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Working paper

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

arxiv.org. math. Cornell University, 2014. No. 1310.2526.
Kolesnikov A., Milman E.
It is well known that by dualizing the Bochner-Lichnerowicz-Weitzenb\"{o}ck formula, one obtains Poincar\'e-type inequalities on Riemannian manifolds equipped with a density, which satisfy the Bakry-\'Emery Curvature-Dimension condition (combining the Ricci curvature with the "curvature" of the density). When the manifold has a boundary, the Reilly formula and its generalizations may be used instead. By systematically dualizing this formula for various combinations of boundary conditions of the domain (convex, mean-convex) and the function (Neumann, Dirichlet), we obtain new Poincar\'e-type inequalities on the manifold and on its boundary. For instance, we may handle Neumann conditions on a mean-convex domain, and obtain generalizations to the weighted-manifold setting of a purely Euclidean inequality of Colesanti, yielding a Brunn-Minkowski concavity result for geodesic extensions of convex domains in the manifold setting. All other previously known Poincar\'e-type inequalities of Lichnerowicz, Brascamp-Lieb, Bobkov-Ledoux, Nguyen and Veysseire are recovered, in some cases improved, and generalized into a single unified formulation, and their appropriate versions in the presence of a boundary are obtained. Finally, a new geometric evolution equation is proposed 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".