# Book Details

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Preface | Table of Contents | Introduction | MARC record | Metadata XML | e-Book PDF (854 KB)*Reto Müller (ETH Zürich)*

#### Differential Harnack Inequalities and the Ricci Flow

ISBN print 978-3-03719-030-2, ISBN online 978-3-03719-530-7DOI 10.4171/030

August 2006, 99 pages, softcover, 17.0 x 24.0 cm.

24.00 Euro

The classical Harnack inequalities play an important role in the
study of parabolic partial differential equations. The idea of
finding 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 of the linear heat
equation on a manifold which leads to a classical Harnack type
inequality if being integrated along a path. Their idea has been
successfully adopted and generalized to (nonlinear) geometric heat
flows such as mean curvature flow or Ricci flow; most of this work
was done by Richard Hamilton. In 2002, Grisha 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. This approach forms the main analytic core of Perelman's
attempt to prove the Poincaré conjecture. It is, however, of
completely independent interest and may as well prove useful in various
other areas, such as, for instance, the theory of Kähler manifolds.

The goal of this book is to explain this analytic tool in full
detail 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
Perelman's entropy formulas and space-time geodesics.

The text is a self-contained, modern introduction to the Ricci flow
and the analytic methods to study it. It is primarily addressed to
students who have a basic introductory knowledge of analysis and of
Riemannian geometry and who are attracted to further study in
geometric analysis. No previous knowledge of differential Harnack
inequalities or the Ricci flow is required.