A Conformal Positive Mass Theorem with Noncompact Boundary

TL;DR

This paper establishes a conformal positive mass theorem for asymptotically flat 3-manifolds with noncompact boundary, extending previous results by removing the positivity assumption on scalar curvature.

Contribution

It introduces an integral inequality for harmonic functions that leads to a positivity result for a convex combination of ADM masses without requiring positive scalar curvature.

Findings

Proves a new integral inequality for harmonic functions on asymptotically flat manifolds.

Shows positivity of a convex combination of ADM masses under relaxed curvature conditions.

Generalizes previous positive mass theorems to broader geometric settings.

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.