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}