REGULARITY OF C1 SURFACES WITH PRESCRIBED MEAN CURVATURE IN THREE-DIMENSIONAL CONTACT SUB-RIEMANNIAN MANIFOLDS MATTEOGALLIANDMANUELRITORÉ 5 1 ABSTRACT. InthispaperweconsidersurfacesofclassC1withcontinuousprescribedmeancurvature 0 inathree-dimensionalcontactsub-Riemannianmanifoldandprovethattheircharacteristiccurves 2 areofclassC2.Thisregularityresultalsoholdsforcriticalpointsofthesub-Riemannianperimeter underavolumeconstraint.AllresultsarevalidinthefirstHeisenberggroupH1. y a M 1 1. INTRODUCTION RecentlyCheng,HwangandYang[9],[10],haveconsideredthefunctional ] G (1.1) F(u)= |∇u+F~|+ fu, D ˆ ˆ Ω Ω h. on a domain Ω⊂R2n, where F~ is a vector field and f ∈ L∞(Ω). In case F~(x,y)=(−y,x), the t integral |∇u+F~| is thesub-Riemannian area of thehorizontal graph ofthefunction uin the a Ω m Heisenbe´rg group Hn. Among several interesting results, they proved in [10, Thm. A] that, in [ case n = 1, u ∈ C1(Ω) is an stationary point of F and f ∈ C0(Ω), the integral curves of the vectorfield((∇u+F~)/|∇u+F~|)⊥,definedintheset|∇u+F~|=6 0,areofclassC2. Thegeometric 3 meaning oftheirresult is thattheprojectionofthecharacteristiccurves ofthegraph ofuareof v 6 class C2. AstationarypointuofF satisfiesweaklytheprescribedmeancurvatureequation 4 ∇u+F~ 2 (1.2) div = f. 7 (cid:18)|∇u+F~|(cid:19) 0 Theorem A in [10] is well-known for C2 minimizers and generalizes a previous result by Pauls . 1 [21, Lemma 3.3] for H-minimal surfaces with components of the horizontal Gauss map in the 0 class W1,1. For lipschitz continuous vanishing viscosity minimal graphs, it was proven by Ca- 5 1 pogna,CittiandManfredini[4,Cor.1.6]. : In order to extend this result to arbitrary surfaces, it is natural to replace F by the sub- v i Riemannianprescribedmeancurvaturefunctional X r (1.3) J(E,B)=P(E,B)+ f, a ˆ E∩B where E is a set of locally finite sub-Riemannian perimeter in Ω, P(E,B) is the relative sub- Riemannian perimeter of E in a bounded open set B ⊂ Ω, and f ∈ L∞(Ω). If E ⊂ Hn is the subgraph of afunction t =u(x,y)intheHeisenberg group Hn,thenJ(E)coincides with (1.1) taking F~(x,y) = (−y,x). The notion of sub-Riemannian perimeter used in sub-Riemannian Date:May4,2015. 2000MathematicsSubjectClassification. 53C17,49Q20. Keywordsandphrases. Sub-Riemanniangeometries. M.GallihasbeensupportedbythePeopleProgramme(MarieCurieActions)oftheEuropeanUnion’sSeventhFrame- workProgrammeFP7/2007-2013/underREAgrantagreementn. 607643. M.RitoréhasbeensupportedbyMec-Feder MTM2010-21206-C02-01andMineco-FederMTM2013-48371-C2-1-Presearchgrants. 2 M.GALLIANDM.RITORÉ geometry was first introduced by Capogna, Danielli and Garofalo [5] for Carnot-Carathéodory spaces. General properties and existence of sets with minimum perimeter were proved later by Garofalo and Nhieu [17]. A rather complete theory of finite perimeter sets in the Heisenberg group Hn following De Giorgi’s original arguments was developed by Franchi, Serapioni and Serra-Cassano [11], and later extended to step 2Carnot groups [12] by thesame authors. The recentmonograph[6]providesaquitecompletesurveyonrecentprogressonthesubject. WehavedefinedtheprescribedmeancurvaturefunctionalfollowingMassari[19],whoconsid- eredminimizers ofJ fortheEuclideanrelativeperimeter. Heobtainedexistenceandregularity results for this problem and observed that, in case E is the subgraph of a Lipschitz function u defined on an open bounded set D ⊂Rn−1, thefunction u satisfies weakly the prescribed mean curvatureequation ∇u div (x)= f(x,u(x)) (cid:18) 1+|∇u|2(cid:19) p for x ∈D. Incase∂E∩Ωisahypersurfaceofclass C2 thenthemeancurvatureof∂E atapoint p∈∂E equals g(p). SeealsoMaggi[18,pp.139–140]. TheaimofthispaperistoextendCheng,HwangandYang’sregularityresultforcharacteristic curves[10,Thm.A]fromC1 horizontalgraphssatisfyingweaklythemeancurvatureequationin thefirstHeisenberggroupH1 tosurfacesofclass C1 withprescribedmeancurvatureinarbitrary three-dimensionalcontactsub-Riemannianmanifolds. In this setting, the Euclidean perimeter is replaced by the sub-Riemannian one and the in- tegral of the function f is computed using Popp’s measure [20, § 10.6], [2]. The minimizing condition will be replaced by a stationary one. Our ambient space will be a three-dimensional contactmanifoldwith asub-Riemannian metricdefinedonitshorizontal distribution. Inpartic- ular, noassumptionontheexistenceofapseudo-hermitian structureismade. Weshallprovein Theorem4.1 Let E⊂Ωbeasetwith C1 boundaryandprescribedmeancurvature f ∈C0(Ω) in a domain Ω⊂ M of a three-dimensional contact sub-Riemannian manifold. Thencharacteristiccurvesin∂E areofclassC2. Weremarkthat[10,Thm.A]statesthattheprojectionofcharacteristiccurvestotheplane t=0 isofclass C2,buttogetherwith[8,(2.22)]thisimpliesthatthecharacteristiccurvesthemselves areC2. WethankJ.-H.Chengforpointingoutthisfact. Whiletheproofof[10,Thm.A]wasbasedontheintegralformula[10,(2.3)],seealso(3.7) in[16,Remark3.4],theproofofTheorem4.1ispurelyvariational andfollowsbylocalizing the first variation ofperimeter along acharacteristic curve. A muchweaker version of Theorem4.1 wasgivenin[16,Thm.3.5],whereitwasproventhattheregularpartofanarea-stationarysur- faceofclass C1 inthesub-Riemannian Heisenberg group H1 is foliatedbyhorizontal geodesics. Theorem4.1providesanewresultevenforthecaseofthefirstHeisenberggroupH1. TheregularityofcharacteristiccurvesproveninTheorem4.1allowsustodefineinSection5a meancurvaturefunctionH intheregularpartof∂E,thatcoincideswith f. Asaconsequenceof thedefinitionofthemeancurvature,weshallproveinProposition5.3thatcharacteristiccurves areofclass Ck+2 incase f isofclass Ck whenrestrictedtoacharacteristicdirection. Thisholds, e.g.,when f ∈Ck(Ω)of f ∈Ck(Ω),thespaceoffunctionswithcontinuoushorizontalderivatives H of order k, k ¾1. This class contains C1(Ω) when k ¾2. Critical points oftheperimeter, even- tuallyunderavolumeconstraint,and C1 boundary, haveconstantprescribedmeancurvatureas REGULARITYOFC1SURFACESWITHPRESCRIBEDMEANCURVATURE 3 shown in Section 3. HenceTheorem 4.1 applies to thesesets and implies that theregular parts oftheirboundariesarefoliatedbyC∞ characteristiccurves,seeProposition5.4. Wehaveorganizedthispaperintoseveralsections. Inthesecondoneweprovidethenecessary backgroundoncontactsub-Riemannian manifoldsandsetsoffiniteperimeter,andwerecallthe firstvariationformulaforC1 surfacesfollowing[13]. InSection3weintroducethedefinitionof setoflocallyfiniteperimeterwithprescribedmeancurvatureandprovethatasetwithC1bound- aryandarea-stationary underavolumeconstrainthasconstantprescribedmeancurvature. The mainresult, Theorem4.1,isproven inSection4. Theconsequences onthemeancurvature and higherregularityforcharacteristiccurveswillappearinSection5. 2. PRELIMINARIES 2.1. Contact sub-Riemannian manifolds. In this paper we shall consider a 3-dimensional C∞ manifold M withcontactformωandasub-Riemannian metric g definedonitshorizontaldis- H tributionH :=ker(ω). Bydefinition, dω| is non-degenerate. Weshall refer to(M,ω,g )as H H a3-dimensionalcontactsub-Riemannianmanifold. Itiswell-known thatω∧dωisanorientation formin M. Since dω(X,Y)=X(ω(Y))−Y(ω(X))−ω([X,Y]), the horizontal distribution H is completely non-integrable. The Reeb vector field T in M is the onlyonesatisfying (2.1) ω(T)=1, L ω=0, T whereL istheLiederivativein M. A canonical contact structure in Euclidean 3-space R3 with coordinates (x,y,t) is given by thecontactone-formω :=dt+xdy− ydx. TheassociatedcontactmanifoldistheHeisenberg 0 group H1. Darboux’s Theorem [3, Thm. 3.1] (see also [15]) implies that, given a point p ∈ M, thereexistsanopenneighborhoodU ofpandadiffeomorphismφ fromU intoanopensetofR3 p satisfying φ∗ω =ω. Such alocalchart will becalled aDarbouxchart. Composing themap φ p 0 p withacontacttransformationofH1alsoprovidesaDarbouxchart. Thisimplieswecanprescribe theimageofapoint p∈U andtheimageofahorizontaldirectionin T M. p The metric g can be extended to a Riemannian metric g on M by requiring T to be a unit H vector orthogonal to H. The Levi-Civita connection associated to g will be denoted by D. The integralcurvesoftheReebvectorfieldT aregeodesicsofthemetric g. Thispropertycanbeeasily checkedsinceconditionL ω=0in(2.1)implies ω([T,X])=0forany X ∈H. Hence,forany T horizontalvectorfieldX,wehave g(X,D T)=−g(D X,T)=−g(D T,T)=0. T T X Wetriviallyhave g(T,D T)=0,andsoweget D T =0,asclaimed. T T The Riemannian volume element in (M,g) will be denoted by dM. It coincides with Popp’s measure [20,§10.6], [2]. Thevolume ofaset E ⊂ M with respecttotheRiemannian metric g willbedenotedby|E|. 2.2. Torsion and the sub-Riemannian connection. Thefollowing is taken from [13, § 3.1.2]. In a contact sub-Riemannian manifold, we can decompose theendomorphism X ∈ TM → D T X intoitsantisymmetricandsymmetricparts,whichwewilldenotedbyJ andτ,respectively, 2g(J(X),Y)=g(D T,Y)−g(D T,X), X Y (2.2) 2g(τ(X),Y)=g(D T,Y)+g(D T,X). X Y 4 M.GALLIANDM.RITORÉ ObservethatJ(X),τ(X)∈H foranyvectorfieldX,andthatJ(T)=τ(T)=0. Alsonotethat (2.3) 2g(J(X),Y)=−g([X,Y],T), X,Y ∈H. We will call τ the (contact) sub-Riemannian torsion. We note that our J differs from the one definedin[14,(2.4)]bytheconstant g([X,Y],T),butplaysthesamegeometricroleandcanbe easilygeneralizedtohigherdimensions,[13,§3.1.2]. Now we define the (contact) sub-Riemannian connection ∇ as the unique metric connection, [7,eq.(I.5.3)],withtorsiontensorTor(X,Y)=∇ Y −∇ X −[X,Y]givenby X Y (2.4) Tor(X,Y):= g(X,T)τ(Y)−g(Y,T)τ(X)+2g(J(X),Y)T. From(2.4)andKoszulformulafortheconnection∇itfollowsthat T isaparallelvectorfieldfor thesub-Riemannian connection. Inparticular,theirintegralcurvesaregeodesicsfortheconnec- tion∇. IfX ∈H,p∈M,andX 6=0,thenJ(X )6=0: asdω| isnon-degenerate,thereexistsY ∈H p p H suchthat dω (X ,Y )6=0. From(2.2)wehave2g(J(X ),Y )=−g([X,Y] ,T ),differentfrom p p p p p p p 0sinceω ([X,Y] )=−dω(X ,Y )6=0. p p p p The standard orientation of M is given by the 3-form ω∧ dω. If X is horizontal, then p the basis {X ,J(X ),T } is positively oriented. To check this, observe first that the sign of p p p (ω∧dω)(X,J(X),T)equalsthesignofdω(X,J(X)),andwehave dω(X,J(X))=−ω([X,J(X)])=−g([X,J(X)],T)=g(Tor(X,J(X)),T)=2g(J(X),J(X))>0. 2.3. Perimeter and C1 surfaces. A set E ⊂ M has locally finite perimeter if, for any bounded opensetB⊂M,wehave P(E,B):=sup divUdM :U horizontal, supp(U)⊂B,||U|| ¶1 <+∞. ˆ ∞ (cid:26) E∩B (cid:27) Thequantity P(E,B)istherelativeperimeterof E inB. Assuming Σ=∂E is a surface of class C1, the relative perimeter of E in a bounded open set B⊂M coincideswiththesub-Riemannian areaofΣ∩B,givenby (2.5) A(Σ∩B)= |N |dΣ. ˆ h Σ∩B HereN istheRiemannianunitnormaltoΣ,N isthehorizontalprojectionofN tothehorizontal h distribution,anddΣistheRiemannianareameasure,allcomputedwithrespecttheRiemannian metric g, see [6]. The quantity |N | vanishes in the singular set Σ ⊂ Σ of points p ∈ Σ where h 0 thetangentspace T ΣcoincideswiththehorizontaldistributionH . Thehorizontalunitnormal p p at p ∈ Σ\Σ is defined by (ν ) := (N ) /|(N ) |. At every point p ∈ Σ\Σ , the intersection 0 h p h p h p 0 H ∩T Σisone-dimensional andgeneratedbythecharacteristicvectorfield Z :=J(ν )/|J(ν )|. p p h h The vector S is defined for p ∈Σ\Σ by S := g(N ,T )(ν ) −|(N ) |T . The tangent space p 0 p p p h p h p p T Σ, p∈Σ\Σ ,isgeneratedby{Z ,S }. p 0 p p 2.4. Thefirstvariationofthesub-Riemannianperimeterfor C1 surfaces. Givenaset E with C1 boundary, we canusetheflow {ϕ } ofa vectorfield U with compactsupport in B to pro- s s∈R duce a variation of Σ∩B. The Riemannian area formula gives the following expression of the sub-Riemannian areaofΣ :=ϕ (Σ∩B), s s A(Σ )= |Ns|Jac(ϕ )dΣ, s ˆ h s Σ where Ns isaunit normal to Σ . Fix p∈Σ\Σ andtheorthonormal basis {e ,e }={Z ,S }in s 0 1 2 p p T Σ. Weconsiderextensions E , E of Z ,S ,respectively, along theintegralcurveof U passing p 1 2 p p REGULARITYOFC1SURFACESWITHPRESCRIBEDMEANCURVATURE 5 through p. The vector fields E (s), E (s) are invariant under the flow of U and generate the 1 2 tangentplanetoΣ atthepointϕ (p). Thevector(E ×E )/|(E ×E )|isnormaltoΣ . Here× s s 1 2 1 2 s denotesthecrossproductwithrespecttoavolumeformηforthemetric g inducingthesameori- entationasω∧dω,i.e. g(w,u×v)=η(w,u,v). Itiseasytocheckthat|(E ×E )|(s)=Jac(ϕ )(p), 1 2 s andthat V(p,s):=(E ×E ) (s)= g(E ,T)(T×E )−g(E ,T)(E ×T) (s). 1 2 h 1 2 2 1 Hence (cid:0) (cid:1) A(Σ )= |V(p,s)|dΣ(p), s ˆ Σ andweget d g(∇U V,Vp) |V(s,p)|= p . ds(cid:12)s=0 |Vp| (cid:12) Since{(ν ) ,Z ,T }ispositivelyorie(cid:12)nted,observethatV =|(N ) |(ν ) . Ontheotherhand, h p p p (cid:12) p h p h p ∇ V =g(∇ E ,T )(T ×(E ) )−g(∇ E ,T )((E ) ×T ) U U 1 p p 2 p U 2 p 1 p p p p p −g((E ) ,T )(∇ E ×T ), 2 p p U 1 p p andso g(∇ V,V ) U p p =−g(∇ E ,T )+|(N ) |g(∇ E ×T ,(ν ) ). |V | Up 2 p h p Up 1 p h p p Since g(∇ E ,T )= g(∇ U+Tor(U ,(E ) ),T ) U 2 p (E ) p 2 p p p 2 p =S (g(U,T))+g(Tor(U ,S ),T ) p p p p =S (g(U,T))+2g(J(U ),S ), p p p and g(∇ E ×T ,(ν ) )=g((∇ U+Tor(U ,(E ) ))×T ,(ν ) ) U 1 p h p (E ) p 1 p p h p p 1 p =η((ν ) ,∇ U+Tor(U ,(E ) ),T ) h p (E ) p 1 p p 1 p =+g(∇ U+Tor(U ,Z ),Z ) Z p p p p =+g(∇ U,Z )+g(U ,T )g(τ(Z ),Z ), Z p p p p p p weconcludethatthefirstvariationofthesub-Riemannian perimeterisgivenby d A(Σ )= −S(g(U,T))−2g(J(U),S)+|N |g(∇ U,Z) (2.6) ds(cid:12)s=0 s ˆΣ∩B h Z (cid:12) (cid:8) (cid:12) +|N |g(U,T)g(τ(Z),Z) dΣ. (cid:12) h Thisformulawasobtainedin[14,Lemma3.4]. (cid:9) 3. SETS WITH PRESCRIBED MEAN CURVATURE The reader is referred to [18, (12.32) and Remark 17.11] for background and references in theEuclideancase. ConsideradomainΩ⊂M,andafunction f :Ω→R. Weshallsaythataset of locally finite perimeter E ⊂Ω has prescribed meancurvature f onΩ if,forany bounded open setB⊂Ω, E isacriticalpointofthefunctional (3.1) P(E,B)− f, ˆ E∩B 6 M.GALLIANDM.RITORÉ where P(E,B) is the relative perimeter of E in B, and the integral on E∩B is computed with respect to the canonical Popp’s measure on M, see [20] and [2]. The admissible variations for thisproblemaretheflowsinducedbyvectorfieldswithcompactsupportinB. IfΣ=∂EisasurfaceofclassC1inΩ,thenΣhasprescribedmeancurvature f ifitisacritical pointofthefunctional (3.2) A(Σ∩B)− f, ˆ E∩B foranyboundedopensetB⊂Ω. If E isacriticalpointoftherelativeperimeter P(E,B)inanyboundedopensetB⊂Ω,then E haszeroorvanishingprescribedmeancurvature. Assumenowthat E⊂ΩisasetoflocallyfiniteperimeterwithC1 boundaryΣ,andthat E isa criticalpointoftheperimeterunderavolumeconstraint. Thismeans(d/ds) A(ϕ (Σ∩B))=0for s=0 s anyflowassociatedtoavectorfieldwithcompactsupportinΩsatisfying(d/ds) |ϕ (E∩B)|=0. s=0 s IftheperimeterofEinΩispositive,thenthereexistsa(horizontal)vectorfieldU withcompact 0 supportinΩsothat divU dM >0. BytheDivergenceTheorem, E∩Ω 0 ´ g(U ,N)dΣ6=0, ˆ 0 Σ∩Ω where N is theouter normal to E. Let {ψ } be theflow associatedto thevectorfield U and s s∈R 0 define d/ds A(ψ (Σ)) (3.3) H := s=0 s . 0 d/d(cid:12)s |ψ (E)| (cid:12) s=0 s Let B⊂ΩbeaboundedopensubsetandW ave(cid:12)ctorfieldwithcompactsupportinB andassoci- (cid:12) atedflow{ϕ } . Chooseλ∈RsothatW−λU satisfies s s∈R 0 d d |ϕ (E)|−λ |ψ (E)|= g(W−λU ,N)dΣ=0. ds(cid:12)s=0 s ds(cid:12)s=0 s ˆΣ 0 (cid:12) (cid:12) Then the flow associ(cid:12)ated to W −λU (cid:12)preserves the volume of E∩(B∪B ), where B ⊂ Ω is a (cid:12) 0(cid:12) 0 0 bounded open set containing supp(U ). LetQ(U) be theintegral expression in (2.6). From our 0 hypothesis andthelinearity of(2.6),Q(W −λU )=0. HenceQ(W)=λQ(U ). From(3.3)we 0 0 get d d Q(W)=λQ(U )=λH |ψ (E)|=H |ϕ (E)|, 0 0ds(cid:12)s=0 s 0ds(cid:12)s=0 s (cid:12) (cid:12) andso E has(constant)prescribedmeancurva(cid:12)tureH . (cid:12) (cid:12) 0 (cid:12) 4. MAIN RESULT InthisSectionweshallproveourmainresult Theorem4.1. Let M bea3-dimensionalcontactsub-Riemannianmanifold,Ω⊂M adomain,and E ⊂Ωa set ofprescribed meancurvature f ∈C0(Ω) with C1 boundary Σ. Then thecharacteristic curvesinΣareofclassC2. Proof. Given any point p∈Σ\Σ , consider aDarboux chart (U ,φ ) such thatφ (p)=0. The 0 p p p metric g canbedescribedinthislocalchartbythematrixofsmoothfunctions H g g g(X,X) g(X,Y) G= 11 12 = . ‚g21 g22Œ ‚g(Y,X) g(Y,Y)Œ REGULARITYOFC1SURFACESWITHPRESCRIBEDMEANCURVATURE 7 AfteraEuclideanrotationaroundthet-axis,whichisacontacttransformationinH1[22,p.640], wemayassumethereexistsanopenneighborhoodB∩Σof p∈Σ\Σ ,whereB⊂H1isanopen 0 setcontaining p,so that B∩Σis theintrinsic graph G ofa C1 functionu: D→R definedona u domain D in thevertical plane y =0. We canalso assume that E∩B is thesubgraph ofu. The graph G canbeparameterizedbythemap f :D→R3 definedby u u f (x,t):=(x,u(x,t),t−xu(x,t)), (x,t)∈D. u ThetangentplanetoanypointinG isgeneratedbythevectors u ∂ 7→(1,u ,−u−xu )=X +u Y −2uT, ∂x x x x ∂ 7→(0,u ,1−xu )=u Y +T, ∂t t t t andsothecharacteristicdirectionisgivenby Z =Z/|Z|,where Z =X +(u +2uu )Y. x e et Ifγ(s)=(x(s),t(s))isaC1 curvein D,then e Γ(s)=(x(s),u(x(s),t(s)),t(s)−x(s)u(x(s),t(s)))⊂G u isalso C1,andso Γ′(s)=x′(X +u Y −2uT)+t′(u Y +T)=x′X +(x′u +t′u )Y +(t′−2ux′)T. x t x t In particular, horizontal curves in G satisfy the ordinary differential equation t′ = 2ux′. Since u u∈C1(D),wehaveuniquenessofcharacteristiccurvesthroughanygivenpointinG . u AunitnormalvectortoΣisgivenbyN/|N|,where N =(X +u Y −2uT)×(u Y +T). ex e t Here × is the cross product with respect to the Riemannian metric g and a given volume form e ηchosensothatη(X,Y,T)>0. Hence g(w,u×v)=η(w,u,v). If{e ,e ,e }is anorthonormal 1 2 3 basis so that η(e ,e ,e )=1 and Ais thematrix whosecolumns are thecoordinates of X, Y, T 1 2 3 inthebasis{e ,e ,e },thenη(X,Y,T)=det(A). Ontheotherhand,as 1 2 3 g g 0 11 12 AtA= g g 0 ,  21 22  0 0 1     wegetdet(A)2=det(G). Sincedet(A)>0weobtaindet(A)=det(G)1/2 andso η(X,Y,T)=det(G)1/2. Let E =X +u Y −2uT, E =u Y +T. Wecomputethescalarproductof N = E ×E with X, 1 x 2 t 1 2 Y, T toobtain e 1 1 0 g(X,E1×E2)=η(X,E1,E2)=det0 ux utη(X,Y,T)=(ux+2uut)det(G)1/2. 0 −2u 1     0 1 0 g(Y,E1×E2)=η(Y,E1,E2)=det1 ux utη(X,Y,T)=−det(G)1/2. 0−2u 1     0 1 0 g(T,E1×E2)=η(T,E1,E2)=det0 ux utη(X,Y,T)=utdet(G)1/2. 1−2u 1     8 M.GALLIANDM.RITORÉ Since g(Y,E ×E ) <0 and E∩B is the subgraph of u, the vector field E ×E points into the 1 2 1 2 interiorof E. IfN =E ×E =αX +βY +γT,then 1 2 g g 0 α u +2uu e 11 12 x t g g 0 β =det(G)1/2 −1 ,  21 22     0 0 1 γ u     t       whence α u +2uu =det(G)1/2G−1 x t , (4.1) ‚βŒ ‚ −1 Œ γ=det(G)1/2u . t Let us compute now the sub-Riemmanian area of the intrinsic graph G . It is easy to check u that dΣ=|E ×E |dxdt,i.e., thatJac(f )=|E ×E |. Since|N |=|(E ×E ) |/|E ×E |, then 1 2 u 1 2 h 1 2 h 1 2 using(4.1)andtheexplicitexpressionoftheinversematrixG−1 weget 1 α 2 |N |Jac(f )=|(E ×E ) |= α β G h u 1 2 h (cid:18) ‚βŒ(cid:19) (cid:0) (cid:1) = (g ◦f )(u +2uu )2+2(g ◦f )(u +2uu )+(g ◦f ) 1/2. 22 u x t 12 u x t 11 u Finally,from(2.5)weo(cid:0)btain (cid:1) (4.2) A(G )= g (u +2uu )2+2g (u +2uu )+g 1/2dxdt, u ˆ 22 x t 12 x t 11 D (cid:0) (cid:1) where,byabuseofnotation,wehavewritten g insteadofthecumbersomenotation(g ◦f ). ij ij u Nowwe consider variations of G by graphs of theform s7→u+sv, where v ∈ C∞(D) and s u 0 is a real parameter close to 0. This variation is obtained by applying the flow associated to the vectorfield ˜vY to thegraph G . Thefunction v˜ is obtainedby extending v tobeconstant along u theintegralcurvesofthevectorfieldY,andmultiplyingbyanappropriatefunctionwithcompact supportequalto1inaneighborhoodofΣ. When F isafunctionof(x,y,t),wehave d ∂F ∂F (F◦f )(x,t)= −x v(x,t)=Y (F)v(x,t). ds(cid:12)(cid:12)s=0 u+sv (cid:18)∂ y ∂t(cid:19)fu(x,t) fu(x,t) Soweget (cid:12) (cid:12) d A(G )= K v+M(v +2uv +2vu ) dxdt, ds(cid:12)s=0 u+sv ˆD 1 x t t (cid:12) (cid:0) (cid:1) wherethefunctionsK(cid:12)and M aregivenby 1(cid:12) 1Y(g )(u +2uu )2+2Y(g )(u +2uu )+Y(g ) K = 22 x t 12 x t 11 , 1 2 (g (u +2uu )2+2g (u +2uu )+g )1/2 22 x t 12 x t 11 and g (u +2uu )+g M = 22 x t 12 . (g (u +2uu )2+2g (u +2uu )+g )1/2 22 x t 12 x t 11 ObservethatthefunctionsK and M arecontinuous. Since 1 X +(u +2uu )Y Z = x t , (g (u +2uu )2+2g (u +2uu )+g )1/2 22 x t 12 x t 11 thefunction M coincideswith g(Z,Y)◦f . Astraightforwardcomputationimplies u 1=|Z|2=det(G)−1 g g(Z,X)2−2g g(Z,X)g(Z,Y)+g g(Z,Y)2 22 12 11 (cid:0) (cid:1) REGULARITYOFC1SURFACESWITHPRESCRIBEDMEANCURVATURE 9 andso g g(Z,y)±(det(G)(g −g(Z,Y)2))1/2 (4.3) g(Z,X)= 12 22 . g 22 By Schwarz’s inequality g(Z,Y)2 ¶ g(Y,Y) = g . Inequality is strict since otherwise Y and Z 22 wouldbecollinear. Hence g(Z,X)hasthesameregularityas g(Z,Y)by(4.3). Thesubgraphofucanbeparameterized bythemap(x,t,s)→(x,s,t−xs). TheJacobianof thismapiseasilyseentobeequaltodet(G). Hence d f = f det(G)vdxdt. ds(cid:12)(cid:12)s=0ˆsubgraphGu+sv ˆD IfΣhasprescribedmeancurv(cid:12)ature f,thisimplies (cid:12) (4.4) Kv+M(v +2uv +2vu dxdt=0, ˆ x t t D (cid:0) (cid:1) forany v ∈C∞(D),where thecontinuous function K is given by K =K − f det(G). By Remark 0 1 4.3below,(4.4)alsoholdsforany v∈C0(D)forwhich v +2uv existsanditiscontinuous. 0 x t Now we proceed as in the proof of Theorem 3.5 in [16]. Assume the point p ∈ G corre- u spondstothepoint(a,b)inthe xt-plane. Thecurves7→(s,t(s))is(areparameterization ofthe projection of) a characteristic curve if and only if the function t(s) satisfies the ordinary differ- ential equation t′(s) = u(s,t(s)). For ǫ small enough, we consider the solution t of equation ǫ t′(s)=2u(s,t (s))withinitialcondition t (a)=b+ǫ,anddefineγ (s):=(s,t (s)),withγ=γ . ǫ ǫ ǫ ǫ ǫ 0 Wemayassumethat,forsmallenoughǫ,thefunctions t aredefinedintheinterval[a−r,a+r] ǫ forsome r>0. Thefunction∂t /∂ǫ satisfies ǫ ∂t ′ ∂t ∂t (4.5) ǫ (s)=2u (s,t (s)) ǫ (s), ǫ(a)=1. ∂ǫ t ǫ ∂ǫ ∂ǫ (cid:18) (cid:19) (cid:18) (cid:19) where′ isthederivativewithrespecttotheparameters. Weconsidertheparameterization F(ξ,ǫ):=(ξ,t (ξ))=(s,t) ǫ nearthecharacteristiccurvethrough(a,b). Thejacobianofthisparameterizationisgivenby 1 t′ ∂t det ǫ = ǫ, ‚0 ∂tǫ/∂ǫŒ ∂ǫ whichispositivebecauseofthechoiceofinitialconditionfor t andthefactthatthecurvesγ (s) ǫ ǫ foliateaneighborhoodof(a,b). Anyfunctionϕ canbeconsideredasafunctionofthevariables (ξ,ǫ) by making ϕ˜(ξ,ǫ) := ϕ(ξ,t (ξ)). Changing variables, and assuming the support of ϕ is ǫ containedinasufficientlysmallneighborhoodof(a,b),wecanexpresstheintegral(4.2)as a+r ∂ϕ˜ ∂t Kϕ˜+M +2ϕ˜u˜ ǫ dξ dǫ, ˆ ˆ ∂ξ t ∂ǫ I(cid:26) a−r (cid:18) (cid:18) (cid:19)(cid:19) (cid:27) whereI isasmallintervalcontaining0. Insteadofϕ˜,wecanconsiderthefunctionϕ˜h/(t −t ), ǫ+h ǫ wherehisasufficientlysmallrealparameter. Wegetthat ∂ hϕ˜ ∂ϕ˜ h u˜(ξ,ǫ+h)−u˜(ξ,ǫ) h = · −2ϕ˜· · ∂ξ t −t ∂ξ t −t t −t t −t (cid:18) ǫ+h ǫ(cid:19) ǫ+h ǫ ǫ+h ǫ ǫ+h ǫ tendsto ∂ϕ˜/∂ξ 2ϕ˜u˜ t − , ∂t /∂ǫ ∂t /∂ǫ ǫ ǫ 10 M.GALLIANDM.RITORÉ whenh→0. SousingthatG isarea-stationarywehavethat u a+r h ∂ϕ˜ u˜(ξ,ǫ+h)−u˜(ξ,ǫ) ∂t Kϕ˜+M ·+2ϕ˜· u˜ − ǫ dξ dǫ ˆ ˆ t −t ∂ξ t t −t ∂ǫ I(cid:26) a−r ǫ+h ǫ(cid:18) (cid:18) (cid:18) ǫ+h ǫ (cid:19)(cid:19)(cid:19) (cid:27) vanishes. Furthermore,lettingh→0weconclude a+r ∂ϕ˜ Kϕ˜+M dξ dǫ=0. ˆ ˆ ∂ξ I(cid:26) a−r (cid:18) (cid:19) (cid:27) Let now η : R → R be a positive function with compact support in the interval I and con- sider the family η (x) := ρ−1η(x/ρ). Inserting a test function of the form η (ǫ)ψ(ξ), where ρ ρ ψ is a C∞ function with compact support in (a−r,a+r), making ρ →0, and using that G is u area-stationary weobtain a+r K(0,ξ)ψ(ξ)+M(0,ξ)ψ′(ξ) dξ=0 ˆ a−r (cid:0) (cid:1) for any ψ∈ C∞((a−r,a+r)). By Lemma 4.2, thefunction M(0,ξ), which is therestriction of 0 g(Z,Y) tothecharacteristic curve, isa C1 functiononthecurve. Byequation (4.3),therestric- tion of g(Z,X) to the characteristic curve is also C1. This proves that horizontal curves are of class C2. (cid:3) Lemma 4.2. Let I ⊂ R be an open interval, k, m ∈ C0(I), and K ∈ C1(I) be a primitive of k. Assume (4.6) kψ+mψ′=0, ˆ I foranyψ∈C∞(I). Thenthefunction−K+misconstanton I. Inparticular,m∈C1(I). 0 Proof. Since(Kψ)′=kψ+Kψ′,integratingbypartsweseethat(4.6)isequivalent to −K+m ψ′=0, ˆ I (cid:0) (cid:1) foranyψ∈C∞(I). Thisimpliesthat−K+misaconstantfunctionon I. (cid:3) 0 Remark 4.3. Let us checkthat (4.4) holds forany w ∈C0(D) such that w +2uw exists and is 0 x t continuous. Letusconsiderasequencew ∈C∞(D),wherew =ρ ∗w,andρ denotethestan- j 0 j j j dardmollifiers. Wehavethatw convergestowandthat(w ) +2u(w ) convergestow +2uw j j x j t x t uniformlyoncompactsubsetsof D,for j→∞. Weconclude 0= lim K(w )+M((w ) +2u(w ) +2w u dxdt= Kw+M(w +2uw +2wu dxdt, j→∞ˆD j j x j t j t ˆD x t t (cid:0) (cid:1) (cid:0) (cid:1) thusprovingtheclaim. Remark 4.4. In case M is the Heisenberg group H1, G is the identity matrix and theexpression forthesub-Riemannian areaofthegraphG givenin(4.2)reads u A(G )= ((u +2uu )2+1)1/2dxdt, u ˆ x t D awell-knownformulaobtainedin[1].