Conformal metrics of prescribed scalar curvature on $4-$manifolds: The degree zero case

TL;DR

This paper establishes new existence criteria for conformal metrics with prescribed positive scalar curvature on 4-manifolds, extending previous results and providing Morse index estimates and inequalities at infinity.

Contribution

It introduces a generalized existence criterion for conformal metrics with prescribed scalar curvature on 4-manifolds, including Morse inequalities at infinity and estimates on Morse indices.

Findings

Existence results for a dense subset of positive functions

Extension of Bahri-Coron and Chang-Gursky-Yang criteria

Morse inequalities at infinity for generic functions

Abstract

In this paper, we consider the problem of existence and multiplicity of conformal metrics on a riemannian compact $4-$dimensional manifold $(M^4,g_0)$ with positive scalar curvature. We prove new exitence criterium which provides existence results for a dense subset of positive functions and generalizes Bahri-Coron and Chang-Gursky-Yang Euler-Hopf type criterium. Our argument gives estimates on the Morse index of the solutions and has the advantage to extend known existence results. Moreover it provides, for generic $K$ {\it Morse Inequalities at Infinity}, which give a lower bound on the number of metrics with prescribed scalar curvature in terms of the topological contribution of its "critical points at Infinity" to the difference of topology between the level sets of the associated Euler-Lagrange functional.