Extremal Hermitian metrics on Riemann surfaces with singularities

Guofang Wang and Xiaohua Zhu

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Submission date: 04. May. 1999
Pages: 30
published in: Duke mathematical journal, 104 (2000) 2, p. 181-210 
Bibtex

Abstract:
In this paper, we discuss the extremal Hermitian metrics with finite energy and area on compact Riemann surfaces with conical singularities. We obtain the following classification theorem of such metrics, which generalizes results of X. X. Chen: Let M be a compact Riemann surface, g an extremal Hermitian metric with finite energy and area on and K the Gaussian curvature of g. Then g is a conical metric with singular angles () (which may include some weak cusps). Furthermore if all singular angles satisfy

then the following classifications hold:

1. If , then .;
2. If and , then .;
3. If and n=2, then there are two cases:
   1. if both singular points are cusp, then there is no extremal Hermitian metric;
   2. if one of singular points is not cusp, then g is a rotationally symmetric extremal Hermitian metric determined uniquely by the total area and two angles .
4. If and n=1, then g is a rotationally symmetric metric determined uniquely by the total area and angle .