1.

Record Nr.

UNINA9910151937303321

Autore

Müller Reto

Titolo

Differential Harnack Inequalities and the Ricci Flow [[electronic resource] /] / Reto Müller

Pubbl/distr/stampa

Zuerich, Switzerland, : European Mathematical Society Publishing House, 2006

ISBN

3-03719-530-4

Descrizione fisica

1 online resource (99 pages)

Collana

EMS Series of Lectures in Mathematics (ELM) ; , 2523-5176

Classificazione

58-xx35-xx53-xx

Soggetti

Differential & Riemannian geometry

Differential equations

Global analysis, analysis on manifolds

Partial differential equations

Differential geometry

Lingua di pubblicazione

Inglese

Formato

Materiale a stampa

Livello bibliografico

Monografia

Sommario/riassunto

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.