The first bifurcation point for Delaunay nodoids
We give two numerical methods for computing the first bifurcation point for Delaunay nodoids. With regard to methods for constructing constant mean curvature surfaces, we conclude that the bifurcation point in the analytic method of Mazzeo-Pacard is the sa
The rstbifurcationpointforDelaunaynodoids
WayneRossman
Abstract:Wegivetwonumericalmethodsforcomputingthe rstbifurcationpointforDelaunaynodoids.Withregardtomethodsforconstructingconstantmeancurvaturesurfaces,weconcludethatthebifurcationpointintheanalyticmethodofMazzeo-PacardisthesameasalimitingpointencounteredintheintegrablesystemsmethodofDorfmeister-Pedit-Wu.
1.Introduction
arXiv:math/0410134v2 [math.DG] 30 Mar 2005
DelaunaysurfacesinEuclidean3-space
areconstantmeancurvature(CMC)surfacesofrevolution,andtheyaretranslationallyperiodic.ByarigidmotionandhomothetyofR3wemayplacetheDelaunaysurfacessothattheiraxisofrevolutionisthex1-axisandtheirconstantmeancurvatureisH=1(henceforthweassumethis).
Weconsiderthepro lecurveinthehalf-plane{(x1,0,x3)∈R3|x3>0}thatgetsrotatedaboutthex1-axistotraceoutaDelaunaysurface.Thiscurvealternatesperiodicallybetweenmaximalandminimalheights(withrespecttothepositivex3direction),whichwerefertoasthebulgeradiusandtheneckradius,respectively,oftheDelaunaysurface.Letusdenotetheneckradiusbyr.
Delaunaysurfacescomeintwo1-parameterfamilies:oneisafamilyofembeddedsurfacescalledunduloidsthatcanbeparametrizedbytheneckradiusr∈(0,1/2];theotherisafamilyofnonembeddedsurfacescallednodoidsthatcanbeparametrizedbytheneckradiusr∈(0,∞).Forunduloids,r=1/2givestheroundcylinder.Forbothunduloidsandnodoids,thelimitingsingularsurfaceasr→0isachainoftangentspheresofradii1centeredalongthex1-axis.
Inthispaperweshallbeconcernedwithnodoids.WewillseethatacommonbifurcationpointforDelaunaynodoidsisencounteredinthefollowingtwodistinctlydi erentapproachesforconstructingCMCsurfaces:
(1)Usinganalytictechniques,MazzeoandPacard[15]showedexistenceofa nitevaluer0sothat
forneckradiir<r0thenodoidsarenonbifurcating,andforr>r0thenodoidscanbifurcate.BifurcatingnodoidsareofinterestbecausetheydeformsmoothlythroughfamiliesofCMCsurfacesthatareofboundeddistancefroma xedlineyetarenotsurfacesofrevolution.Beforethework[15],suchexampleswereunknown.
Tostudythisbifurcation,aparticularJacobioperatorassociatedtothesecondvariationformulaforDelaunaysurfacesisused,alongwithafunctionspacethatcontainsonlyfunctionswiththesametranslationalperiodicityastheDelaunaysurfacesthemselves.Becausethesefunctionsaretranslationallyperiodic,theydonothave niteL2normsontheentiretyofthesurfaces.Hencebifurcationisadi erentnotionfromthatof”nondegeneracy”ofCMCsurfaces,i.e.CMCsurfaceswithnononzeroJacobi eldsof niteL2normontheentiresurfaces.However,thetwonotionshavethecommontraitofbeinghighlyusefultoolsforproducingpreviouslyunknownCMCsurfaces.(TherehasbeenmuchworkdonerelatedtothenondegeneracyofCMCsurfacesandconstructionofnewCMCexamples,seeforexampletheworksofKapouleas,Kusner,Mazzeo,Pacard,Pollack,Ratzkin[10],[11],[14],[16],[17],[21].)AsourinteresthereisinthebifurcationofDelaunaynodoids,fromtheoutsetweconsideronlyperiodicfunctionsonDelaunaysurfaces.
MazzeoandPacardgaveaclearreasonfortheexistenceofthisbifurcationpointr0,intermsoftheexistenceofnontrivialnullityfortheJacobioperator,buttheydidnotcomputetheprecisevalueofr0.
(2)UsingintegrablesystemstechniquesdevelopedbyDorfmeister,PeditandWuin[8],Dorfmeister,
Wu[9]andSchmitt[23](seealso[12])constructedgenus0CMCsurfaceswiththreeasymp-toticallyDelaunayends.In[9]theconstructionwasrestrictedtosurfaceswithasymptoticallyunduloidalends,becausesuchendsareembedded.However,theconstructionin[23]and[12]
1
R3={(x1,x2,x3)|xj∈R}