Optimal Transport: A Semi-Discrete Approach

Certain conditions for a Riemannian manifold to be isometric with a sphere:Dedicated to Professor Kentaro Yano on his fiftieth birthday

Sets of Finite Perimeter and Geometric Variational Problems: Equilibrium shapes of liquids and sessile drops

It is known that by dualizing the Bochner-Lichnerowicz-Weitzenbock formula, one obtains Poincare-type inequalities on Riemannian manifolds equipped with a density, which satisfy the Bakry-Emery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalized dimension). When the manifold has a boundary, an appropriate generalization of the Reilly formula 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 Brascamp-Lieb type inequalities on the manifold. All previously known inequalities of Lichnerowicz, Brascamp-Lieb, Bobkov-Ledoux and Veysseire are recovered, extended to the Riemannian setting and generalized into a single unified formulation, and their appropriate versions in the presence of a boundary are 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 generalized dimension.

Brascamp-Lieb type inequalities on weighted Riemannian manifolds with boundary

Stability of the Bakry-Émery theorem on Rn

We establish a quantitative isoperimetric inequality for weighted Riemannian manifolds with $\mathrm{Ric}_{\infty} \ge 1$. Precisely, we give an upper bound of the volume of the symmetric difference between a Borel set and a sub-level (or super-level) set of the associated guiding function (arising from the needle decomposition), in terms of the deficit in Bakry-Ledoux's Gaussian isoperimetric inequality. This is the first quantitative isoperimetric inequality on noncompact spaces besides Euclidean and Gaussian spaces. Our argument makes use of Klartag's needle decomposition (also called localization), and is inspired by a recent work of Cavalletti, Maggi and Mondino on compact spaces. Besides the quantitative isoperimetry, a reverse Poincare inequality for the guiding function that we have as a key step, as well as the way we use it, are of independent interest.

Cong Hung Mai

Papers

Quantitative estimates for the Bakry-Ledoux isoperimetric inequality