The main result of this article states that the (K, N )-cone over some metric measure space satisfies the reduced Riemannian curvature-dimension condition RCD * (KN, N +1) if and only if the underlying space satisfies RCD * (N −1, N ). The proof uses a characterization of reduced Riemannian curvature-dimension bounds by Bochner's inequality that was established for general metric measure spaces by Erbar, Kuwada and Sturm in [21] (independently, the same result has been announced by Ambrosio, Mondino and Savaré). As a corollary of our result and the Gigli-Cheeger-Gromoll splitting theorem [25] we obtain a maximal diameter theorem in the context of metric measure spaces that satisfy the condition RCD * .

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

Abstract:We prove Obata's rigidity theorem for metric measure spaces that satisfy a Riemannian curvaturedimension condition. Additionally, we show that a lower bound K for the generalized Hessian of a su ciently regular function u holds if and only if u is K-convex. A corollary is also a rigidity result for higher order eigenvalues.

We give two examples of metric measure spaces satisfying the measure contraction property MCP(K, N ) but having different topological dimensions at different regions of the space. The first one satisfies MCP(0, 3) and contains a subset isometric to R, but does not topologically split. The second space satisfies MCP(2, 3) and has diameter π, which is the maximal possible diameter for a space satisfying MCP (N − 1, N ), but is not a topological spherical suspension. The latter example gives an answer to a question by Ohta.2000 Mathematics Subject Classification. Primary 53C23. Secondary 28A33, 49Q20.

The goal of the paper is to give an optimal transport characterization of sectional curvature lower (and upper) bounds for smooth n-dimensional Riemannian manifolds. More generally we characterize, via optimal transport, lower bounds on the so called p-Ricci curvature which corresponds to taking the trace of the Riemann curvature tensor on p-dimensional planes, 1 ≤ p ≤ n. Such characterization roughly consists on a convexity condition of the p-Renyi entropy along L 2 -Wasserstein geodesics, where the role of reference measure is played by the p-dimensional Hausdorff measure. As application we establish a new Brunn-Minkowski type inequality involving p-dimensional submanifolds and the p-dimensional Hausdorff measure.