Prescribed Gauss curvature problem on singular surfaces

TL;DR

This paper investigates the existence of conformal metrics with prescribed Gaussian curvature on surfaces with conical singularities, especially when the curvature changes sign, by solving a singular Liouville equation using advanced variational methods.

Contribution

It introduces new perturbative existence results for conformal metrics with prescribed Gaussian curvature on singular surfaces, especially near critical topological values.

Findings

Established existence results when hi()+\u2212sum lpha_i approaches a positive even integer

Developed a min-max scheme combined with finite dimensional reduction techniques

Extended the understanding of prescribed curvature problems on singular surfaces

Abstract

We study the existence of at least one conformal metric of prescribed Gaussian curvature on a closed surface $\Sigma$ admitting conical singularities of orders $\alpha_i$'s at points $p_i$'s. In particular, we are concerned with the case where the prescribed Gaussian curvature is sign-changing. Such a geometrical problem reduces to solving a singular Liouville equation. By employing a min-max scheme jointly with a finite dimensional reduction method, we deduce new perturbative results providing existence when the quantity $\chi(\Sigma)+\sum_i \alpha_i$ approaches a positive even integer, where $\chi(\Sigma)$ is the Euler characteristic of the surface $\Sigma$.