HALF SPACE THEOREMS AND THE EMBEDDED CALABI YAU PROBLEM IN LIE GROUPS

Hubert Curien

Description

Niveau: Supérieur, Doctorat, Bac+8
HALF-SPACE THEOREMS AND THE EMBEDDED CALABI-YAU PROBLEM IN LIE GROUPS BENOIT DANIEL, WILLIAM H. MEEKS III, AND HAROLD ROSENBERG Abstract. We study the embedded Calabi-Yau problem for complete embedded constant mean curvature surfaces of finite topology or of posi- tive injectivity radius in a simply-connected three-dimensional Lie group X endowed with a left-invariant Riemannian metric. We first prove a half-space theorem for constant mean curvature surfaces. This half-space theorem applies to certain properly immersed constant mean curvature surfaces of X contained in the complements of normal R2 subgroups F of X. In the case X is a unimodular Lie group, our results imply that every minimal surface in X ? F that is properly immersed in X is a left translate of F and that every complete embedded minimal surface of finite topology or of positive injectivity radius in X ? F is also a left translate of F . 1. Introduction. A natural question in the global theory of minimal surfaces, first raised by Calabi in 1965 [1] and later revisited by Yau [31, 32], asks whether or not there exists a complete immersed minimal surface ? in a bounded domain of R3, or more generally, the question asks: If ? is contained in a half-space of R3, then is it a plane parallel to the boundary of the half-space? For complete immersed minimal surfaces these questions were answered by the existence results of Jorge and Xavier [10] and by Nadirashvili [25].

• complete embedded


• constant curvature


• any metric


• invariant metric


• has positive


• mean curvature


• injectivity radius


• simply-connected homogeneous

Sujets

• Hubert Curien
• Constant curvature
• Mean curvature
• Glossary of Riemannian and metric geometry

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 19
• Langue: English

Extrait

HALF-SPACETHEOREMSANDTHEEMBEDDED CALABI-YAUPROBLEMINLIEGROUPS BENOI ˆ TDANIEL,WILLIAMH.MEEKSIII,ANDHAROLDROSENBERG Abstract.WestudytheembeddedCalabi-Yauproblemforcomplete embeddedconstantmeancurvaturesurfacesoffinitetopologyorofposi- tiveinjectivityradiusinasimply-connectedthree-dimensionalLiegroup Xendowedwithaleft-invariantRiemannianmetric.Wefirstprovea half-spacetheoremforconstantmeancurvaturesurfaces.Thishalf-space theoremappliestocertainproperlyimmersedconstantmeancurvature surfacesofXcontainedinthecomplementsofnormal R2 subgroupsF ofX.InthecaseXisaunimodularLiegroup,ourresultsimplythat everyminimalsurfaceinX − FthatisproperlyimmersedinXisa lefttranslateofFandthateverycompleteembeddedminimalsurface offinitetopologyorofpositiveinjectivityradiusinX − Fisalsoaleft translateofF. 1.Introduction. Anaturalquestionintheglobaltheoryofminimalsurfaces,firstraised byCalabiin1965[1]andlaterrevisitedbyYau[31,32],askswhetherornot thereexistsacompleteimmersedminimalsurfaceΣinaboundeddomainof R 3 ,ormoregenerally,thequestionasks:IfΣiscontainedinahalf-spaceof R 3 ,thenisitaplaneparalleltotheboundaryofthehalf-space?Forcomplete immersedminimalsurfacesthesequestionswereansweredbytheexistence resultsofJorgeandXavier[10]andbyNadirashvili[25].Closelyrelated totheCalabi-YauprobleminEuclidean3-spaceisthehalf-spacetheorem byHoffmanandMeeks[9]:IfSisaproperlyimmersedminimalsurfacein R 3 thatliesononesideofsomeplaneP,thenSisaplaneparalleltoP. Date:December17,2010. 2000MathematicsSubjectClassification.Primary:53A10.Secondary:53C42,53A35. Keywordsandphrases.Minimalsurface,constantmeancurvature,H-surface,homoge- neousmanifold,half-spacetheorem,maximumprinciple,embeddedCalabi-Yauproblem. ThefirstauthorispartiallysupportedbyMiniste`redesAffairese´trange`reseteu- rope´ennes(France),PartenariatHubertCurienSTAR,projetn◦23889UG. ThismaterialisbaseduponworkfortheNSFunderAwardNo.DMS-1004003bythe secondauthor.Anyopinions,findings,andconclusionsorrecommendationsexpressedin thispublicationarethoseoftheauthorsanddonotnecessarilyreflecttheviewsofthe .FSN 1