A new notion of angle between three points in a metric space

We give a new notion of angle in general metric spaces; more precisely, given a triple a points $p,x,q$ in a metric space $(X,d)$, we introduce the notion of angle cone ${\angle_{pxq}}$ as being an interval ${\angle_{pxq}}:=[\angle^-_{pxq},\angle^+_{pxq}]$, where the quantities $\angle^\pm_{pxq}$ are defined in terms of the distance functions from $p$ and $q$ via a duality construction of differentials and gradients holding for locally Lipschitz functions on a general metric space. Our definition in the Euclidean plane gives the standard angle between three points and in a Riemannian manifold coincides with the usual angle between the geodesics, if $x$ is not in the cut locus of $p$ or $q$. We show that in general the angle cone is not single valued (even in case the metric space is a smooth Riemannian manifold, if $x$ is in the cut locus of $p$ or $q$), but if we endow the metric space with a positive Borel measure $m$ obtaining the metric measure space $(X,d,m)$ then under quite general assumptions (which include many fundamental examples as Riemannian manifolds, finite dimensional Alexandrov spaces with curvature bounded from below, Gromov-Hausdorff limits of Riemannian manifolds with Ricci curvature bounded from below, and normed spaces with strictly convex norm), fixed $p,q \in X$, the angle cone at $x$ is single valued for $m$-a.e. $x \in X$. We prove some basic properties of the angle cone (such as the invariance under homotheties of the space) and we analyze in detail the case $(X,d,m)$ is a measured-Gromov-Hausdorff limit of a sequence of Riemannian manifolds with Ricci curvature bounded from below, showing the consistency of our definition with a recent construction of Honda.Comment: 19 page

Immersed Spheres of Finite Total Curvature into Manifolds

We prove that a sequence of possibly branched, weak immersions of the two-sphere $S^2$ into an arbitrary compact riemannian manifold $(M^m,h)$ with uniformly bounded area and uniformly bounded $L^2-$norm of the second fundamental form either collapse to a point or weakly converges as current, modulo extraction of a subsequence, to a Lipschitz mapping of $S^2$ and whose image is made of a connected union of finitely many, possibly branched, weak immersions of $S^2$ with finite total curvature. We prove moreover that if the sequence belongs to a class $\gamma$ of $\pi_2(M^m)$ the limiting lipschitz mapping of $S^2$ realizes this class as well.Comment: 33 pages. Original preprint (2011). This is the final version to appear in Adv. Calc. Va

On the universal cover and the fundamental group of an RCD∗(K,N)RCD^*(K,N)-space

The main goal of the paper is to prove the existence of the universal cover for $RCD^*(K,N)$-spaces. This generalizes earlier work of C. Sormani and the second named author on the existence of universal covers for Ricci limit spaces. As a result, we also obtain several structure results on the (revised) fundamental group of such spaces. These are the first topological results for $RCD^{*}(K,N)$-spaces without extra structural-topological assumptions (such as semi-local simple connectedness).Comment: Final version to appear in Journal f\"ur die Reine und Angewandte Mathemati

Polya-Szego inequality and Dirichlet pp-spectral gap for non-smooth spaces with Ricci curvature bounded below

We study decreasing rearrangements of functions defined on (possibly non-smooth) metric measure spaces with Ricci curvature bounded below by $K>0$ and dimension bounded above by $N\in (1,\infty)$ in a synthetic sense, the so called $CD(K,N)$ spaces. We first establish a Polya-Szego type inequality stating that the $W^{1,p}$-Sobolev norm decreases under such a rearrangement and apply the result to show sharp spectral gap for the $p$-Laplace operator with Dirichlet boundary conditions (on open subsets), for every $p\in (1,\infty)$. This extends to the non-smooth setting a classical result of B\'erard-Meyer and Matei; remarkable examples of spaces fitting out framework and for which the results seem new include: measured-Gromov Hausdorff limits of Riemannian manifolds with Ricci$\geq K>0$, finite dimensional Alexandrov spaces with curvature$\geq K>0$, Finsler manifolds with Ricci$\geq K>0$. In the second part of the paper we prove new rigidity and almost rigidity results attached to the aforementioned inequalities, in the framework of $RCD(K,N)$ spaces, which seem original even for smooth Riemannian manifolds with Ricci$\geq K>0$.Comment: 33 pages. Final version published in Journal de Math\'ematiques Pures et Appliqu\'ee

New formulas for the Laplacian of distance functions and applications

The goal of the paper is to prove an exact representation formula for the Laplacian of the distance (and more generally for an arbitrary 1-Lipschitz function) in the framework of metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense (more precisely in essentially non-branching MCP(K,N)-spaces). Such a representation formula makes apparent the classical upper bounds and also some new lower bounds, together with a precise description of the singular part. The exact representation formula for the Laplacian of 1-Lipschitz functions (in particular for distance functions) holds also (and seems new) in a general complete Riemannian manifold. We apply these results to prove the equivalence of CD(K,N) and a dimensional Bochner inequality on signed distance functions. Moreover we obtain a measure-theoretic Splitting Theorem for infinitesimally Hilbertian essentially non-branching spaces verifying MCP(0,N).Comment: Final version to appear in Analysis and PD

Sectional and intermediate Ricci curvature lower bounds via Optimal Transport

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, $1\leq p\leq n$. 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.Comment: Final version, published by Advances in Mathematic

A Gap Theorem for Willmore Tori and an application to the Willmore Flow

In 1965 Willmore conjectured that the integral of the square of the mean curvature of a torus immersed in $R^3$ is at least $2\pi^2$ and attains this minimal value if and only if the torus is a M\"obius transform of the Clifford torus. This was recently proved by Marques and Neves. In this paper, we show for tori there is a gap to the next critical point of the Willmore energy and we discuss an application to the Willmore flow. We also prove an energy gap from the Clifford torus to surfaces of higher genus.Comment: 9 pages. In this new version we performed some small changes to improve the exposition. To appear in Nonlinear Analysis: Theory Methods & Application