Prescribing scalar curvatures: loss of minimizability

TL;DR

This paper investigates the conditions under which scalar curvature can be prescribed on closed manifolds, highlighting when minimizers exist or fail, and constructing examples with saddle point solutions.

Contribution

It analyzes the loss of minimizability in scalar curvature prescription problems and constructs specific functions where saddle point solutions still exist.

Findings

Loss of minimizability under certain conditions.

Existence of saddle point solutions despite non-minimizable cases.

Conditions for uniqueness and non-uniqueness of solutions.

Abstract

Prescribing conformally the scalar curvature on a closed manifold with negative Yamabe invariant as a given function $K$ is possible under smallness assumptions on $K_{+}=\max\{K,0\}$ and in particular, when $K<0$. In addition, while solutions are unique in case $K\leq 0$, non uniqueness generally holds, when $K$ is sign changing and $K_{+}$ sufficiently small and flat around its critical points. These solutions are found variationally as minimizers. Here we study, what happens, when the relevant arguments fail to apply, describing on one hand the loss of minimizability generally, while on the other we construct a function $K$, for which saddle point solutions to the conformally prescribed scalar curvature problem still exist.