Softcover ISBN:  9780821893937 
Product Code:  AMSIP/47.S 
List Price:  $133.00 
MAA Member Price:  $119.70 
AMS Member Price:  $106.40 
Electronic ISBN:  9781470417505 
Product Code:  AMSIP/47.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 

Book DetailsAMS/IP Studies in Advanced MathematicsVolume: 47; 2009; 482 ppMSC: Primary 58; Secondary 31; 35; 47; 53;
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 spectraltheoretic, 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.ReadershipGraduate students and research mathematicians interested in geometric analysis; heat kernel methods in geometry and analysis.

Table of Contents

Chapters

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

Appendix A. Reference material


Additional Material

Request Review Copy
 Book Details
 Table of Contents
 Additional Material

 Request Review Copy
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 spectraltheoretic, 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.
Graduate students and research mathematicians interested in geometric analysis; heat kernel methods in geometry and analysis.

Chapters

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

Appendix A. Reference material