Recent developments about Geometric Analysis on RCD(K,N) spaces

This thesis is about some recent developments on Geometric Analysis and Geometric Measure Theory on RCD(K,N) metric measure spaces that have been obtained in [8,48,49,51,52,171]. After the preliminary Chapter 1, where we collect the basic notions of the theory relevant for our purposes, Chapter 2 is dedicated to the presentation of a simplified approach to the structure theory of RCD(K,N) spaces via δ- splitting maps developed in collaboration with Brué and Pasqualetto. The strategy is similar to the one adopted by Cheeger-Colding in the theory of Ricci limit spaces and it is suitable for adaptations to codimension one. Chapter 3 is devoted to the proof of the constancy of the dimension conjecture for RCD(K,N) spaces. This has been obtained in a joint work with Brué, where we proved that dimension of the tangent space is the same almost everywhere with respect to the reference measure, generalizing a previous result obtained by Colding-Naber for Ricci limits. The strategy is based on the study of regularity of flows of Sobolev vector fields on spaces with Ricci curvature bounded from below, which we find of independent interest. In Chapters 4 and 5 we present the structure theory for boundaries of sets of finite perimeter in this framework, as developed in collaboration with Ambrosio, Brué and Pasqualetto. An almost complete generalization of De Giorgi’s celebrated theorem is given, opening to further developments for Geometric Measure Theory in the setting of synthetic lower bounds on Ricci curvature. In Chapter 6 we eventually collect some results about sharp lower bounds on the first Dirichlet eigenvalue of the p-Laplacian based on a joint work with Mondino. We also address the problems of rigidity and almost rigidity, heavily relying on the compactness and stability properties of RCD spaces.