Uniqueness of conformal metrics with prescribed total curvature in $R^2$

Abstract.

In this paper, we investigate the solution structure of solutions of

\begin{equation} \label{0.1} \Delta u(x)+K(x) e^{2u}=0 \mbox{in} \mathbb{R}^2, \end{equation}

where K(x) is a Hölder function in $\mathbb{R}^2$. For a given positive total curvature, we consider the problem of the uniqueness of solutions with this prescribed total curvature. We apply various methods such as the method of moving spheres and the isoperimetric inequality to show the uniqueness for several classes of K.