The Ricci Flow : techniques and applications. Part II, Analytic aspects / Bennett Chow [and nine others].

Format:

Book

Author/Creator:

Chow, Bennett, author.

Series:

Mathematical surveys and monographs ; volume 144.

Mathematical surveys and monographs ; volume 144

Language:

English

Subjects (All):

Global differential geometry.

Ricci flow.

Riemannian manifolds.

Physical Description:

1 online resource (489 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2008]

Language Note:

English

Summary:

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.

Contents:

""Contents""; ""Preface""; ""What Part II is about""; ""Highlights and interdependencies of Part II""; ""Acknowledgments""; ""Contents of Part II of Volume Two""; ""Notation and Symbols""; ""Chapter 10. Weak Maximum Principles for Scalars, Tensors, and Systems""; ""1. Weak maximum principles for scalars and symmetric 2-tensors""; ""2. Vector bundle formulation of the weak maximum principle for systems""; ""3. Spatial maximum function and its Dini derivatives""; ""4. Convex sets, support functions, ODEs preserving convex sets""

""5. Proof of the WMP for systems: time-dependent sets and avoidance sets""""6. Maximum principles for weak solutions of heat equations""; ""7. Variants of maximum principles""; ""8. Notes and commentary""; ""Chapter 11. Closed Manifolds with Positive Curvature""; ""1. Multilinear algebra related to the curvature operator""; ""2. Algebraic curvature operators and Rm""; ""3. A family of linear transformations and their effect on R[sup(2)] + R[sup(#)]""; ""4. Proof of the main formula for D[sub(a,b)](R)""; ""5. The convex cone of 2-nonnegative algebraic curvature operators""

""6. A pinching family of convex cones in the space of algebraic curvature operators""""7. Obtaining a generalized pinching set from a pinching family and the proof of Theorem 11.2""; ""8. Summary of the proof of the convergence of Ricci flow""; ""9. Notes and commentary""; ""Chapter 12. Weak and Strong Maximum Principles on Noncompact Manifolds""; ""1. Weak maximum principles for scalar heat-type equations""; ""2. Mollifying distance functions on Riemannian manifolds""; ""3. Weak maximum principle for parabolic systems""; ""4. Strong maximum principle for parabolic systems""

""5. Applications to the curvature operator under the Ricci flow""""6. Notes and commentary""; ""Chapter 13. Qualitative Behavior of Classes of Solutions""; ""1. Curvature conditions that are not preserved""; ""2. Real analyticity in the space variables for solutions of the Ricci flow""; ""Chapter 14. Local Derivative of Curvature Estimates""; ""1. Introduction�fine versus coarse estimates""; ""2. A quick review of the global derivative estimates""; ""3. Shi's local derivative estimates""

""4. Modified Shi's local derivative estimates assuming bounds on some derivatives of curvatures of the initial metrics""""5. Some applications of the local derivative estimates""; ""6. Local heat equation and local Ricci flow""; ""7. Notes and commentary""; ""Chapter 15. Differential Harnack Estimates of LYH-type""; ""1. Deriving the Harnack expression using Ricci solitons""; ""2. Statement of the matrix Harnack estimate""; ""3. Proofs: getting started with surfaces""; ""4. Proof of the matrix Harnack estimate""; ""5. A variant on Hamilton's proof of the matrix Harnack estimate""

""6. Ricci solitons and ancient solutions attaining R[sub(max)]""

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on online resource; title from PDF title page (ebrary, viewed May 28, 2014).

ISBN:

1-4704-1371-X