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

Abstract

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

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