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Positive mass theorems for manifolds with ALH toroidal ends

Gregory J. Galloway, Tin-Yau Tsang

TL;DR

The paper extends positive mass results to asymptotically locally hyperbolic manifolds with toroidal ends in the boundary-free setting, under scalar curvature $R \ge -n(n-1)$ and a noncompact complement that is asymptotically retractible. It combines MOTS techniques with μ-bubble methods to prove nonnegativity of the end mass for dimensions $4 \le n \le 7$, using a contradiction argument that constructs a DEC-satisfying initial data set and analyzes a barrier-driven MOTS to force a violation of outermostness. The argument relies on a local MOTS foliation (via a rigidity lemma) and cohomology constraints associated with toroidal ends to preclude metrics of positive scalar curvature on certain MOTS components. The work clarifies the role of the mass aspect, illustrates with Birmingham-Kottler-type metrics, and situates the result among known ALH PMTs, including discussions of sharpness and possible extensions to boundary cases.

Abstract

In work with P. Chruściel, L. Nguyen and T.-T. Paetz [8], a positive mass theorem was obtained for asymptotically locally hyperbolic manifolds with boundary, having a toroidal end. The proof made use of properties of marginally outer trapped surfaces (MOTS). Here we present some new PMT results for such manifolds, but without boundary, which allow for other more general ends. The proofs, while still MOTS-based, involve a more elaborate technique (related to $μ$-bubbles) introduced in work of D. A. Lee, M. Lesourd, and R. Unger [20] for manifolds with an asymptotically flat end, and further developed in [23] for manifolds with an asymptotically hyperbolic end.

Positive mass theorems for manifolds with ALH toroidal ends

TL;DR

The paper extends positive mass results to asymptotically locally hyperbolic manifolds with toroidal ends in the boundary-free setting, under scalar curvature and a noncompact complement that is asymptotically retractible. It combines MOTS techniques with μ-bubble methods to prove nonnegativity of the end mass for dimensions , using a contradiction argument that constructs a DEC-satisfying initial data set and analyzes a barrier-driven MOTS to force a violation of outermostness. The argument relies on a local MOTS foliation (via a rigidity lemma) and cohomology constraints associated with toroidal ends to preclude metrics of positive scalar curvature on certain MOTS components. The work clarifies the role of the mass aspect, illustrates with Birmingham-Kottler-type metrics, and situates the result among known ALH PMTs, including discussions of sharpness and possible extensions to boundary cases.

Abstract

In work with P. Chruściel, L. Nguyen and T.-T. Paetz [8], a positive mass theorem was obtained for asymptotically locally hyperbolic manifolds with boundary, having a toroidal end. The proof made use of properties of marginally outer trapped surfaces (MOTS). Here we present some new PMT results for such manifolds, but without boundary, which allow for other more general ends. The proofs, while still MOTS-based, involve a more elaborate technique (related to -bubbles) introduced in work of D. A. Lee, M. Lesourd, and R. Unger [20] for manifolds with an asymptotically flat end, and further developed in [23] for manifolds with an asymptotically hyperbolic end.
Paper Structure (3 sections, 5 theorems, 20 equations)

This paper contains 3 sections, 5 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm:pm1']} and further comments

Key Result

Theorem 1.1

Let $(M^n,g)$, $4 \le n \le 7$, be a complete orientable ALH manifold with a conformally compactifiable toroidal end ${\mathcal{E}}$, and with scalar curvature $R \ge -n(n-1)$. Suppose further that $N = \overline{M\setminus{\mathcal{E}}}$ is noncompact and asymptotically retractible onto $\partial\c

Theorems & Definitions (5)

  • Theorem 1.1
  • Lemma 2.1: AndMetEic09Eic10
  • Lemma 2.2: motsv4
  • Lemma 2.3: EGM
  • Theorem 3.1