Deforming the Scalar Curvature of the De Sitter–Schwarzschild Space

Abstract

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 ≥ 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.

Brown–York Mass and Compactly Supported Conformal Deformations of Scalar Curvature

“…Although people made many attempts trying to solve Min-Oo's conjecture, unfortunately it was disproved in 2010. Brendle, Marques and Neves constructed a counter example for Min-Oo's Conjecture by combining technics of perturbation and gluing ( [4], see also [13] for a generalization).…”

On scalar curvature rigidity of vacuum static spaces

“…Min-Oo's conjecture attracted a lot of attentions among geometric analysts. It was remarkable that Brendle, Marques and Neves in [6] (see also [14] for a later developement) discovered that there is even no local scalar curvature rigidity of the round hemispheres and constructed counter-examples to Min-Oo's conjecture. Later, in a subsequent paper [7], Brendle and Marques established the local scalar curvature rigidity of round spherical caps of some appropriate size (cf.…”