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Mass from an Extrinsic Point of View

Alexandre de Sousa, Frederico Girão

TL;DR

The paper develops an extrinsic perspective on the Gauss-Bonnet-Chern mass $m_q$ by introducing asymptotically Euclidean immersions and proving a version of the Hsiung-Minkowski identities for noncompact submanifolds. It then expresses $m_q$ as a linear combination of intrinsic curvature data, specifically $(n-2q)\int S_{2q}\,dM$ and $(2q+1)\int \langle S_{2q+1}, \bar{Z}\rangle \, dM$, via an explicit mass formula $m_q = a(n,q)[(n-2q)\int S_{2q} + (2q+1)\int \langle S_{2q+1}, \bar{Z}\rangle]$, under asymptotic and integrability assumptions. The authors connect this to the Lovelock tensor $G_{(q)}$ and Newton transformations, providing a flux-type expression and a gradient-field refinement, and derive a geometric inequality that must hold if the PMT for the GBC mass is valid. They also propose conjectures on asymptotically Euclidean immersions and their implications for the PMT, highlighting a path to deducing PMT results from intrinsic-extrinsic relations. Overall, the work links extrinsic geometric data to global mass-type invariants, offering a framework to study positivity and rigidity in higher-order curvature settings.

Abstract

We express the $q$-th Gauss-Bonnet-Chern mass of an immersed submanifold of Euclidean space as a linear combination of two terms: the total $(2q)$-th mean curvature and the integral, over the entire manifold, of the inner product between the $(2q+1)$-th mean curvature vector and the position vector of the immersion. As a consequence, we obtain, for each $q$, a geometric inequality that holds whenever the positive mass theorem (for the $q$-th Gauss-Bonnet-Chern mass) holds.

Mass from an Extrinsic Point of View

TL;DR

The paper develops an extrinsic perspective on the Gauss-Bonnet-Chern mass by introducing asymptotically Euclidean immersions and proving a version of the Hsiung-Minkowski identities for noncompact submanifolds. It then expresses as a linear combination of intrinsic curvature data, specifically and , via an explicit mass formula , under asymptotic and integrability assumptions. The authors connect this to the Lovelock tensor and Newton transformations, providing a flux-type expression and a gradient-field refinement, and derive a geometric inequality that must hold if the PMT for the GBC mass is valid. They also propose conjectures on asymptotically Euclidean immersions and their implications for the PMT, highlighting a path to deducing PMT results from intrinsic-extrinsic relations. Overall, the work links extrinsic geometric data to global mass-type invariants, offering a framework to study positivity and rigidity in higher-order curvature settings.

Abstract

We express the -th Gauss-Bonnet-Chern mass of an immersed submanifold of Euclidean space as a linear combination of two terms: the total -th mean curvature and the integral, over the entire manifold, of the inner product between the -th mean curvature vector and the position vector of the immersion. As a consequence, we obtain, for each , a geometric inequality that holds whenever the positive mass theorem (for the -th Gauss-Bonnet-Chern mass) holds.
Paper Structure (5 sections, 10 theorems, 33 equations)

This paper contains 5 sections, 10 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Mass of asymptotically Euclidean spaces
  3. Mass in terms of the Lovelock tensor
  4. Imersions in Euclidean spaces
  5. Asymptotically Euclidean imersions

Key Result

Theorem 2.3

If $(M^n,g)$ is an asymptotically flat Riemannian manifold of order $\tau > (n-2)/2$ whose scalar curvature is nonnegative and integrable, then each of its ends has nonnegative ADM mass. Moreover, if the ADM mass of at least one of its ends is zero, then $(M,g)$ is isometric to the Euclidean space $

Theorems & Definitions (17)

  • Definition 2.1: asymptotically Euclidean end
  • Definition 2.2: ADM mass
  • Theorem 2.3
  • Definition 2.4: GBC mass
  • Conjecture 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Corollary 2.8
  • Proposition 3.1
  • Proposition 3.2
  • ...and 7 more