lectures on the ricci flow

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1、lecturesonthericciflow1PeterToppingMarch9,20061cPeterTopping2004,2005,2006.Contents1Introduction61.1Ricciflow:whatisit,andfromwherediditcome?......61.2Examplesandspecialsolutions.................81.2.1Einsteinmanifolds....................81.2.2Riccisolitons...................

2、....81.2.3ParabolicrescalingofRicciflows............111.3GettingafeelforRicciflow...................121.3.1Twodimensions.....................121.3.2Threedimensions.....................131.4Thetopologyandgeometryofmanifoldsinlowdimensions.171.5UsingRicciflowtoprovetopologicaland

3、geometricresults.212Riemanniangeometrybackground242.1Notationandconventions....................242.2Einsteinmetrics..........................282.3DeformationofgeometricquantitiesastheRiemannianmet-ricisdeformed..........................282.3.1Theformulae.....................

4、..282.3.2Thecalculations.....................322.4Laplacianofthecurvaturetensor................392.5EvolutionofcurvatureandgeometricquantitiesunderRicciflow................................413Themaximumprinciple443.1Statementofthemaximumprinciple..............443.2Basiccontrol

5、ontheevolutionofcurvature...........453.3Globalcurvaturederivativeestimates..............494CommentsonexistencetheoryforparabolicPDE534.1LinearscalarPDE........................534.2Theprincipalsymbol.......................544.3GeneralisationtoVectorBundles................564

6、.4Propertiesofparabolicequations................5815ExistencetheoryfortheRicciflow595.1Ricciflowisnotparabolic....................595.2Short-timeexistenceanduniqueness:TheDeTurcktrick...605.3Curvatureblow-upatfinite-timesingularities.........636Ricciflowasagradientflow676.1Gradie

7、ntoftotalscalarcurvatureandrelatedfunctionals..676.2TheF-functional.........................686.3Theheatoperatoranditsconjugate..............706.4Agradientflowformulation...................706.5Theclassicalentropy.......................746.6Thezerotheigenvalueof−4∆+R.........

8、......767CompactnessofRiemannianmanifoldsandflows787.1Convergenceandcompactn