Deforming the Scalar Curvature of the De Sitter–Schwarzschild Space

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 $$n\ge 3$$ satisfying the dominant energy condition and agreeing with the standard metric along the event and cosmological horizons, which remain totally geodesic. Our results 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 rigidity statement 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.