Prescribing scalar curvatures: on the negative Yamabe case

TL;DR

This paper investigates the problem of prescribing scalar curvature on negatively curved manifolds, providing new variational insights and conditions for existence and non-uniqueness of solutions when the prescribed function changes sign.

Contribution

It offers a new variational approach to the negative Yamabe problem, recovering and quantifying previous existence results and establishing conditions for non-uniqueness of solutions.

Findings

Solvability when the prescribed function is strictly negative.

Recovery and quantification of Rauzy's existence principle.

Existence of multiple solutions under certain conditions.

Abstract

The problem of prescribing conformally the scalar curvature on a closed Riemannian manifold of negative Yamabe invariant is always solvable, when the function $K$ to be prescribed is strictly negative, while sufficient and necessary conditions are known for $K\leq 0$. For sign changing $K$ Rauzy showed solvability, if $K$ is not too positive. We revisit this problem in a different variational context, thereby recovering and quantifying the principle existence result of Rauzy and show under additional assumptions, that for a sign changing $K$ solutions to the conformally prescribed scalar curvature problem, while existing, are not unique.