Mass, Staticity, and a Riemannian Penrose Inequality for Weighted Manifolds

TL;DR

This paper introduces a geometric approach to weighted mass and static metrics, establishing a Riemannian Penrose inequality for weighted manifolds with uniqueness results for static solutions.

Contribution

It derives a natural geometric mass invariant for weighted manifolds, introduces weighted static metrics, and proves a Riemannian Penrose inequality with uniqueness for these metrics.

Findings

Weighted mass can be derived as a geometric invariant.

Weighted static metrics are characterized as minimizers of weighted mass.

A Riemannian Penrose inequality is established with unique static solutions.

Abstract

In this note, we show that the weighted mass of Baldauf and Ozuch (2022) can be derived as a natural geometric mass invariant following Michel (2011), 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 (2025) 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.