Heat Kernel and Analysis on Manifolds

by Alexander Grigor'yan

The heat kernel has long been an essential tool in both classical and modern mathematics but has become especially important in geometric analysis as a result of major innovations beginning in the 1970s. The methods based on heat kernels have been used in areas as diverse as analysis, geometry, and probability, as well as in physics. This book is a comprehensive introduction to heat kernel techniques in the setting of Riemannian manifolds, which inevitably involves analysis of the Laplace-Beltrami operator and the associated heat equation. The first ten chapters cover the foundations of the subject, while later chapters deal with more advanced results involving the heat kernel in a variety of settings. The exposition starts with an elementary introduction to Riemannian geometry, proceeds with a thorough study of the spectral-theoretic, Markovian, and smoothness properties of the Laplace and heat equations on Riemannian manifolds, and concludes with Gaussian estimates of heat kernels. Grigor'yan has written this book with the student in mind, in particular by including over 400 exercises. The text will serve as a bridge between basic results and current research. Titles in this series are co-published with International Press, Cambridge, MA.

482 pages, Paperback

First published November 20, 2012

Reviews

[HikBook]

10.18 In this problem, the circle S^1 is identified with R/2piZ
(This will be useful for the stationary problem on the circle)

13.
Definition 13.3 A function h localy integrable on a manifold M is called a fundamental solution of the laplace operator at a point x in M if -laplacian_mu h=delta_x