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Heat Kernel and Analysis on Manifolds

Alexander Grigor’yan University of Bielefeld, Bielefeld, Germany
A co-publication of the AMS and International Press of Boston
Available Formats:
Softcover ISBN: 978-0-8218-9393-7
Product Code: AMSIP/47.S
List Price: $133.00 MAA Member Price:$119.70
AMS Member Price: $106.40 Electronic ISBN: 978-1-4704-1750-5 Product Code: AMSIP/47.E List Price:$125.00
MAA Member Price: $112.50 AMS Member Price:$100.00
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This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $199.50 MAA Member Price:$179.55
AMS Member Price: $159.60 Click above image for expanded view Heat Kernel and Analysis on Manifolds Alexander Grigor’yan University of Bielefeld, Bielefeld, Germany A co-publication of the AMS and International Press of Boston Available Formats:  Softcover ISBN: 978-0-8218-9393-7 Product Code: AMSIP/47.S  List Price:$133.00 MAA Member Price: $119.70 AMS Member Price:$106.40
 Electronic ISBN: 978-1-4704-1750-5 Product Code: AMSIP/47.E
 List Price: $125.00 MAA Member Price:$112.50 AMS Member Price: $100.00 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$199.50 MAA Member Price: $179.55 AMS Member Price:$159.60
• Book Details

Volume: 472009; 482 pp
MSC: 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 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.

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

• Request Review Copy
Volume: 472009; 482 pp
MSC: 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 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.

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
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