Prescribing Morse scalar curvatures: incompatibility of non existence

TL;DR

This paper investigates the conditions under which prescribed scalar curvatures can be realized on closed manifolds, revealing that non-existence of solutions for certain functions implies bounded solutions for related functions.

Contribution

It provides a quantitative analysis of the prescribed scalar curvature problem, highlighting the incompatibility of non-existence results with the structure of candidate functions.

Findings

Non-existence of solutions implies bounded solutions for related functions.

The shape and structure of functions influence the existence of conformal metrics.

The results connect geometric analysis with variational methods.

Abstract

Given a closed manifold of positive Yamabe invariant and for instance positive Morse functions upon it, the conformally prescribed scalar curvature problem raises the question, whether or not such functions can by conformally changing the metric be realised as the scalar curvature of this manifold. As we shall quantify depending on the shape and structure of such functions, every lack of a solution for some candidate function leads to existence of energetically uniformly bounded solutions for entire classes of related candidate functions.