Limit theorems for the total scalar curvature

TL;DR

This paper investigates the stability of lower bounds of total scalar curvature, especially Perelman's $ ext{F}$-functional, under specific convergence conditions of Riemannian metrics and potential functions, using Ricci flow stability.

Contribution

It establishes the preservation of the lower bound of weighted total scalar curvature under $W^{1,p}$ and $C^{0}$ convergence, expanding understanding of scalar curvature behavior under metric limits.

Findings

Lower bounds of weighted total scalar curvature are preserved under specified convergence.

Stability results for Ricci flow and heat flow are utilized in the proofs.

Examples suggest potential minimal topologies for such preservation phenomena.

Abstract

We study some preservation phenomena for lower bound of total scalar curvatures on a smooth manifold. In particular, we prove that the lower bound of the weighted total scalar curvature (which is known as Perelman's $\mathcal{F}$-functional) on a closed $n$-manifold is preserved under the $W^{1, p}~(p > n^{2}/2)$-convergence of Riemannian metrics and uniformly $C^{0}$-convergence of potential functions, provided that each scalar curvature is nonnegative. In the proof, we used a certain stability of the Ricci flow and the heat flow with the Ricci flow background. We also give some examples that may provide clues to identify the weakest topology for such a preservation phenomenon of the lower bound.