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The positive mass theorem and distance estimates in the spin setting

Simone Cecchini, Rudolf Zeidler

Abstract

Let $\mathcal{E}$ be an asymptotically Euclidean end in an otherwise arbitrary complete and connected Riemannian spin manifold $(M,g)$. We show that if $\mathcal{E}$ has negative ADM-mass, then there exists a constant $R > 0$, depending only on $\mathcal{E}$, such that $M$ must become incomplete or have a point of negative scalar curvature in the $R$-neighborhood around $\mathcal{E}$ in $M$. This gives a quantitative answer to Schoen and Yau's question on the positive mass theorem with arbitrary ends for spin manifolds. Similar results have recently been obtained by Lesourd, Unger and Yau without the spin condition in dimensions $\leq 7$ assuming Schwarzschild asymptotics on the end $\mathcal{E}$. We also derive explicit quantitative distance estimates in case the scalar curvature is uniformly positive in some region of the chosen end $\mathcal{E}$. Here we obtain refined constants reminiscent of Gromov's metric inequalities with scalar curvature.

The positive mass theorem and distance estimates in the spin setting

Abstract

Let be an asymptotically Euclidean end in an otherwise arbitrary complete and connected Riemannian spin manifold . We show that if has negative ADM-mass, then there exists a constant , depending only on , such that must become incomplete or have a point of negative scalar curvature in the -neighborhood around in . This gives a quantitative answer to Schoen and Yau's question on the positive mass theorem with arbitrary ends for spin manifolds. Similar results have recently been obtained by Lesourd, Unger and Yau without the spin condition in dimensions assuming Schwarzschild asymptotics on the end . We also derive explicit quantitative distance estimates in case the scalar curvature is uniformly positive in some region of the chosen end . Here we obtain refined constants reminiscent of Gromov's metric inequalities with scalar curvature.

Paper Structure

This paper contains 9 sections, 18 theorems, 45 equations.

Table of Contents

  1. Introduction and main results
  2. Callias operators in asymptotically Euclidean manifolds
  3. Asymptotically Euclidean manifolds and mass
  4. Weighted Sobolev spaces on AE manifolds
  5. Callias operators
  6. Mass formulas
  7. Fredholm properties of the Callias operator on AE manifolds
  8. The neck of AE manifolds of nonpositive mass
  9. The positive mass theorem with arbitrary ends

Key Result

Theorem 1.1

Let $(M,g)$ be a complete asymptotically Euclidean manifold (without boundary) of dimension $n \geq 3$ which has non-negative scalar curvature. Then the ADM-mass of each end of $(M,g)$ is non-negative. Furthermore, if $M$ has an end of zero mass, then $(M,g)$ is isometric to Euclidean space.

Theorems & Definitions (37)

  • Theorem 1.1: Riemannian Positive Mass Theorem
  • Conjecture 1.2: Positive Mass Conjecture with Arbitrary Ends lesourd2020positiveSY88
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 27 more