On the Minkowski-type inequality for outward minimizing hypersurfaces in Schwarzschild space

Abstract

Using the weak solution of Inverse mean curvature flow, we prove the sharp Minkowski-type inequality for outward minimizing hypersurfaces in Schwarzschild space.2010 Mathematics Subject Classification. 53C44, 53C42.

On a Minkowski-like inequality for asymptotically flat static manifolds

“…The proof of Theorem 1.4 follows the main proof of [19] very closely, replacing the Schwarzschild potential with a general static potential. The inequality follows from the monotonicity of a quantity Q(t) (defined below) under weak inverse mean curvature flow (IMCF).…”

“…In recent years, there has been interest in generalisations of the Minkowski inequality to surfaces embedded in manifolds other than Euclidean space. For example, Minkowski inequalities are known for surfaces in hyperbolic space [8,5], Schwarzschild manifolds [5,19], and Schwarzschild-AdS manifolds [5]. In this note, we prove a Minkowski inequality for general static asymptotically flat manifolds, generalising the classical inequality and the known inequality for Schwarzschild manifolds.…”

“…This monotonicity has been used previously (eg. [5,19]) to prove Minkowski inequalities in Schwarzschild and Schwarzschild-AdS manifolds.…”

“…The key contribution here is the observation that using the weak formulation of inverse mean curvature flow allows one to prove the inequality on a general static manifold, rather than working only within a fixed manifold where the smooth flow is known to be well-behaved. The proof adapts an analysis of Wei [19] that is used to prove a Minkowski inequality for outer-minimizing surfaces in the Schwarzschild manifold.…”

A Penrose type inequality for graphs over Reissner-Nordström-anti-deSitter manifold

“…Remark 1.9. By use of the weak solution of inverse mean curvature flow (see [23]) and the techniques of [41], it is an interesting problem to prove the Minkowski-type inequality for outward minimizing hypersurfaces in Reissner-Nordström-AdS manifold (P,ḡ).…”