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arXiv · 2406.18997

On distance estimates for complete manifolds with lower scalar curvature bounds

Daoqiang Liu
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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.

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Daoqiang Liu. 2024-06-27. On distance estimates for complete manifolds with lower scalar curvature bounds. https://arxiv.org/abs/2406.18997

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