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Mass, staticity, and a Riemannian Penrose inequality for weighted manifolds

Stephen McCormick

TL;DR

The paper addresses mass notions in weighted manifolds by embedding Baldauf–Ozuch's weighted mass into Michel's geometric mass framework via the densitised curvature map $\Phi(g,f)$ with $S_f=R_f+\frac{1}{n-1}|\nabla f|^2$. It shows a natural conformal reduction to the unweighted ADM mass through $\widetilde{g}=e^{-\frac{2}{n-1}f}g$, enabling Penrose-type results to be derived from the standard Riemannian theory. The authors define $f$-staticity via the adjoint of the linearised map, prove a uniqueness theorem identifying the $f$-static case with the $f$-Schwarzschild family of metrics, and establish a weighted Riemannian Penrose inequality with rigidity when equality occurs. Overall, the work demonstrates that weighted geometric mass problems can be effectively translated into the unweighted setting through a natural conformal change, yielding sharp positivity and rigidity statements for weighted manifolds.

Abstract

In this note, we show that the weighted mass of Baldauf and Ozuch can be derived as a natural geometric mass invariant following Michel, for a certain weighted curvature map. An associated weighted centre of mass definition is also derived from this. The adjoint of the linearisation of this curvature map leads to a notion of weighted static metrics, which are natural candidates for weighted mass minimisers. This weighted curvature quantity is essentially the scalar curvature of a conformally related metric that Law, Lopez and Santiago used to considerably simplify the proof of the weighted positive mass theorem. We show an equivalence between static metrics and weighted static metrics via the conformal relationship, from which we show that a uniqueness theorem holds for weighted static manifolds with weighted minimal surface boundaries. Furthermore, we show that weighted manifolds satisfy a Riemannian Penrose inequality whose equality case holds precisely for these unique weighted static metrics.

Mass, staticity, and a Riemannian Penrose inequality for weighted manifolds

TL;DR

The paper addresses mass notions in weighted manifolds by embedding Baldauf–Ozuch's weighted mass into Michel's geometric mass framework via the densitised curvature map with . It shows a natural conformal reduction to the unweighted ADM mass through , enabling Penrose-type results to be derived from the standard Riemannian theory. The authors define -staticity via the adjoint of the linearised map, prove a uniqueness theorem identifying the -static case with the -Schwarzschild family of metrics, and establish a weighted Riemannian Penrose inequality with rigidity when equality occurs. Overall, the work demonstrates that weighted geometric mass problems can be effectively translated into the unweighted setting through a natural conformal change, yielding sharp positivity and rigidity statements for weighted manifolds.

Abstract

In this note, we show that the weighted mass of Baldauf and Ozuch can be derived as a natural geometric mass invariant following Michel, for a certain weighted curvature map. An associated weighted centre of mass definition is also derived from this. The adjoint of the linearisation of this curvature map leads to a notion of weighted static metrics, which are natural candidates for weighted mass minimisers. This weighted curvature quantity is essentially the scalar curvature of a conformally related metric that Law, Lopez and Santiago used to considerably simplify the proof of the weighted positive mass theorem. We show an equivalence between static metrics and weighted static metrics via the conformal relationship, from which we show that a uniqueness theorem holds for weighted static manifolds with weighted minimal surface boundaries. Furthermore, we show that weighted manifolds satisfy a Riemannian Penrose inequality whose equality case holds precisely for these unique weighted static metrics.
Paper Structure (4 sections, 10 theorems, 56 equations)

This paper contains 4 sections, 10 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. The weighted mass
  3. Weighted staticity
  4. The weighted Riemannian Penrose inequality

Key Result

Theorem 1.1

Let $(M^n,g,e^{-f}dV_g)$ be an asymptotically flat weighted manifold with $R_f\in L^1(M)$. Assume either $3\le n\le 7$ or $M$ is spin. Then if $R_f\ge 0$, $\mathfrak{m}_f(g)\ge 0$ with equality if and only if $(M^n,g)$ is isometric to Euclidean space and $f\equiv0$.

Theorems & Definitions (20)

  • Theorem 1.1: Baldauf--Ozuch BaldaufOzuch22; Chu--Zhu ChuZhu24
  • Theorem 1.2: Law--Lopez--Santiago LawLopezSantiago24
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Definition 3.1: $f$-static potential
  • ...and 10 more