CMC-trinoids with properly embedded annular ends [Elektronische Ressource] / Philipp Lang

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TECHNISCHE UNIVERSITAT MUNCHENZentrum MathematikCMC-Trinoids with Properly Embedded Annular EndsPhilipp LangVollst andiger Abdruck der von der Fakult at fur Mathematik der Technischen Universit at Munc hen zurErlangung des akademischen Grades einesDoktors der Naturwissenschaften (Dr. rer. nat.)genehmigten Dissertation.Vorsitzender: Univ.-Prof. Dr. J. ScheurlePrufer der Dissertation:1. Hon.-Prof. Dr. J. Dorfmeister2. Univ.-Prof. Dr. T. N. Ho mann3. Dr. F. Pedit, Eberhard Karls Universit at TubingenDie Dissertation wurde am 28.01.2010 bei der Technischen Universit at Munc hen eingereicht und durchdie Fakult at fur Mathematik am 08.06.2010 angenommen.AbstractWe consider CMC-trinoids in Euclidian three-space with properly embedded annular ends. Startingwith a holomorphic potential ~ and a special solution to the di erential equation d = ~ , we char-acterize all solutions to this di erential equation which produce CMC-trinoids with properly embeddedannular ends via the loop group method. Moreover, we give a classi cation of CMC-trinoids with properlyembedded annular ends with respect to their symmetry properties in terms of the monodromy matricesof the solution associated with the trinoid ends.3Contents1 Introduction 62 Outline of the loop group method 112.1 Loop Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Iwasawa decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Sujets

• Mathematics

Informations

• Publié par: technische_universitat_munchen
• Publié le: 01 janvier 2010
• Nombre de lectures: 179
• Langue: English
• Poids de l'ouvrage: 2 Mo

Extrait

TECHNISCHEUNIVERSIT¨ATM¨UNCHEN
MathematiktrumZen

CMC-TrinoidswithProperlyEmbeddedAnnularEnds

angLPhilipp

Vollst¨andigerAbdruckdervonderFakult¨atf¨urMathematikderTechnischenUniversit¨atM¨unchenzur
ErlangungdesakademischenGradeseines

DoktorsderNaturwissenschaften(Dr.rer.nat.)

Dissertation.genehmigten

Vorsitzender:Univ.-Prof.Dr.J.Scheurle
Pr¨uferderDissertation:
DorfmeisterJ.Dr.Hon.-Prof.1.2.Univ.-Prof.Dr.T.N.Hoffmann
3.Univ.-Prof.Dr.F.Pedit,EberhardKarlsUniversit¨atT¨ubingen

DieDissertationwurdeam28.01.2010beiderTechnischenUniversit¨atM¨uncheneingereichtunddurch
dieFakult¨atf¨urMathematikam08.06.2010angenommen.

Abstract

WeconsiderCMC-trinoidsinEuclidianthree-spacewithproperlyembeddedannularends.Starting

withaholomorphicpotentialη˜andaspecialsolutionΨtothedifferentialequationdΨ=Ψη˜,wechar-

acterizeallsolutionstothisdifferentialequationwhichproduceCMC-trinoidswithproperlyembedded

annularendsviatheloopgroupmethod.Moreover,wegiveaclassificationofCMC-trinoidswithproperly

embeddedannularendswithrespecttotheirsymmetrypropertiesintermsofthemonodromymatrices

ofthesolutionΨassociatedwiththetrinoidends.

3

tstenCon6ductiontroIn12Outlineoftheloopgroupmethod11
2.1LoopGroups...........................................11
2.2Iwasawadecomposition.....................................11
2.3Holomorphicpotentials.....................................11
2.4Theloopgroupmethod.....................................12
2.5Monodromy............................................14
2.6Delaunaysurfaces.........................................15
19rinoidsT33.1TrinoidsonthedomainM=C\{0,1}.............................19
˜3.2TheuniversalcoverMofM...................................20
˜3.3ThefundamentalgroupΓofManditsmonodromyactiononM..............23
33.4Thesu(2)modelofR......................................26
3.5Thetrinoidpotential.......................................27
3.6Thestandardizedtrinoidpotential...............................32
3.7TheFuchsianODE........................................34
3.8SolvingdΦ=Φη.........................................36
3.9Simultaneousunitarizationofthemonodromymatrices....................43
52symmetriesrinoidT44.1Definitions.............................................52
4.2Theextendedframe.......................................52
4.3Trinoidswithproperlyembeddedannularends........................55
4.4Theextendedframesymmetrytransformations........................56
4.5Theextendedframemonodromyrelations...........................68
4.6Trinoidsymmetries........................................69
5Rotationalsymmetrywithrespecttothetrinoidnormal80
5.1Definition.............................................80
5.2Implicationsofrotationalsymmetrywithrespecttothetrinoidnormal...........80
5.3Monodromymatricesoftrinoidswithproperlyembeddedannularends,whicharerota-
tionallysymmetricwithrespecttothetrinoidnormal....................83
5.4Normalizedtrinoidswithproperlyembeddedannularends,whicharerotationallysym-
metricwithrespecttothetrinoidnormal...........................87
2005.5Solvingζζ=4sin(πµ)−1..................................93
6Rotationalsymmetrywithrespecttoatrinoidaxis104
6.1Definition.............................................104
6.2Implicationsofrotationalsymmetrywithrespecttoatrinoidaxis.............104
6.3Monodromymatricesoftrinoidswithproperlyembeddedannularends,whicharerota-
tionallysymmetricwithrespecttoatrinoidaxis.......................106
6.4Normalizedtrinoidswithproperlyembeddedannularends,whicharerotationallysym-
metricwithrespecttoatrinoidaxis..............................111
7Reflectionalsymmetrywithrespecttothetrinoidplane121
7.1Definition.............................................121
7.2Implicationsofreflectionalsymmetrywithrespecttothetrinoidplane...........121
7.3Monodromymatricesoftrinoidswithproperlyembeddedannularends,whicharereflec-
tionallysymmetricwithrespecttothetrinoidplane.....................122
7.4Normalizedtrinoidswithproperlyembeddedannularends,whicharereflectionallysym-
metricwithrespecttothetrinoidplane............................125
4

8Reflectionalsymmetrywithrespecttoatrinoidnormalplane131
8.1Definition.............................................131
8.2Implicationsofreflectionalsymmetrywithrespecttoatrinoidnormalplane........131
8.3Monodromymatricesoftrinoidswithproperlyembeddedannularends,whicharereflec-
tionallysymmetricwithrespecttoatrinoidnormalplane..................134
8.4Normalizedtrinoidswithproperlyembeddedannularends,whicharerefletionallysym-
metricwithrespecttoatrinoidnormalplane.........................138
9Rotoreflectionalsymmetrywithrespecttothetrinoidnormal150
9.1Definition.............................................150
9.2Implicationsofrotoreflectionalsymmetrywithrespecttothetrinoidnormal........150
9.3Monodromymatricesoftrinoidswithproperlyembeddedannularends,whicharerotore-
flectionallysymmetricwithrespecttothetrinoidnormal...................153
AAppendix:BasicTopology157
A.1Topologicalspaces,continuousmappingsandpaths......................157
A.2Thefundamentalgroup.....................................157
A.3Theautomorphismgroup....................................158
A.4Themonodromyactionofthefundamentalgroup.......................159
BAppendix:Thefunctionµ=XX164
jjjCAppendix:Proofoflemma3.37168
DAppendix:OnthenecessityoftheunitarizingmatrixT172
EAppendix:Amendmentstotheproofoftheorem3.53173
FAppendix:Proofofremark3.55177
GAppendix:Proofofremark3.56178
HAppendix:Proofoftheorem5.16180
IAppendix:Proofsoflemma5.21andlemma5.24183
186References

5

ductiontroIn1AmongthesurfacesofconstantmeancurvatureH=0,CMC-surfacesforshort,onlyafewsubclasses
havDelaunaebeenysurfaces.classified.TheyThewerefirstfoundoneshavalmostebee200nytheearsagosurfaces[7],ofandrevareolutionstillofinamongterest,thesinceevCMC-surfaces,eryproperlythe
embMoreeddedanngenerallyular,endofaCMC-immersionsCMC-surfaceofisroundasympcylinderstoticallyintoaR3Delaunaareyfairlysurfacewellu[25].nderstood.Theclassof
CMC-torihasbeeninvestigatedextensivelyusingdifferentmathematicaltechniques[31],[3],[23]andis
clearlysofarthebestinvestigatedoneamongallCMC-immersions.
groupsAllthearepsurfaceerhapsthclassesosewhimenchtionedaresofreefarandhavhaeveanabonlyeliantwofundamengenerators.talThgroup.usitTheseemstosimplestbenon-abparticularlyelian
impsphereortanTtintotoR3.understandtheCMC-trinoids,i.e.CMC-immersionsofthethrice-puncturedRiemannian
3AmongtheCMC-trinoidsT3→R3clearlytheembeddingsareofparticularinterest.Itseemsto
beBraucdifficultkmann,toKusnerclassifyandthisSuclassllivanofhaveCMC-immersions.classifiedtheHowAlexandroever,vinemabbeddedeautifulpieceCMC-surfacesofwork,T3→Große-R3
[21].In[27]itwasshown,however,thatthereareCMC-trinoidsT3→R3,whichhaveproperlyembedded
annularendsbutarenotAlexandrovembedded.Furtherexamplesofsuchsurfaceshavebeengivenin
[17]CMC-trinoidsusingthelowithoppropgrouperlymethoembdedded[15]forannaularcertainendsclassofencompassesstartingthepotenclasstials.oftheNaturally(globally),thepropclasserlyof
embeddedtrinoids.Inthissense,theinvestigationofCMC-trinoidswithproperlyembeddedannular
endsseemstobeanaturalnextstepfortheunderstandingofallCMC-trinoids.
In[8]itisshownthatallCMC-trinoidsT3→R3withproperlyembeddedannularendscanbe
aobtainedclassificationviatheofloallopgroupCMC-trinoidsmethodwhicfromhcanthebpeotenobtainedtialsofvia[17].theloBasedopongroupthismethoresult,dthisfromthethesispprootenvidestials
of[17],andthusinparticularaclassificationofallCMC-trinoidswithproperlyembeddedannularends,
intermsofthemonodromymatricesassociatedwiththetrinoidends.
WegiveallpossibletriplesofmonodromymatricesassociatedwiththeendsofaCMC-trinoidT3→R3
awhicgivhencanCMbeC-triobtainednoidwithfromproptheperlyotenembtialsedofded[17].annularMoreovendser,weunderinvestigateEuclideanthepmotionsossibleinR3symmetriesandchar-of
acterizethesesymmetriesintermsofthecorrespondingmonodromymatrices.I.e.,westatenecessary
andsufficientconditionsonthemonodromymatricesofagivenCMC-trinoidT3→R3withproperly
embeddedannularends,3suchthatthe(imageofthe)givenCMC-trinoidisinvariantunderaspecific
.RinmotionEuclideanInholomorphicsectionp2otenwetials.reviewAtheloholomorphicopgrouppotenmethotialdη˜isfromasl(2[15],Cfor)-valuedconstructingdifferentialCMC-immersionsone-form,whichfromis
definedontheuniversalcoverM˜ofaRiemann˜surfaceM∗.Furthermore,η˜involvesaloopparameter
λanddependsholomorphicallyonbothz∈Mandλ∈C.Givenaholomorphicpotentialη˜,thefirst
stepmappingoftheΨloonopM˜,groupsatisfyimethongdsomeconsistsinitialinsolvingconditiontheΨ(z∗)differen=Ψ0tial.ΨalsoequationdepdΨends=onΨη˜λforandatheSL(2,formC)-vofaluedthis
dependenceisdeterminedbytheinitialconditionΨ0.AssumingthatΨ0(andthusΨ)isdefinedforallλ
fromsomer-circleC(r),0<r≤1,onecanproceedwiththesecondstepoftheloopgroupmethod.This
involves(foreachz0∈M˜)anr-Iwasawadecomposition(rof)theλ-dependentloopΨ(z0):C(r)→SL(2,C),
opi.e.ena(pannointuluswise)r<|λ|factorization<1andofisΨintounitaryaloonopFtheonunitC,circlewhicS1h,canandbaeloopextendedB+onC(r),holomorphicallywhichcantobthee
extendedholomorphicallyrtothedisc|λ|<r,Ψ=FB+.ThefactorFproducesinthethirdandfinalstep
ofthelo˜opgroupmethod,byevaluatingthesocalledSym-Bobenkoformulaforλ=1,aCMC-immersion
ψonM.ψ“descends”toaCMC-immersionφonM˜ifandonlyifthemonodromymatricesM(γ˜,λ)of
Ψ2.11).assoInciatedparticular,withtheallcovmonoeringdromymtransformationsatrices