On a Minkowski-like inequality for asymptotically flat static manifolds

Abstract

Abstract. The Minkowski inequality is a classical inequality in differential geometry, giving a bound from below, on the total mean curvature of a convex surface in Euclidean space, in terms of its area. Recently there has been interest in proving versions of this inequality for manifolds other than R n ; for example, such an inequality holds for surfaces in spatial Schwarzschild and AdSSchwarzschild manifolds. In this note, we adapt a recent analysis of Y. Wei to prove a Minkowski-like inequality for general static asymptotically flat manifolds.

A Minkowski type inequality in space forms

“…with rigidity for a coordinate sphere. Recently, McCormick [10] showed that the inequality (1.3) holds for asymptotically flat static spacetimes of dimension 3 ≤ n ≤ 7. Note that all of these inequalities are for static manifolds with spherical infinity.…”

Minkowski Inequalities via Nonlinear Potential Theory

“…This means that, albeit the level set flow is possibly subject to jumps, our monotonicity formulas are strong enough to survive them. Finally, from a theoretical point of view, these formulas can be seen as the crucial step towards the completion of a program initiated in the series of works [1,3,4,26] and intended to link the monotonicity formulas employed by Huisken, Ilmanen and other authors in studying the geometric implications of the IMCF (see e.g., [8][9][10]21,27,[29][30][31]34,36,37,45,51,64] to the monotonicity formulas discovered by Colding and Minicozzi in [16][17][18] for the level set flow of the Green's functions on complete manifolds with nonnegative Ricci curvature. In fact, as explained in Sect.…”

“…In [37] this is achieved by means of an elliptic regularisation procedure in which the weak solution of the IMCF is approximated by a family of smooth functions whose level sets obey a slightly modified version of the desired monotonicity. To the best of our knowledge, such a spectacular though technically demanding construction has never been replied beyond the original context of asymptotically flat Riemannian manifolds [27,36,37,51,64], with the only exception of [43,Theorem 3.2], where the authors have checked that Huisken-Ilmanen theory applies to the case under consideration. Hence, the expected extensions of the results in [10,21,[29][30][31] to the case of outward minimising hypersufaces are missing so far.…”

Minkowski inequalities and constrained inverse curvature flows in warped spaces

“…It can be shown however, that the result is also true for outward minimizing hypersurfaces, which follows from Huisken's and Ilmanen's weak inverse mean curvature flow [22] or also from [1]. See [27,38] for extensions of this approach to some asymptotically flat manifolds. In the other spaceforms, including de Sitter space, lower bounds for W 2 (M ) were given in [10,26,29,32,36,39].…”