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 $K:\Sigma \to \mathbb{R}$ is a Hölder 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 $\lambda \le 0$, 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.

15 pages

References in corpus (1)

• Extremal functions for singular Trudinger-Moser inequalities in the entire Euclidean space