LEADER 03442nam 22004575a 450 001 9910151937303321 005 20091109150325.0 010 $a3-03719-530-4 024 70$a10.4171/030 035 $a(CKB)3710000000953802 035 $a(CH-001817-3)48-091109 035 $a(PPN)178155136 035 $a(EXLCZ)993710000000953802 100 $a20091109j20060830 fy 0 101 0 $aeng 135 $aurnn|mmmmamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aDifferential Harnack Inequalities and the Ricci Flow$b[electronic resource] /$fReto Mu?ller 210 3 $aZuerich, Switzerland $cEuropean Mathematical Society Publishing House$d2006 215 $a1 online resource (99 pages) 225 0 $aEMS Series of Lectures in Mathematics (ELM) ;$x2523-5176 330 $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 Poincare? 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 Ka?hler 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. 606 $aDifferential & Riemannian geometry$2bicssc 606 $aDifferential equations$2bicssc 606 $aGlobal analysis, analysis on manifolds$2msc 606 $aPartial differential equations$2msc 606 $aDifferential geometry$2msc 615 07$aDifferential & Riemannian geometry 615 07$aDifferential equations 615 07$aGlobal analysis, analysis on manifolds 615 07$aPartial differential equations 615 07$aDifferential geometry 686 $a58-xx$a35-xx$a53-xx$2msc 700 $aMu?ller$b Reto$0471659 801 0$bch0018173 906 $aBOOK 912 $a9910151937303321 996 $aDifferential Harnack inequalities and the Ricci flow$9229438 997 $aUNINA