Gap phenomenon for scalar curvature

TL;DR

This paper develops new estimates for scalar curvature without assuming nonnegative curvature operator and explores their implications for extremal metrics and boundary extension problems on manifolds.

Contribution

It introduces novel scalar curvature estimates inspired by previous work, applicable to manifolds with and without boundary, and addresses Gromov's boundary extension question.

Findings

Scalar curvature estimates without nonnegativity assumption

Any metric on certain even-dimensional manifolds is an epsilon-gap extremal

Applications to boundary extension problems and Gromov's question

Abstract

Inspired by Goette-Semmelmann \cite{GSSU2002}, we derive an estimate for the scalar curvature without a nonnegativity assumption on curvature operator. As an application, we show that, on an even dimensional closed manifold with nonzero Euler characteristic, any Riemannian metric $g$ is $\epsilon$-gap distance extremal for some $\epsilon \geq 0$. For manifolds with boundary, inspired by Lott \cite{JL2021}, we obtained a similar estimate for scalar curvature and mean curvature. We apply the estimate on certain Euclidean domains to study a Gromov's question in \cite{GM20233} concerning the extension problem of metric on the boundary to the interior.