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.

Readership

Graduate students and research mathematicians interested in geometric analysis; heat kernel methods in geometry and analysis.

Table of Contents

• Laplace operator and the heat equation in \(\mathbb{R}^n\)
• Function spaces in \(\mathbb{R}^n\)
• Laplace operator on a Riemannian manifold
• Laplace operator and heat equation in \(L^{2}(M)\)
• Weak maximum principle and related topics
• Regularity theory in \(\mathbb{R}^n\)
• The heat kernel on a manifold
• Positive solutions
• Heat kernel as a fundamental solution
• Spectral properties
• Distance function and completeness
• Gaussian estimates in the integrated form
• Green function and Green operator
• Ultracontractive estimates and eigenvalues
• Pointwise Gaussian estimates I
• Pointwise Gaussian estimates II
• Reference material
• Bibliography
• Some notation
• Index