Geometry of spaces with a synthetic lower curvature bound

Abstract

In recent years, several major breakthroughs in the field of geometry have been closely linked to the investigation of two pivotal topics: curvature and singularities. The objective of this thesis is to explore the geometric and structural properties of singular spaces with a synthetic lower curvature bound. The crucial tool employed in this research field is the theory of optimal transport, which allows to develop an intrinsic approach to curvature bounds and to study many singular spaces from an abstract viewpoint. <br /> In their seminal papers, Sturm and Lott–Villani introduced a synthetic notion of curvature-dimension bounds in the non-smooth setting of metric measure spaces. This condition, called CD(K,N), is formulated in terms of the optimal transport interpolation of measures and consists in a convexity property of the Rényi entropy functionals along Wasserstein geodesics. The CD(K,N) condition represents a lower (Ricci) curvature bound by K and an upper bound on the dimension by N , and it is coherent with the Riemannian setting. <br /> The CD(K,N) spaces enjoy different metric and geometric features, nevertheless, their study by means of classical analytic tools presents non-trivial difficulties, mainly due to their complex geometric structure. In Paper 1 we present several examples of singular CD(K,N) spaces, having different dimensions in different regions. As a consequence, we show how basic rigidity properties, such as weak non-branching conditions, may fail in this setting, despite the curvature-dimension bound. <br /> One of the main merits of the CD(K,N) condition is that it is sufficient to deduce geometric inequalities that hold in the smooth setting. A notable example is the generalized Brunn–Minkowski inequality, called BM(K,N). In Papers 2 and 3, we obtain two partial results in the direction of proving the equivalence between BM(K,N) and CD(K,N). Firstly, we prove it in the setting of weighted Riemannian manifolds. Secondly, we show that, in the general framework of essentially non-branching metric measure spaces, the CD(K,N) condition is equivalent to a more stringent version of the BM(K,N) inequality, that we call strong Brunn-Minkowski SBM(K,N). <br /> While the CD(K,N) condition is equivalent to a lower curvature bound in the Riemannian and Finsler settings, a similar result does not hold for sub-Riemannian and sub-Finsler manifolds. In Papers 4 and 5, we show how the CD(K,N) condition is not well-suited to characterize curvature in these frameworks. On the one hand, we show that every almost-Riemannian manifold, with dimension 2 or strongly regular, equipped with a smooth measure, does not satisfy the CD(K,N) condition for every K and N. On the other hand, we prove the failure of the CD(K,N) condition for smooth sub-Finsler manifolds and in the specific case of the Heisenberg group, under weaker regularity assumptions. <br /> In Papers 6 and 7, we study the extension of the CD(K,N) condition where the dimensional bound N is negative (introduced by Ohta), considering metric measure spaces with quasi-Radon reference measures, as a natural framework for its analysis. In particular, we prove the stability of the CD(K,N) condition with respect to a suitably refined measured Gromov-Hausdorff convergence, that controls the behavior of singular points of the reference measures. Moreover, we prove two remarkable features for the CD(K,N) condition (with negative N),namely the existence of optimal transport maps and the local-to-global property.