Recent zbMATH articles in MSC 30Fhttps://www.zbmath.org/atom/cc/30F2022-05-16T20:40:13.078697ZWerkzeugSpecial cases of the orbifold version of Zvonkine's \(r\)-ELSV formulahttps://www.zbmath.org/1483.140922022-05-16T20:40:13.078697Z"Borot, Gaëtan"https://www.zbmath.org/authors/?q=ai:borot.gaetan"Kramer, Reinier"https://www.zbmath.org/authors/?q=ai:kramer.reinier"Lewanski, Danilo"https://www.zbmath.org/authors/?q=ai:lewanski.danilo"Popolitov, Alexandr"https://www.zbmath.org/authors/?q=ai:popolitov.aleksandr"Shadrin, Sergey"https://www.zbmath.org/authors/?q=ai:shadrin.sergeySummary: We prove the orbifold version of Zvonkine's \(r\)-ELSV formula in two special cases: the case of \(r=2\) (completed 3-cycles) for any genus \(g\geq 0\) and the case of any \(r\geq 1\) for genus \(g=0\).Matrix orthogonality in the plane versus scalar orthogonality in a Riemann surfacehttps://www.zbmath.org/1483.300142022-05-16T20:40:13.078697Z"Charlier, Christophe"https://www.zbmath.org/authors/?q=ai:charlier.christopheSummary: We consider a non-Hermitian matrix orthogonality on a contour in the complex plane. Given a diagonalizable and rational matrix valued weight, we show that the Christoffel-Darboux (CD) kernel, which is built in terms of matrix orthogonal polynomials, is equivalent to a scalar valued reproducing kernel of meromorphic functions in a Riemann surface. If this Riemann surface has genus \(0\), then the matrix valued CD kernel is equivalent to a scalar reproducing kernel of polynomials in the plane. Interestingly, this scalar reproducing kernel is not necessarily a scalar CD kernel. As an application of our result, we show that the correlation kernel of certain doubly periodic lozenge tiling models admits a double contour integral representation involving only a scalar CD kernel. This simplifies a formula of Duits and Kuijlaars.On the number of linearly independent solutions of the Riemann boundary value problem on the Riemann surface of an algebraic functionhttps://www.zbmath.org/1483.300762022-05-16T20:40:13.078697Z"Kruglov, V. E."https://www.zbmath.org/authors/?q=ai:kruglov.vladislav-e|kruglov.viktor-eSummary: We suggest a modified solution to the Riemann boundary value problem on a Riemann surface of an algebraic function of genus \(\rho \). This allows us to to reduce the problem of finding the number \(l\) of linearly independent algebraic functions (LIAF), that are multiples of a fractional divisor \(Q\), to finding the number of LIAF that are multiples of an effective divisor \(J \) (\(\operatorname{ord}J = \rho\)); this provides a solution to the Jacobi inversion problem given in this paper. We study the case, where the exponents of the normal basis elements coincide, and solve the problem of finding the number of LIAF, multiples of an effective divisor. The definitions of conjugate points of Riemann surface and hyperorder of an effective divisor are introduced. Depending on the structure of divisor \(J\), exact formulas are obtained for number \(l\); they are expressed in terms of the order of divisor \(Q\), the hyperorder of divisor \(J\), and numbers \(\rho\) and \(n\), where \(n\) is the number of sheets of the algebraic Riemann surface.Uniformization of branched surfaces and Higgs bundleshttps://www.zbmath.org/1483.300782022-05-16T20:40:13.078697Z"Biswas, Indranil"https://www.zbmath.org/authors/?q=ai:biswas.indranil"Bradlow, Steven"https://www.zbmath.org/authors/?q=ai:bradlow.steven-b"Dumitrescu, Sorin"https://www.zbmath.org/authors/?q=ai:dumitrescu.sorin"Heller, Sebastian"https://www.zbmath.org/authors/?q=ai:heller.sebastian-gregorOn \(s\)-extremal Riemann surfaces of even genushttps://www.zbmath.org/1483.300792022-05-16T20:40:13.078697Z"Kozłowska-Walania, Ewa"https://www.zbmath.org/authors/?q=ai:kozlowska-walania.ewaSummary: We consider Riemann surfaces of even genus \(g\) with the action of the group \(\mathcal{D}_n\times \mathbb{Z}_2\), with \(n\) even. These surfaces have the maximal number of 4 non-conjugate symmetries and shall be called \textit{\(s\)-extremal}. We show various results for such surfaces, concerning the total number of ovals, topological types of symmetries, hyperellipticity degree and the minimal genus problem. If in addition an \(s\)-extremal Riemann surface has the maximal total number of ovals, then it shall simply be called \textit{extremal}. In the main result of the paper we find all the families of extremal Riemann surfaces of even genera, depending on if one of the symmetries is fixed-point free or not.The dual volume of quasi-Fuchsian manifolds and the Weil-Petersson distancehttps://www.zbmath.org/1483.300802022-05-16T20:40:13.078697Z"Mazzoli, Filippo"https://www.zbmath.org/authors/?q=ai:mazzoli.filippoSummary: Making use of the dual Bonahon-Schläfli formula, we prove that the dual volume of the convex core of a quasi-Fuchsian manifold \(M\) is bounded by an explicit constant, depending only on the topology of \(M\), times the Weil-Petersson distance between the hyperbolic structures on the upper and lower boundary components of the convex core of \(M\).Quadratic differentials and signed measureshttps://www.zbmath.org/1483.300812022-05-16T20:40:13.078697Z"Baryshnikov, Yuliy"https://www.zbmath.org/authors/?q=ai:baryshnikov.yuliy-m"Shapiro, Boris"https://www.zbmath.org/authors/?q=ai:shapiro.boris-zalmanovichSummary: In this paper, motivated by the classical notion of a Strebel quadratic differential on a compact Riemann surface without boundary, we introduce several more general classes of quadratic differentials (called non-chaotic, gradient, and positive gradient) which possess certain properties of Strebel differentials and often appear in applications. We discuss the relation between gradient differentials and special signed measures supported on their set of critical trajectories. We provide a characterization of gradient differentials for which there exists a positive measure in the latter class.On existence of quasi-Strebel structures for meromorphic \(k\)-differentialshttps://www.zbmath.org/1483.300822022-05-16T20:40:13.078697Z"Shapiro, Boris"https://www.zbmath.org/authors/?q=ai:shapiro.boris-zalmanovich"Tahar, Guillaume"https://www.zbmath.org/authors/?q=ai:tahar.guillaumeA meromorphic differential \(\Psi\) of order \(k\geq 2\) on a compact orientable Riemann surface \(Y\) without boundary is a meromorphic section of the \(k\)-th tensor power \((T^{*}_{\mathbb{C}}Y)^{\otimes k}\) of the holomorphic cotangent bundle of \(Y\). Zeros and poles of \(\Psi\) constitute the set of critical points.
For a differential of order \(k\) given locally by \(f(z)dz^{k}\) in a neighborhood of a non-critical point, there are \(k\) locally distinct directions, called \textit{horizontal}, which are given by the conditions that \(f(z)dz^{k}\) is real and positive.
Recall that a quadratic differential is called \textit{Strebel} if almost all horizontal trajectories are closed. Such a phenomenon can never happen for a \(k\)-differential of order \(k\geq 3\) unless it is a power of a \(1\)-form or a quadratic differential. The authors introduce thus the notion of quasi-Strebel structure and give sufficient conditions for a meromorphic \(k\)-differential to be quasi-Strebel. The main result is the following:
\textbf{Theorem 1.} Let \(\Psi\) be a meromorphic \(k\)-differential such that
\begin{itemize}
\item no poles have of order smaller than \(-k\);
\item at a pole of order \(-k\), the residue belongs to \(i^{k}\mathbb{R}\).
\end{itemize}
Then,
\begin{itemize}
\item[1.] if \(k>2\) is even, then \(\Psi\) has a quasi-Strebel structure;
\item[2.] if \(k>2\) is odd, and, up to a common factor, the period of \(\sqrt[k]{\Psi}\) along every path connecting any two singularities of \(\Psi\) belongs to \(\mathbb{Q}[e^{\frac{2i\pi}{k}}]\), then \(\Psi\) has a quasi-Strebel structure.
\end{itemize}
Reviewer: Andrea Tamburelli (Houston)On the discreteness of states accessible via right-angled paths in hyperbolic spacehttps://www.zbmath.org/1483.300832022-05-16T20:40:13.078697Z"Lessa, Pablo"https://www.zbmath.org/authors/?q=ai:lessa.pablo"Garcia, Ernesto"https://www.zbmath.org/authors/?q=ai:garcia.ernestoSummary: We consider the control problem where, given an orthonormal tangent frame in the hyperbolic plane or three dimensional hyperbolic space, one is allowed to transport the frame a fixed distance \(r>0\) along the geodesic in direction of the first vector, or rotate it in place a right angle. We characterize the values of \(r>0\) for which the set of orthonormal frames accessible using these transformations is discrete.
In the hyperbolic plane this is equivalent to solving the discreteness problem (see [\textit{J.Gilman}, Geom. Dedicata 201, 139--154 (2019; Zbl 1421.30056)] and the references therein) for a particular one parameter family of two-generator subgroups of \(\mathrm{PSL}_2(\mathbb{R})\). In the three dimensional case we solve this problem for a particular one parameter family of subgroups of the isometry group which have four generators.The induced metric on the boundary of the convex hull of a quasicircle in hyperbolic and anti-de Sitter geometryhttps://www.zbmath.org/1483.300842022-05-16T20:40:13.078697Z"Bonsante, Francesco"https://www.zbmath.org/authors/?q=ai:bonsante.francesco"Danciger, Jeffrey"https://www.zbmath.org/authors/?q=ai:danciger.jeffrey"Maloni, Sara"https://www.zbmath.org/authors/?q=ai:maloni.sara"Schlenker, Jean-Marc"https://www.zbmath.org/authors/?q=ai:schlenker.jean-marcA theorem by Alexandrov and Pogorelov says that any smooth Riemannian metric on the 2-sphere with curvature \(K>-1\) coincides with the induced metric on the boundary of some compact convex subset of hyperbolic 3-space with smooth boundary and, furthermore, that this compact convex subset is unique up to a global isometry of hyperbolic 3-space. In the paper under review, the authors study a generalization of this result to unbounded convex subsets of hyperbolic 3-space, more especially to convex subsets bounded by two properly embedded disks which meet at infinity along a Jordan curve in the ideal boundary. In this setting, they supplement the notion of induced metric on the boundary of the convex set so that it includes a gluing map at infinity which records how the asymptotic geometries of the two surfaces fit together near the limiting Jordan curve. They restrict their study to the case where the induced metrics on the two bounding surfaces have constant curvature \(K \in [-1, 0)\) and were the Jordan curve at infinity is a quasicircle. In this case the gluing map becomes a quasisymmetric homeomorphism of the circle and the authors prove that for \(K\) in the given interval, any quasisymmetric map can be obtained as the gluing map at infinity along some quasicircle. They also obtain Lorentzian analogous of these results, in which hyperbolic 3-space is replaced by the 3-dimensional anti-de Sitter space \(\mathbb{A}d\mathbb{S}^3\), whose natural boundary is the Einstein space \(\mathrm{Ein}^{1,1}\), a conformal Lorentzian analogue of the Riemannian sphere. The authors say that their results may be viewed as universal versions of a conjecture of Thurston about the realization of metrics on boundaries of convex cores of quasifuchsian hyperbolic manifolds and of an analogue of this conjecture, due to Mess, in the setting of globally hyperbolic anti-de Sitter spacetimes.
Reviewer: Athanase Papadopoulos (Strasbourg)From hierarchical to relative hyperbolicityhttps://www.zbmath.org/1483.300852022-05-16T20:40:13.078697Z"Russell, Jacob"https://www.zbmath.org/authors/?q=ai:russell.jacobSummary: We provide a simple, combinatorial criteria for a hierarchically hyperbolic space to be relatively hyperbolic by proving a new formulation of relative hyperbolicity in terms of hierarchy structures. In the case of clean hierarchically hyperbolic groups, this criteria characterizes relative hyperbolicity. We apply our criteria to graphs associated to surfaces and prove that the separating curve graph of a surface is relatively hyperbolic when the surface has zero or two punctures. We also recover a celebrated theorem of Brock and Masur on the relative hyperbolicity of the Weil-Petersson metric on Teichmüller space for surfaces with complexity three.Intersection pairings for higher laminationshttps://www.zbmath.org/1483.300862022-05-16T20:40:13.078697Z"Le, Ian"https://www.zbmath.org/authors/?q=ai:le.ianSummary: One can realize higher laminations as positive configurations of points in the affine building [the author, Geom. Topol. 20, No. 3, 1673--1735 (2016; Zbl 1348.30023)]. The duality pairings of \textit{V. Fock} and \textit{A. Goncharov} [Publ. Math., Inst. Hautes Étud. Sci. 103, 1--211 (2006; Zbl 1099.14025)] give pairings between higher laminations for two Langlands dual groups \(G\) and \(G^{\vee}\). These pairings are a generalization of the intersection pairing between measured laminations on a topological surface.
We give a geometric interpretation of these intersection pairings in a wide variety of cases. In particular, we show that they can be computed as the minimal weighted length of a network in the building. Thus we relate the intersection pairings to the metric structure of the affine building. This proves several of the conjectures from [the author and \textit{E. O'Dorney}, Doc. Math. 22, 1519--1538 (2017; Zbl 1383.51009)]. We also suggest the next steps toward giving geometric interpretations of intersection pairings in general.
The key tools are linearized versions of well-known classical results from combinatorics, like Hall's marriage lemma, König's theorem, and the Kuhn-Munkres algorithm, which are interesting in themselves.On an Enneper-Weierstrass-type representation of constant Gaussian curvature surfaces in 3-dimensional hyperbolic spacehttps://www.zbmath.org/1483.300872022-05-16T20:40:13.078697Z"Smith, Graham"https://www.zbmath.org/authors/?q=ai:smith.graham-a|smith.graham-mSummary: For all \(k\in ]0,1[\), we construct a canonical bijection between the space of ramified coverings of the sphere of hyperbolic type and the space of complete immersed surfaces in 3-dimensional hyperbolic space of finite area and of constant extrinsic curvature equal to \(k\). We show, furthermore, that this bijection restricts to a homeomorphism over each stratum of the space of ramified coverings of the sphere.
For the entire collection see [Zbl 1473.53006].On Bloch seminorm of finite Blaschke products in the unit diskhttps://www.zbmath.org/1483.301002022-05-16T20:40:13.078697Z"Baranov, Anton D."https://www.zbmath.org/authors/?q=ai:baranov.anton-d"Kayumov, Ilgiz R."https://www.zbmath.org/authors/?q=ai:kayumov.ilgiz-rifatovich"Nasyrov, Semen R."https://www.zbmath.org/authors/?q=ai:nasyrov.semen-rSummary: We prove that, for any finite Blaschke product \(w = B(z)\) in the unit disk, the corresponding Riemann surface over the \(w\)-plane contains a one-sheeted disk of the radius 0.5. Moreover, it contains a unit one-sheeted disk with a radial slit. We apply this result to obtain a universal lower estimate of the Bloch seminorm for finite Blaschke products.The translating soliton equationhttps://www.zbmath.org/1483.351782022-05-16T20:40:13.078697Z"López, Rafael"https://www.zbmath.org/authors/?q=ai:lopez.rafael-beltran|lopez-camino.rafaelSummary: We give an analytic approach to the translating soliton equation with a special emphasis in the study of the Dirichlet problem in convex domains of the plane.
For the entire collection see [Zbl 1473.53006].Anti-de Sitter geometry and Teichmüller theoryhttps://www.zbmath.org/1483.530022022-05-16T20:40:13.078697Z"Bonsante, Francesco"https://www.zbmath.org/authors/?q=ai:bonsante.francesco"Seppi, Andrea"https://www.zbmath.org/authors/?q=ai:seppi.andreaSummary: The aim of this chapter is to provide an introduction to Anti-de Sitter geometry, with special emphasis on dimension three and on the relations with Teichmüller theory, whose study has been initiated by the seminal paper of Geoffrey Mess in 1990. In the first part we give a broad introduction to Anti-de Sitter geometry in any dimension. The main results of Mess, including the classification of maximal globally hyperbolic Cauchy compact manifolds and the construction of the Gauss map, are treated in the second part. Finally, the third part contains related results which have been developed after the work of Mess, with the aim of giving an overview on the state-of-the-art.
For the entire collection see [Zbl 1470.57002].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)].