Table Of ContentAn-MinLi,UdoSimon,GuosongZhao,ZejunHu
GlobalAffineDifferentialGeometryofHypersurfaces
De Gruyter Expositions
in Mathematics
|
Edited by
Victor P. Maslov, Moscow, Russia
Walter D. Neumann, New York City, New York, USA
Markus J. Pflaum, Boulder, Colorado, USA
Dierk Schleicher, Bremen, Germany
Raymond O. Wells, Bremen, Germany
Volume 11
An-Min Li, Udo Simon, Guosong Zhao, Zejun Hu
Global Affine
Differential Geometry
of Hypersurfaces
|
2nd revised and extended edition
MathematicsSubjectClassification2010
Primary:53A15,53C21,53C24,53C40,53C42;Secondary:17C40,26B20,35J35,35J60,53B21,
53B25,58E30,58J32,58J60.
Authors
Prof.Dr.An-MinLi Prof.Dr.UdoSimon
SichuanUniversity TechnischeUniversitätBerlin
SchoolofMathematics InstitutfürMathematik,MA8–3
Chengdu610064,Sichuan Straẞedes17.Juni136
People’sRepublicofChina D-10623Berlin,Germany
[email protected] [email protected]
Prof.GuosongZhao Prof.Dr.ZejunHu
SichuanUniversity ZhengzhouUniversity
SchoolofMathematics SchoolofMathematicsandStatistics
Chengdu610064,Sichuan Zhengzhou450001,Henan
People’sRepublicofChina People’sRepublicofChina
[email protected] [email protected]
ISBN978-3-11-026667-2
e-ISBN(PDF)978-3-11-026889-8
e-ISBN(EPUB)978-3-11-039090-2
Set-ISBN978-3-11-026890-4
ISSN0938-6572
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Contents
Introduction|1
1 Preliminariesandbasicstructuralaspects|7
1.1 Affinespaces|7
1.1.1 Basicnotationsforaffinespaces|7
1.1.2 Volumeandorientation|8
1.1.3 Dualdeterminantforms|8
1.1.4 Dualityandcross-productconstruction|9
1.1.5 Affinemappingsandtransformationgroups|9
1.1.6 Affineinvarianceandduality|11
1.1.7 Directionalderivatives|11
1.1.8 Frames|12
1.2 Euclideanspaces|15
1.2.1 BasicnotationsforEuclideanspaces|15
1.2.2 Normeddeterminantforms|15
1.2.3 DifferentEuclideanstructures|15
1.2.4 Isometriesandorthogonaltransformations|16
1.3 DifferentialgeometricstructuresofaffineandEuclideanspace|16
1.3.1 Differentialgeometricequiaffinestructures|16
1.3.2 DifferentialgeometricEuclideanstructures|18
1.3.3 ℝn+1asastandardspace|18
1.4 Klein’sErlangenprogram|19
1.5 Motivation:AshortsketchoftheEuclideanhypersurfacetheory|20
1.6 Hypersurfacesinequiaffinespace|24
1.6.1 Definitionandnotation|24
1.6.2 Bundles|24
1.7 Structuralmotivationforfurtherinvestigations|25
1.8 Transversalfieldsandinducedstructures|26
1.8.1 Volumeform|26
1.8.2 Weingartenstructureequation|26
1.8.3 Gaußstructureequation|26
1.8.4 Compatibilityoftheinducedvolumeformandtheconnection|27
1.8.5 Relativenormalfields|27
1.9 Conormalfieldsandinducedstructures|28
1.9.1 Volumeform|28
1.9.2 Bilinearform|28
1.10 Relativenormalizations|29
1.11 Nondegeneratehypersurfaces|29
1.12 Gaußstructureequationsforconormalfields|30
vi | Contents
1.12.1 Compatibilityofconnectionsandvolumeformsintheconormal
bundle|31
1.12.2 Compatibilityofbilinearformsandconnections;conjugate
connections|31
1.12.3 Cubicform|32
1.12.4 Tchebychevform|32
1.12.5 Localnotation|32
1.12.6 RelativeGaußmaps|33
1.13 Affineinvarianceoftheinducedstructures|35
1.14 Asummaryofrelativehypersurfacetheory|36
1.14.1 Structureequationsintermsof∇̃ |36
1.14.2 Integrabilityconditionsintermsof∇̃ |37
1.15 Specialrelativenormalizations|38
1.15.1 Euclideannormalizationasarelativenormalization|38
1.15.2 Equiaffinenormalization|38
1.15.3 Centroaffinenormalization|40
1.16 Comparisonofrelativenormalizations|41
1.16.1 Comparisonofcorrespondinggeometricproperties|42
1.16.2 Gaugeinvariance|43
2 Localequiaffinehypersurfacetheory|45
2.1 Blaschke–Berwaldmetricandstructureequations|45
2.2 TheaffinenormalandtheFubini–Pickform|48
2.2.1 Affinenormal|49
2.2.2 Fubini–Pickform|52
2.2.3 Affinecurvatures|54
2.2.4 Geometricmeaningoftheaffinenormal|56
2.3 Theequiaffineconormal|59
2.3.1 Propertiesoftheequiaffineconormal|59
2.3.2 Theaffinesupportfunction|62
2.4 Hyperquadrics|63
2.4.1 Hyperquadrics|63
2.4.2 HypersurfaceswithvanishingPickinvariant|68
2.5 Integrabilityconditionsandthelocalfundamentaltheorem|72
2.5.1 Relationsbetweenthecoefficients|73
2.5.2 Integrabilityconditions|73
2.5.3 Thefundamentaltheorem|76
2.6 Euclideanboundarypointsoflocallyconveximmersed
hypersurfaces|80
2.7 Graphimmersions|85
2.7.1 Graphimmersionswithequiaffinenormalization|85
2.7.2 GraphImmersionswithCalabimetric|89
Contents | vii
3 Affinehyperspheres|93
3.1 Definitionsandbasicresultsforaffinehyperspheres|95
3.1.1 Definitionofaffinehyperspheres|95
3.1.2 Differentialequationsforaffinehyperspheres|96
3.1.3 Calabicompositions|100
3.2 Affinehypersphereswithconstantsectionalcurvature|104
3.2.1 Examples|104
3.2.2 Localclassificationoftwo-dimensionalaffinesphereswithconstant
scalarcurvature|107
3.2.3 Generalizationtohigherdimensions|109
3.3 AffinehypersurfaceswithparallelFubini–Pickform|116
3.3.1 ImplicationsfromaparallelFubini–Pickform|116
3.3.2 ClassificationofaffinehypersurfaceswithparallelFubini–Pick
form|117
3.4 Affinecompleteness,Euclideancompleteness,andCalabi
completeness|131
3.4.1 Euclideancompletenessandaffinecompleteness|131
3.4.2 TheequivalencebetweenCalabimetricsandEuclideanmetrics|137
3.4.3 Remarks|143
3.5 Affinecompleteellipticaffinehyperspheres|144
3.6 AdifferentialinequalityonacompleteRiemannianmanifold|147
3.7 EstimatesoftheRiccicurvaturesofaffinecompleteaffinehyperspheres
ofparabolicorhyperbolictype|151
3.8 Qualitativeclassificationofcompletehyperbolicaffine
hyperspheres|154
3.8.1 Euclideancompletehyperbolicaffinehyperspheres|155
3.8.2 Affinecompletehyperbolicaffinehyperspheres|159
3.8.3 ProofofthesecondpartoftheCalabiconjecture|166
3.9 Completehyperbolicaffine2-spheres|172
3.9.1 AsplittingoftheLevi–Civitaoperator|173
4 Rigidityanduniquenesstheorems|181
4.1 Integralformulasforaffinehypersurfacesandtheir
applications|181
4.1.1 Minkowski-typeintegralformulasforaffinehypersurfaces|182
4.1.2 Characterizationofellipsoids|183
4.1.3 Somefurthercharacterizationsofellipsoids|186
4.1.4 GlobalsolutionsofadifferentialequationofSchrödingertype|190
4.1.5 Rigiditytheoremsforovaloids|191
4.1.6 Hypersurfaceswithboundary|194
4.2 Theindexmethod|201
4.2.1 Fieldsoflineelementsandnets|201
viii | Contents
4.2.2 Vekua’ssystemofpartialdifferentialequations|206
4.2.3 AffineWeingartensurfaces|207
4.2.4 AnaffineanalogueoftheCohn–Vossentheorem|214
5 Variationalproblemsandaffinemaximalsurfaces|217
5.1 Variationalformulasforhigheraffinemeancurvatures|218
5.2 Affinemaximalhypersurfaces|223
5.2.1 Definitionsandfundamentalresults|223
5.2.2 AnaffineanalogueoftheWeierstraßrepresentation|226
5.2.3 Constructionofaffinemaximalsurfaces|227
5.2.4 Constructionofimproper(parabolic)affinespheres|229
5.2.5 AffineBernsteinproblems|231
5.3 DifferentialinequalitiesforΔ(B)J,Δ(B)Φ,andΔ(C)Φ |232
5.3.1 NotationsfromE.Calabi|232
5.3.2 ComputationofΔ(B)J|234
5.3.3 ComputationofΔ(B)(‖B̂‖2)|235
5.3.4 EstimationsofΔ(C)Φ andΔ(B)Φ |236
5.4 ProofofCalabi’sconjectureindimension2|240
5.5 Chern’saffineBernsteinconjecture|246
5.5.1 Sometoolsfromp.d.e.theory|246
5.5.2 ApartialresultonChern’sconjectureinarbitrarydimensions|248
5.5.3 ProofofChern’sconjectureindimension2|250
5.5.4 EstimatesforthedeterminantoftheHessian|252
5.5.5 FirstproofofLemma5.29|257
5.6 AnanalyticproofofChern’sconjectureindimension2|259
5.6.1 Technicalestimates|259
5.6.2 Estimatesfor∑f |263
ii
5.6.3 Estimatesforthethirdorderderivatives|266
5.6.4 SecondproofofLemma5.29|271
5.7 AnaffineBernsteinproblemwithrespecttotheCalabimetric|272
6 HypersurfaceswithconstantaffineGauß–Kroneckercurvature|277
6.1 TheaffineGauß–Kroneckercurvature|277
6.1.1 Motivation|277
6.1.2 Mainresults|278
6.2 SplittingofthefourthorderPDES =constintotwo(secondorder)
n
Monge–Ampèreequations|280
6.3 ConstructionofEuclideancompletehypersurfaceswithconstantaffine
G-Kcurvature|282
6.4 CompletenesswithrespecttotheBlaschkemetric|290
Contents | ix
7 Geometricinequalities|303
7.1 Theaffineisoperimetricinequality|303
7.1.1 Steinersymmetrization|303
7.1.2 Acharacterizationofellipsoids|306
7.1.3 Theaffineisoperimetricinequality|308
7.2 Inequalitiesforhigheraffinemeancurvatures|310
7.2.1 Mixedvolumes|310
7.2.2 Integralinequalitiesforcurvaturefunctions|312
7.2.3 Totalcentroaffinearea|314
A Basicconceptsfromdifferentialgeometry|317
A.1 Tensorsandexterioralgebra|317
A.1.1 Tensors|317
A.1.2 Exterioralgebra|319
A.2 Differentiablemanifolds|321
A.2.1 Differentiablemanifoldsandsubmanifolds|321
A.2.2 Tensorfieldsonmanifolds|324
A.2.3 Integrationonmanifolds|326
A.3 AffineconnectionsandRiemanniangeometry:Basicfacts|328
A.3.1 Affineconnections|328
A.3.2 Riemannmanifolds|332
A.3.3 Manifoldsofconstantcurvature,Einsteinmanifolds|335
A.3.4 Examples|336
A.3.5 Exponentialmappingandcompleteness|336
A.4 Green’sformula|338
B Laplaciancomparisontheorem|341
Bibliography|345
Index|362
