Preprint 34/1999

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 tex2html_wrap_inline24 and K the Gaussian curvature of g. Then g is a conical metric with singular angles tex2html_wrap_inline32 (tex2html_wrap_inline34) (which may include some weak cusps). Furthermore if all singular angles satisfy
displaymath36
then the following classifications hold:

  1. If tex2html_wrap_inline38, then tex2html_wrap_inline40.;
  2. If tex2html_wrap_inline42 and tex2html_wrap_inline44, then tex2html_wrap_inline40.;
  3. If tex2html_wrap_inline42 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 tex2html_wrap_inline54.
  4. If tex2html_wrap_inline42 and n=1, then g is a rotationally symmetric metric determined uniquely by the total area and angle tex2html_wrap_inline62.

 

03.07.2017, 01:40