Date of Award
Spring 2015
Degree Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematics
First Advisor
Fabrice Baudoin
Committee Chair
Fabrice Baudoin
Committee Member 1
Rodrigo Bañuelos
Committee Member 2
Laszlo Lempert
Committee Member 3
Sai Kee Yeung
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.
Recommended Citation
Kim, Bumsik, "Functional inequalities and the curvature dimension inequality on totally geodesic foliations" (2015). Open Access Dissertations. 487.
https://docs.lib.purdue.edu/open_access_dissertations/487