Calculus of Variations and Geometric Measure Theory
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Isoperimetric Problems Between Analysis and Geometry

Brunn-Minkowski and Poincaré-type inequalities on weighted Riemannian manifolds with boundary

Emanuel Milman (Israel Institute of Technology)

created by paolini on 02 Jun 2014
modified on 03 Jun 2014

18 jun 2014 -- 11:00   [open in google calendar]

Abstract.

By systematically dualizing a generalized version of the Reilly formula (an integrated form of Bochner’s formula in the presence of boundary), we obtain new Poincar ́e-type inequalities on a Riemannian manifold equipped with a density and on its boundary under a Curvature-Dimension condition, for various combinations of boundary conditions of the domain (convex, mean-convex) and the function (Neumann, Dirichlet). In particular, we extend and refine the Poincar’e inequalities of Lichnerowicz, Brascamp–Lieb, Bobkov–Ledoux, Nguyen and Colesanti to the weighted Rieman- nian setting, in a single unified framework. We then propose a new geometric evolution equation, which extends to the Riemannian setting the Minkowski addition operation of convex domains, a notion previously confined to the linear setting, and for which a novel Brunn–Minkowski inequality in the weighted-Riemannian setting is obtained. Our framework allows to encompass the entire class of Borell’s convex measures, including heavy-tailed measures, and extends the latter class to weighted-manifolds having negative “dimension”. Based on joint work with Alexander Kolesnikov.

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