Prescribing Gaussian curvature on closed Riemann surface with conical singularity in the negative case

Abstract

The problem of prescribing Gaussian curvature on Riemann surface with conical singularity is considered. Let $(\Sigma,\beta)$ be a closed Riemann surface with a divisor $\beta$, and $K_\lambda=K+\lambda$, where $K:\Sigma\rightarrow\mathbb{R}$ is a H\"older continuous function satisfying $\max_\Sigma K= 0$, $K\not\equiv 0$, and $\lambda\in\mathbb{R}$. If the Euler characteristic $\chi(\Sigma,\beta)$ is negative, then by a variational method, it is proved that there exists a constant $\lambda^\ast>0$ such that for any $\lambda\leq 0$, there is a unique conformal metric with the Gaussian curvature $K_\lambda$; for any $\lambda$, $0<\lambda<\lambda^\ast$, there are at least two conformal metrics having $K_\lambda$ its Gaussian curvature; for $\lambda=\lambda^\ast$, there is at least one conformal metric with the Gaussian curvature $K_{\lambda^\ast}$; for any $\lambda>\lambda^\ast$, there is no certain conformal metric having $K_{\lambda}$ its Gaussian curvature. This result is an analog of that of Ding and Liu \cite{Ding-Liu}, partly resembles that of Borer, Galimberti and Struwe \cite{B-G-Stru}, and generalizes that of Troyanov \cite{Troyanov} in the negative case.