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Positive mass and Dirac operators on weighted manifolds and smooth metric measure spaces

Michael B. Law, Isaac M. Lopez, Daniel Santiago

Abstract

We establish a weighted positive mass theorem which unifies and generalizes results of Baldauf--Ozuch and Chu--Zhu. Our result is in fact equivalent to the usual positive mass theorem, and can be regarded as a positive mass theorem for smooth metric measure spaces. We also study Dirac operators on certain warped product manifolds associated to smooth metric measure spaces. Applications of this include, among others, an alternative proof for a special case of our positive mass theorem, eigenvalue bounds for the Dirac operator on closed spin manifolds, and a new way to understand the weighted Dirac operator using warped products.

Positive mass and Dirac operators on weighted manifolds and smooth metric measure spaces

Abstract

We establish a weighted positive mass theorem which unifies and generalizes results of Baldauf--Ozuch and Chu--Zhu. Our result is in fact equivalent to the usual positive mass theorem, and can be regarded as a positive mass theorem for smooth metric measure spaces. We also study Dirac operators on certain warped product manifolds associated to smooth metric measure spaces. Applications of this include, among others, an alternative proof for a special case of our positive mass theorem, eigenvalue bounds for the Dirac operator on closed spin manifolds, and a new way to understand the weighted Dirac operator using warped products.
Paper Structure (17 sections, 21 theorems, 101 equations, 1 table)

This paper contains 17 sections, 21 theorems, 101 equations, 1 table.

Table of Contents

  1. Introduction
  2. Equivalence between weighted and unweighted positive mass theorems
  3. A positive mass theorem for smooth metric measure spaces
  4. Warped product Dirac operators and applications
  5. Positive mass theorems
  6. Asymptotically Euclidean manifolds and mass
  7. A weighted positive mass theorem
  8. The mass of a smooth metric measure space
  9. A positive mass theorem for smooth metric measure spaces
  10. Spin geometry on warped products
  11. Generalities on spin geometry
  12. The spinor bundle of a warped product
  13. The warped product spin connection and Dirac operator
  14. Applications of spinors on the warped product, and more
  15. A spin proof of our positive mass theorem
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $(M^n,g)$, $n \geq 3$ be an AE manifold of order $\tau > \frac{n-2}{2}$, and assume that $3 \leq n \leq 7$ or $M$ is spin. If $(M^n,g)$ has nonnegative scalar curvature $R \geq 0$, then it has nonnegative ADM mass $\mathfrak{m}(g) \geq 0$, with equality if and only if $(M^n,g)$ is isometric to $

Theorems & Definitions (46)

  • Theorem 1.1: schoen1979PMT1schoen1979PMT2witten1981new
  • Theorem 1.2: baldauf2022spinorschu2023nonspin
  • Theorem 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Remark
  • Theorem 1.7
  • Corollary 1.8
  • Definition 2.1
  • ...and 36 more