Functional inequalities and the curvature dimension inequality on totally geodesic foliations

<p>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.</p>

Name thesis:degree_name

Doctor of Philosophy (PhD)

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Year

2015

Author dc:creator

• Kim, Bumsik

Contributors dc:contributor

• Fabrice Baudoin
• Rodrigo Bañuelos
• Laszlo Lempert
• Sai Kee Yeung

• Mathematics

Repository record dc:identifier

https://docs.lib.purdue.edu/open_access_dissertations/487

OAI identifier oai:identifier

oai:docs.lib.purdue.edu:open_access_dissertations-1383