Gap phenomenon for scalar curvature
Yukai Sun, Changliang Wang
TL;DR
The paper addresses how scalar curvature can increase under metric deformations without assuming a nonnegative curvature operator, establishing an upper bound for $\inf_{x\in N}[{\rm Sc}_{g}(x)-{\rm Sc}_{g_{0}}(f(x))]$ for area nonincreasing maps $f:N\to M$ with ${\rm deg}(f)\neq 0$ and $\chi(M)\neq 0$, namely $\inf_{x\in N}[{\rm Sc}_{g}(x)-{\rm Sc}_{g_{0}}(f(x))] \le -2n(2n-1){\mathcal R}_{\min}+2n(2n-1)|{\mathcal R}_{\min}|$. This result, derived via a twisted Dirac operator on $S(N)\otimes f^{*}S(M)$ and the Lichnerowicz-Weitzenböck-Bochner formula with a Goette-Semmelmann-type curvature-term estimate (without requiring nonnegativity of the target’s curvature operator), yields an $\varepsilon$-gap distance extremality for even-dimensional closed manifolds with nonzero Euler characteristic: for any $\varepsilon \ge 2n(2n-1)(|{\mathcal R}_{\min}|- {\mathcal R}_{\min})$, the metric $g_{0}$ is $\varepsilon$-gap distance extremal. The paper also extends the analysis to manifolds with boundary, producing a boundary mean-curvature estimate, and applies these results to a Euclidean-domain corollary related to Gromov’s extension problem for prescribing lower bounds of scalar curvature and boundary mean curvature. Overall, the work broadens scalar-curvature rigidity phenomena beyond nonnegative curvature operators, highlighting the role of topological data and index theory.
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 $ε$-gap distance extremal for some $ε\geq 0$. 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.