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

TL;DR

This paper studies a specific nonlinear differential equation on compact Riemann surfaces, providing conditions for the associated functional to attain its minimum, which is relevant for understanding solutions to geometric PDEs.

Contribution

It establishes a sufficient condition for the functional related to the differential equation to achieve its minimum on a compact Riemann surface.

Findings

Derived a sufficient condition for the functional to attain its minimum.

Connected the minimization problem to solutions of the differential equation.

Provided insights into the existence of solutions on Riemann surfaces.

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.