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A Conformal Positive Mass Theorem with Noncompact Boundary

Alex Freire, Mohammad Tariquel Islam

TL;DR

The paper extends conformal positive mass-type inequalities to asymptotically flat half-spaces with noncompact boundary by using a one-parameter conformal family and harmonic-function techniques. It proves a convex-combination inequality $(1-\lambda)m_\Sigma(g)+\lambda m_\Sigma(g')\ge0$ under combined curvature conditions, with explicit expressions for $R_\lambda$ and $H_\lambda$, and establishes rigidity when equality holds. The results generalize previous work to settings without assuming positive scalar curvature and provide a framework for comparing conformally related metrics in AF half-spaces, including a spinor-free Ul-Alam application that yields a unique flat geometry in the equality case. An appendix connects these ideas to Ul-Alam’s static Lorentzian system, illustrating the versatility of the conformal-analytic approach in gravitational contexts.

Abstract

We obtain an integral inequality for asymptotically linear harmonic functions on asymptotically flat 3-manifolds with noncompact boundary, which implies positivity of a convex combination of ADM masses of two conformally related metrics under a positivity condition on a corresponding convex combination of their scalar curvatures and boundary mean curvatures. This generalizes a result of Batista and Lopes de Lima, under conditions that do not assume positivity of scalar curvature.

A Conformal Positive Mass Theorem with Noncompact Boundary

TL;DR

The paper extends conformal positive mass-type inequalities to asymptotically flat half-spaces with noncompact boundary by using a one-parameter conformal family and harmonic-function techniques. It proves a convex-combination inequality under combined curvature conditions, with explicit expressions for and , and establishes rigidity when equality holds. The results generalize previous work to settings without assuming positive scalar curvature and provide a framework for comparing conformally related metrics in AF half-spaces, including a spinor-free Ul-Alam application that yields a unique flat geometry in the equality case. An appendix connects these ideas to Ul-Alam’s static Lorentzian system, illustrating the versatility of the conformal-analytic approach in gravitational contexts.

Abstract

We obtain an integral inequality for asymptotically linear harmonic functions on asymptotically flat 3-manifolds with noncompact boundary, which implies positivity of a convex combination of ADM masses of two conformally related metrics under a positivity condition on a corresponding convex combination of their scalar curvatures and boundary mean curvatures. This generalizes a result of Batista and Lopes de Lima, under conditions that do not assume positivity of scalar curvature.

Paper Structure

This paper contains 4 sections, 4 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Simon's conformal positive mass theorem via harmonic functions
  3. Proof of Theorem 1 and Corollary 2.
  4. Appendix: application to a theorem of ul-Alam

Key Result

Theorem 1

Let $(M,g)$ be an asymptotically flat 3-dimensional half-space. Suppose $g'=f^4g$ is a conformally related metric on $M$, where $f>0$ and $f\rightarrow 1$ at infinity, so that $g'$ is also asymptotically flat. Assume also $\partial_nf=0$ on $\partial M_{ext}$. Let $\lambda\in [0,1]$. The metric $g_{ (Subscripts $\lambda$ denote quantities computed in the metric $g_{\lambda}$.)

Theorems & Definitions (10)

  • Definition
  • Theorem 1
  • Corollary 2
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • proof
  • proof
  • proof