Heat Kernel and Analysis on Manifolds (Ams/Ip Studies in Advanced Mathematics)

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. This Volume Contains The Expanded Lecture Notes Of Courses Taught At The Emile Borel Centre Of The Henri Poincaré Institute (paris). In The Book, Leading Experts Introduce Recent Research In Their Fields. The Unifying Theme Is The Study Of Heat Kernels In Various Situations Using Related Geometric And Analytic Tools. Topics Include Analysis Of Complex-coefficient Elliptic Operators, Diffusions On Fractals And On Infinite-dimensional Groups, Heat Kernel And Isoperimetry On Riemannian Manifolds, Heat Kernels And Infinite Dimensional Analysis, Diffusions And Sobolev-type Spaces On Metric Spaces, Quasi-regular Mappings And P -laplace Operators, Heat Kernel And Spherical Inversion On Sl 2 (c) , Random Walks And Spectral Geometry On Crystal Lattices, Isoperimetric And Isocapacitary Inequalities, And Generating Function Techniques For Random Walks On Graphs.--publisher's Website. Some Questions On Elliptic Operators / P. Auscher -- Heat Kernels And Sets With Fractal Structure / M. T. Barlow -- Brownian Motions On Compact Groups Of Infinite Dimension / A. Bendikov And L. Saloff-coste -- Heat Kernel And Isoperimetry On Non-compact Riemannian Manifolds / T. Coulhon -- Heat Kernels Measures And Infinite Dimensional Analysis / B. K. Driver -- Heat Kernels And Function Theory On Metric Measure Spaces / A. Grigor'yan -- Sobolev Spaces On Metric-measure Spaces / P. Hajłasz -- Quasiregular Mappings And The P -laplace Operator / I. Holopainen -- Spherical Inversion On Sl 2 (c) / J. Jorgenson And S. Lang -- Spectral Geometry Of Crystal Lattices / M. Kotani And T. Sunada -- Lectures On Isoperimetric And Isocapacitary Inequalities In The Theory Of Sobolev Spaces / V. Maz'ya -- Some Topics Related To Analysis On Metric Spaces / S. Semmes -- Probability Measures On Metric Spaces Of Nonpositive Curvature / K.-t. Sturm -- Generating Function Techniques For Random Walks On Graphs / W. Woess. Alexander Grigorʹyan. Includes Bibliographical References (p. 457-473) And Index. Cover Title page Dedication Contents Preface Laplace operator and the heat equation in Rn Function spaces in Rn Laplace operator on a Riemannian manifold Laplace operator and heat equation in L2(M) Weak maximum principle and related topics Regularity theory in Rn 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 Bibliography Some notation Index Back Cover 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. This title offers an introduction to heat kernel techniques in the setting of Riemannian manifolds.