:li.yingke"Teng, Zhidong"https://www.zbmath.org/authors/?q=ai:teng.zhi-dong"Zhang, Tailei"https://www.zbmath.org/authors/?q=ai:zhang.tailei"Lu, Jingjing"https://www.zbmath.org/authors/?q=ai:lu.jingjingSummary: An SVEIR epidemic model with continuous age-dependent vaccination, latency, infection, and disease relapse is proposed and analyzed in this paper. The dynamical behaviors including the derivation of basic reproduction number \(\mathcal{R}_0\), the existence and the stability of steady states, and the uniform persistence of the model are investigated. The results indicate that if \(\mathcal{R}_0 \leq 1\), the disease-free steady state is globally asymptotically stable, and the disease dies out, whereas if \(\mathcal{R}_0 > 1\), the disease is uniformly persistent, and the endemic steady state is also globally asymptotically stable, and the disease remains at the endemic level. The research shows the global dynamics of the model are sharply determined by its basic reproduction number \(\mathcal{R}_0\). Finally, numerical examples support our main theoretical results.The nonlinear fractional relativistic Schrödinger equation: existence, multiplicity, decay and concentration resultshttps://www.zbmath.org/1483.353142022-05-16T20:40:13.078697Z"Ambrosio, Vincenzo"https://www.zbmath.org/authors/?q=ai:ambrosio.vincenzoSummary: In this paper we study the following class of fractional relativistic Schrödinger equations:
\[
\begin{cases}
(-\Delta +m^2)^s u + V(\varepsilon x) u = f(u) & \text{in } \mathbb{R}^N, \\
u\in H^s (\mathbb{R}^N), \quad u>0 & \text{in } \mathbb{R}^N,
\end{cases}
\]
where \(\varepsilon >0\) is a small parameter, \(s\in (0, 1), \, m>0, N>2s, (-\Delta+m^2)^s\) is the fractional relativistic Schrödinger operator, \(V: \mathbb{R}^N \rightarrow \mathbb{R}\) is a continuous potential satisfying a local condition, and \(f: \mathbb{R} \rightarrow \mathbb{R}\) is a continuous subcritical nonlinearity. By using a variant of the extension method and a penalization technique, we first prove that, for \(\varepsilon >0\) small enough, the above problem admits a weak solution \(u_{\varepsilon}\) which concentrates around a local minimum point of \(V\) as \(\varepsilon \rightarrow 0\). We also show that \(u_{\varepsilon}\) has an exponential decay at infinity by constructing a suitable comparison function and by performing some refined estimates. Secondly, by combining the generalized Nehari manifold method and Ljusternik-Schnirelman theory, we relate the number of positive solutions with the topology of the set where the potential \(V\) attains its minimum value.Approximation of the fractional Schrödinger propagator on compact manifoldshttps://www.zbmath.org/1483.353172022-05-16T20:40:13.078697Z"Chen, Jie Cheng"https://www.zbmath.org/authors/?q=ai:chen.jiecheng"Fan, Da Shan"https://www.zbmath.org/authors/?q=ai:fan.dashan"Zhao, Fa You"https://www.zbmath.org/authors/?q=ai:zhao.fayouSummary: Let \(\mathcal{L}\) be a second order positive, elliptic differential operator that is self-adjoint with respect to some \(C^\infty\) density \(dx\) on a compact connected manifold \(\mathbb{M}\). We proved that if \(0<\alpha<1\), \(\alpha/2<s<\alpha\) and \(f\in H^s(\mathbb{M})\) then the fractional Schrödinger propagator \(\mathrm{e}^{\mathrm{i}t\mathcal{L}^{\alpha/2}}\) on \(\mathbb{M}\) satisfies \(\mathrm{e}^{\mathrm{i}t\mathcal{L}^{\alpha/2}}f(x)-f(x)=o(t^{s/\alpha-\varepsilon})\) almost everywhere as \(t\rightarrow 0^+\), for any \(\varepsilon>0\).Invariant states on noncommutative torihttps://www.zbmath.org/1483.370092022-05-16T20:40:13.078697Z"Bambozzi, Federico"https://www.zbmath.org/authors/?q=ai:bambozzi.federico"Murro, Simone"https://www.zbmath.org/authors/?q=ai:murro.simone"Pinamonti, Nicola"https://www.zbmath.org/authors/?q=ai:pinamonti.nicolaAuthors' abstract: For any number \(h\) such that \(\hbar :=h/2\pi\) is irrational and any skew-symmetric, non-degenerate bilinear form \(\sigma : \mathbb{Z}^{2g}\times \mathbb{Z}^{2g} \to \mathbb{Z}\), let be \(\mathcal{A}^h_{g,\sigma}\) be the twisted group \(\ast\)-algebra \(\mathbb{C}[\mathbb{Z}^{2g}]\) and consider the ergodic group of \(\ast\)-automorphisms of \(\mathcal{A}^h_{g,\sigma }\) induced by the action of the symplectic group \(\mathrm{Sp} (\mathbb{Z}^{2g}, \sigma)\). We show that the only \(\text{Sp} (\mathbb{Z}^{2g}, \sigma)\)-invariant state on \(\mathcal{A}^h_{g,\sigma}\) is the trace state \(\tau\).
Reviewer: Michael Skeide (Campobasso)Almost-periodic response solutions for a forced quasi-linear Airy equationhttps://www.zbmath.org/1483.370922022-05-16T20:40:13.078697Z"Corsi, Livia"https://www.zbmath.org/authors/?q=ai:corsi.livia"Montalto, Riccardo"https://www.zbmath.org/authors/?q=ai:montalto.riccardo"Procesi, Michela"https://www.zbmath.org/authors/?q=ai:procesi.michelaThis paper is mainly concerned with the existence of almost-periodic solutions for quasi-linear perturbations of the Airy equation. By combining the approach developed by
\textit{W. Craig} and \textit{C. E. Wayne} [Commun. Pure Appl. Math. 46, No. 11, 1409--1498 (1993; Zbl 0794.35104)] with a KAM reducibility scheme and pseudo-differential calculus on \(\mathbb{T}^{\infty}\), the authors establish some new results on the existence of this type of solutions for a quasi-linear PDE.
Reviewer: Chao Wang (Kunming)The dynamics of the angular and radial density correlation scaling exponents in fractal to non-fractal morphodynamicshttps://www.zbmath.org/1483.371152022-05-16T20:40:13.078697Z"Nicolás-Carlock, J. R."https://www.zbmath.org/authors/?q=ai:nicolas-carlock.j-r"Solano-Altamirano, J. M."https://www.zbmath.org/authors/?q=ai:solano-altamirano.j-m"Carrillo-Estrada, J. L."https://www.zbmath.org/authors/?q=ai:carrillo-estrada.j-lSummary: Fractal/non-fractal morphological transitions allow for the systematic study of the physics behind fractal morphogenesis in nature. In these systems, the fractal dimension is considered a non-thermal order parameter, commonly and equivalently computed from the scaling of the two-point radial- or angular-density correlations. However, these two quantities lead to discrepancies during the analysis of basic systems, such as in the diffusion-limited aggregation fractal. Hence, the corresponding clarification regarding the limits of the radial/angular scaling equivalence is needed. In this work, considering three fundamental fractal/non-fractal transitions in two dimensions, we show that the unavoidable emergence of growth anisotropies is responsible for the breaking-down of the radial/angular equivalence. Specifically, we show that the angular scaling behaves as a critical power-law, whereas the radial scaling as an exponential that, under the fractal dimension interpretation, resemble first- and second-order transitions, respectively. Remarkably, these and previous results can be unified under a single fractal dimensionality equation.On weakly reflective PF submanifolds in Hilbert spaceshttps://www.zbmath.org/1483.460762022-05-16T20:40:13.078697Z"Morimoto, Masahiro"https://www.zbmath.org/authors/?q=ai:morimoto.masahiroSummary: A weakly reflective submanifold is a minimal submanifold of a Riemannian manifold which has a certain symmetry at each point. In this paper we introduce this notion into a class of proper Fredholm (PF) submanifolds in Hilbert spaces and show that there exist many infinite dimensional weakly reflective PF submanifolds in Hilbert spaces. In particular each fiber of the parallel transport map is shown to be weakly reflective. These imply that in infinite dimensional Hilbert spaces there exist many homogeneous minimal submanifolds which are not totally geodesic, unlike in the finite dimensional Euclidean case.\(S\)-iterative algorithm for solving variational inequalitieshttps://www.zbmath.org/1483.490142022-05-16T20:40:13.078697Z"Ertürk, Müzeyyen"https://www.zbmath.org/authors/?q=ai:erturk.muzeyyen"Gürsoy, Faik"https://www.zbmath.org/authors/?q=ai:gursoy.faik"Şimşek, Necip"https://www.zbmath.org/authors/?q=ai:simsek.necipSummary: In this paper we propose an \(S\)-iterative algorithm for finding the common element of the set of solutions of the variational inequalities and the set of fixed points of nonexpansive mappings. We study the convergence criteria of the mentioned algorithm under some mild conditions. As an application, a modified algorithm is suggested to solve convex minimization problems. Numerical examples are given to validate the theoretical findings obtained herein. Our results may be considered as an improvement, refinement and complement of the previously known results.Variational time discretization of Riemannian splineshttps://www.zbmath.org/1483.490172022-05-16T20:40:13.078697Z"Heeren, Behrend"https://www.zbmath.org/authors/?q=ai:heeren.behrend"Rumpf, Martin"https://www.zbmath.org/authors/?q=ai:rumpf.martin"Wirth, Benedikt"https://www.zbmath.org/authors/?q=ai:wirth.benediktSummary: We investigate a generalization of cubic splines to Riemannian manifolds. Spline curves are defined as minimizers of the spline energy -- a combination of the Riemannian path energy and the time integral of the squared covariant derivative of the path velocity -- under suitable interpolation conditions. A variational time discretization for the spline energy leads to a constrained optimization problem over discrete paths on the manifold. Existence of continuous and discrete spline curves is established using the direct method in the calculus of variations. Furthermore, the convergence of discrete spline paths to a continuous spline curve follows from the \(\Gamma \)-convergence of the discrete to the continuous spline energy. Finally, selected example settings are discussed, including splines on embedded finite-dimensional manifolds, on a high-dimensional manifold of discrete shells with applications in surface processing and on the infinite-dimensional shape manifold of viscous rods.Geometry of lightlike locus on mixed type surfaces in Lorentz-Minkowski 3-space from a contact viewpointhttps://www.zbmath.org/1483.530342022-05-16T20:40:13.078697Z"Honda, Atsufumi"https://www.zbmath.org/authors/?q=ai:honda.atsufumi"Izumiya, Shyuichi"https://www.zbmath.org/authors/?q=ai:izumiya.shyuichi"Saji, Kentaro"https://www.zbmath.org/authors/?q=ai:saji.kentaro"Teramoto, Keisuke"https://www.zbmath.org/authors/?q=ai:teramoto.keisukeSummary: A surface in the Lorentz-Minkowski 3-space is generally a mixed type surface, namely, it has the lightlike locus. We study local differential geometric properties of such a locus on a mixed type surface. We define a frame field along a lightlike locus, and using it, we define two lightlike ruled surfaces along a lightlike locus which can be regarded as lightlike approximations of the surface along the lightlike locus. We study a relationship of singularities of these lightlike surfaces and differential geometric properties of the lightlike locus. We also consider the intersection curve of two lightlike approximations, which gives a model curve of the lightlike locus.Model Higgs bundles in exceptional components of the \(\mathrm{Sp}(4,\mathbb{R})\)-character varietyhttps://www.zbmath.org/1483.530422022-05-16T20:40:13.078697Z"Kydonakis, Georgios"https://www.zbmath.org/authors/?q=ai:kydonakis.georgiosLet \(\Sigma\) be a closed connected oriented surface of genus \(g\geq 2.\) The character variety \(\mathcal R(G)\) consists of reductive representations of \(\pi_1(\Sigma)\) into \(G\) modulo conjugation. By the non-abelian Hodge correspondence, the character variety \(\mathcal R(G)\) is isomorphic to the moduli space \(\mathcal M(G)\) of polystable \(G\)-Higgs bundles over a Riemann surface \(X=(\Sigma,J)\). When \(G=\mathrm{Sp}(4,\mathbb R)\), the subspace \(\mathcal M^{\max}(\mathrm{Sp}(4,\mathbb R))\) consists of maximal Higgs bundles, that is, with extremal Toledo invariant. It has been shown that \(\mathcal M^{\max}(\mathrm{Sp}(4,\mathbb R))\) contains \(3\cdot 2^{2g}+2g-4\) components. Among them, there are \(2g-3\) exceptional components solely consisting of Zariski dense representations. By the work of Guichard-Wienhard, representations in such components are continuous deformations of hybrid representations which involves a gluing construction for fundamental group representations over a connected sum of surfaces. The representations in the remaining \(3\cdot 2^{2g}-1\) components are deformations of standard representations, which are compositions of \(\mathrm{SL}(2,\mathbb R)\)-representations and embeddings of \(\mathrm{SL}(2,\mathbb R)\rightarrow \mathrm{Sp}(4,\mathbb R).\) The Higgs bundles corresponding to standard representations are also embeddings of \(\mathrm{SL}(2,\mathbb R)\)-Higgs bundles.
The main goal of the current paper is to establish a gluing construction for Higgs bundles over a connected sum of Riemann surfaces in terms of solutions to the \(\mathrm{Sp}(4,\mathbb R)\)-Hitchin equations, in analogy with the construction of hybrid representations.
The gluing techniques used here generalize the work of Swoboda. Start with initial parabolic Higgs bundles over two distinct Riemann surfaces \(X_1, X_2\) and model solutions to Hitchin's equation on annuli around the points in the divisors. First, perturb the initial data into model solutions by appropriate complex gauge transformations and construct a pair \((A_R^{\mathrm{app}}, \Phi_R^{\mathrm{app}})\) over \(X_{\sharp}=X_1\sharp X_2\) that coincides with initial data over \(X_1\) and \(X_2\) away from the divisors. Next, find a complex gauge transformation \(g\) such that \(g^*(A_R^{\mathrm{app}}, \Phi_R^{\mathrm{app}})\) is an exact solution to Hitchin's equation and thus finish the gluing construction. The argument showing the existence of \(g\) is translated into a Banach fixed point theorem argument and involves the study of linearization of a relevant elliptic operator.
Also, the author finds an additive formula of topological invariants under the complex connected sum operation of Higgs bundles, in analogous to the additivity property for the Toledo invariant established by Burger et al. With this formula, one can determine the topological invariant for hybrid Higgs bundles from the initial data. With an appropriate choice of initial data, the gluing constructions provide model Higgs bundles in each component of the \(2g-3\) exceptional components of maximal representations.
Reviewer: Qiongling Li (Tianjin)Liouville type theorem for transversally harmonic mapshttps://www.zbmath.org/1483.530442022-05-16T20:40:13.078697Z"Fu, Xueshan"https://www.zbmath.org/authors/?q=ai:fu.xueshan"Jung, Seoung Dal"https://www.zbmath.org/authors/?q=ai:jung.seoung-dalSummary: Let \((M,\mathcal{F})\) be a complete foliated Riemannian manifold and all leaves be compact. Let \((M',\mathcal{F}')\) be a foliated Riemannian manifold of non-positive transversal sectional curvature. Assume that the transversal Ricci curvature \(\mathrm{Ric}^Q\) of \(M\) satisfies \(\mathrm{Ric}^Q\geq -\lambda_0\) at all point \(x\in M\) and \(\mathrm{Ric}^Q>-\lambda_0\) at some point \(x_0\), where \(\lambda_0\) is the infimum of the spectrum of the basic Laplacian acting on \(L^2\)-basic functions on \(M\). Then every transversally harmonic map \(\phi:M \rightarrow M'\) of finite transversal energy is transversally constant.On \(L^2\)-harmonic forms of complete almost Kähler manifoldhttps://www.zbmath.org/1483.530472022-05-16T20:40:13.078697Z"Huang, Teng"https://www.zbmath.org/authors/?q=ai:huang.tengSummary: In this article, we study the \(L^2\)-harmonic forms on the complete \(2n\)-dimensional almost Käher manifold \(X\). We observe that the \(L^2\)-harmonic forms can decomposition into Lefschetz powers of primitive forms. Therefore we can extend vanishing theorems of \(d\)(bounded) (resp. \(d\)(sublinear)) Kähler manifold proved by Gromov (resp. Cao-Xavier, Jost-Zuo) to almost Kählerian case, that is, the spaces of all harmonic \((p, q)\)-forms on \(X\) vanishing unless \(p+q=n\). We also give a lower bound on the spectra of the Laplace operator to sharpen the Lefschetz vanishing theorem on \(d\)(bounded) case.A de Rham decomposition type theorem for contact sub-Riemannian manifoldshttps://www.zbmath.org/1483.530542022-05-16T20:40:13.078697Z"Grochowski, Marek"https://www.zbmath.org/authors/?q=ai:grochowski.marekThe author proves a result which can be regarded as a sub-Riemannian version of de Rham decomposition theorem. More precisely, suppose that \((M, H, g)\) is a contact and oriented sub-Riemannian manifold such that the Reeb vector field \(\xi\) is an infinitesimal isometry. Under such assumptions there exists a unique metric and torsion-free connection on \(H\). Suppose that there exists a point \(q\in M \) such that the holonomy group \(\Psi(q)\) acts reducibly on \(H(q)\) yielding a decomposition \(H(q)=H_1(q)\oplus\cdots\oplus H_m(q) \) into \(\Psi(q)\)-irreducible factors. Using parallel transport the authors obtain the decomposition \(H = H_1 \oplus\cdots\oplus H_m\) of \(H\) into sub-distributions \(H_i\). Unlike the Riemannian case, the distributions \(H_i\) are not integrable, however they induce integrable distributions \(\delta_i\) on \(M/\xi\), which is locally a smooth manifold. As a result, every point in \(M\) has a neighborhood \(U\) such that \(T (U/\xi) = \delta_1\oplus\cdots\oplus\delta_m\), and the latter decomposition of \(T(U/\xi)\) induces the decomposition of \(U/\xi\) into the product of Riemannian manifolds. One can restate this as follows: every contact sub-Riemannian manifold whose holonomy group acts reducibly has, at least locally, the structure of a fiber bundle over a product of Riemannian manifolds. The authors also give a version of the theorem for indefinite metrics.
Reviewer: Peibiao Zhao (Nanjing)An ODE reduction method for the semi-Riemannian Yamabe problem on space formshttps://www.zbmath.org/1483.530592022-05-16T20:40:13.078697Z"Fernández, Juan Carlos"https://www.zbmath.org/authors/?q=ai:fernandez.juan-carlos"Palmas, Oscar"https://www.zbmath.org/authors/?q=ai:palmas.oscarThe authors prove the existence of blowing-up and globally defined solutions of Yamabe-type partial differential equations on semi-Euclidean space and on the pseudosphere of dimension at least 3. In the proof they use isoparametric functions which allow the reduction to a generalized Emden-Fowler ordinary differential equation.
Reviewer: Hans-Bert Rademacher (Leipzig)Lower bounds for Cauchy data on curves in a negatively curved surfacehttps://www.zbmath.org/1483.530602022-05-16T20:40:13.078697Z"Galkowski, Jeffrey"https://www.zbmath.org/authors/?q=ai:galkowski.jeffrey"Zelditch, Steve"https://www.zbmath.org/authors/?q=ai:zelditch.steveSummary: We prove a uniform lower bound on Cauchy data on an arbitrary curve on a negatively curved surface using the Dyatlov-Jin(-Nonnenmacher) observability estimate on the global surface. In the process, we prove some further results about defect measures of restrictions of eigenfunctions to a hypersurface.Decompositions of the space of Riemannian metrics on a compact manifold with boundaryhttps://www.zbmath.org/1483.530622022-05-16T20:40:13.078697Z"Hamanaka, Shota"https://www.zbmath.org/authors/?q=ai:hamanaka.shotaLet \(M\) is a compact connected oriented smooth \(n\)-manifold, \(n \geq 3\), with smooth non-empty boundary \(\partial M\). The main goal of the paper is to prove analogues of the classical slice theorem of \textit{D. G. Ebin} [Proc. Sympos. Pure Math. 15, 11--40 (1970; Zbl 0205.53702)] and decomposition theorem of \textit{N. Koiso} [Osaka J. Math. 16, 423--429 (1979; Zbl 0416.58007)] for the space of all Riemannian metrics on \(M\) endowed with a fixed conformal class on \(\partial M\). We recall that the theorems of Ebin and Koiso apply to closed manifolds. The precise statement of the boundary condition used in the paper is somewhat technical but important for the proofs and results.
As an application, the author obtains rigidity results for relative constant scalar curvature metrics and gives a characterization of relative Einstein metrics.
Reviewer: Mikhail Belolipetsky (Rio de Janeiro)Canonical identification at infinity for Ricci-flat manifoldshttps://www.zbmath.org/1483.530692022-05-16T20:40:13.078697Z"Park, Jiewon"https://www.zbmath.org/authors/?q=ai:park.jiewonSummary: We give a natural way to identify between two scales, potentially arbitrarily far apart, in a non-compact Ricci-flat manifold with Euclidean volume growth when a tangent cone at infinity has smooth cross section. The identification map is given as the gradient flow of a solution to an elliptic equation.A formula for the heat kernel coefficients of the Dirac Laplacians on spin manifoldshttps://www.zbmath.org/1483.530722022-05-16T20:40:13.078697Z"Nagase, Masayoshi"https://www.zbmath.org/authors/?q=ai:nagase.masayoshi"Shirakawa, Takumi"https://www.zbmath.org/authors/?q=ai:shirakawa.takumiSummary: Based on Getzler's rescaling transformation, we obtain a formula for the heat kernel coefficients of the Dirac Laplacian on a spin manifold. One can compute them explicitly up to an arbitrarily high order by using only a basic knowledge of calculus added to the formula.Around Efimov's differential test for homeomorphismhttps://www.zbmath.org/1483.530782022-05-16T20:40:13.078697Z"Alexandrov, Victor"https://www.zbmath.org/authors/?q=ai:alexandrov.victor-aThere is a famous result due to Efimov, more precisely the following Theorem: No surface can be \(C^2\)-immersed in Euclidean 3-space so as to be complete in the induced Riemannian metric, with Gauss curvature \(K \le \) constant \(< 0\).
The paper under review starts with a mini-survey of results related to the previous theorem.
Among other things, Efimov established that the condition \(K \le\) constant \(< 0\) is not the only obstacle for the immersibility of a complete surface of negative curvature; he showed that a rather slow change of Gauss curvature is another obstacle. In all those numerous articles, he used to a large extent one and the same method based on the study of the spherical image of a surface. At that study, an essential role belongs to statements that, under some conditions, a locally homeomorphic mapping \(f : \mathbb{R}^2 \to \mathbb{R}^2\) is a global homeomorphism and \(f(\mathbb{R}^2)\) is a convex domain in \(\mathbb{R}^2\).
Two other theorems of Efimov are recalled in the present paper and the author gives an overview on the analogues of these theorems, their generalizations and applications. The article is devoted to presentation of results motivated by the theory of surfaces, the theory of global inverse function, the Jacobian Conjecture, and the global asymptotic stability of dynamical systems, respectively.
Reviewer: Adela-Gabriela Mihai (Bucureşti)On free boundary minimal hypersurfaces in the Riemannian Schwarzschild spacehttps://www.zbmath.org/1483.530802022-05-16T20:40:13.078697Z"Barbosa, Ezequiel"https://www.zbmath.org/authors/?q=ai:barbosa.ezequiel-r"Espinar, José M."https://www.zbmath.org/authors/?q=ai:espinar.jose-mariaAuthors' abstract: In contrast with the three-dimensional case (cf. Montezuma in Bull Braz Math Soc), where rotationally symmetric totally geodesic free boundary minimal surfaces have Morse index one; we prove in this work that the Morse index of a free boundary rotationally symmetric totally geodesic hypersurface of the \(n\)-dimensional Riemannnian Schwarzschild space with respect to variations that are tangential along the horizon is zero, for \(n\geq 4\). Moreover, we show that there exist non-compact free boundary minimal hypersurfaces which are not totally geodesic, \(n\geq 8\), with Morse index equal to 0. In addition, it is shown that, for \(n\geq 4\), there exist infinitely many non-compact free boundary minimal hypersurfaces, which are not congruent to each other, with infinite Morse index. We also study the density at infinity of a free boundary minimal hypersurface with respect to a minimal cone constructed over a minimal hypersurface of the unit Euclidean sphere. We obtain a lower bound for the density in terms of the area of the boundary of the hypersurface and the area of the minimal hypersurface in the unit sphere. This lower bound is optimal in the sense that only minimal cones achieve it.
Reviewer: Mohammad Nazrul Islam Khan (Buraidah)\(\Phi\)-harmonic maps and \(\Phi\)-superstrongly unstable manifoldshttps://www.zbmath.org/1483.530832022-05-16T20:40:13.078697Z"Han, Yingbo"https://www.zbmath.org/authors/?q=ai:han.yingbo"Wei, Shihshu Walter"https://www.zbmath.org/authors/?q=ai:wei.shihshu-walterSummary: We motivate and define \(\Phi\)-energy density, \(\Phi\)-energy, \(\Phi\)-harmonic maps and stable \(\Phi\)-harmonic maps. Whereas harmonic maps or \(p\)-harmonic maps can be viewed as critical points of the integral of the first symmetric function \(\sigma_1\) of a pull-back tensor, \(\Phi\)-harmonic maps can be viewed as critical points of the integral of the second symmetric function \(\sigma_2\) of a pull-back tensor. By an extrinsic average variational method in the calculus of variations
(cf.
[\textit{R. Howard} and \textit{S. W. Wei}, Trans. Am. Math. Soc. 294, 319--331 (1986; Zbl 0588.58015);
\textit{S. W. Wei} and \textit{C.-M. Yau}, J. Geom. Anal. 4, No. 2, 247--272 (1994; Zbl 0851.58014);
\textit{S. W. Wei}, Indiana Univ. Math. J. 47, No. 2, 625--670 (1998; Zbl 0930.58010);
\textit{R. Howard} and \textit{S. W. Wei}, Contemp. Math. 646, 127--167 (2015; Zbl 1361.53047)]
),
we derive the average second variation formulas for \(\Phi\)-energy functional, express them in orthogonal notation in terms of the differential matrix, and find \(\Phi\)-superstrongly unstable \((\Phi\)-SSU manifolds. We prove, in particular that every compact \(\Phi\)-SSU manifold must be \(\Phi\)-strongly unstable (\(\Phi\)-SU), i.e., (a) A compact \(\Phi\)-SSU manifold cannot be the target of any nonconstant stable \(\Phi\)-harmonic maps from any manifold, (b) The homotopic class of any map from any manifold into a compact \(\Phi\)-SSU manifold contains elements of arbitrarily small \(\Phi\)-energy, (c) A compact \(\Phi\)-SSU manifold cannot be the domain of any nonconstant stable \(\Phi\)-harmonic map into any manifold, and (d) The homotopic class of any map from a compact \(\Phi\)-SSU manifold into any manifold contains elements of arbitrarily small \(\Phi\)-energy [cf. Theorem 1.1(a),(b),(c), and (d).] We provide many examples of \(\Phi\)-SSU manifolds, which include but not limit to spheres or some unstable Yang-Mills fields
(cf.
[\textit{J.-P. Bourguignon} et al., Proc. Natl. Acad. Sci. USA 76, 1550--1553 (1979; Zbl 0408.53023);
\textit{J.-P. Bourguignon} and \textit{H. B. Lawson jun.}, Commun. Math. Phys. 79, 189--230 (1981; Zbl 0475.53060);
\textit{S. Kobayashi} et al., Math. Z. 193, 165--189 (1986; Zbl 0634.53022);
\textit{S. W. Wei}, Indiana Univ. Math. J. 33, 511--529 (1984; Zbl 0559.53027);
\textit{L. Wu} et al., ``Discovering geometric and topological properties of ellipsoids by curvatures'', Br. J. Math. Comput. Sci. 8, No. 4, 318--329 (2015)]), and examples of \(\Phi\)-harmonic, or \(\Phi\)-unstable map from or into \(\Phi\)-SSU manifold that are not constant. We establish a link of \(\Phi\)-SSU manifold to \(p\)-SSU manifold and topology. The extrinsic average variational method in the calculus of variations, employed is in contrast to an average method in PDE that we applied in
[\textit{B.-Y. Chen} and \textit{S. W. Wei}, J. Geom. Symmetry Phys. 52, 27--46 (2019; Zbl 1427.53046)] to obtain sharp growth estimates for warping functions in multiply warped product manifolds.On the generalized of \(p\)-harmonic and \(f\)-harmonic mapshttps://www.zbmath.org/1483.530842022-05-16T20:40:13.078697Z"Remli, Embarka"https://www.zbmath.org/authors/?q=ai:remli.embarka"Cherif*, Ahmed Mohammed"https://www.zbmath.org/authors/?q=ai:cherif.ahmed-mohammedSummary: In this paper, we extend the definition of \(p\)-harmonic maps between two Riemannian manifolds. We prove a Liouville type theorem for generalized \(p\)-harmonic maps. We present some new properties for the generalized stress \(p\)-energy tensor. We also prove that every generalized \(p\)-harmonic map from a complete Riemannian manifold into a Riemannian manifold admitting a homothetic vector field satisfying some condition is constant.On center of mass and foliations by constant spacetime mean curvature surfaces for isolated systems in general relativityhttps://www.zbmath.org/1483.530862022-05-16T20:40:13.078697Z"Cederbaum, Carla"https://www.zbmath.org/authors/?q=ai:cederbaum.carla"Sakovich, Anna"https://www.zbmath.org/authors/?q=ai:sakovich.annaThe authors define a new total center of mass for an ``isolated system'': the ``Universe'' is a 4-dimensional Lorentzian manifold \((\mathfrak M^{1,3}, \mathfrak{g})\), endowed with an energy-momentum tensor field \(\mathfrak T\), with an ``initial data set'' given by a spacelike hypersuface \((M^3,g)\), with the second fundamental form \(K\), the scalar local energy density \(\mu\) and the (1-form) local momentum density \(J\). When this configuration is ``asymptotically Euclidean'' and with non-vanishing energy, it gives rise to a (unique) foliation by 2-spheres of constant spacetime mean curvature. This foliation is the main tool for constructing the total center of mass. It is shown that this center of mass behaves as a point particle in Special Relativity (i.e. it transforms equivariantly under the asymptotic Poincaré group of \({\mathfrak M^{1,3}}\)). In particular, it evolves in time under the Einstein evolution equations like a point particle in Special Relativity.
Reviewer: Gabriel Teodor Pripoae (Bucureşti)Evolution of the first eigenvalue of the Laplace operator and the \(p\)-Laplace operator under a forced mean curvature flowhttps://www.zbmath.org/1483.531102022-05-16T20:40:13.078697Z"Qi, Xuesen"https://www.zbmath.org/authors/?q=ai:qi.xuesen"Liu, Ximin"https://www.zbmath.org/authors/?q=ai:liu.ximinSummary: In this paper, we discuss the monotonicity of the first nonzero eigenvalue of the Laplace operator and the \(p\)-Laplace operator under a forced mean curvature flow (MCF). By imposing conditions associated with the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced MCF, and some special pinching conditions on the second fundamental form of the initial hypersurface, we prove that the first nonzero closed eigenvalues of the Laplace operator and the \(p\)-Laplace operator are monotonic under the forced MCF, respectively, which partially generalize Mao and Zhao's work. Moreover, we give an example to specify applications of conclusions obtained above.Geometry of surfaces in \(\mathbb{R}^5\) through projections and normal sectionshttps://www.zbmath.org/1483.570352022-05-16T20:40:13.078697Z"Deolindo-Silva, J. L."https://www.zbmath.org/authors/?q=ai:silva.jorge-luiz-deolindo"Sinha, R. Oset"https://www.zbmath.org/authors/?q=ai:sinha.raul-osetThe main focus of the article is the study of geometry of surfaces in \(\mathbb{R}^5\). The first approach used by the authors is to relate the geometry of a surface in \(\mathbb{R}^5\) to that of corresponding (both regular and singular) surfaces in \(\mathbb{R}^4\) obtained by orthogonal projections. In particular, relations between the asymptotic directions of the original surface and those of the projected surface are obtained. It is interesting to note here that the asymptotic directions for surfaces in \(\mathbb{R}^5\), unlike those in \(\mathbb{R}^4\), do not depend only on the second order geometry of the surface.
The authors also establish relations between the umbilical curvatures for surfaces in \(\mathbb{R}^5\) [\textit{S. I. R. Costa} et al., Differ. Geom. Appl. 27, No. 3, 442--454 (2009; Zbl 1176.53015)] and their projected surfaces in \(\mathbb{R}^4\). The authors study the contact between the projected surfaces with spheres in \(\mathbb{R}^4\) and show that there exists a unique umbilical focal hypersphere at a point of the surface if and only if there exists a unique umbilic focal hypersphere at the corresponding point on the projected surface.
Surfaces in \(\mathbb{R}^5\) can also be obtained as normal sections of 3-manifolds in \(\mathbb{R}^6\), so the authors then go on to consider the geometry of surfaces in \(\mathbb{R}^5\) by relating the asymptotic directions at a point in the 3-manifold with asymptotic directions at the corresponding point in the normal section. By introducing an appropriate umbilic curvature for 3-manifolds, they then study the contact with spheres using this invariant and relate it to the contact between spheres and the surface in \(\mathbb{R}^5\) obtained as a normal section.
Reviewer: Graham Reeve (Liverpool)Fubini-study metrics and Levi-Civita connections on quantum projective spaceshttps://www.zbmath.org/1483.580012022-05-16T20:40:13.078697Z"Matassa, Marco"https://www.zbmath.org/authors/?q=ai:matassa.marcoIn the study of noncommutative spaces, it is desirable to quantize important notions of commutative spaces. In classical Riemannian geometry, the Riemannian metric and the Levi-Civita connection are fundamental notions. The idea to quantize the Riemannian metric and the Levi-Civita connection to noncommutative algebra is used as algebraic approach, which is explained for instance in [\textit{E.J. Beggs} and \textit{S. Majid}, Quantum Riemannian geometry. Cham: Springer (2020; Zbl 1436.53001)]. In this paper under review, the author introduces analogues of the Fubini-Study metrics and the corresponding Levi-Civita connections on the quantum projective spaces.
Let \(\mathcal{B}\) be the algebra of a generic quantum projective space (see Section 4 for the definition) and \(\Omega\) the degree-one part of a differential calculus over \(\mathcal{B}\). Then a quantum metric can be defined as an element \(g \in \Omega \otimes_{\mathcal{B}} \Omega\) satisfying an appropriate condition. The first main result in this paper is a quantization of the Fubini-Study metric.
Theorem (see Theorem 6.11). Any quantum projective space \(\mathcal{B}\) admits a quantum metric \(g \in \Omega \otimes_{\mathcal{B}} \Omega\). Moreover, in the classical limit it reduces to the Fubini-Study metric.
Let \(\nabla : \Omega \to \Omega \otimes_{\mathcal{B}} \Omega\) be a connection on \(\Omega\) (the notion of the connection on \(\Omega\) and its tortion can be defined in the standard algebraic sense). To formulate an analogue of the compatibility of \(\nabla\) with \(g\), there are two possibilities: 1) (weak sense) use ``cotortion'', 2) (strong sense) require \(\nabla\) to be a bimodule connection. In this paper, the author defines a quantization of the Levi-Civita connection which is compatible with the quantum Fubini-Study metric \(g\) in the weak and strong sense.
Theorem (see Theorem 7.7). Any quantum projective space \(\mathcal{B}\) admits a connection \(\nabla : \Omega \to \Omega \otimes_{\mathcal{B}} \Omega\) which is torsion free and cotorsion free. Moreover, in the classical limit it reduces to the Levi-Civita connection for the Fubini-Study metric on the cotangent bundle.
Theorem (see Theorem 8.4). The connection \(\nabla : \Omega \to \Omega \otimes_{\mathcal{B}} \Omega\) is a bimodule connection and is compatible with the quantum metric, in the sense that \(\nabla g = 0\).
Reviewer: Tatsuki Seto (Tokyo)On solving semilinear singularly perturbed Neumann problems for multiple solutionshttps://www.zbmath.org/1483.580022022-05-16T20:40:13.078697Z"Xie, Ziqing"https://www.zbmath.org/authors/?q=ai:xie.ziqing"Yuan, Yongjun"https://www.zbmath.org/authors/?q=ai:yuan.yongjun"Zhou, Jianxin"https://www.zbmath.org/authors/?q=ai:zhou.jianxinOn traces of operators associated with actions of compact Lie groupshttps://www.zbmath.org/1483.580032022-05-16T20:40:13.078697Z"Savin, A. Yu."https://www.zbmath.org/authors/?q=ai:savin.anton-yu"Sternin, B. Yu."https://www.zbmath.org/authors/?q=ai:sternin.boris-yuLet \((X,M)\) be a smooth pair consisting of a manifold \(M\) and its sub-manifold \(X\), \(G\) be a compact Lie group acting on manifold \(M\), \(T_g\) be a shift operator
\[
T_gu=g^{-1^*}u
\]
which is induced by the diffeomorhism \(g\). The \(G\)-operator \(D\) under consideration is the following
\[
D=\int\limits_GD_gT_gdg, \tag{1}
\]
where \(dg\) is Haar measure on the group \(G\), \(T_g\) is a smooth family of pseudo-differential operators.
The authors study the trace operator
\[
i^*Di_*: H^s(X)\rightarrow H^{s-d-\nu}(X),
\]
where \(i^*: H^s(M)\rightarrow H^{s-\nu/2}(X)\) and \( i_*: H^s(X)\rightarrow H^{s-\nu/2}(X)\) are boundary and co-boundary operators induced by the imbedding \(i: X\rightarrow M\), \(d\) is an order of the operator \(D\), \(\nu\) is a codimension of \(X\). They introduce the set
\[
X_G=\{x\in X: Gx\subset X\}
\]
and prove their main result ``on a localization''. It asserts that the trace operator (1) is supported on the set \(X_G\subset X\), and particularly if \(X_G\) is an empty set then the operator (1) is compact.
There are a lot of examples in the paper illustrating these conclusions.
Reviewer: Vladimir Vasilyev (Belgorod)A weighted Trudinger-Moser inequality on a closed Riemann surface with a finite isometric group actionhttps://www.zbmath.org/1483.580042022-05-16T20:40:13.078697Z"Yang, Jie"https://www.zbmath.org/authors/?q=ai:yang.jie.4|yang.jie.3|yang.jie.1|yang.jie.2Summary: Let \((\Sigma, g)\) be a closed Riemann surface, \(G\) be a finite isometric group acting on \((\Sigma, g)\) and \(H^{1, 2}(\Sigma)\) be the standard Sobolev space. Taking a positive smooth function \(f\) which is \(G\)-invariant, we define a function space \(\mathcal{H}_f^G\) by
\[
\mathcal{H}_f^G=\left\{ u\in H^{1,2}(\Sigma)\left| u(\sigma(x))=u(x), \int_\Sigma uf dv_g=0,\, \forall x\in \Sigma ,\, \forall \sigma \in G \right.\right\}.
\]
Using blow-up analysis, we prove that for any \(\alpha <\lambda_1^f\), the supremum
\[
\sup_{u\in\mathcal{H}_f^G, \int_\Sigma |\nabla_g u|^2fdv_g-\alpha \int_\Sigma u^2fdv_g\le 1}\int_\Sigma e^{4\pi \ell u^2f}dv_g
\]
is attained, where \(\lambda_1^f\) is the first eigenvalue of the \(f\)-Laplacian \(\Delta_f=-\operatorname{div}_g(f\nabla_g)\) on the space \(\mathcal{H}_f^G\), \(\ell =\min_{x\in \Sigma}\sharp G(x)\) and \(\sharp G(x)\) denotes the number of all distinct points of \(G(x)\). Moreover, we consider the case of higher order eigenvalues. Our results generalized those of \textit{Y. Yang} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 65, No. 3, 647--659 (2006; Zbl 1095.58005); J. Differ. Equations 258, No. 9, 3161--3193 (2015; Zbl 1339.46041)] and \textit{Y. Fang} and \textit{Y. Yang} [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 20, No. 4, 1295--1324 (2020; Zbl 1471.30005)].Fredholm conditions and index for restrictions of invariant pseudodifferential operators to isotypical componentshttps://www.zbmath.org/1483.580052022-05-16T20:40:13.078697Z"Baldare, Alexandre"https://www.zbmath.org/authors/?q=ai:baldare.alexandre"Côme, Rémi"https://www.zbmath.org/authors/?q=ai:come.remi"Lesch, Matthias"https://www.zbmath.org/authors/?q=ai:lesch.matthias"Nistor, Victor"https://www.zbmath.org/authors/?q=ai:nistor.victorSummary: Let \(\Gamma\) be a compact group acting on a smooth, compact manifold \(M\), let \(P\in\psi^m(M;E_0,E_1)\) be a \(\Gamma\)-invariant, classical pseudodifferential operator acting between sections of two equivariant vector bundles \(E_i\to M\), \(i=0,1\), and let \(\alpha\) be an irreducible representation of the group \(\Gamma\). Then \(P\) induces a map \(\pi_\alpha(P):H^s(M;E_0)_\alpha\to H^{s-m}(M;E_1)_\alpha\) between the \(\alpha\)-isotypical components of the corresponding Sobolev spaces of sections. When \(\Gamma\) is finite, we explicitly characterize the operators \(P\) for which the map \(\pi_\alpha(P)\) is Fredholm in terms of the principal symbol of \(P\) and the action of \(\Gamma\) on the vector bundles \(E_i\). When \(\Gamma=\{1\}\), that is, when there is no group, our result extends the classical characterization of Fredholm (pseudo)differential operators on compact manifolds. The proof is based on a careful study of the symbol \(C^*\)-algebra and of the topology of its primitive ideal spectrum. We also obtain several results on the structure of the norm closure of the algebra of invariant pseudodifferential operators and their relation to induced representations. As an illustration of the generality of our results, we provide some applications to Hodge theory and to index theory of singular quotient spaces.Eigenvalue asymptotics for weighted Laplace equations on rough Riemannian manifolds with boundaryhttps://www.zbmath.org/1483.580062022-05-16T20:40:13.078697Z"Bandara, Lashi"https://www.zbmath.org/authors/?q=ai:bandara.lashi"Nursultanov, Medet"https://www.zbmath.org/authors/?q=ai:nursultanov.medet"Rowlett, Julie"https://www.zbmath.org/authors/?q=ai:rowlett.julieSummary: Our topological setting is a smooth compact manifold of dimension two or higher with smooth boundary. Although this underlying topological structure is smooth, the Riemannian metric tensor is only assumed to be bounded and measurable. This is known as a \textit{rough Riemannian manifold}. For a large class of boundary conditions we demonstrate a Weyl law for the asymptotics of the
eigenvalues of the Laplacian associated to a rough metric. Moreover, we obtain eigenvalue asymptotics for weighted Laplace equations associated to a rough metric. Of particular novelty is that the weight function is not assumed to be of fixed sign, and thus the eigenvalues may be both positive and negative. Key ingredients in the proofs were demonstrated by Birman and Solomjak nearly fifty
years ago in their seminal work on eigenvalue asymptotics. In addition to determining the eigenvalue asymptotics in the rough Riemannian manifold setting for weighted Laplace equations, we also wish to promote their achievements which may have further applications to modern problems.Compact manifolds with fixed boundary and large Steklov eigenvalueshttps://www.zbmath.org/1483.580072022-05-16T20:40:13.078697Z"Colbois, Bruno"https://www.zbmath.org/authors/?q=ai:colbois.bruno"El Soufi, Ahmad"https://www.zbmath.org/authors/?q=ai:el-soufi.ahmad"Girouard, Alexandre"https://www.zbmath.org/authors/?q=ai:girouard.alexandreSummary: Let \((M,g)\) be a compact Riemannian manifold with boundary. Let \(b>0\) be the number of connected components of its boundary. For manifolds of dimension \(\geq 3\), we prove that for \(j=b+1\) it is possible to obtain an arbitrarily large Steklov eigenvalue \(\sigma_j(M,e^\delta g)\) using a conformal perturbation \(\delta \in C^\infty (M)\) which is supported in a thin neighbourhood of the boundary, with \(\delta =0\) on the boundary. For \(j\leq b\), it is also possible to obtain arbitrarily large eigenvalues, but the conformal factor must spread throughout the interior of \(M\). In fact, when working in a fixed conformal class and for \(\delta =0\) on the boundary, it is known that the volume of \((M,e^\delta g)\) has to tend to infinity in order for some \(\sigma _j\) to become arbitrarily large. This is in stark contrast with the situation for the eigenvalues of the Laplace operator on a closed manifold, where a conformal factor that is large enough for the volume to become unbounded results in the spectrum collapsing to 0. We also prove that it is possible to obtain large Steklov eigenvalues while keeping different boundary components arbitrarily close to each other, by constructing a convenient Riemannian submersion.Geometric hypoelliptic Laplacian and orbital integrals [after Bismut, Lebeau and Shen]https://www.zbmath.org/1483.580082022-05-16T20:40:13.078697Z"Ma, Xiaonan"https://www.zbmath.org/authors/?q=ai:ma.xiaonanSummary: Hodge theory for a hypoelliptic Laplacian acting on the total space of the cotangent bundle of a Riemannian manifold. This operator interpolates between the classical elliptic Laplacian on the base and the generator of the geodesic flow. We will describe recent developments of the theory of hypoelliptic Laplacians, in particular the explicit formula obtained by Bismut for orbital integrals and the recent solution by Shen of Fried's conjecture (dating back to 1986) for locally symmetric spaces. The conjecture predicts the equality of the analytic torsion and the value at 0 of the dynamic zeta function.
For the entire collection see [Zbl 1416.00029].Renormalization of stochastic continuity equations on Riemannian manifoldshttps://www.zbmath.org/1483.580092022-05-16T20:40:13.078697Z"Galimberti, Luca"https://www.zbmath.org/authors/?q=ai:galimberti.luca"Karlsen, Kenneth H."https://www.zbmath.org/authors/?q=ai:karlsen.kenneth-hvistendahlThe authors study initial value problems for stochastic continuity equations on smooth closed Riemannian manifolds \(M\) with metric \(h\), of the form \[ \partial_{t}\rho + \operatorname{div}_{h}\left[ \rho \left( u(t,x) + \sum_{i=1}^{N} a_{i}(x) \circ \frac{d W^{i}}{dt} \right) \right]=0, \tag{1} \] for Sobolev velocity fields \(u\), perturbed by Gaussian noise terms driven by that independent Wiener processes \(W^{i}\), where \(a_{i}\) are smooth spatially dependent vector fields on M (with the stochastic integrals interpreted in the Stratonovich sense), supplemented with initial data \(\rho(0)=\rho_{0} \in L^2(M)\).
This type of equation is very interesting both from the mathematical point of view as well as from the point of view of applications (e.g. in fluid mechanics). The deterministic case (\(a_{i}=0\)) has been studied and existence of weak solution was shown using the DiPerna-Lions theory of renormalized solutions [\textit{R. J. DiPerna} and \textit{P. L. Lions}, Invent. Math. 98, No. 3, 511--547 (1989; Zbl 0696.34049)], both in the Euclidean and the smooth closed manifold case and there are important extensions by \textit{L. Ambrosio} [ibid. 158, No. 2, 227--260 (2004; Zbl 1075.35087)] in the case of BV velocity fields). We recall that a renormalized solution \(\rho\) is a weak solution such that \(S(\rho)\) is also a weak solution for any ``reasonable'' \(S : {\mathbb R} \to {\mathbb R}\). The stochastic case for Lipschitz type coefficients has been studied by \textit{H. Kunita} [Stochastic flows and stochastic differential equations. Cambridge etc.: Cambridge University Press (1990; Zbl 0743.60052)] in the Euclidean case, whereas results by \textit{S. Attanasio} and \textit{F. Flandoli} [Commun. Partial Differ. Equations 36, No. 7--9, 1455--1474 (2011; Zbl 1237.60048)] establish the renormalization property for BV velocity field and constant \(a_{i}\), revealing an interesting regularization by noise property of the equation (in the sense that the renormalization property implies uniqueness without the usual \(L^{\infty}\) assumption on the divergence of \(u\)) which has become a recurrent theme in the study of stochastic transport or continuity equations. Extensions in the case of stochastic continuity equations in Itō form in the Euclidean domain with spatially dependent noise coefficients were obtained in [\textit{S. Punshon-Smith} and \textit{S. Smith}, Arch. Ration. Mech. Anal. 229, No. 2, 627--708 (2018; Zbl 1394.35313)] and [\textit{J. A. Rossmanith} et al., J. Comput. Phys. 199, No. 2, 631--662 (2004; Zbl 1126.76350)].
In this paper the authors study problem (1) in the generalized setting already mentioned above, and their main result is the renormalization property for weak \(L^2\) solutions of (1). As a corollary they deduce the uniqueness of weak solutions and an a priori estimate under the usual condition that \(\operatorname{div}_{h} u \in L_{t}^{1}L^{\infty}\). The proof consists of a systematic procedure for regularizing tensor fields on a manifold, a convenient choice of atlas to simplify technical computations linked to the Christoffel symbols, and several DiPerna-Lions type commutators between (first/second order) geometric differential operators and the regularization device.
Reviewer: Athanasios Yannacopoulos (Athína)Singularities of singular solutions of first-order differential equations of clairaut typehttps://www.zbmath.org/1483.580102022-05-16T20:40:13.078697Z"Saji, Kentaro"https://www.zbmath.org/authors/?q=ai:saji.kentaro"Takahashi, Masatomo"https://www.zbmath.org/authors/?q=ai:takahashi.masatomoThe work of this paper is a part of an ongoing research on understanding singularities of envelopes for differential equations of Clairaut type. Let us explain the main notions and concepts. First, consider the ordinary differential equation \[ F(x,y,p)=0, \tag{1} \] where \(p\) stands for the derivative \(dy/dx\) and \(F\) is defined in a domain of the space \(J^1(\mathbb{R},\mathbb{R})\) that consists of 1-jets of functions \(y(x)\), i.e., the space with coordinates \(x,y,p\) equipped with the contact 1-form \(pdx - dy = 0\). Assume that equation (1) defines a smooth surface in \(J^1(\mathbb{R},\mathbb{R})\), then the contact structure cuts the vector field \[ X = F_p \partial_x + pF_p \partial_y + (F_x+pF_y) \partial_p \] on this surface. Integral curves of the field \(X\) are 1-jet extensions of solutions of equation (1). The canonical projection \(\pi(x,y,p) = (x,y)\) restricted to the surface \(\{F=0\}\) has singular points on the set \(\{F=F_p=0\}\) called the criminant, and the projection of the criminant is the discriminant set of equation (1). Generically, the criminant and the discriminant set are curves, the field \(X\) vanishes at isolated points of the criminant, and the discriminant set is the locus of singularities of solution of (1) (almost all of which are 3:2-cusps). However, there exist a special class of equations (1) called Clairaut type. It is defined by the condition that the function \(F_x+pF_y\) vanishes on the criminant identically. Under light additional conditions, in this case the discriminant set is the envelop of solutions of (1), and consequently, it is a solution as well. The basic examples are \(p^2 = y\) and classical Clairaut's equation itself: \[ f(p) = xp-y, \ \ f''(p) \not\equiv 0. \tag{2} \] The discriminant set of (2) is the dual Legendrian curve to the graph \(y=f(x)\), it is the envelop of its tangent lines \(xc-y=f(c)\), \(c=const\), which are also solutions of (2). It is regular at points where \(f''(p) \neq 0\) are it is singular if \(f''(p)=0\). For instance, it has 3:2-cusps at points where \(f''(p)=0\), \(f'''(p) \neq 0\). This is the simplest example of singularities of envelopes.
Second, the authors investigate the partial differential equation \[ F(x_1, x_2,y,p_1, p_2)=0, \tag{3} \] where \(p_i\) stands for the derivative \(dy/dx_i\) and \(F\) is defined in a domain of the space \(J^1(\mathbb{R}^2,\mathbb{R})\) that consists of 1-jets of functions \(y(x_1,x_2)\), i.e., the space with coordinates \(x_1, x_2,y,p_1, p_2\) equipped with the contact 1-form \(p_1 dx_1+ p_2 dx_2 - dy = 0\). Similarly to the above, there exists a special class of equations (3) called Clairaut type. The authors show that (under light additional conditions) Clairaut type equations (3) have envelops of solutions (which are solutions as well) and establish the list of typical singularities of their envelops: cuspidal edge, swallowtail, cuspidal butterfly, cuspidal lips/beaks, etc (frontal singularities).
Third, the authors investigate the system of equations \[ F(x_1, x_2,y,p_1, p_2)=0, \ \ G(x_1, x_2,y,p_1, p_2)=0, \tag{4} \] where the functions \(F, G\) are defined in a domain of the space \(J^1(\mathbb{R}^2,\mathbb{R})\), all notations are similar to (3) and the Poisson bracket \([F,G]\) on the manifold \(\{F=G=0\}\) is identically zero. The authors introduce a natural notion of Clairaut type systems (4) and establish the list of typical singularities of their envelops similar to those for (3).
Reviewer: Alexey O. Remizov (Moskva)Preserving invariance properties of reaction-diffusion systems on stationary surfaceshttps://www.zbmath.org/1483.651552022-05-16T20:40:13.078697Z"Frittelli, Massimo"https://www.zbmath.org/authors/?q=ai:frittelli.massimo"Madzvamuse, Anotida"https://www.zbmath.org/authors/?q=ai:madzvamuse.anotida"Sgura, Ivonne"https://www.zbmath.org/authors/?q=ai:sgura.ivonne"Venkataraman, Chandrasekhar"https://www.zbmath.org/authors/?q=ai:venkataraman.chandrasekharSummary: We propose and analyse a lumped surface finite element method for the numerical approximation of reaction-diffusion systems on stationary compact surfaces in \(\mathbb{R}^3\). The proposed method preserves the invariant regions of the continuous problem under discretization and, in the special case of scalar equations, it preserves the maximum principle. On the application of a fully discrete scheme using the implicit-explicit Euler method in time, we prove that invariant regions of the continuous problem are preserved (i) at the spatially discrete level with no restriction on the meshsize and (ii) at the fully discrete level under a timestep restriction. We further prove optimal error bounds for the semidiscrete and fully discrete methods, that is, the convergence rates are quadratic in the meshsize and linear in the timestep. Numerical experiments are provided to support the theoretical findings. We provide examples in which, in the absence of lumping, the numerical solution violates the invariant region leading to blow-up.Local finite element approximation of Sobolev differential formshttps://www.zbmath.org/1483.651812022-05-16T20:40:13.078697Z"Gawlik, Evan"https://www.zbmath.org/authors/?q=ai:gawlik.evan-s"Holst, Michael J."https://www.zbmath.org/authors/?q=ai:holst.michael-j"Licht, Martin W."https://www.zbmath.org/authors/?q=ai:licht.martin-wernerSummary: We address fundamental aspects in the approximation theory of vector-valued finite element methods, using finite element exterior calculus as a unifying framework. We generalize the Clément interpolant and the Scott-Zhang interpolant to finite element differential forms, and we derive a broken Bramble-Hilbert lemma. Our interpolants require only minimal smoothness assumptions and respect partial boundary conditions. This permits us to state local error estimates in terms of the mesh size. Our theoretical results apply to curl-conforming and divergence-conforming finite element methods over simplicial triangulations.Bohr-Sommerfeld levels for quantum completely integrable systemshttps://www.zbmath.org/1483.810702022-05-16T20:40:13.078697Z"Guillemin, Victor"https://www.zbmath.org/authors/?q=ai:guillemin.victor-w"Wang, Zuo Qin"https://www.zbmath.org/authors/?q=ai:wang.zuoqinSummary: In this paper we will show how the Bohr-Sommerfeld levels of a quantum completely integrable system can be computed modulo \(O(\hbar^{\infty})\) by an inductive procedure starting at stage zero with the Bohr-Sommerfeld levels of the corresponding classical completely integrable system.Schrödinger dynamics and optimal transport of measureshttps://www.zbmath.org/1483.810712022-05-16T20:40:13.078697Z"Zanelli, Lorenzo"https://www.zbmath.org/authors/?q=ai:zanelli.lorenzoOne parameter family of rationally extended isospectral potentialshttps://www.zbmath.org/1483.810812022-05-16T20:40:13.078697Z"Yadav, Rajesh Kumar"https://www.zbmath.org/authors/?q=ai:yadav.rajesh-kumar"Banerjee, Suman"https://www.zbmath.org/authors/?q=ai:banerjee.suman"Kumari, Nisha"https://www.zbmath.org/authors/?q=ai:kumari.nisha"Khare, Avinash"https://www.zbmath.org/authors/?q=ai:khare.avinash"Mandal, Bhabani Prasad"https://www.zbmath.org/authors/?q=ai:mandal.bhabani-prasadSummary: We start from a given one dimensional rationally extended shape invariant potential associated with \(X_m\) exceptional orthogonal polynomials and using the idea of supersymmetry in quantum mechanics, we obtain one continuous parameter \((\lambda)\) family of rationally extended strictly isospectral potentials. We illustrate this construction by considering three well known rationally extended potentials, two with pure discrete spectrum (the extended radial oscillator and the extended Scarf-I) and one with both the discrete and the continuous spectrum (the extended generalized Pöschl-Teller) and explicitly construct the corresponding one continuous parameter family of rationally extended strictly isospectral potentials. Further, in the special case of \(\lambda=0\) and \(-1\), we obtain two new exactly solvable rationally extended potentials, namely the rationally extended Pursey and the rationally extended Abraham-Moses potentials respectively. We illustrate the whole procedure by discussing in detail the particular case of the \(X_1\) rationally extended one parameter family of potentials including the corresponding Pursey and the Abraham Moses potentials.From 2d droplets to 2d Yang-Millshttps://www.zbmath.org/1483.811002022-05-16T20:40:13.078697Z"Chattopadhyay, Arghya"https://www.zbmath.org/authors/?q=ai:chattopadhyay.arghya"Dutta, Suvankar"https://www.zbmath.org/authors/?q=ai:dutta.suvankar"Mukherjee, Debangshu"https://www.zbmath.org/authors/?q=ai:mukherjee.debangshu"Neetu"https://www.zbmath.org/authors/?q=ai:neetu.babbarSummary: We establish a connection between time evolution of free Fermi droplets and partition function of \textit{generalised q}-deformed Yang-Mills theories on Riemann surfaces. Classical phases of \((0 + 1)\) dimensional unitary matrix models can be characterised by free Fermi droplets in two dimensions. We quantise these droplets and find that the modes satisfy an abelian Kac-Moody algebra. The Hilbert spaces \(\mathcal{H}_+\) and \(\mathcal{H}_-\) associated with the upper and lower free Fermi surfaces of a droplet admit a Young diagram basis in which the phase space Hamiltonian is diagonal with eigenvalue, in the large \(N\) limit, equal to the quadratic Casimir of \(u(N)\).
We establish an exact mapping between states in \(\mathcal{H}_\pm\) and geometries of droplets. In particular, coherent states in \(\mathcal{H}_\pm\) correspond to classical deformation of upper and lower Fermi surfaces. We prove that correlation between two coherent states in \(\mathcal{H}_\pm\) is equal to the chiral and anti-chiral partition function of \(2d\) Yang-Mills theory on a cylinder. Using the fact that the full Hilbert space \(\mathcal{H}_+ \otimes \mathcal{H}_-\) admits a \textit{composite} basis, we show that correlation between two classical droplet geometries is equal to the full \(U(N)\) Yang-Mills partition function on cylinder. We further establish a connection between higher point correlators in \(\mathcal{H}_\pm\) and higher point correlators in \(2d\) Yang-Mills on Riemann surface. There are special states in \(\mathcal{H}_\pm\) whose transition amplitudes are equal to the partition function of 2\textit{d q}-deformed Yang-Mills and in general character expansion of Villain action. We emphasise that the \(q\)-deformation in the Yang-Mills side is related to special deformation of droplet geometries without deforming the gauge group associated with the matrix model.Batalin-Vilkovisky quantization of fuzzy field theorieshttps://www.zbmath.org/1483.811062022-05-16T20:40:13.078697Z"Nguyen, Hans"https://www.zbmath.org/authors/?q=ai:nguyen.hans"Schenkel, Alexander"https://www.zbmath.org/authors/?q=ai:schenkel.alexander"Szabo, Richard J."https://www.zbmath.org/authors/?q=ai:szabo.richard-jSummary: We apply the modern Batalin-Vilkovisky quantization techniques of Costello and Gwilliam to noncommutative field theories in the finite-dimensional case of fuzzy spaces. We further develop a generalization of this framework to theories that are equivariant under a triangular Hopf algebra symmetry, which in particular leads to quantizations of finite-dimensional analogues of the field theories proposed recently through the notion of `braided \(L_{\infty}\)-algebras'. The techniques are illustrated by computing perturbative correlation functions for scalar and Chern-Simons theories on the fuzzy 2-sphere, as well as for braided scalar field theories on the fuzzy 2-torus.Chern-Simons perturbative series revisitedhttps://www.zbmath.org/1483.811082022-05-16T20:40:13.078697Z"Lanina, E."https://www.zbmath.org/authors/?q=ai:lanina.elena|lanina.e-g"Sleptsov, A."https://www.zbmath.org/authors/?q=ai:sleptsov.alexey"Tselousov, N."https://www.zbmath.org/authors/?q=ai:tselousov.nSummary: A group-theoretical structure in a perturbative expansion of the Wilson loops in the 3d Chern-Simons theory with \(SU(N)\) gauge group is studied in symmetric approach. A special basis in the center of the universal enveloping algebra \(ZU(\mathfrak{sl}_N)\) is introduced. This basis allows one to present group factors in an arbitrary irreducible finite-dimensional representation. Developed methods have wide applications, the most straightforward and evident ones are mentioned. Namely, Vassiliev invariants of higher orders are computed, a conjecture about existence of new symmetries of the colored HOMFLY polynomials is stated, and the recently discovered tug-the-hook symmetry of the colored HOMFLY polynomial is proved.Implications for colored HOMFLY polynomials from explicit formulas for group-theoretical structurehttps://www.zbmath.org/1483.811092022-05-16T20:40:13.078697Z"Lanina, E."https://www.zbmath.org/authors/?q=ai:lanina.e-g|lanina.elena"Sleptsov, A."https://www.zbmath.org/authors/?q=ai:sleptsov.alexey"Tselousov, N."https://www.zbmath.org/authors/?q=ai:tselousov.nSummary: We have recently proposed [Phys. Lett., B 823, Article ID 136727, 8 p. (2021; Zbl 1483.81108)] a powerful method for computing group factors of the perturbative series expansion of the Wilson loop in the Chern-Simons theory with \(SU(N)\) gauge group. In this paper, we apply the developed method to obtain and study various properties, including nonperturbative ones, of such vacuum expectation values.
First, we discuss the computation of Vassiliev invariants. Second, we discuss the Vogel theorem of not distinguishing chord diagrams by weight systems coming from semisimple Lie (super)algebras. Third, we provide a method for constructing linear recursive relations for the colored Jones polynomials considering a special case of torus knots \(T[2, 2 k + 1]\). Fourth, we give a generalization of the one-hook scaling property for the colored Alexander polynomials. And finally, for the group factors we provide a combinatorial description, which has a clear dependence on the rank \(N\) and the representation \(R\).Wilson loops in SYM \(\mathcal{N}=4\) do not parametrize an orientable spacehttps://www.zbmath.org/1483.811302022-05-16T20:40:13.078697Z"Agarwala, Susama"https://www.zbmath.org/authors/?q=ai:agarwala.susama"Marcott, Cameron"https://www.zbmath.org/authors/?q=ai:marcott.cameronSummary: We explore the geometric space parametrized by (tree level) Wilson loops in SYM \(\mathcal{N}=4\). We show that this space can be seen as a vector bundle over a totally non-negative subspace of the Grassmannian, \(\mathcal{W}_{k,n}\). Furthermore, we explicitly show that this bundle is non-orientable in the majority of the cases, and conjecture that it is non-orientable in the remaining situation. Using the combinatorics of the Deodhar decomposition of the Grassmannian, we identify subspaces \(\Sigma(W)\subset\mathcal{W}_{k,n}\) for which the restricted bundle lies outside the positive Grassmannian. Finally, while probing the combinatorics of the Deodhar decomposition, we give a diagrammatic algorithm for reading equations determining each Deodhar component as a semialgebraic set.Spacetimes with continuous linear isotropies. I: Spatial rotationshttps://www.zbmath.org/1483.830062022-05-16T20:40:13.078697Z"MacCallum, M. A. H."https://www.zbmath.org/authors/?q=ai:maccallum.malcolm-a-hSummary: The weakest known criterion for local rotational symmetry (LRS) in spacetimes of Petrov type D is due to \textit{S. W. Goode} and \textit{J. Wainwright} [ibid. 18, 315--331 (1986; Zbl 0584.53029)]. Here it is shown, using methods related to the Cartan-Karlhede procedure, to be equivalent to local spatial rotation invariance of the Riemann tensor and its first derivatives. Conformally flat spacetimes are similarly studied and it is shown that for almost all cases the same criterion ensures LRS. Only for conformally flat accelerated perfect fluids are three curvature derivatives required to ensure LRS, showing that Ellis's original condition for that case is necessary as well as sufficient.Three-dimensional Maxwellian Carroll gravity theory and the cosmological constanthttps://www.zbmath.org/1483.830112022-05-16T20:40:13.078697Z"Concha, Patrick"https://www.zbmath.org/authors/?q=ai:concha.patrick"Peñafiel, Diego"https://www.zbmath.org/authors/?q=ai:penafiel.diego-m"Ravera, Lucrezia"https://www.zbmath.org/authors/?q=ai:ravera.lucrezia"Rodríguez, Evelyn"https://www.zbmath.org/authors/?q=ai:rodriguez.evelynSummary: In this work, we present the three-dimensional Maxwell Carroll gravity by considering the ultra-relativistic limit of the Maxwell Chern-Simons gravity theory defined in three spacetime dimensions. We show that an extension of the Maxwellian Carroll symmetry is necessary in order for the invariant tensor of the ultra-relativistic Maxwellian algebra to be non-degenerate. Consequently, we discuss the origin of the aforementioned algebra and theory as a flat limit. We show that the theoretical setup with cosmological constant yielding the extended Maxwellian Carroll Chern-Simons gravity in the vanishing cosmological constant limit is based on a new enlarged extended version of the Carroll symmetry. Indeed, the latter exhibits a non-degenerate invariant tensor allowing the proper construction of a Chern-Simons gravity theory which reproduces the extended Maxwellian Carroll gravity in the flat limit.Spacetimes with continuous linear isotropies. III: Null rotationshttps://www.zbmath.org/1483.830202022-05-16T20:40:13.078697Z"MacCallum, M. A. H."https://www.zbmath.org/authors/?q=ai:maccallum.malcolm-a-hSummary: It is shown that in many of the possible cases local null rotation invariance of the curvature and its first derivatives is sufficient to ensure that there is an isometry group \(G_r\) with \(r\ge 3\) acting on (a neighbourhood of) the spacetime and containing a null rotation isotropy. The exceptions where invariance of the second derivatives is additionally required to ensure this conclusion are Petrov type N Einstein spacetimes, spacetimes containing ``pure radiation'' (a Ricci tensor of Segre type [(11,2)]), and conformally flat spacetimes with a Ricci tensor of Segre type [1(11,1)] (a ``tachyon fluid'').
For Parts I and II, see [the author, ibid. 53, No. 6, Paper No. 57, 21 p. (2021; Zbl 1483.83006); ibid. 53, No. 6, Paper No. 61, 12 p. (2021; Zbl 1483.83019)].Entropy of Reissner-Nordström-like black holeshttps://www.zbmath.org/1483.830362022-05-16T20:40:13.078697Z"Blagojević, M."https://www.zbmath.org/authors/?q=ai:blagojevic.milutin"Cvetković, B."https://www.zbmath.org/authors/?q=ai:cvetkovic.branislavSummary: In Poincaré gauge theory, black hole entropy is defined canonically by the variation of a boundary term \(\Gamma_H\), located at horizon. For a class of static and spherically symmetric black holes in vacuum, the explicit formula reads \(\delta \Gamma_H = T \delta S\), where \(T\) is black hole temperature and \(S\) entropy. Here, we analyze a new member of the same class, the Reissner-Nordström-like black hole with torsion [\textit{ J. A.R. Cembranos} and [\textit{J. G. Valcarcel}, ``New torsion black hole solutions in Poincaré gauge theory,'' J. Cosmol. Astropart. Phys., 01, Article 014 (2017; \url{doi:10.1088/1475-7516/2017/01/014})], where the electric charge of matter is replaced by a gravitational parameter, induced by the existence of torsion. This parameter affects \(\delta \Gamma_H\) in a way that ensures the validity of the first law.Chiral and non-chiral spinning string dynamo instability from quantum torsion sourceshttps://www.zbmath.org/1483.830422022-05-16T20:40:13.078697Z"Garcia de Andrade, L. C."https://www.zbmath.org/authors/?q=ai:garcia-de-andrade.l-cSummary: In this paper dynamo instability is investigated by making use of a condensed matter screw dislocation model of spinning cosmic strings in the framework of teleparallel \(T_4\) spacetime, The magnetic field is encoded into the \(T_4\) metric tensor. By using quantum torsion from quantisation flux dependence on the spin angular momentum of strings, one is able to obtain dynamo instability where chiral currents grow exponentially as in Witten superconducting cosmic string. Our results are compared to [\textit{V. Galitsky} et al., Phys. Rev. Lett. 121, No. 17, Article ID 176603, 6 p. (2018; \url{doi:10.1103/PhysRevLett.121.176603})], who have investigated dynamo instability from Weyl semimetals, showing that chiral anomaly term reduces the Reynolds number for dynamo instability. String dynamos in the framework of Einstein-Cartan (EC) [the author, Cosmol. Astropart. Phys. 2014, No. 8, Paper No. 23, 9 p. (2014; \url{doi:10.1088/1475-7516/2014/08/023})], ER - [\textit{L. C. Garcia de Andrade}, Cosmic magnetism in modified theories of gravity. ChisinauÉditions universitaires europ'eennes (2017)] cosmology have been previously addressed. Moreover, it is shown that dynamo strings possess a monopole singularity at Big Bang. Topological defects dynamos, such as torsion spin-polarised fermions orthogonal to the wall [the author, Classical Quantum Gravity 38, No. 6, Article ID 065005, 9 p. (2021; Zbl 1479.83135)] have been obtained from torsional anomalies sources. Galitski et al. also discovered that chiral anomalies help dynamo effect in real magnetic fields. Non-chiral solutions where the current is along the string but the magnetic field is not, such as in pion string dynamo [\textit{R. Gwyn} et al., ``Magnetic fields from heterotic cosmic strings'', Phys. Rev. D (3) 79, No. 8, Article ID 083502, 13 p. (2009; \url{doi:10.1103/PhysRevD.79.083502})] are obtained from the Heaviside step function which tells us that the magnetic field grows, in the plane above \(z=0\) where the spinning string crosses the plane, and vanishes below the crossing plane.An alternative to the Teukolsky equationhttps://www.zbmath.org/1483.830472022-05-16T20:40:13.078697Z"Hatsuda, Yasuyuki"https://www.zbmath.org/authors/?q=ai:hatsuda.yasuyukiSummary: We conjecture a new ordinary differential equation exactly isospectral to the radial component of the homogeneous Teukolsky equation. We find this novel relation by a hidden symmetry implied from a four-dimensional \(\mathcal{N}=2\) supersymmetric quantum chromodynamics. Our proposal is powerful both in analytical and in numerical studies. As an application, we derive high-order perturbative series of quasinormal mode frequencies in the slowly rotating limit. We also test our result numerically by comparing it with a known technique.Holographic complexity in charged accelerating black holeshttps://www.zbmath.org/1483.830492022-05-16T20:40:13.078697Z"Jiang, Shun"https://www.zbmath.org/authors/?q=ai:jiang.shun"Jiang, Jie"https://www.zbmath.org/authors/?q=ai:jiang.jie.2Summary: Using the ``complexity equals action'' (CA) conjecture, for an ordinary charged system, it has been shown that the late-time complexity growth rate is given by a difference between the value of \(\Phi_H Q + \Omega_H J\) on the inner and outer horizons. In this paper, we investigate the complexity of the boundary quantum system with conical deficits. From the perspective of holography, we consider charged accelerating black holes which contain conical deficits on the north and south poles in the bulk gravitational theory and evaluate the complexity growth rate using the CA conjecture. As a result, the late-time growth rate of complexity is different from the ordinary charged black holes. It implies that complexity can carry the information of conical deficits on the boundary quantum system.Geometrothermodynamics of black holes with a nonlinear sourcehttps://www.zbmath.org/1483.830632022-05-16T20:40:13.078697Z"Sánchez, Alberto"https://www.zbmath.org/authors/?q=ai:rivadulla-sanchez.albertoSummary: We study thermodynamics and geometrothermodynamics of a particular black hole configuration with a nonlinear source. We use the mass as fundamental equation, from which it follows that the curvature radius must be considered as a thermodynamic variable, leading to an extended equilibrium space. Using the formalism of geometrothermodynamics, we show that the geometric properties of the thermodynamic equilibrium space can be used to obtain information about thermodynamic interaction, critical points and phase transitions. We show that these results are compatible with the results obtained from classical black hole thermodynamics.Can magnetogenesis driven by dynamo instabilities in chiral fermion plasmas, favor Einstein-Cartan AdS cosmology?https://www.zbmath.org/1483.830682022-05-16T20:40:13.078697Z"de Andrade, L. C. Garcia"https://www.zbmath.org/authors/?q=ai:garcia-de-andrade.l-cSummary: In this letter we show that the magnetic field (MF) bounds induced by Einstein-Cartan-de Sitter (EC) with AdS anti-de Sitter spacetime geometry and fermionic torsion with chiral dynamo plasmas instabilities, with inhomogeneous uniform chiral chemical potential, indicates that there is a possibility, as far as magnetogenesis is concerned, that we may obtain more stringent results within ECAdS gravity, compared with general relativity (GR) limits obtained by Tsagas using torsionless magnetogenesis. Spinorial fermionic source with torsion induces a chiral plasmas oscillation. The time dependent chiral chemical potential, sometimes associated formally with torsion, [the author, ``Metric-torsion decay of non-adiabatic chiral helical magnetic fields against chiral dynamo action in bouncing cosmological models'', Eur. Phys. J. C, Part. Fields 78, No. 6, Paper No. 530, 7 p. (2018; \url{doi:10.1140/epjc/s10052-018-5983-x})], is obtained from chiral flipping equation. The MFs obtained from the present universe ECdS chiral dynamo model, meet the criteria of galactic seed fields with better limits within Tsagas range [\textit{C. G. Tsagas}, ``Relaxing the limits on inflationary magnetogenesis'', Phys. Rev. D (3) 92, No. 10--15, Article ID 101301, 5 p. (2015; \url{doi:10.1103/PhysRevD.92.101301})] seed galactic magnetic fields for dynamo mechanisms, than the bound obtained by chiral dynamos in torsionless AdS cosmology. Majorana neutrinos are used as spatial components of axial spin-torsion source of dynamo instability from neutrino asymmetry.Holographic superconductors in 4D Einstein-Gauss-Bonnet gravity with backreactionshttps://www.zbmath.org/1483.830722022-05-16T20:40:13.078697Z"Pan, Jie"https://www.zbmath.org/authors/?q=ai:pan.jie"Qiao, Xiongying"https://www.zbmath.org/authors/?q=ai:qiao.xiongying"Wang, Dong"https://www.zbmath.org/authors/?q=ai:wang.dong.7|wang.dong|wang.dong.3|wang.dong.8|wang.dong.2|wang.dong.4|wang.dong.1|wang.dong.6|wang.dong.5"Pan, Qiyuan"https://www.zbmath.org/authors/?q=ai:pan.qiyuan"Nie, Zhang-Yu"https://www.zbmath.org/authors/?q=ai:nie.zhang-yu"Jing, Jiliang"https://www.zbmath.org/authors/?q=ai:jing.jiliangSummary: We construct the holographic superconductors away from the probe limit in the consistent \(D \to 4\) Einstein-Gauss-Bonnet gravity. We observe that, both for the ground state and excited states, the critical temperature first decreases then increases as the curvature correction tends towards the Chern-Simons limit in a backreaction dependent fashion. However, the decrease of the backreaction, the increase of the scalar mass, or the increase of the number of nodes will weaken this subtle effect of the curvature correction. Moreover, for the curvature correction approaching the Chern-Simons limit, we find that the gap frequency \(\omega_g/T_c\) of the conductivity decreases first and then increases when the backreaction increases in a scalar mass dependent fashion, which is different from the finding in the \((3 + 1)\)-dimensional superconductors that increasing backreaction increases \(\omega_g/T_c\) in the full parameter space. The combination of the Gauss-Bonnet gravity and backreaction provides richer physics in the scalar condensates and conductivity in the \((2 + 1)\)-dimensional superconductors.Non-singular non-flat universeshttps://www.zbmath.org/1483.830942022-05-16T20:40:13.078697Z"Salamanca, Andrés Felipe Estupiñán"https://www.zbmath.org/authors/?q=ai:salamanca.andres-felipe-estupinan"Medina, Sergio Bravo"https://www.zbmath.org/authors/?q=ai:bravo-medina.sergio"Nowakowski, Marek"https://www.zbmath.org/authors/?q=ai:nowakowski.marek"Batic, Davide"https://www.zbmath.org/authors/?q=ai:batic.davideSummary: The quest to understand better the nature of the initial cosmological singularity is with us since the discovery of the expanding universe. Here, we propose several non-flat models, among them the standard cosmological scenario with a critical cosmological constant, the Einstein-Cartan cosmology, the Milne-McCrea universe with quantum corrections and a non-flat universe with bulk viscosity. Within these models, we probe into the initial singularity by using different techniques. Several nonsingular universes emerge, one of the possibilities being a static non-expanding and stable Einstein universe.Inflation with Gauss-Bonnet and Chern-Simons higher-curvature-corrections in the view of GW170817https://www.zbmath.org/1483.830972022-05-16T20:40:13.078697Z"Venikoudis, S. A."https://www.zbmath.org/authors/?q=ai:venikoudis.s-a"Fronimos, F. P."https://www.zbmath.org/authors/?q=ai:fronimos.f-pSummary: Inflationary era of our Universe can be characterized as semi-classical because it can be described in the context of four-dimensional Einstein's gravity involving quantum corrections. These string motivated corrections originate from quantum theories of gravity such as superstring theories and include higher gravitational terms as, Gauss-Bonnet and Chern-Simons terms. In this paper we investigated inflationary phenomenology coming from a scalar field, with quadratic curvature terms in the view of GW170817. Firstly, we derived the equations of motion, directly from the gravitational action. As a result, formed a system of
differential equations with respect to Hubble's parameter and the inflaton field which was very complicated and cannot be solved analytically, even in the minimal coupling case. Based on the observations from GW170817 event, which have shown that the speed of the primordial gravitational wave is equal to the speed of light, \(c_{\mathcal{T}}^2=1\) in natural units, our equations of motion where simplified after applying the constraint \(c_{\mathcal{T}}^2=1\), the slow-roll approximations and neglecting the string corrections. We described the dynamics of inflationary phenomenology and proved that theories with Gauss-Bonnet term can be compatible with recent observations. Also, the Chern-Simons term leads to asymmetric generation and evolution of the two circular polarization states of gravitational wave. Finally, viable inflationary models are presented, consistent with the observational constraints. The possibility of a blue tilted tensor spectral index is briefly investigated.