The Equation Δu+∇φ•∇u=8πc(1-heu) on a Riemann Surface

Abstract

Let M be a compact Riemann surface, h(x) a positive smooth function on M, and f(x) a smooth function on M which satisfies that $∫_Me^φdV_g=1$. In this paper, we consider the functional  $J(u)=½∫_M|∇u|^2e^φdV_g+8πc∫_Mue^φdV_g-8πclog∫_Mhe^{u+φ}dV_g$. We give a sufficient condition under which J achieves its minimum for $c≤inf_{x∈M^{e^φ(x)}}$.