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...

The study of stable minimal surfaces in Riemannian 3-manifolds (M, g) with non-negative scalar curvature has a rich history. In this paper, we prove rigidity of such surfaces when (M, g) is asymptotically flat and has horizon boundary. As a consequence, we obtain an effective version of the positive mass theorem in terms of isoperimetric or, more generally, closed volume-preserving stable CMC surfaces that is appealing from both a physical and a purely geometric point of view. We also include a proof of the following conjecture of Schoen: An asymptotically flat Riemannian 3-manifold with non-negative scalar curvature that contains an unbounded area-minimizing surface is isometric to flat R 3 .

We show the existence of isoperimetric regions of sufficiently large volumes in general asymptotically hyperbolic three manifolds. Furthermore, we show that large coordinate spheres in compact perturbations of Schwarzschild-anti-deSitter are uniquely isoperimetric. This is relevant in the context of the asymptotically hyperbolic Penrose inequality.Our results require that the scalar curvature of the metric satisfies Rg ≥ −6, and we construct an example of a compact perturbation of Schwarzschild-anti-deSitter without Rg ≥ −6 so that large centered coordinate spheres are not isoperimetric. The necessity of scalar curvature bounds is in contrast with the analogous uniqueness result proven by Bray for compact perturbations of Schwarzschild, where no such scalar curvature assumption is required.This demonstrates that from the point of view of the isoperimetric problem, mass behaves quite differently in the asymptotically hyperbolic setting compared to the asymptotically flat setting. In particular, in the asymptotically hyperbolic setting, there is an additional quantity, the "renormalized volume," which has a strong effect on the large-scale geometry of volume.I am very grateful to my advisor, Simon Brendle, for suggesting this problem, as well as for his invaluable assistance, support, and many suggestions on a preliminary version of this paper. Additionally, I would like to thank Michael Eichmair and Brian White for patiently answering numerous questions, as well as for their continued encouragement. Finally, I would like to acknowledge

We prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index (and volume if the ambient dimension is greater than three) in a Riemannian manifold (M n , g), 3 ≤ n ≤ 7, can degenerate. Loosely speaking, our results show that embedded minimal hypersurfaces with bounded index behave qualitatively like embedded stable minimal hypersurfaces, up to controlled errors. Several compactness/finiteness theorems follow from our local picture.