Deforming the scalar curvature of the de Sitter-Schwarzschild space

TL;DR

This paper constructs specific deformations of the de Sitter-Schwarzschild space that satisfy the dominant energy condition and match the standard metric at horizons, challenging the existence of a Penrose inequality analogue in positive cosmological constant scenarios.

Contribution

It demonstrates the existence of deformations of de Sitter-Schwarzschild space satisfying energy conditions and matching horizons, extending previous counterexamples to Min-Oo's conjecture.

Findings

Deformations satisfy dominant energy condition.

Deformations match standard metric at horizons.

No Penrose inequality analogue for positive cosmological constant.

Abstract

Building upon the work of Brendle, Marques and Neves on the construction of counterexamples to Min-Oo's conjecture, we exhibit deformations of the de Sitter-Schwarzschild space of dimension $n\geq 3$ satisfying the dominant energy condition and agreeing with the standard metric along the event and cosmological horizons, which remain totally geodesic. Our results actually hold for generalized Kottler-de Sitter-Schwarzschild spaces whose cross sections are compact rank one symmetric spaces and indicate that there exists no analogue of the Penrose inequality in the case of positive cosmological constant. As an application we construct solutions of Einstein field equations satisfying the dominant energy condition and being asymptotic to (or agreeing with) the de Sitter-Schwarzschild space-time both at the event horizon and at spatial infinity.