By works of Schoen-Yau and Gromov-Lawson any Riemannian manifold with nonnegative scalar curvature and diffeomorphic to a torus is isometric to a flat torus. Gromov conjectured subconvergence of tori with respect to a weak Sobolev type metric when the scalar curvature goes to 0. We prove flat and intrinsic flat subconvergence to a flat torus for sequences of 3-dimensional tori M j that can be realized as graphs of certain functions defined over flat tori satisfying a uniform upper diameter bound, a uniform lower bound on the area of the smallest closed minimal surface, and scalar curvature bounds of the form R gM j ≥ −1/j. We also show that the volume of the manifolds of the convergent subsequence converges to the volume of the limit space. We do so adapting results of Huang-Lee, Huang-Lee-Sormani and Sormani. Furthermore, our results also hold when the condition on the scalar curvature of a torus (M, g M ) is replaced by a bound on the quantity − T min{R gM , 0}dvol gT , where M = graph(f ), f : T → R and (T, g T ) is a flat torus.1 the mean curvature convention is that spheres have positive mean curvature with respect to the inner pointing normal vector A 0 arising from (T i , f i ) and m(f i ) → 0 subconverges in intrinsic flat sense to some integral current space (X ∞ , d X∞ , S X∞ ). By adapting Sormani's Arzela-Ascoli Theorem 8.1 in [Sor14] we obtain an Arzela-Ascoli limit function that can be extended to a covering map p∞ : R 3 → X ∞ . Studying the properties of X ∞ and p∞ we conclude that (X ∞ , d X∞ ) is isometric to T ∞ . Contents

Given a pair of metric tensors g1 ≥ g0 on a Riemannian manifold, M , it is well known that Vol1(M ) ≥ Vol0(M ). Furthermore one has rigidity: the volumes are equal if and only if the metric tensors are the same g1 = g0. Here we prove the that if gj ≥ g0 and Volj(M ) → Vol0(M ) then (M, gj) converge to (M, g0) in the volume preserving intrinsic flat sense. Well known examples demonstrate that one need not obtain smooth, C 0 , Lipschitz, or even Gromov-Hausdorff convergence in this setting. To complete our proof, we provide a new way of estimating the intrinsic flat distance between Riemannian manifolds.

We consider sequences of open Riemannian manifolds with boundary that have no regularity conditions on the boundary. To define a reasonable notion of a limit of such a sequence, we examine ı-inner regions, that avoid the boundary by a distance ı. We prove Gromov-Hausdorff compactness theorems for sequences of these ı-inner regions. We then build "glued limit spaces" out of the Gromov-Hausdorff limits of ı-inner regions and study the properties of these glued limit spaces. Our main applications assume the sequence is noncollapsing and has nonnegative Ricci curvature. We include open questions.

For n-dimensional Riemannian manifolds M with Ricci curvature bounded below by −(n − 1), the volume entropy is bounded above by n − 1. If M is compact, it is known that the equality holds if and only if M is hyperbolic. We extend this result to RCD * (−(N − 1), N) spaces. While the upper bound is straightforward, the rigidity case is quite involved due to the lack of a smooth structure in RCD * spaces. As an application, we obtain an almost rigidity result which partially recovers a result by Chen-Rong-Xu for Riemannian manifolds.

Abstract. This survey reviews precompactness theorems for classes of Riemannian manifolds with boundary. We begin with the works of Kodani, Anderson-Katsuda-KurylevLassas-Taylor and Wong. We then present new results of Knox and the author with Sormani.