Functional inequalities and the curvature dimension inequality on totally geodesic foliations
Abstract
We discover following analytic / geometric properties on Riemannian foliations with bundle-like metric and totally geodesic leaves, or shortly, totally geodesic foliations. Under a certain curvature condition, we obtain (1) Sobolev-isoperimetric inequalities, global Poincar\'e inqualities, and a lower bound for Cheeger's isoperimetric constant, (2) Poincar\'e inequalities on balls and uniqueness of positive(or $L p,p\geq 1$) solutions for the subelliptic heat equation, (3) A lower bound for the first non-zero eigenvalue of sub-Laplacians (Lichnerowicz theorem), and Obata's sphere theorem. In this context, the curvature condition is a sub-Riemannian analogue of lower bounds for Ricci curvature tensor. Earlier, it is given by Baudoin-Garofalo's curvature dimension inequality, or Baudoin's Weitzenb"ock formulas for one forms. Our framework includes CR Sasakian manifolds with Tanaka-Webster (or pseudo-Hermitian) Ricci tensor bounds, K-contact manifolds, and Carnot group of step 2.
Degree
Ph.D.
Advisors
Baudoin, Purdue University.
Subject Area
Mathematics
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