Abstract: Our setting is a simply connected bounded domain with a smooth connected boundary, which arises as an initial data set for the general relativistic constraint equations satisfying the dominant energy condition. Assuming the domain to be admissible in a certain precise sense, we prove a localized spacetime Penrose inequality for the Liu-Yau and Wang-Yau quasi-local masses and the area of an outermost marginally outer trapped surface (MOTS). On the basis of this inequality, we obtain sufficient conditions for the existence and non-existence of a MOTS (along with outer trapped surfaces) in the domain, and for the existence of a minimal surface in its Jang graph, expressed in terms of various quasi-local mass quantities and the boundary geometry of the domain.
Title: Stability of a quasi-local positive mass theorem for graphical hypersurfaces of Euclidean space
Aghil Alaee, Armando J. Cabrera Pacheco, Stephen McCormick
Abstract: We present a quasi-local version of the stability of the positive mass theorem. We work with the Brown–York quasi-local mass as it possesses positivity and rigidity properties, and therefore the stability of this rigidity statement can be studied. Specifically, we ask if the Brown–York mass of the boundary of some compact manifold is close to zero, must the manifold be close to a Euclidean domain in some sense?
Here we consider a class of compact n-manifolds with boundary that can be realized as graphs in ℝn+1, and establish the following. If the Brown–York mass of the boundary of such a compact manifold is small, then the manifold is close to a Euclidean hyperplane with respect to the Federer–Fleming flat distance.