哈密顿 Ricci 流 Hamilton s Ricci Flow 英文版[DJVU]

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资源信息： 中文名: 哈密顿 Ricci 流 原名: Hamilton's Ricci Flow 作者: Bennett Chow 图书分类: 科技 资源格式: DJVU 版本: 英文版 出版社: 科学出版社 书号: 7030177991 发行时间: 2006年 地区: 大陆 语言: 英文 概述:

【原 书 名】 Hamilton's Ricci Flow 【作　　者】Bennett Chow, Peng Lu, Lei Ni 丛书名：当代数学讲座丛书 【出 版 社】 科学出版社 / AMS 【书 号】 7030177991 【出版日期】 2006 年12月 【开 本】 16开 【页 码】 608 【版 次】1 扫描分辨率：600 dpi ; 646 Scans djvu 阅读器： http://windjview.sourceforge.net/ Product Description Ricci flow is a powerful analytic method for studying the geometry and topology of manifolds. This book is an introduction to Ricci flow for graduate students and mathematicians interested in working in the subject. To this end, the first chapter is a review of the relevant basics of Riemannian geometry. For the benefit of the student, the text includes a number of exercises of varying difficulty. The book also provides brief introductions to some general methods of geometric analysis and other geometric flows. Comparisons are made between the Ricci flow and the linear heat equation, mean curvature flow, and other geometric evolution equations whenever possible. Several topics of Hamilton's program are covered, such as short time existence, Harnack inequalities, Ricci solitons, Perelman's no local collapsing theorem, singularity analysis, and ancient solutions. A major direction in Ricci flow, via Hamilton's and Perelman's works, is the use of Ricci flow as an approach to solving the Poincaré conjecture and Thurston's geometrization conjecture.

目录: Preface Acknowledgments A Detailed Guide for the Reader Notation and Symbols Chapter 1．Riemannian Geometry §1．Introduction §2．Metrics，connections，curvatures and covariant differentiation §3．Basic formulas and identities in Riemannian geometry §4．Exterior differential calculus and Bochner formulas §5．Integration and Hodge theory §6．Curvature decomposition and locally conformally flat manifolds §7．Moving frames and the Gauss—Bonnet formula §8．Variation of arc length，energy and area §9．Geodesics and the exponential map §10．Second fundamental forms of geodesic spheres §11．Laplacian，volume and Hessian comparison theorems §12．Proof of the comparison theorems §13．Manifolds with nonnegative curvature §14．Lie groups and left—invariant metrics §15．Notes and commentary Chapter 2．Fundamentals of the Ricci Flow Equation §1．Geometric flows and geometrization §2．Ricci flow and the evolution of scalar curvature §3．The maximum principle for heat—type equations §4．The Einstein—Hilbert functional §5．Evolution of geometric quantities §6．DeTurck’S trick and short time existence §7．Reaction—diffusion equation for the curvature tensor §8．Notes and commentary Chapter 3．Closed 3一manifolds with Positive Ricci Curvature §1．Hamilton’S 3-manifolds with positive Ricci curvature theorem §2．The maximum principle for tensors §3．Curvature pinching estimates §4．Gradient bounds for the scalar curvature §5．Curvature tends to constant §6．Exponential convergence of the normafized flow §7．Notes and commentary Chapter 4．Ricci Solitons and Special Solutions §1．Gradient Ricci solitons §2．Ganssian and cylinder solitons §3．Cigar steady soliton §4．Rosenau solution §5．An expanding soliton §6．Bryant soliton §7．Homogeneous solutions §8．The isometry group §9．Notes and commentary Chapter 5．Isoperimetric Estimates and No Local Collapsing §1．Sobo~v and logarithmic Sobolev inequalities §2．Evolution of the length of a geodesic §3．Isoperimetric estimate for surfaces §4．Perelman’S no local collapsing theorem §5．Geometric applications of no local collapsing §6．3-manifolds with positive Ricci curvature revisited §7．Isoperimetric estimate for 3-dimensional Type I solutions §8．Notes and commentary Chapter 6．Preparation for Singularity Analysis §1．Derivative estimates and long time existence §2．Proof of Shi’S local first and second derivative estimates §3．Cheeger—Gromov—type compactness theorem for Ricci flow §4．Long time existence of solutions with bounded Ricci curvature §5．The Hamilton—Ivey curvature estimate §6．Strong maximum principles and metric splitting §7．Rigidity of 3-manifolds with nonnegative curvature §8．Notes and commentary Chapter 7．High-dimensional and Noncompact Ricci Flow §1．Spherical space form theorem of Huisken-Margerin—Nishikawa §2．4-manifolds with pos~ive curvature operator §3．Manifolds with nonnegative curvature operator §4．The maximum principle on noncompact manifolds §5．Complete solutions of the Ricci flow on noncompact manifolds §6．Notes and commentary Chapter 8．Singularity Analysis §1．SingulariW dilations and Wpes §2．Point picking and types of singularity models §3．Geometric invaxiants of ancient solutions §4．Dimension reduction §5．Notes and commentary Chapter 9．Ancient Solutions §1．Classification of ancient solutions on surfaces §2．Properties of ancient solutions that relate to their type §3．Geometry at infinity of gradient Ricci solitons §4．Injectivity radius of steady gradient Ricci solitons §5．Towards a classification of 3一dimensional ancient solutions §6．Classification of 3一dimensional shrinking Ricci solitons §7．Summary and open problems Chapter 10．Differential Harnack Estimates §1．Harnack estimates for the heat and Laplace equations §2．Harnack estimate on surfaces with x>0 §3．Linear trace and interpolated Harnack estimates on surfaces §4．Hamilton’S matrix Harnack estimate for the Ricci flow §5．Proof of the matrix Harnack estimate §6．Harnack and pinching estimates for linearized Ricci flow §7．Notes and commentary Chapter 11．Space-time Geometry §1．Space-time solution to the Pdcci flow for degenerate metrics §2．Space-time curvature is the matrix Harnack quadratic §3．Potentially infinite metrics and potentially infinite dimensions §4．Renormalizing the space-time length yields the g-length §5．Space-time DeTurck’S trick and fixing the measure §6．Notes and commentary Appendix A．Geometric Analysis Related to Ricci Flow §1．Compendium of inequalities §2．Comparison theory for the heat kernel §3．Green’S function §4．The Liouville theorem revisited §5．Eigenvalues and eigenfunctions of the Laplacian §6．The determinant of the Laplacian §7．Parametrix for the heat equation §8．Monotonicity for harmonic functions and maps §9．Bieberbach theorem §10．Notes and commentary Appendix B．Analytic Techniques for Geometric Flows §1．Riemannian surfaces §2．Kazdan-Warner—type identities and solitons §3．Andrews’Poincare-type inequality §4．The Yamabe flow and Aleksandrov reflection §5．The cross curvature flow §6．Time derivative of the sup function §7．Notes and commentary Appendix S．Solutions to Selected Exercises Bibliography Index