The aim of this paper is to prove a sharp inequality for the area of a four dimensional compact Einstein manifold (Σ,gΣ) embedded into a complete five dimensional manifold (M5,g) with positive scalar curvatureRand nonnegative Ricci curvature. Under a suitable choice, we have$area(\Sigma)^{\frac{1}{2}}\inf_{M}R \leq 8\sqrt{6}\pi$. Moreover, occurring equality we deduce that (Σ,gΣ) is isometric to a standard sphere ($\mathbb{S}$4,gcan) and in a neighbourhood of Σ, (M5,g) splits as ((-ϵ, ϵ) ×$\mathbb{S}$4,dt2+gcan) and the Riemannian covering of (M5,g) is isometric to$\Bbb{R}$×$\mathbb{S}$4.
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.
We obtain some estimates on the area of the boundary and on the volume of a certain free boundary hypersurface Σ with nonpositive Yamabe invariant in a Riemannian n-manifold with bounds for the scalar curvature and the mean curvature of the boundary. Assuming further that Σ is locally volume-minimizing in a manifold M n with scalar curvature bounded below by a nonpositive constant and mean convex boundary, we conclude that locally M splits along Σ. In the case that the scalar curvature of M is at least −n(n − 1) and Σ locally minimizes a certain functional inspired by [30], a neighborhood of Σ in M is isometric to ((−ε, ε) × Σ, dt 2 + e 2t g), where g is Ricci flat.2000 Mathematics Subject Classification. Primary 53C42, 53C21; Secondary 58J60.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.