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1、Chapter2MinimalSurfacesSincethelastcentury,thenameminimalsurfaceshasbeenappliedtosurfacesofvanishingmeancurvature,becausetheconditionH=0willnecessarilybesatisfiedbysurfaceswhichminimizeareawithinagivenboundaryconfiguration.ThiswasimplicitlyprovedbyLagrangefornonpara-metricsurfacesin1760,andthenb
2、yMeusnierin1776whousedtheanalyticexpressionforthemeancurvatureanddeterminedtwominimalsurfaces,thecatenoidandthehelicoid.(ThenotionofmeancurvaturewasintroducedbyYoung[1]andLaplace[1],butusuallyitisascribedtoSophieGermain[1].)InSection2.1weshallderiveanexpressionforthefirstvariationofareawithresp
3、ecttogeneralvariationsofagivensurface.Fromthisexpressionweob-taintheequationH=0asnecessaryconditionforstationarysurfacesoftheareafunctional,andwealsodemonstratethatsolutionsofthefreeboundaryproblemmeettheirsupportingsurfacesatarightangle.InSection2.2,weparticularlyinvestigatenonparametricsurfa
4、ces,andwestatetheminimalsurfaceequationindivergenceandnondivergenceformwhichhastobesatisfiedbytheheightfunction.Finallyweprovethat,foranonparametricminimalsurfaceX,the1-formN∧dXisclosed.InSection2.3itisshownthatanonparametricminimalsurfaceX(x,y)=(x,y,z(x,y))hasarealanalyticheightfunctionz(x,y)a
5、nd,moreover,thatXcanbecon-formallymappedontosomeplanardomain.ThisconformalmappingcanbeconstructedexplicitlyifthedomainofdefinitionΩofthesurfaceXisconvex.ThereafterweproveinSection2.4thecelebratedBernsteintheoremfornonparametricminimalsurfacesandalsoaquantitativelocalversionofthistheoremwhichwas
6、discoveredbyE.Heinz.ThenweshowinSection2.5thateveryregularsurfaceX:Ω→R3satisfiestheequationΔXX=2HNU.Dierkes,S.Hildebrandt,F.Sauvigny,MinimalSurfaces,GrundlehrendermathematischenWissenschaften339,DOI10.1007/978-3-642-11698-82,cSpringer-VerlagBerlinHeidelberg201053542MinimalSurfacesand,therefore
7、,minimalsurfacesarecharacterizedbytheequationΔXX=0.IfXisgivenbyconformalparameters,thisrelationisequivalenttoΔX=0.ThisobservationisusedinSection2.6toenlargetheclassofminimalsur-faces.Wecannowadmitsurfaceswithisolatedsingularitiesbydefini
