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Differential Harnack inequalities and the Ricci flow. (English) Zbl 1103.58014

EMS Series of Lectures in Mathematics. Zürich: European Mathematical Society Publishing House (ISBN 978-3-03719-030-2/pbk). vii, 92 p. (2006).
The very interesting book under review is a revised and extended version of the author’s diploma thesis written in the winter semester 2004/05 at ETH Zürich under the guidance of Michael Struwe. Its main goal is to explain some of the theory of Perelman’s first Ricci flow paper as well as its thematic context, presenting in details the underlying analytic methods to non-experts or students who are new to the subject. In this sense, the author does not claim any original work beside some simplifications of Perelman’s result for the easier case of a heat equation on a static manifold.
The classical Harnack inequalities play an extremely important role in the study of nonlinear partial differential equations. The idea to find a differential version of such a classical Harnack inequality goes back to Peter Li and Shing-Tung Yau, who introduced a pointwise gradient estimate for a solution to the linear heat equation on a manifold which leads to a classical Harnack inequality if being integrated along a path. Their idea has been adopted and successfully generalized to nonlinear geometric heat flows such as the mean curvature flow or Ricci flow; most of that work was done by Richard Hamilton. In 2002, Grigory Perelman presented a new kind of differential Harnack inequality which involves both the adjoint linear heat equation and the Ricci flow. This led to a completely new approach to the Ricci flow that allowed interpretation as a gradient flow which maximizes different entropy functionals. The technique sets up the main analytic core of Perelman’s approach in proving the Poincaré conjecture. Moreover, it is also of independent interest and may be useful in various other areas such as the theory of Kähler manifolds.
The book explains these analytic tools in full details for the two examples of the linear heat equation and the Ricci flow. It begins with the original Li-Yau result, presents Hamilton’s Harnack inequalities for the Ricci flow, and ends with the Perelman entropy formulas and space-time geodesics.
The book is very well written and self-contained. Highly recommended text!

MSC:

58J35 Heat and other parabolic equation methods for PDEs on manifolds
58-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to global analysis
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
35K55 Nonlinear parabolic equations
58E11 Critical metrics
58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)
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