Springer Monographs in Mathematics Luis J. Alías Paolo Mastrolia Marco Rigoli Maximum Principles and Geometric Applications Springer Monographs in Mathematics Moreinformationaboutthisseriesathttp://www.springer.com/series/3733 Luis J. Alías (cid:129) Paolo Mastrolia (cid:129) Marco Rigoli Maximum Principles and Geometric Applications 123 LuisJ.Alías PaoloMastrolia DepartamentodeMatemáticas DipartimentodiMatematica UniversidaddeMurcia UniversitàdegliStudidiMilano Murcia,Spain Milan,Italy MarcoRigoli DipartimentodiMatematica UniversitàdegliStudidiMilano Milan,Italy ThisworkhasbeenpartiallysupportedbyMINECO/FEDERprojectMTM2012-34037and Fundación Sénecaproject 04540/GERM/06, Spain.Thisresearchisaresultof theactivity developed within the framework of the Programme in Support of Excellence Groups of the Región de Murcia, Spain, by Fundación Séneca, Regional Agency for Science and Technology. M.RigolihasbeenpartiallysupportedbyMECGrantSAB2010-0073andFundaciónSéneca Grant18883/IV/13,ProgramaJiménezdelaEspada. ISSN1439-7382 ISSN2196-9922 (electronic) SpringerMonographsinMathematics ISBN978-3-319-24335-1 ISBN978-3-319-24337-5 (eBook) DOI10.1007/978-3-319-24337-5 LibraryofCongressControlNumber:2016930292 MathematicsSubjectClassification:58-02,35B50,53C20,53C42,35R01 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Printedonacid-freepaper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com) Contents 1 ACrashCourseinRiemannianGeometry................................ 1 1.1 Moving Frames, Levi-Civita Connection Forms andtheFirstStructureEquation........................................ 2 1.2 CovariantDerivativeof Tensor Fields, Connection andMeaningoftheFirstStructureEquation.......................... 4 1.3 LieDerivatives,theSecondStructureEquationandCurvature(s).... 11 1.4 DecompositionsoftheCurvatureTensor.............................. 22 1.5 CommutationRules..................................................... 31 1.6 SomeFormulasforImmersedSubmanifolds.......................... 35 1.7 TheGeometryofSmoothMaps........................................ 44 1.8 WarpedProducts........................................................ 49 1.9 ComparisonResults..................................................... 58 1.9.1 TheLaplacianComparisonTheorem.......................... 59 1.9.2 TheBishop-GromovComparisonTheorem................... 67 1.9.3 TheHessianComparisonTheorem............................ 71 2 TheOmori-YauMaximumPrinciple...................................... 77 2.1 SomePreliminaryConsiderations...................................... 78 2.2 TheGeneralizedOmori-YauMaximumPrinciple..................... 86 2.2.1 TwoSignificantExamples...................................... 88 2.3 StochasticCompletenessandtheWeakMaximumPrinciple......... 99 2.4 TwoApplicationsofStochasticCompleteness........................ 120 2.5 Parabolicity.............................................................. 128 3 NewFormsoftheMaximumPrinciple.................................... 141 3.1 NewFormsoftheWeakandOmori-YauMaximumPrinciples ...... 142 3.1.1 ProofofTheorem3.1andRelatedResults.................... 146 3.1.2 ProofofTheorem3.2andSomeRelatedResults............. 152 3.2 AnAPrioriEstimate.................................................... 153 v vi Contents 3.3 TheNonlinearCase..................................................... 161 3.3.1 AnalyticPreliminaries.......................................... 163 3.3.2 TheMaximumPrinciple........................................ 189 4 SufficientConditionsfortheValidityoftheWeak MaximumPrinciple.......................................................... 203 4.1 VolumeGrowthConditionsandAnotherAPrioriEstimate.......... 204 4.2 AControlledGrowthWeakMaximumPrinciple...................... 217 4.3 AnEquivalentOpenFormoftheWeakMaximumPrinciple......... 231 4.3.1 AFirstApplicationtoPDE’s................................... 238 4.4 StrongParabolicity...................................................... 242 4.5 ALiouville-TypeTheorem............................................. 254 5 MiscellanyResultsforSubmanifolds...................................... 271 5.1 ImmersionsintoNondegenerateConesinEuclideanSpace.......... 272 5.2 MapsintoNondegenerateConesinEuclideanSpace................. 279 5.3 BoundedSubmanifoldsandJorge-KoutroufiotisTypeResults....... 288 5.4 CylindricallyBoundedSubmanifolds.................................. 295 5.4.1 SectionalCurvatureEstimates ................................. 295 5.4.2 MeanCurvatureEstimatesandStochasticCompleteness .... 304 5.5 Consequenceson the Gauss Map of Submanifolds ofEuclideanSpace...................................................... 308 5.6 AnApplicationoftheOpenWeakMaximumPrinciple .............. 320 6 ApplicationstoHypersurfaces.............................................. 325 6.1 ConstantMeanCurvatureHypersurfacesinSpaceForms............ 326 6.1.1 ProofoftheMainResults ...................................... 331 6.1.2 AlternativeApproachestoCorollary6.2 ...................... 345 6.2 ConstantScalarCurvatureHypersurfaces ............................. 348 6.2.1 HypersurfacesandNewtonOperators......................... 351 6.2.2 SomePreliminaryResults...................................... 356 6.2.3 AnOmori-YauMaximumPrinciplefortheCheng andYauOperator ............................................... 361 6.2.4 ProofoftheMainResults ...................................... 366 6.3 HypersurfacesintoNondegenerateEuclideanCones ................. 371 6.4 HigherOrderMeanCurvatureEstimates.............................. 378 7 HypersurfacesinWarpedProducts........................................ 385 7.1 Preliminaries ............................................................ 386 7.2 CurvatureEstimatesforHypersurfacesinWarpedProducts.......... 392 7.3 HypersurfaceswithConstant2-MeanCurvature...................... 397 7.4 HypersurfaceswithConstantHigherOrderMeanCurvature......... 405 7.4.1 FurtherResultsforHypersurfaceswithConstant HigherOrderMeanCurvatures................................ 408 7.5 HeightEstimates........................................................ 420 7.5.1 WarpedProductSpaces......................................... 420 7.5.2 Products......................................................... 428 7.6 KillingGraphs........................................................... 439 Contents vii 8 ApplicationstoRicciSolitons............................................... 443 8.1 BasicFormulasforGenericRicciSolitons............................ 444 8.2 TheValidityoftheMaximumPrincipleonSolitons.................. 456 8.3 StatementsandProofsoftheMainResults............................ 463 8.3.1 TheGenericCase............................................... 463 8.3.2 GradientSolitons ............................................... 473 8.3.3 ATopologicalResultonWeightedManifolds ................ 486 8.4 AFurtherResultonGenericRicciSolitons ........................... 493 9 SpacelikeHypersurfacesinLorentzianSpacetimes...................... 499 9.1 FoundationsofLorentzianGeometry.................................. 500 9.1.1 Levi-CivitaConnectionandGeodesics........................ 504 9.1.2 CurvatureofaLorentzianManifold........................... 506 9.2 SpacelikeHypersurfacesinLorentzianSpacetimes................... 508 9.2.1 Maximal Hypersurfaces as Solutions ofaVariationalProblem........................................ 511 9.2.2 SpacelikeHypersurfacesandGeneralRelativity.............. 512 9.3 SpacelikeHypersurfacesinLorentz-MinkowskiSpace............... 514 9.3.1 AlternativeApproachesinDimensionm D 2 UsingParabolicity.............................................. 521 9.4 ComparisonTheoryfortheLorentzianDistanceFunction fromaPoint ............................................................. 527 9.4.1 HessianandLaplacianComparisonTheorems................ 530 9.5 SpacelikeHypersurfacesContainedintheChronological FutureofaPoint ........................................................ 534 9.6 GeneralizedRobertson-WalkerSpacetimes............................ 542 References......................................................................... 553 Index............................................................................... 565 List of Symbols .˙;g;A/ initialdatasetfortheEinsteinequation,page513 .M;h; i;e(cid:2)fdx/ weightedRiemannianmanifold,page142 .M;h; i;X/ Riccisolitonstructure,page443 .U;'/ localchart,page2 Œ ; (cid:2) Liebracket,page10 (cid:3)s;t characteristicfunctionoftheannulusBt.o/nBs.o/,page69 (cid:4)u Laplacianofthefunctionu,page32 ıj suggestivewayofwritingtheKroneckersymbol,page2 i (cid:4) f-Laplacian,page142 f (cid:4) X-Laplacian,page142 X ı Diracdeltacenteredaty,page100 y ı Kroneckersymbol,page2 ij (cid:5)P tangentvectorofthecurve(cid:5),page59 ` h; i .1;1/-versionofL h; i,page448 Y Y (cid:6) cutofffunction,page117 (cid:5) thematrixof1-forms.(cid:7)i/onU,page4 j (cid:8) solitonconstant,page443 (cid:9)2.U/ spaceofskew-symmetric2-formsontheopensetU,page13 log.j/ j-thiteratedlogarithm,page88 A m-dimensional area function in the Lorentzian setting, page511 Co;(cid:10);(cid:7) nondegenerate cone of Rn with vertex o, direction (cid:10) and width(cid:7),page273 GL GreenkerneloftheoperatorLonthedomain˝ ,page127 k k L ! Liederivativeofthe1-form! inthedirectionofX,page11 X L f Liederivativeofthefunctionf inthedirectionofX,page11 X L Y Lie derivative of the vector field Y in the direction of X, X page11 L h; i Liederivativeofthemetrich; iinthedirectionofX,page11 X T unitnormal @ tothesliceP,page52 @t t ix