Recent zbMATH articles in MSC 30F10https://www.zbmath.org/atom/cc/30F102022-05-16T20:40:13.078697ZWerkzeugMatrix 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 \(M\).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)].