A scalar curvature flow in low dimensions

TL;DR

This paper studies a scalar curvature flow on low-dimensional closed manifolds with positive Yamabe invariant, proving global existence and convergence under certain conditions, thus advancing the understanding of prescribed scalar curvature problems.

Contribution

It introduces a scalar curvature flow approach for prescribing scalar curvature on low-dimensional manifolds and establishes conditions for its convergence and solvability.

Findings

Global existence of the flow on 3-5 dimensional manifolds.

Convergence of the flow under certain conditions.

Solvability of the prescribed scalar curvature problem.

Abstract

Let $(M^{n},g_{0})$ be a $n=3,4,5$ dimensional, closed Riemannian manifold of positive Yamabe invariant. For a smooth function $K>0$ on $M$ we consider a scalar curvature flow, that tends to prescribe $K$ as the scalar curvature of a metric $g$ conformal to $g_{0}$. We show global existence and in case $M$ is not conformally equivalent to the standard sphere smooth flow convergence and solubility of the prescribed scalar curvature problem under suitable conditions on $K$.