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

## TL;DR

This paper studies the problem of prescribing Gaussian curvature on closed Riemann surfaces with conical singularities in the negative Euler characteristic case, establishing existence, uniqueness, and multiplicity results for conformal metrics based on a variational approach.

## Contribution

It provides new existence and multiplicity results for conformal metrics with prescribed Gaussian curvature on singular surfaces, extending previous work to the negative Euler characteristic case.

## Key findings

- Existence of a unique conformal metric for certain negative curvature parameters.
- Multiple conformal metrics exist for a range of curvature parameters.
- No conformal metric exists beyond a critical curvature threshold.

## 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.