Mass-type invariants in the presence of a cosmological constant

TL;DR

This paper introduces new mass-type invariants for space-times with a positive cosmological constant, offering refined characterizations of de Sitter space and extending positive mass theorems.

Contribution

It presents a novel family of invariants that improve understanding of space-time geometries with a cosmological constant, including a positive mass theorem and Penrose inequality.

Findings

New invariants characterize de Sitter space effectively.

Established a positive mass theorem for the 1-harmonic Mass.

Proved a Penrose-type inequality for these invariants.

Abstract

In this paper, we introduce a new family of mass-type invariants for time-symmetric initial data in space-times satisfying the Dominant Energy Condition. For positive cosmological constant, these invariants, unlike the total Hawking mass, turn out to be genuinely effective in providing new characterizations of the de Sitter solution. From a theoretical standpoint, this opens a new perspective on how one might refine the rigidity statement originally proposed by Min-Oo in his well known conjecture, later refuted by the counterexamples of Brendle, Marques, and Neves. Via a formal limiting procedure, we also define another invariant, the 1-harmonic Mass, for which we independently prove a positive mass theorem and a Penrose-type inequality, thereby extending tools for probing space-time geometries in the presence of a positive cosmological constant.