03271nam a22003855a 450000100100000000300120001000500170002200600190003900700150005800800410007302000180011402400210013204000140015307200170016707200170018408400290020110000280023024500960025826000820035430000330043633600260046933700260049533800360052134700240055749000610058150600660064252018770070865000470258565000350263265000480266765000400271565000310275585600320278685600670281848-091109CH-001817-320091109150325.0a fot ||| 0|cr nn mmmmamaa091109e20060830sz fot ||| 0|eng d a978303719530770a10.4171/0302doi ach0018173 7aPBMP2bicssc 7aPBKJ2bicssc a58-xxa35-xxa53-xx2msc1 aMüller, Reto,eauthor.10aDifferential Harnack Inequalities and the Ricci Flowh[electronic resource] /cReto Müller3 aZuerich, Switzerland :bEuropean Mathematical Society Publishing House,c2006 a1 online resource (99 pages) atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier atext filebPDF2rda0 aEMS Series of Lectures in Mathematics (ELM) ;x2523-51761 aRestricted to subscribers:uhttps://www.ems-ph.org/ebooks.php aThe 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.07aDifferential & Riemannian geometry2bicssc07aDifferential equations2bicssc07aGlobal analysis, analysis on manifolds2msc07aPartial differential equations2msc07aDifferential geometry2msc40uhttps://doi.org/10.4171/030423cover imageuhttps://www.ems-ph.org/img/books/mueller_mini.jpg