The equality case of the Penrose inequality for asymptotically flat graphs

Abstract

We prove the equality case of the Penrose inequality in all dimensions for asymptotically flat hypersurfaces. It was recently proven by G. Lam [19] that the Penrose inequality holds for asymptotically flat graphical hypersurfaces in Euclidean space with non-negative scalar curvature and with a minimal boundary. Our main theorem states that if the equality holds, then the hypersurface is a Schwarzschild solution. As part of our proof, we show that asymptotically flat graphical hypersurfaces with a minimal boundary and non-negative scalar curvature must be mean convex, using the argument that we developed in [15]. This enables us to obtain the ellipticity for the linearized scalar curvature operator and to establish the strong maximum principles for the scalar curvature equation . 1 THE EQUALITY CASE OF THE PENROSE INEQUALITY 2 where ω n−1 is the volume of the unit (n − 1)-sphere in Euclidean space. Moreover, the equality holds if and only if (M, g) is isometric to the region of a Schwarzschild metric outside its minimal hypersurface.G. Huisken and T. Ilmanen proved the conjecture in dimension three for a connected minimal boundary [17]. H. Bray used a different approach and proved the conjecture in dimension three for any number of components of the minimal boundary [2]. In dimensions less than 8, the inequality was proved by H. Bray and D. Lee, with the extra spin assumption for the equality case [5]. In the case that (M, g) is conformally flat, H. Bray and K. Iga derived new properties of superharmonic functions in R n and proved the Penrose inequality with a suboptimal constant for n = 3 [4], F. Schwartz obtained a lower bound of the ADM mass in terms of the Euclidean volume of the region enclosed by the minimal boundary [25], and J. Jauregui proved a Penrose-like inequality [18]. For the Penrose inequality (with the sharp constant) in dimensions higher than 8, the only result that we know, other than the spherically symmetric case, is the result of G. Lam [19] (cf. [8]), where he proved that the Penrose inequality for graphical asymptotically flat hypersurfaces. (Note some related work regarding the Penrose inequality for asymptotically hyperbolic graphs in [7,9].)

An Alexandrov–Fenchel-Type Inequality in Hyperbolic Space with an Application to a Penrose Inequality

“…This Penrose inequality improves recent results by Dahl-Gicquaud-Sakovich [DGS] and settles, for this class of initial data sets, the conjectured Penrose inequality for time-symmetric space-times with negative cosmological constant [BC] [Ma]. We remark that the proof of the rigidity is based on a recent preprint by Huang and Wu [HW1]; see also [dLG3].…”

“…As remarked above, the rigidity statement requires a separate argument and is based on results in a recent preprint by Huang and Wu [HW1].…”

“…First, the Alexandrov-Fenchel type inequality (1.5) holds for mean convex hypersurfaces as well, since any such hypersurface can be arbitrarily approximated, in the CThis is achieved in [HW1] for asymptotically flat graphs with a minimal horizon in Euclidean space by first observing that, by Gauss equation, the points where H M = 0 are precisely those where the shape operator vanishes. These are the socalled geodesic points and a well-known structure result by Sacksteder [Sa] provides a precise description of the subset M * of interior geodesic points.…”

“…As in [HW1], Sacksteder's result can be used to reach a contradiction if we assume that M ′ * is bounded. Once we know that M ′ * is necessarily unbounded, we can proceed as in [HW1] by examining the geometry of the level sets M t of the graph with respect to the t-coordinate, which are shown to be compact as one approaches infinity. For t close enough to its limiting value, which is finite due to the unboundedness of M ′ * , a geometric inequality essentially following from (4.62) implies that the mean curvature of M t (viewed as a hypersurface of the hyperplane at level t and computed with respect to the inward unit normal) is non-positive everywhere.…”

“…Thus, by Proposition 4.1, M is elliptic indeed and the uniqueness result in Theorem 1.2 follows from a standard application of the Maximum Principle. We emphasize that in [HW1] an extra condition, constraining the oscillation of the graph at infinity, is imposed in dimension n = 3 and 4 due to the fact that, as the Schwarzschild examples show, asymptotic flatness is compatible with the graph being unbounded as one approaches infinity. However, as it is apparent from (4.60), the adSS solutions are always uniformly bounded in a neighbourhood of infinity regardless of the dimension, so that this somewhat more involved case does not occur in our analysis.…”