New York Journal of Mathematics
Volume 14 (2008) 459-494


Melanie Pivarski and Laurent Saloff-Coste

Small time heat kernel behavior on Riemannian complexes

Published: September 30, 2008
Keywords: Poincaré inequality, heat kernel, heat equation, polynomial growth group, Euclidean complex, Riemannian complex, polytopal complex, polyhedral complex
Subject: 26D10, 35B40, 43A85, 57S, 58J35

We study how bounds on the local geometry of a Riemannian polyhedral complex yield uniform local Poincaré inequalities. These inequalities have a variety of applications, including bounds on the heat kernel and a uniform local Harnack inequality. We additionally consider the example of a complex, X, which has a finitely generated group of isomorphisms, G, such that X/G =Y is a complex consisting of a finite number of polytopes. We show that when this group, G, has polynomial volume growth, there is a uniform global Poincaré inequality on the complex, X.


Pivarski's research was partially supported by NSF Grant DMS 0306194; Saloff-Coste's research was partially supported by NSF Grant DMS 0603886.

Author information

Melanie Pivarski:
Texas A&M University, College Station, TX 77843 USA

Laurent Saloff-Coste:
Cornell University, Ithaca, NY 15843 USA