Sharp Isoperimetric Inequalities and Model Spaces for Curvature-Dimension-Diameter Conditions
speaker: Emanuel Milman (Technion - Israel Institute of Technology)abstract: In recent years, thanks to the works of Sturm, Lott--Villani and others, the relevance of optimal-transport to the characterization of metric-measure spaces admitting a generalized Ricci curvature lower-bound and generalized dimension upper-bound, has become apparent. Various properties of a space satisfying such a Curvature-Dimension condition may be studied. In this work, we obtain new sharp isoperimetric inequalities in the more traditional setting of a Riemannian manifold equipped with a probability measure, whose generalized Ricci curvature is bounded from below (possibly negatively), and generalized dimension and diameter of the convex support are bounded from above (possibly infinitely).
Our inequalities are sharp for sets of any given measure and with respect to all parameters (curvature, dimension and diameter). Moreover, for each choice of parameters, we identify the model spaces which are extremal for the isoperimetric problem. In particular, we recover the Gromov--Lévy and Bakry--Ledoux isoperimetric inequalities, which state that whenever the curvature is strictly positively bounded from below, these model spaces are the \(n\)-sphere and Gauss space, corresponding to generalized dimension being \(n\) and \(\infty\), respectively. In all other cases, which seem new even for the classical Riemannian-volume measure, it turns out that there is no single model space to compare to, and that a simultaneous comparison to a natural one parameter family of model spaces is required, nevertheless yielding a sharp result.
Our method, going back to Gromov, builds on tools provided by Geometric Measure Theory, and is thus currently confined to the Riemannian setting. Even in this "classical" setting, other methods (such as optimal-transport and heat semi-group methods) face various challenges, which we will try to outline (time-permitting).
timetable:
Tue 6 Nov, 9:30 - 10:20, Aula Dini
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