lecture notes on geometric analysis

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1、LECTURENOTESONGEOMETRICANALYSISPeterLiDepartmentofMathematicsUniversityofCaliforniaIrvine,CA92697-3875USApli@math.uci.eduJuly16,1992;Revised-August15,1996TableofContents§0Introduction§1FirstandSecondVariationalFormulasforArea§2BishopComparisonTheorem§3Bochner-Weitzenb¨ockFormulas§4LaplacianCompar

2、isonTheorem§5Poincar´eInequalityandtheFirstEigenvalue§6GradientEstimateandHarnackInequality§7MeanValueInequality§8Reilly’sFormulaandApplications§9IsoperimetricInequalitiesandSobolevInequalities§10LowerBoundsofIsoperimetricInequalities§11HarnackInequalityandRegularityTheoryofDeGiorgi-Nash-MoserRef

3、erences§0IntroductionThissetoflecturenotesoriginatedfromaseriesoflecturesgivenbytheauthorataGeometrySummerProgramin1990attheMathematicalSciencesResearchInstituteinBerkeley.DuringtheFallquarterof1990,theauthoralsotaughtacourseinGeometricAnalysisattheUniversityofArizona.Forthatpurpose,thelecturenot

4、eswererevisedandexpanded.Duringtheauthor’svisitwiththeGlobalAnalysisResearchInstituteatSeoulNationalUniversity,hewasencouragedtosubmitthesenotesinthepresent,butstillrathercrude,formforpublicationintheirlecturenotesseries.Thereadersshouldbeawarethatthesenotesaremeanttoaddresstheentrylevelgeometric

5、analystsbyintroducingthebasictechniquesingeometricanalysisinthemosteconomicalway.12PETERLIThetheoremsdiscussedarechosensometimesfortheirfundamentalusefulnessandsometimesforpurposeofdemonstratingvarioustechniques.Inmanycases,theydonotrepresentthebestpossibleresultswhichareavailable.Moreover,little

6、timewasspentonhistoricalaccountsandchronologicalordering.TheauthorwouldliketoexpresshisgratitudetotheGlobalAnalysisResearchInstituteandtheMathematicsDepartmentofSeoulNationalUniversityfortheirhos-pitality.Inparticular,specialthankstoProfessorDongPyoChi,ProfessorHyeongInChoi,ProfessorHyukKim,andth

7、egeometrygraduatestudentsformakinghisvisitamemorableone.§1.FirstandSecondVariationalFormulasforAreaLetMbeaRiemannianmanifoldofdimensionmwithmetricdenotedbyds2.Intermsoflocalcoordinates{x1,...,xm}themetriciswritteninthe