1st Edition

Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincare Conjecture

By Qi S. Zhang Copyright 2010

    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




    Qi S. Zhang is a professor of mathematics at the University of California, Riverside.

    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.
    Zentralblatt MATH

    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