Recent zbMATH articles in MSC 30Fhttps://www.zbmath.org/atom/cc/30F2021-04-16T16:22:00+00:00WerkzeugConformal invariants associated with quadratic differentials.https://www.zbmath.org/1456.300472021-04-16T16:22:00+00:00"Schippers, Eric"https://www.zbmath.org/authors/?q=ai:schippers.eric-dSummary: We associate a functional of pairs of simply-connected regions \(D_2\subseteq D_1\) to any quadratic differential on \(D_1\) with specified singularities. This functional is conformally invariant, monotonic, and negative. Equality holds if and only if the inner domain is the outer domain minus trajectories of the quadratic differential. This generalizes the simply-connected case of results of \textit{Z. Nehari} [Trans. Am. Math. Soc. 75, 256--286 (1953; Zbl 0051.31204)], who developed a general technique for obtaining inequalities for conformal maps and domain functions from contour integrals and the Dirichlet principle for harmonic functions. Nehari's method corresponds to the special case that the quadratic differential is of the form \((\partial q)^2\) for a singular harmonic function \(q\) on \(D_1\).
As an application we give a one-parameter family of monotonic, conformally invariant functionals which correspond to growth theorems for bounded univalent functions. These generalize and interpolate the Pick growth theorems, which appear in a conformally invariant form equivalent to a two-point distortion theorem of \textit{W. Ma} and \textit{D. Minda} [Ann. Acad. Sci. Fenn., Math. 22, No. 2, 425--444 (1997; Zbl 0908.30016)].Characterizations of circle patterns and finite convex polyhedra in hyperbolic 3-space.https://www.zbmath.org/1456.520252021-04-16T16:22:00+00:00"Huang, Xiaojun"https://www.zbmath.org/authors/?q=ai:huang.xiaojun.1"Liu, Jinsong"https://www.zbmath.org/authors/?q=ai:liu.jinsongSummary: The aim of this paper is to study finite convex polyhedra in three dimensional hyperbolic space \({\mathbb {H}}^3\). We characterize the quasiconformal deformation space of each finite convex polyhedron. As a corollary, we obtain some results on finite circle patterns in the Riemann sphere with \textit{dihedral angle} \(0\leq \Theta < \pi \). That is, for any circle pattern on \(\hat{\mathbb {C}}\), its quasiconformal deformation space can be naturally identified with the product of the Teichmüller spaces of its interstices.The Riemann-Roch theorem.https://www.zbmath.org/1456.300752021-04-16T16:22:00+00:00"A'Campo, Norbert"https://www.zbmath.org/authors/?q=ai:acampo.norbert"Alberge, Vincent"https://www.zbmath.org/authors/?q=ai:alberge.vincent"Frenkel, Elena"https://www.zbmath.org/authors/?q=ai:frenkel.elenaSummary: We sketch here a proof of the Riemann-Roch theorem.
For the entire collection see [Zbl 1381.01003].Conformal embeddings of an open Riemann surface into another -- a counterpart of univalent function theory.https://www.zbmath.org/1456.300802021-04-16T16:22:00+00:00"Shiba, Masakazu"https://www.zbmath.org/authors/?q=ai:shiba.masakazuSummary: We study conformal embeddings of a noncompact Riemann surface of finite genus into compact Riemann surfaces of the same genus and show some of the close relationships between the classical theory of univalent functions and our results. Some new problems are also discussed. This article partially intends to introduce our results and to invite the function-theorists on plane domains to the topics on Riemann surfaces.Loewner theory for quasiconformal extensions: old and new.https://www.zbmath.org/1456.300412021-04-16T16:22:00+00:00"Hotta, Ikkei"https://www.zbmath.org/authors/?q=ai:hotta.ikkeiSummary: This survey article gives an account of quasiconformal extensions of univalent functions with its motivational background from Teichmüller theory and classical and modern approaches based on Loewner theory.On the Hardy number of a domain in terms of harmonic measure and hyperbolic distance.https://www.zbmath.org/1456.300242021-04-16T16:22:00+00:00"Karafyllia, Christina"https://www.zbmath.org/authors/?q=ai:karafyllia.christinaThe author establishes the Hardy number of a domain in terms of harmonic measure and hyperbolic distance. For a domain \(D\subset\mathbb C\) and \(z\in D\) and a Borel subset \(E\) of the closure \(\overline D\) of \(D\), let \(\omega_D(z,E)\) be the harmonic measure at \(z\) of \(\overline E\) with respect to the component of \(D\setminus\overline E\) containing \(z\), and let \(d_D(z,w)\) be the hyperbolic distance between \(z\) and \(w\) in \(E\) for a simply connected domain \(D\ne\mathbb C\). Denote by \(\psi\) a conformal map on the unit disk \(\mathbb D=\{z\in\mathbb C:|z|<1\}\), \(\psi(0)=0\), and \(F_{\alpha}=\{z\in\mathbb D:|\psi(z)|=\alpha\}\), \(\alpha>0\). The Hardy number of \(\psi\) is given by \(h(\psi)=\sup\{p>0: \psi\in H^p(\mathbb D)\}\), where \(H^p(\mathbb D)\) is the Hardy space on \(\mathbb D\). The following theorem expresses \(h(\psi)\) in terms of the hyperbolic distance.
Theorem 1.1. If \(h(\psi)\) denotes the Hardy number of \(\psi\), then \[h(\psi)=\liminf_{\alpha\to+\infty}\frac{d_{\mathbb D}(0,F_{\alpha})}{\log\alpha}.\]
Denote \[L=\lim_{\alpha\to+\infty}(\log\omega_{\mathbb D}(0,F_{\alpha})^{-1}/\log\alpha),\;\;\;\mu=\lim_{\alpha\to+\infty}(d_{\mathbb D}(0,F_{\alpha})/\log\alpha).\]
Theorem 1.3. If \(\mu\) exists, then \(L\) exists and \(L=\mu\).
Let \(N(\alpha)\in\mathbb N\cup\{\infty\}\) denote the number of components of \(F_{\alpha}\), \(\alpha>0\), and let \(F_{\alpha}^i\) denote each of these components, \(i=1,\dots,N(\alpha)\). Denote by \(F_{\alpha}^*\) a component of \(F_{\alpha}\) such that \(\omega_{\mathbb D}(0,F_{\alpha}^*)=\max_i\{\omega_{\mathbb D}(0,F_{\alpha}^i)\}\).
Theorem 1.5. If \(L\) exists, then \(\mu\) exists if and only if \[\limsup_{\alpha\to+\infty}\frac{\log_{\mathbb D}\omega(0,F_{\alpha}^*)^{-1}}{\log\alpha}=L.\] If \(\mu\) exists, then \(\mu=L\).
Reviewer: Dmitri V. Prokhorov (Saratov)On the non-smoothness of the vector fields for the dynamically invariant Beltrami coefficients.https://www.zbmath.org/1456.300422021-04-16T16:22:00+00:00"Huo, Shengjin"https://www.zbmath.org/authors/?q=ai:huo.shengjin"Guo, Hui"https://www.zbmath.org/authors/?q=ai:guo.huiSummary: For \(\mu \in L^{\infty}(\Delta )\), the vector fields on the unit circle determined by \(\mu \) play an important role in the theory of the universal Teichmüller space. The aim of this paper is to give some characterizations of the vector fields induced by dynamically invariant \(\mu \). We show that those vector fields are not contained in the Sobolev class \(H^{3/2}\). At last, we give some results on dynamically invariant vectors to show that the vector fields, the quasi-symmetric homeomorphisms, and the quasi-circles are closely related.Schiffer operators and calculation of a determinant line in conformal field theory.https://www.zbmath.org/1456.300262021-04-16T16:22:00+00:00"Radnell, David"https://www.zbmath.org/authors/?q=ai:radnell.david"Schippers, Eric"https://www.zbmath.org/authors/?q=ai:schippers.eric-d"Shirazi, Mohammad"https://www.zbmath.org/authors/?q=ai:shirazi.mohammad"Staubach, Wolfgang"https://www.zbmath.org/authors/?q=ai:staubach.wolfgangSummary: We consider an operator associated to compact Riemann surfaces endowed with a conformal map, \(f\), from the unit disk into the surface, which arises in conformal field theory. This operator projects holomorphic functions on the surface minus the image of the conformal map onto the set of functions \(h\) so that the Fourier series \(h \circ f\) has only negative powers. We give an explicit characterization of the cokernel, kernel, and determinant line of this operator in terms of natural operators in function theory.Families of elliptic functions, realizing coverings of the sphere, with branch-points and poles of arbitrary multiplicities.https://www.zbmath.org/1456.300762021-04-16T16:22:00+00:00"Nasyrov, S."https://www.zbmath.org/authors/?q=ai:nasyrov.samyon|nasyrov.s-rSummary: We investigate smooth one-parameter families of complex tori over the Riemann sphere. The main problem is to describe such families in terms of projections of their branch-points. Earlier we investigated the problem for the case where, for every torus of the family, there is only one point lying over infinity. Here we consider the general case. We show that the uniformizing functions satisfy a partial differential equation and derive a system of differential equations for their critical points, poles, and moduli of tori. Based on the system we suggest an approximate method allowing to find an elliptic function uniformizing a given genus one ramified covering of the Riemann sphere.Morrey type Teichmüller space and higher Bers maps.https://www.zbmath.org/1456.300782021-04-16T16:22:00+00:00"Hu, Guangming"https://www.zbmath.org/authors/?q=ai:hu.guangming"Liu, Yutong"https://www.zbmath.org/authors/?q=ai:liu.yutong"Qi, Yi"https://www.zbmath.org/authors/?q=ai:qi.yi"Shi, Qingtian"https://www.zbmath.org/authors/?q=ai:shi.qingtianSummary: In this paper, we focus on the set of univalent analytic functions \(f\) with \(\log f' \in H_K^2\). Motivated by the study of BMO-Teichmüller spaces and Morrey type spaces, we establish serval equivalent characterizations of Morrey type domains. Furthermore, we show that the higher Bers maps, induced by the higher Schwarzian differential operators, are holomorphic in Morrey type Teichmüller spaces. Finally, one of connected components in the small pre-logarithmic derivative model of the Morrey type Teichmüller space is also obtained.Pressure metrics and Manhattan curves for Teichmüller spaces of punctured surfaces.https://www.zbmath.org/1456.300792021-04-16T16:22:00+00:00"Kao, Lien-Yung"https://www.zbmath.org/authors/?q=ai:kao.lien-yungSummary: In this paper, we extend the construction of pressure metrics to Teichmüller spaces of surfaces with punctures. This construction recovers Thurston's Riemannian metric on Teichmüller spaces. Moreover, we prove the real analyticity and convexity of Manhattan curves of finite area type-preserving Fuchsian representations, and thus we obtain several related entropy rigidity results. Lastly, relating the two topics mentioned above, we show that one can derive the pressure metric by varying Manhattan curves.A generalized Hurwitz metric.https://www.zbmath.org/1456.300772021-04-16T16:22:00+00:00"Arstu"https://www.zbmath.org/authors/?q=ai:arstu."Sahoo, Swadesh Kumar"https://www.zbmath.org/authors/?q=ai:sahoo.swadesh-kumarSummary: In 2016, the Hurwitz metric was introduced by D. Minda in arbitrary proper subdomains of the complex plane and he proved that this metric coincides with the Poincaré's hyperbolic metric when the domains are simply connected. In this paper, we provide an alternate definition of the Hurwitz metric through which we could define a generalized Hurwitz metric in arbitrary subdomains of the complex plane. This paper mainly highlights various important properties of the Hurwitz metric and the generalized metric including the situations where they coincide with each other.Unveiling the fractal structure of Julia sets with Lagrangian descriptors.https://www.zbmath.org/1456.320062021-04-16T16:22:00+00:00"García-Garrido, Víctor J."https://www.zbmath.org/authors/?q=ai:garcia-garrido.victor-jSummary: In this paper we explore by means of the method of Lagrangian descriptors the Julia sets arising from complex maps, and we analyze their underlying dynamics. In particular, we take a look at two classical examples: the quadratic mapping \(z_{n+1}=z_n^2+c\), and the maps generated by applying Newton's method to find the roots of complex polynomials. To achieve this goal, we provide an extension of this scalar diagnostic tool that is capable of revealing the phase space of open maps in the complex plane, allowing us to avoid potential issues of orbits escaping to infinity at an increasing rate. The simple idea is to compute the \(p\)-norm version of Lagrangian descriptors, not for the points on the complex plane, but for their projections onto the Riemann sphere in the extended complex plane. We demonstrate with several examples that this technique successfully reveals the rich and intricate dynamical features of Julia sets and their fractal structure.Polyakov-Alvarez type comparison formulas for determinants of Laplacians on Riemann surfaces with conical singularities.https://www.zbmath.org/1456.580242021-04-16T16:22:00+00:00"Kalvin, Victor"https://www.zbmath.org/authors/?q=ai:kalvin.victorSummary: We present and prove Polyakov-Alvarez type comparison formulas for the determinants of Friederichs extensions of Laplacians corresponding to conformally equivalent metrics on a compact Riemann surface with conical singularities. In particular, we find how the determinants depend on the orders of conical singularities. We also illustrate these general results with several examples: based on our Polyakov-Alvarez type formulas we recover known and obtain new explicit formulas for determinants of Laplacians on singular surfaces with and without boundary. In one of the examples we show that on the metrics of constant curvature on a sphere with two conical singularities and fixed area \(4 \pi\) the determinant of Friederichs Laplacian is unbounded from above and attains its local maximum on the metric of standard round sphere. In another example we deduce the famous Aurell-Salomon formula for the determinant of Friederichs Laplacian on polyhedra with spherical topology, thus providing the formula with mathematically rigorous proof.