NicholasSheridanHonoursThesis英文版

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1、TheUniversityofMelbourne,DepartmentofMathematicsandStatisticsHamilton’sRicciFlowNicholasSheridanSupervisor:AssociateProfessorCraigHodgsonSecondReader:ProfessorHyamRubinsteinHonoursThesis,November2006.AbstractTheaimofthisprojectistointroducethebasicsofHamilton’sRicciFlow.TheRicciflowisapdeforevolv

2、ingthemetrictensorinaRiemannianmanifoldtomakeit“rounder”,inthehopethatonemaydrawtopologicalconclusionsfromtheexistenceofsuch“round”metrics.Indeed,theRicciflowhasrecentlybeenusedtoprovetwoverydeeptheoremsintopology,namelytheGeometrizationandPoincar´eConjectures.Webeginwithabriefsurveyofthedifferent

3、ialgeometrythatisneededintheRicciflow,thenproceedtointroduceitsbasicpropertiesandthebasictechniquesusedtounderstandit,forexample,provingexistenceanduniquenessandboundsonderivativesofcurvatureundertheRicciflowusingthemaximumprinciple.Weusetheseresultstoprovethe“original”Ricciflowtheorem–the1982theor

4、emofRichardHamiltonthatclosed3-manifoldswhichadmitmetricsofstrictlypositiveRiccicurvaturearediffeomorphictoquotientsoftheround3-spherebyfinitegroupsofisometriesactingfreely.WeconcludewithaqualitativediscussionoftheideasbehindtheproofoftheGeometrizationConjectureusingtheRicciflow.Mostoftheprojectisb

5、asedonthebookbyChowandKnopf[6],thenotesbyPeterTopping[28](whichhaverecentlybeenmadeintoabook,see[29]),thepapersofRichardHamilton(inparticular[9])andthelecturecourseonGeometricEvolutionEquationspresentedbyBenAndrewsatthe2006ICE-EMGraduateSchoolheldattheUniversityofQueensland.Wehavereformulatedand

6、expandedtheargumentscontainedinthesereferencesinsomeplaces.Inparticular,theproofofTheorem7.19isoriginal,basedonasuggestionbyGerhardHuisken.WealsodivergefromtheexistingreferencesbyemphasisingtheanalogybetweenthetechniquesappliedtotheRicciflowandthoseappliedtothecurve-shorteningflow,whichwefeelhelps

7、clarifytheimportantideasbehindthetechnicaldetailsoftheRicciflow.Chapter6isbasedon[6,Chap.6,7],butwehavesignificantlyreformulatedthematerialandelaboratedontheproofs.WefeelthatourorganizationiseasiertofollowthanChowandKnopf’sboo