Extremal Hermitian metrics on Riemann surfaces
The paper under review is concerned with the following variational problem. Given a compact Riemann surface \(M\) and a Hermitian netric \(g_0\) on \(M\), minimize the energy functional \(E(g)=\int_MK^2_gdg\) with the area constraint \(A(g)=\int_Mdg=\) constant over the space of metrics \({\mathcal G}(M)\) consisting of the closure of the set of smooth metrics on \(M\) that are pointwise conformally related to \(g_0\) and agree with \(g_0\) in a neighborhood of \(\partial M\), where \(K_g\) is the Gauss curvature of \(g\). Critical points are called extremal Hermitian metrics. Then, the main goal is to classify the extremal Hermitian metrics on a punctured Riemann surface having a finite number of weak cusp singularities as in the following: Theorem A. Let \(g\) be any extremal Hermitian metric on \(M\setminus\{p_1,p_2, \dots, p_n\}\) \((\partial M=\emptyset)\) with finite energy and area (i.e., \(E(g)\) and \(A(g)\) are finite), and suppose \(g\) has only weak cusp singularities at all the singular points, then 1. If \(\chi (M)\leq 0\), then \(K\) is a negative constant. 2. If \(n\geq 3\) and \(\chi (M)=2\), then \(K\) is a negative constant. 3. If \(n=2\) and \(\chi(M)=2\), there exists no extremal Hermitian metric. 4. If \(n=1\) and \(\chi(M)=2\), there exists a unique extremal Hermitian metric determined by the total area.
scientific article; zbMATH DE number 1299952