We prove a regularity result for Lagrangian flows of Sobolev vector fields over RCD(K, N ) metric measure spaces, regularity is understood with respect to a newly defined quasi-metric built from the Green function of the Laplacian. Its main application is that RCD(K, N ) spaces have constant dimension. In this way we generalize to such abstract framework a result proved by Colding-Naber for Ricci limit spaces, introducing ingredients that are new even in the smooth setting. * Scuola Normale Superiore, elia.brue@sns.it. † Scuola Normale Superiore, daniele.semola@sns.it.
This note is dedicated to the study of the asymptotic behaviour of sets of finite perimeter over RCD(K, N ) metric measure spaces. Our main result asserts existence of a Euclidean tangent half-space almost everywhere with respect to the perimeter measure and it can be improved to an existence and uniqueness statement when the ambient is non collapsed. As an intermediate tool, we provide a complete characterization of the class of RCD(0, N ) spaces for which there exists a nontrivial function satisfying the equality in the 1-Bakry-Émery inequality. This result is of independent interest and it is new, up to our knowledge, even in the smooth framework.
The goal of the paper is four-fold. In the setting of non-smooth spaces with Ricci curvature lower bounds (more precisely RCD(K, N ) metric measure spaces):• We establish a new principle relating lower Ricci curvature bounds to the preservation of Laplacian lower bounds under the evolution via the p-Hopf-Lax semigroup, for general exponents p ∈ [1, ∞).• We develop an instrinsic viscosity theory of Laplacian bounds.• We prove Laplacian bounds on the distance function from a set (locally) minimizing the perimeter; this corresponds to vanishing mean curvature in the smooth setting.• We initiate a regularity theory for boundaries of sets (locally) minimizing the perimeter. The class of RCD(K, N ) metric measure spaces includes as remarkable sub-classes: measured Gromov-Hausdorff limits of smooth manifolds with lower Ricci curvature bounds and finite dimensional Alexandrov spaces with lower sectional curvature bounds. Most of the results are new also in these frameworks.
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 ∈ (1, ∞) 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 ∈ (1, ∞). This extends to the non-smooth setting a classical result of Bérard-Meyer [BM92] and Matei [Ma00]; remarkable examples of spaces fitting out framework and for which the results seem new include: measured-Gromov Hausdorff limits of Riemannian manifolds with Ricci≥ K > 0, finite dimensional Alexandrov spaces with curvature≥ K > 0, Finsler manifolds with Ricci≥ 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≥ K > 0.
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