Prescribing Morse scalar curvatures: pinching and Morse theory

TL;DR

This paper investigates the problem of prescribing scalar curvature on certain compact manifolds, establishing new existence and non-existence results using Morse theory and analysis of blow-up solutions.

Contribution

It introduces novel existence results under pinching conditions and demonstrates the sharpness of these conditions with non-existence results, advancing understanding of scalar curvature prescription.

Findings

New existence results for prescribed scalar curvature under pinching conditions

Non-existence results showing sharpness of assumptions

Analysis of blow-up solutions in the prescribed curvature problem

Abstract

We consider the problem of prescribing conformally the scalar curvature on compact manifolds of positive Yamabe class in dimension $n \geq 5$. We prove new existence results using Morse theory and some analysis on blowing-up solutions, under suitable pinching conditions on the curvature function. We also provide new non-existence results showing the sharpness of some of our assumptions, both in terms of the dimension and of the Morse structure of the prescribed function.