Geometric analysis has become one of the most important tools in geometry and topology. In their books on the Ricci flow, the authors reveal the depth and breadth of this flow method for understanding the structure of manifolds. With the present book, the authors focus on the analytic aspects of Ricci flow. Some highlights of the presentation are weak and strong maximum principles for scalar heat-type equations and systems on manifolds, the classification by Bohm and Wilking of closed manifolds with 2-positive curvature operator, Bando's result that solutions to the Ricci flow are real analytic in the space variables, Shi's local derivative of curvature estimates and some variants, and differential Harnack estimates of Li-Yau-type including Hamilton's matrix estimate for the Ricci flow and Perelman's estimate for fundamental solutions of the adjoint heat equation coupled to the Ricci flow. The authors have tried to make some advanced material accessible to graduate students and nonexperts. The book gives a rigorous introduction to some of Perelman's work and explains some technical aspects of Ricci flow useful for singularity analysis. They have also attempted to give the appropriate references so that the reader may further pursue the statements and proofs of the various results. See also: The Ricci Flow: An Introduction The Ricci Flow: Techniques and Applications. Part I: Geometric Aspects The Ricci Flow: Techniques and Applications. Part III: Geometric-Analytic Aspects (forthcoming)
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"Just like in the first part this book is meant to be a text book as well as a reference book and it includes exercises as well as the solutions for some of them. The authors took a great care in making this a self contained book which compiles a great amount of data related to Ricci flow, organizing the information in a very clear way giving very often information on the content of each chapter and relating it to other parts of the book. An example is the sumary given in the preface, another more detailed sumary given in a special kind of chapter zero, also in the beginning of each chapter and sometimes introducing each section. Besides all this, each chapter includes a special section in the end with notes and commentaries. This approach gives the reader permanently a global view on the subject while the details are being explained. The book also includes a large number of exercises very often with suggestions for its resolutions." ---- Zentralblatt MATH
"This book and this series is necessarily quite technical in parts, but it does reveal the full beauty and technical details of the Ricci flow. The authors should be congratulated on presenting such material in a very readable style, creating an excellent resource with thorough discussion of many aspects of Ricci flow for the benefit of mathematical researchers today and in the future." ---- Mathematical Reviews
"This book and this series is necessarily quite technical in parts, but it does reveal the full beauty and technical details of the Ricci flow. The authors should be congratulated on presenting such material in a very readable style, creating an excellent resource with thorough discussion of many aspects of Ricci flow for the benefit of mathematical researchers today and in the future." ---- Mathematical Reviews
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In this chapter we present (starting in Section 2) the statements and proofs of various versions of the weak maximum principle for systems. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
algebraic curvature operators, fiberwise convex sets, local heat equation, local derivative estimates, local first derivative estimate, local collapsing theorem, nonnegative curvature operator, pinching set, minimal positive fundamental solution, heat kernel parametrix, curvature pinching estimates, maximum principles for scalars, scalar maximum principle, positive isotropic curvature, adjoint heat equation, manifolds with positive curvature, positive curvature operator, pinching family, gradient soliton, complete noncompact manifolds, constant positive sectional curvature, nonnegative sectional curvature, linear entropy, invariant under parallel translation, weak maximum principle Key Phrases - Capitalized Phrases (CAPs): (learn more)
Volume One, Hamilton Ivey, Weinberger Hamilton, Cheeger Gromov, Bishop Gromov, Second Derivative Test Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
In this chapter we present (starting in Section 2) the statements and proofs of various versions of the weak maximum principle for systems. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
algebraic curvature operators, fiberwise convex sets, local heat equation, local derivative estimates, local first derivative estimate, local collapsing theorem, nonnegative curvature operator, pinching set, minimal positive fundamental solution, heat kernel parametrix, curvature pinching estimates, maximum principles for scalars, scalar maximum principle, positive isotropic curvature, adjoint heat equation, manifolds with positive curvature, positive curvature operator, pinching family, gradient soliton, complete noncompact manifolds, constant positive sectional curvature, nonnegative sectional curvature, linear entropy, invariant under parallel translation, weak maximum principle Key Phrases - Capitalized Phrases (CAPs): (learn more)
Volume One, Hamilton Ivey, Weinberger Hamilton, Cheeger Gromov, Bishop Gromov, Second Derivative Test Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
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