Ph.D Thesis | |

Ph.D Student | Calderon Eran |
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Subject | Functional Inequalities on Weighted Riemannian Manifolds Subject to Curvature-Dimension Conditions |

Department | Department of Mathematics |

Supervisor | PROF. Emanuel Milman |

We establish new sharp inequalities of Poincaré or log-Sobolev type, on geodesically-convex weighted Riemannian manifolds (M,g,μ) whose (generalized) Ricci curvature Ricg,μ,Nwith effective dimension parameter N∈(−∞,∞] is bounded from below by a constant K∈R, and whose diameter is bounded above by D∈(0,∞] (Curvature-Dimension-Diameter conditions CDD(K,N,D)). 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. In this work we firstly develop a general approach which leads to a reduction of this optimization problem into a simpler optimization problem, on a subclass of `model measures'. This reduction is based on functional analytic techniques, in particular a classification of extreme points of a specific subset of measures, and showing that the solution to the optimization problem is attained on this set of extreme points. Finally we solve the optimization problems associated with the Poincaré, p-Poincaré and the log-Sobolev inequalities subject to specific CDD(K,N,D)conditions. Notably, we prove new sharp Poincaré inequalities for N∈(−∞,0]. We find that for N∈(−1,0] the characterization of the sharp lower bound on the Poincaré constant is of different nature; in addition we derive new lower bounds on the log-Sobolev constant under CDD(K,∞,D) conditions where K∈R and D∈(0,∞], which up to numeric constants are best possible.