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Field name	Details
Title	Differential Harnack Inequalities and the Ricci Flow ([EBook] /) / Reto Müller
Author	Müller, Reto
Publication	Zuerich, Switzerland : : European Mathematical Society Publishing House, , 2006
Physical Details	1 online resource (99 pages)
Series	EMS Series of Lectures in Mathematics (ELM) 2523-5176
ISBN	9783037195307
Summary Note	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.:
Mode of acces to digital resource	Mode of access: World Wide Web. System requirements: Internet Explorer 6.0 (or higher) or Firefox 2.0 (or higher). Available as searchable text in PDF format.
System details note	Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users).
Internet Site	https://doi.org/10.4171/030
See Also	https://www.ems-ph.org/img/books/mueller_mini.jpg
Links to Related Works	Subject References: Differential Equations . Differential Geometry . Differential & Riemannian geometry . Global analysis, analysis on manifolds . Partial differential equations . Authors: author . Müller, Reto . Series: EMS Series of Lectures in Mathematics (ELM) .

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Differential Harnack Inequalities and the Ricci Flow

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