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A sharp inequality between scalar curvature and the bottom spectrum on complete manifolds

Daoqiang Liu

Abstract

In this paper, we generalize the notion of relative $\widehat{A}$-cowaist, introduced by Cecchini and Zeidler, and establish a sharp inequality linking it to scalar curvature and the bottom spectrum. This yields a number of geometric applications, including progress on the generalized Geroch conjecture and estimates for the bottom spectrum under scalar curvature lower bounds. Our approach is based on deformed Dirac operators.

A sharp inequality between scalar curvature and the bottom spectrum on complete manifolds

Abstract

In this paper, we generalize the notion of relative -cowaist, introduced by Cecchini and Zeidler, and establish a sharp inequality linking it to scalar curvature and the bottom spectrum. This yields a number of geometric applications, including progress on the generalized Geroch conjecture and estimates for the bottom spectrum under scalar curvature lower bounds. Our approach is based on deformed Dirac operators.
Paper Structure (7 sections, 19 theorems, 96 equations)

This paper contains 7 sections, 19 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Applications of the main theorems
  3. Generalized Geroch conjecture
  4. Sharp bottom spectrum upper bounds
  5. Preliminaries
  6. Callias operators and spectral flow
  7. Proofs of main theorems

Key Result

Theorem 1.1

Let $(M,g_M)$ be an $m$-dimensional closed Riemannian spin manifold. Then For the definition of $K$-cowaist, see Gro96*p. 21 for even $m$ and Shi25+*p. 24 for odd $m$. where $\operatorname{scal}_{g_M}$ denotes the scalar curvature of $g_M$ and $c(m)>0$ is a constant depending only on $m$.

Theorems & Definitions (50)

  • Theorem 1.1: Dav03
  • Definition 1.2
  • Definition 1.3
  • Theorem A
  • Remark 1.4
  • Example 1.5
  • Theorem B
  • Theorem 2.1
  • proof
  • Remark 2.2
  • ...and 40 more