Static potentials and area minimizing hypersurfaces

Abstract

We show that if an asymptotically flat manifold with horizon boundary admits a global static potential, then the static potential must be zero on the boundary. We also show that if an asymptotically flat manifold with horizon boundary admits an unbounded static potential in the exterior region, then the manifold must contain a complete non-compact area minimizing hypersurface. Some results related to the Riemannian positive mass theorem and Bartnik's quasi-local mass are obtained.

On a Minkowski-like inequality for asymptotically flat static manifolds

“…The following property of static potentials is very recent result of Huang, Martin and Miao [11], reformulated slightly such that it is directly applicable for us here. The above theorem is used in the proof of our main inequality in order to allow for manifolds with boundary components outside of the one we flow from, provided they are minimal surfaces.…”

Second-order mass estimates for static vacuum metrics with small Bartnik data

“…Bartnik conjectured ( [11][12][13]) that his quasilocal mass should be realized by a unique asymptotically flat static vacuum extension inducing the given boundary metric and mean curvature. The necessity of the boundary and static vacuum conditions for a minimizer have been established (see [7], [23][24][25] and [32], and [41], as well as [2,33] for the spacetime version), but the existence of a minimizer is known only (at least to the author) for (i) apparent horizons satisfying a natural nondegenracy condition, by virtue of [37], and (ii) for data which can be realized as an outer-minimizing embedded sphere (required for our definition, (1.5)) enclosing the horizon in a time-symmetric Schwarzschild slice (or any outer-minimizing embedded sphere in Euclidean space), by virtue of [34] (or [49,54]). Although the most aggressive formulations of the conjecture are now known to be false (see [7,37,42] and for counterexamples in the spacetime setting in higher dimensions also [33]), it does suggest a strategy, pursued in this article, for seeking a tighter upper bound than the Brown-York mass affords in (1.11).…”

Embeddings, Immersions and the Bartnik Quasi-Local Mass Conjectures

“…It is clear this infimum is strictly positive, so that λ t 0 > 0 for t sufficiently small. 27 Thus −L t is a positive operator for t sufficiently small; it is then standard, cf. [34] for instance, that the Green's functionḠ t exists and is strictly negative in the interior of M t and hence the Poisson kernel is strictly positive (sinceP t (x, y) = −N xḠt (x, y) andḠ t (x, y) = 0 for x ∈ ∂M t ).…”

“…Unfortunately, this does not apply to the present setting, since it is not clear that a static vacuum metric is asymptotically Schwarzschild without knowing in advance its potential has a sign at infinity. Huang-Martin-Miao [27,Theorem 10] show that the static potential of a Bartnik mass minimizer approaches 1 at infinity (though this argument does not show u > 0 globally). Note that they use a different version of the Bartnik mass.…”