Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincare Conjecture
Focusing on Sobolev inequalities and their applications to analysis on manifolds and Ricci flow, Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincaré Conjecture introduces the field of analysis on Riemann manifolds and uses the tools of Sobolev imbedding and heat kernel estimates to study Ricci flows, especially with surgeries. The author explains key ideas, difficult proofs, and important applications in a succinct, accessible, and unified manner.
The book first discusses Sobolev inequalities in various settings, including the Euclidean case, the Riemannian case, and the Ricci flow case. It then explores several applications and ramifications, such as heat kernel estimates, Perelman’s W entropies and Sobolev inequality with surgeries, and the proof of Hamilton’s little loop conjecture with surgeries. Using these tools, the author presents a unified approach to the Poincaré conjecture that clarifies and simplifies Perelman’s original proof.
Since Perelman solved the Poincaré conjecture, the area of Ricci flow with surgery has attracted a great deal of attention in the mathematical research community. Along with coverage of Riemann manifolds, this book shows how to employ Sobolev imbedding and heat kernel estimates to examine Ricci flow with surgery.
Sobolev Inequalities in the Euclidean Space
Weak derivatives and Sobolev space Wk,p(D), D subset Rn
Main imbedding theorem for W01,p(D)
Poincaré inequality and log Sobolev inequality
Best constants and extremals of Sobolev inequalities
Basics of Riemann Geometry
Riemann manifolds, connections, Riemann metric
Second covariant derivatives, curvatures
Common differential operators on manifolds
Geodesics, exponential maps, injectivity radius etc.
Integration and volume comparison
Conjugate points, cut-locus, and injectivity radius
Bochner–Weitzenbock type formulas
Sobolev Inequalities on Manifolds
A basic Sobolev inequality
Sobolev, log Sobolev inequalities, heat kernel
Sobolev inequalities and isoperimetric inequalities
Parabolic Harnack inequality
Maximum principle for parabolic equations
Gradient estimates for the heat equation
Basics of Ricci Flow
Local existence, uniqueness and basic identities
Maximum principles under Ricci flow
Qualitative properties of Ricci flow
Solitons, ancient solutions, singularity models
Perelman’s Entropies and Sobolev Inequality
Perelman’s entropies and their monotonicity
(Log) Sobolev inequality under Ricci flow
Critical and local Sobolev inequality
Harnack inequality for the conjugate heat equation
Fundamental solutions of heat type equations
Ancient κ Solutions and Singularity Analysis
Heat kernel and κ solutions
Backward limits of κ solutions
Qualitative properties of κ solutions
Singularity analysis of 3-dimensional Ricci flow
Sobolev Inequality with Surgeries
A brief description of the surgery process
Sobolev inequality, little loop conjecture, and surgeries
Applications to the Poincaré Conjecture
Evolution of regions near surgery caps
Canonical neighborhood property with surgeries
Summary and conclusion
The approach here is somewhat different from that of Perelman. The author shows that the W-entropy and its monotonicity imply certain uniform Sobolev inequalities along Ricci flows. These are used in the proofs of the two steps mentioned above, bypassing the use of the reduced volume and reduced distance, which simplifies Perelman’s proof considerably.
—John Urbas, Mathematical Reviews, Issue 2011m
This is a very good book on Ricci flows. Anyone who is interested to know the most recent development in Ricci flows and the Poincaré conjecture should take a look at the book.
It is clear as vodka that, as Zhang advertises in the Preface, ‘the first half of the book is aimed at graduate students and the second half is intended for researchers.’ With some good timing, the same reader can start as one and continue as the other. … a very important contribution to the genre.
—MAA Reviews, September 2010