Faculty Activities[ Edit ]
Advisor: Emanuel Milman
Abstract: We establish new sharp inequalities of Poincare or log-Sobolev type, on weighted Riemannian manifolds whose (generalized) Ricci curvature is bounded from below. To this end we establish a general method which complements the 'localization' theorem which has recently been established by B. Klartag. Klartag's theorem is based on optimal transport techniques, leading to a disintegration of the manifold measure into marginal measures supported on geodesics of the manifold. This leads to a reduction of the problem of proving a n-dimensional inequality into an optimization problem over a class of measures with 1-dimensional supports. Our method is based on functional analytic techniques, and leads to a further reduction of the optimization problem into a simpler problem over a sub-class of model-space measures. By employing ad-hoc analytical techniques we solve the optimization problems associated with the Poincare and the log-Sobolev inequalities. Quiet unexpectedly the solution to the problem of characterizing the sharp Poincare constant reveals anomalous behavior within a certain domain of the generalized-dimension parameter, hinting on a new phenomena.