On distance estimates for complete manifolds with lower scalar curvature bounds

TL;DR

This paper develops new distance estimates for complete manifolds with boundary under scalar curvature bounds, introducing a relative index concept and deriving inequalities and width estimates related to scalar curvature.

Contribution

It introduces a novel relative index for Gromov-Lawson pairs using a deformed Dirac operator and applies it to derive geometric inequalities and width estimates.

Findings

Proves a short neck inequality for manifolds with positive scalar curvature.

Establishes a quantitative shielding result for nonnegative scalar curvature.

Generalizes the relative -area concept to manifolds with boundary.

Abstract

In this paper, we focus on the distance estimate problem on complete manifolds with compact boundary and with lower scalar curvature bounds. On these manifolds, relative to a background manifold with nonnegative curvature operator, we introduce a definition of the relative index of relative Gromov-Lawson pairs via a deformed Dirac operator trick in \cite{Zh20}. We prove that the relative index coincides with the index of associated Callias operators of the relative Gromov-Lawson pairs. As applications, we prove a short neck inequality with uniformly positive scalar curvature and the corresponding quantitative shielding result with nonnegative scalar curvature. Moreover, we generalize the concept of relative $\widehat{A}$-area in \cite{CZ24} to complete manifolds with compact boundary and then investigate width estimates of geodesic collar neighborhoods.