# A gradient flow for the prescribed Gaussian curvature problem on a closed Riemann surface with conical singularity

## TL;DR

This paper establishes the well-posedness, long-term existence, and convergence of a gradient flow on conical Riemann surfaces, solving the prescribed Gaussian curvature problem for surfaces with non-positive singular Euler characteristic.

## Contribution

It extends the gradient flow method to surfaces with conical singularities, providing new solutions to the prescribed Gaussian curvature problem in this setting.

## Key findings

- Flow is well-posed on conical surfaces.
- Long-term existence and convergence are proven.
- Prescribed Gaussian curvature problem is solved for non-positive singular Euler characteristic.

## Abstract

In this note, we prove that the abstract gradient flow introduced by Baird-Fardoun-Regbaoui \cite{BFR}is well-posed on a closed Riemann surface with conical singularity. Long time existence and convergence of the flow are proved under certain assumptions. As an application, the prescribed Gaussian curvature problem is solved when the singular Euler characteristic of the conical surface is non-positive.