Compactness issues and bubbling phenomena for the prescribed Gaussian curvature equation on the Torus

TL;DR

This paper investigates the behavior of conformal metrics with prescribed Gaussian curvature on the torus, focusing on their compactness and bubbling phenomena as parameters approach boundary values.

Contribution

It extends previous work on higher genus surfaces to the torus, analyzing the bubbling behavior of solutions to the prescribed Gaussian curvature equation.

Findings

Characterization of bubbling phenomena on the torus.

Analysis of compactness issues for conformal metrics.

Behavior of solutions as parameters approach boundary points.

Abstract

In the spirit of the paper "Large conformal metrics of prescribed Gauss curvature on surfaces of higher genus" by Borer-Galimberti-Struwe, where we dealt with the case of a closed Riemann surface $(M,g_0)$ of genus greater than one, here we study the behaviour of the conformal metrics $g_\lambda$ of prescribed Gauss curvature $K_{g_\lambda} = f_0 + \lambda$ on the torus, when the parameter $\lambda$ tends to one of the boundary points of the interval of existence of $g_\lambda$, and we characterize their "bubbling behaviour".