Extremal Hermitian metrics on Riemann surfaces with singularities
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 \(2\pi\alpha\).
scientific article; zbMATH DE number 1505822
scientific article