Isoperimetric inequalities on weighted manifolds with boundary

It is well known that Poincaré-type inequalities on Riemannian manifolds with measure satisfying the generalized Bakry–Émery condition can be obtained by using the Bochner–Lichnerowicz–Weitzenböck formula. In the case of manifolds with boundary, a suitable generalization is Reilly’s formula. New Poincarétype inequalities for manifolds with measure are obtained by systematically using this formula combined with various conditions on the boundary of the manifold and boundary conditions for elliptic equations. Among other results, a generalization of Colesanti’s inequality, proved earlier in Euclidean space, is presented. It implies a generalization of Brunn–Minkowski-type inequalities for manifolds. A new evolution equation for surfaces on Riemannian manifolds is studied, which determines the Minkowski addition of convex sets in the Euclidean case. The proposed approach covers a large class of convex measures, including measures with heavy tails, which correspond to negative analytic dimension.