Rigidity results in CD(K, N) spaces with negative N

The goal of this project is to study the analytic and geometric properties of metric measure spaces with a lower bound on the Ricci curvature and negative effective dimension. Admitting the effective dimension to be negative may sound strange if one thinks to it as an upper bound on the topological one; however, in the setting of weighted Riemannian manifolds with certain concave weights, it has been shown that it is very useful to introduce this notion of negative effective dimension in order to better understand the geometry of these manifolds. In particular, this concept plays a role in the physics of scalar tensor gravitation theories and low-energy approximations to string theory. A prototypical example of these structures is given by the n-dimensional unit sphere equipped with the harmonic measure. Therefore it is possible and meaningful to generalize this new notion also in the setting of metric measure spaces, which include structures which are very far from being Euclidean. The so-called CD spaces are metric measure structures in which a lower bound on the curvature and an upper bound on the dimension, formulated in terms of optimal transport, hold. The aim of this project is to prove new results in this framework: while the properties of the CD spaces for positive values of the dimension have been a central object of investigation in the last years, very little is known for metric measure spaces satisfying the curvature-dimension condition for negative values of the effective dimension. This class of spaces covers wider/wilder structures than the ones with positive dimension; therefore we will obtain new results in the context of the analysis of CD spaces and provide new techniques to study geometry on very general metric measure spaces. In particular, in this framework we aim at proving: (1) the possibility to define a suitable energy functional in this setting and consequentially a version of the Bochners inequality, (2) regularizing and contraction properties for the heat flow, (3) rigidity properties of these structures. This will be done thanks to an original interaction between optimal transport and metric geometry techniques.