Extremal Hermitian metrics on Riemann surfaces with singularities
type: Item
instance of: scholarly article
title: Extremal Hermitian metrics on Riemann surfaces with singularities
zbMATH Open document ID: 0980.58009
DOI: 10.1215/S0012-7094-00-10421-8
author: Guofang Wang
author: Xiaohua Zhu
published in: Duke Mathematical Journal
publication date: 2002-02-27T00:00:00Z
review text: In this paper the authors discuss the extremal Hermitian metrics with finite energy and area on Riemannian surfaces with conical singularities. After establishing the preliminary technical results they prove the following main theorem: Let \(M\) be a compact Riemann surface, let \(g\) be an extremal Hermitian metric with finite energy and area on \(M\setminus \{p_j\}_{j=1,\ldots ,n}\), and let \(K\) be the Gaussian curvature of \(g\). Then \(g\) is a conical metric with singular angle \(\alpha _j\), \((j=1,\ldots , n)\) (which may include some weak cusps). Furthermore, if all singular angles satisfy \(2\pi\alpha_j\leq\pi\), then the following classification holds: (i) If \(\text{genus} (M)\geq 1\), then \(K\equiv \text{const.}\) (ii) If \(M=S^2\) and \(n\geq 3\), then \(K\equiv \text{const.}\) (iii) If \(M=S^2\) and \(n=2\), there are two cases: (a) If both singular points are cusps, then there is no extremal Hermitian metric; (b) If one of the singular points is not a cusp, then \(g\) is a rotationally symmetric, extremal Hermitian metric that is determined uniquely by the total area and two angles \(2\pi\alpha_j\). (iv) If \(M=S^2\) and \(n=1\), then \(g\) is a rotationally symmetric metric that is determined uniquely by the total area and angle \(2\pi\alpha\).
Mathematics Subject Classification ID: 58E11
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 1505822
zbMATH Keywords: conical singularity
zbMATH Keywords: energy functional
zbMATH Keywords: extremal Hermitian metric
zbMATH Keywords: cusp
reviewed by: Witold Mozgawa
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1505822