The differential equation $\Delta u = 8\pi - 8\pi h\exp {u}$ on a compact Riemann surface

W. Ding; J. Jost; J. Li; G. Wang

The differential equation $\Delta u = 8\pi - 8\pi h\exp {u}$ on a compact Riemann surface

dg-ga 2007-05-23 v1 Differential Geometry

Authors: W. Ding , J. Jost , J. Li , G. Wang

Abstract

Let $M$ be a compact Riemann surface, $h(x)$ a positive smooth function on $M$ . In this paper, we consider the functional $J(u)={1/2}\int \sb{M}|\bigtriangledown u|\sp 2 + 8\pi \int\sb{M}u -8\pi \log\int\sb{M}h\exp {u}$ . We give a sufficient condition under which $J$ achieves its minimum.

Keywords

differential equation compact Riemann surface functional minimum smooth function

Cite

@article{arxiv.dg-ga/9710005,
  title  = {The differential equation $\Delta u = 8\pi - 8\pi h\exp {u}$ on a compact Riemann surface},
  author = {W. Ding and J. Jost and J. Li and G. Wang},
  journal= {arXiv preprint arXiv:dg-ga/9710005},
  year   = {2007}
}

Comments

26 pages, Latex, to appear in Asian J. Math. 1(1997) No. 2

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