In this paper we characterize the intrinsic geometry of apparent horizons (outermost marginally outer trapped surfaces) in asymptotically flat spacetimes; that is, the Riemannian metrics on the two sphere which can arise. Furthermore we determine the minimal ADM mass of a spacetime containing such an apparent horizon. The results are conveniently formulated in terms of the quasi-local mass introduced by Bartnik [Bar89] in 1989. The Hawking mass provides a lower bound for Bartnik's quasilocal mass on apparent horizons by way of Penrose's conjecture on time symmetric slices, proven in 1997 by Huisken and Ilmanen [HI01] and in full generality in 1999 by Bray [Bra01]. We compute Bartnik's mass for all non-degenerate apparent horizons and show that it coincides with the Hawking mass. As a corollary we disprove a conjecture due to Gibbons in the spirit of Thorne's hoop conjecture [Gib09], and construct a new large class of examples of apparent horizons with the integral of the negative part of the Gauss curvature arbitrarily large.

The Allen-Cahn equation is a semilinear PDE which is deeply linked to the theory of minimal hypersurfaces via a singular limit. We prove curvature estimates and strong sheet separation estimates for stable solutions (building on recent work of Wang-Wei [WW19]) of the Allen-Cahn equation on a 3-manifold. Using these, we are able to show for generic metrics on a 3-manifold, minimal surfaces arising from Allen-Cahn solutions with bounded energy and bounded Morse index are two-sided and occur with multiplicity one and the expected Morse index. This confirms, in the Allen-Cahn setting, a strong form of the multiplicity one conjecture and the index lower bound conjecture of Marques-Neves [Mar14,Nev14] in 3-dimensions regarding min-max constructions of minimal surfaces.Allen-Cahn min-max constructions were recently carried out by Guaraco [Gua18] and Gaspar-Guaraco [GG18]. Our resolution of the multiplicity one and the index lower bound conjectures shows that these constructions can be applied to give a new proof of Yau's conjecture on infinitely many minimal surfaces in a 3-manifold with a generic metric (recently proven by Irie-Marques-Neves [IMN18]) with new geometric conclusions. Namely, we prove that a 3-manifold with a generic metric contains, for every p = 1, 2, 3, . . ., a two-sided embedded minimal surface with Morse index p and area ∼ p OTIS CHODOSH AND CHRISTOS MANTOULIDIS Appendix E. An interpolation lemma 76 References 77Here, h 0 > 0 is a constant that is canonically associated with W (see Section 1.3). A deep result of Hutchinson-Tonegawa [HT00, Theorem 1] ensures that V limits to a varifold with a.e. integer density as ε ց 0. If, in addition, one assumes that the solutions are stable, Tonegawa-Wickramasekera [TW12] have shown that the limiting varifold is stable and satisfies the conditions of Wickramasekera's deep regularity theory [Wic14]; thus the limiting varifold is a smooth stable minimal hypersurface (outside of a codimension 7 singular set). In two dimensions, this was shown by Tonegawa [Ton05].1 Added in proof: There has been dramatic progress in Almgren-Pitts theory since we first posted this article. In particular, we note that A. Song [Son18] has proved the full Yau conjecture in dimensions 3 through 7, and X. Zhou [Zho19] proved the multiplicity one conjecture in the Almgren-Pitts setting, also in dimensions 3 through 7.1.2.3. The multiplicity one-conjecture for limits of the Allen-Cahn equation in 3-manifolds. In their recent work [MN16a], Marques-Neves make the following conjecture:Conjecture 1.5 (Multiplicity one conjecture). For generic metrics on (M n , g), 3 ≤ n ≤ 7, two-sided unstable components of closed minimal hypersurfaces obtained by min-max methods must have multiplicity one.In [MN16a], Marques-Neves confirm this in the case of a one parameter Almgren-Pitts sweepout. The one parameter case had been previously considered for metrics of positive Ricci curvature by Marques-Neves [MN12] and subsequently by Zhou [Zho15]. See also [Gua18, Corollary E] and [GG18, Theorem 1]for results c...

We study positive scalar curvature on the regular part of Riemannian manifolds with singular, uniformly Euclidean (L ∞ ) metrics that consolidate Gromov's scalar curvature polyhedral comparison theory and edge metrics that appear in the study of Einstein manifolds. We show that, in all dimensions, edge singularities with cone angles ≤ 2π along codimension-2 submanifolds do not affect the Yamabe type. In three dimensions, we prove the same for more general singular sets, which are allowed to stratify along 1-skeletons, exhibiting edge singularities (angles ≤ 2π) and arbitrary L ∞ isolated point singularities. We derive, as an application of our techniques, Positive Mass Theorems for asymptotically flat manifolds with analogous singularities. arXiv:1708.08211v3 [math.DG] 18 Sep 2018 Edge singularitiesThe starting point of our discussion is the classical example of isolated conical singularities on two-dimensional Riemannian manifolds.

Abstract. We study the supremum of the total mean curvature on the boundary of compact, mean-convex 3-manifolds with nonnegative scalar curvature, and a prescribed boundary metric. We establish an additivity property for this supremum and exhibit rigidity for maximizers assuming the supremum is attained. When the boundary consists of 2-spheres, we demonstrate that the finiteness of the supremum follows from the previous work of Shi-Tam and Wang-Yau on the quasi-local mass problem in general relativity. In turn, we define a variational analog of Brown-York quasilocal mass without assuming that the boundary 2-sphere has positive Gauss curvature.

We derive new inequalities between the boundary capacity of an asymptotically flat 3-manifold with nonnegative scalar curvature and boundary quantities that relate to quasi-local mass; one relates to Brown-York mass and the other is new. We argue by recasting the setup to the study of mean-convex fill-ins with nonnegative scalar curvature and, in the process, we consider fillins with singular metrics, which may have independent interest. Among other things, our work yields new variational characterizations of Riemannian Schwarzschild manifolds and new comparison results for surfaces in them.