The classification of complete stable area-stationary surfaces in the Heisenberg group $mathbb{H}^1$

THE CLASSIFICA TION O F COMPLETE ST ABLE AREA-ST A TIONAR Y SURF A CES IN THE HEISENBERG GR OUP H 1 ANA HUR T ADO, M ANUEL RITOR ´ E, AND C ´ ESAR R OSALES Abstract. W e prov e that any C 2 complete, orien table, connected, stable area-stationary surface in the sub-Riemannian Heisenberg group H 1 is either a Euclidean plane or con- gruen t to the hyperb olic parab oloid t = xy . 1. Introduction Minimal surfaces in Euclidean space are a rea-sta tionary , a condition which is equiv alent, by the E uler-Lagr ange equation, to ha ve mea n cur v a ture zero. An impor tant q ue s tion for such a v a riational problem is the classiﬁcatio n of global min imizers. Hence is natural to consider the second v ariation. Minimal s ur faces with non-ne g ative second v ariation of the area are called stabl e minimal surfac es . It is w ell-known that minimal graphs are stable minimal sur faces (in fact ar ea-minimizing by a standard calibratio n argument). A complete minimal g raph m ust be a plane by the classical Bernstein’s Theo rem [6]. Ber ns tein r esult was later extended by do Carmo a nd Peng [1 8], and Fisc her-Co lbrie and Scho en [21], who prov ed that a complete stable or iented minimal surface in R 3 m ust b e a plane. The pr o of in [21] follo ws fro m mor e general results for 3-manifolds of no n- negative scalar curv ature. Non existence of no n-orientable complete stable minimal s urfaces in R 3 has b een proved by Ros [35]. A similar analys is of the v ariational prop erties of area- minimizing surfaces is also of grea t int erest in some sp ecial spaces , such as the three-dimensional Heisenberg group H 1 . This is the simples t mo del of a sub-Riemannian spa ce and of a Carnot group. It is also the lo cal mo del of a ny 3-dimensional pse udo-hermitian manifold. F or background on H 1 we refer the reader to Section 2 and [8]. Area-statio na ry surfaces of class C 2 in H 1 are w ell understo o d. It is well-known [1 0], [33] that, outside the sing ular set given by the p oints where the tangent plane is ho rizontal, suc h a surface is ruled by characteristic horizontal segments. Moreo ver, based on the description of the singular set for t -graphs of class C 2 given by Cheng, Hwang, Malc hio di and Y a ng [10], Ritor´ e and Rosa les [33] proved that a C 2 surface Σ immersed in H 1 is ar ea-stationa ry if and only if its mean curv ature is zero and the characteristic segmen ts in Σ meet o rthogo- nally the singular curves. A similar result was indep endently obtained for area -minimizing t -graphs by Cheng, Hwang, a nd Y ang [11]. F urthermore, the classiﬁca tion of C 2 complete, connected, orientable, area-stationar y surfa ces with non-empt y singular set w as provided Date : October 11, 2018. 2000 Mathematics Subje ct Classiﬁc ation. 53C17,49Q20. Key wor ds and phr ases. Heisenberg group, s ingular set, stable area-stationary surfaces, second v ariation, area-minimizing surfaces. The ﬁrst author has b een supp orted by MCyT-F eder grant MTM2007-62344 and the Caixa-Castell´ o F oundation. The s econd and third authors hav e b een suppor ted by MCyT-F eder grant MTM 2007-61919 and Jun ta de Andaluc ´ ıa grant P06-FQM-01642. 2 A. HUR T ADO, M. RITOR ´ E, AND C. ROSALES in [3 3]: the only examples a re, modulo cong r uence, non-vertical Euclidea n pla nes, the h y- per b olic parab oloid t = xy , and the cla ssical left-handed minimal helicoids. Though some results for complete area -stationar y s urfaces with empty singular set have b een prov ed, see for example [32 , Thm. 5.4], [9] and [33, Pr op. 6.1 6], a deta iled descriptio n o f such surface s seems far fr o m b eing established. This provides an additional motiv ation for the study of second order minima of the ar ea in H 1 . As in the Euclidean case, we de ﬁne a stable ar e a-stationary surface in H 1 as a C 2 area- stationary surface with non- negative s econd deriv ativ e of the a rea under compactly sup- po rted v aria tions. These s urfaces ha ve b een consider ed in prev ious paper s in co nnection with some Ber nstein t yp e pr oblems in H 1 . Let us describ e so me r elated works. In [10], a classiﬁcatio n of all the complete C 2 solutions to the minimal surfaces equa tion for t -g raphs in H 1 is given. In [3 3], this classiﬁcation was reﬁned by showing that the only complete area- stationary t -gr aphs ar e Euclidean non-vertical planes or those congruent to the hyperb olic parab oloid t = xy . By means of a calibration argument it is also proved in [33] that they a re all ar ea-minimizing. In [13] and [4] the Bernstein problem for intrinsic gr aphs in H 1 was studied. The no tion of in trinsic gra ph is the one used by F ranchi, Serapioni and Ser ra Cassano in [23]. Geomet- rically , an in trinsic graph is a nor ma l gra ph o ver some Euclidean v ertical pla ne with resp e ct to the left in v a riant Riemannian metric g in H 1 deﬁned in Section 2. A C 1 int rinsic gr aph has empty singular set. Ex a mples of C 2 complete area- stationary int rinsic gra phs diﬀerent from v ertical Euclidean planes were found in [13]. So a n atura l q ue s tion is to study c om- plete a rea-minimizing intrinsic gr aphs. A rema r k able diﬀerence with res pec t to the case of the t - g raphs is the existence o f co mplete C 2 area-s tationary in trinsic gr aphs which are not area-minimizing , see [13]. In [4 ], Barone, Serra Cas sano and Vittone clas s iﬁed complete C 2 area-s tationary intrinsic gr a phs. Then they computed the s e cond v ariation formula o f the area for such g raphs to establish that the only stable ones are the E uc lide a n vertical planes. An interesting ca libr ation a rgument, also g iven in [4], yields that the vertical planes ar e in fact are a-minimizing surfaces in H 1 . In the interesting paper [15], it is proven tha t C 2 complete stable ar ea-stationa ry Eu- clidean gr aphs with e mpty sing ular set m ust be vertical planes. This is done by s howing that if such a gra ph is diﬀerent from a vertical plane then it contains a particular ex a mple of unstable surfaces called st rict gr aphic al s t rips . F rom the g eometrical p o int of view, a graphical s trip is a C 2 surface given by the union o f a family of hor iz ontal lines L t passing through and ﬁlling a vertical segment so that the angle function of the horiz ontal pro jection of L t is a monotonic function. The graphica l s trip is strict if the angle function is s trictly monotonic. If the angle function is constant we ha ve a piece of a vertical plane. W e would like to remark that there are examples of complete area -stationar y surfaces w ith empty singular set which do not co ntain a graphical strip, such as the sub- Rie ma nnian cateno ids t 2 = λ 2 ( x 2 + y 2 − λ 2 ), λ 6 = 0. Hence the main result in [15] do e s not apply to general surfaces. The following natura l step is to consider complete sta ble s urfaces in H 1 . In fact, all the aforementioned res ults leave op en the existence of stable examples diﬀerent from intrinsic graphs or Euclidean g raphs with empt y singular set. The pur p o se of the pr esent pap er is to cla ssify c omplete sta ble ar ea-statio nary surfaces in H 1 with empty sing ular set or no t. In Theorem 6.1 we prov e the following result The only c omplete, orientable, c onne cte d, stable ar e a-stationary surfac es in H 1 of class C 2 ar e the Euclide an planes and t he surfac es c ongruent to the hyp erb olic p ar ab oloid t = xy . COMPLETE ST ABLE SURF ACES IN THE HE ISENBERG GROUP H 1 3 In pa rticular, this result provides the classiﬁcatio n of all the co mplete C 2 orientable area- minimizing surfac e s in H 1 . In order to pro ve Theorem 6 .1 we compute the second der iv a tive of t he area for some compactly supp orted v ariations o f a C 2 area-s tationary surface Σ by means of Riemannian geo desics. In Theor em 3.7, v ariatio ns of a p or tion Σ ′ of the regula r part of Σ in the direc- tion of v N + w T , where N is the unit nor mal to Σ and T is the Reeb vector ﬁeld in H 1 , will be co ns idered. Here v , w are assumed to hav e c o mpact supp ort in Σ, but not on Σ ′ . Hence the bounda r y of Σ ′ is moving a long the v ar iation. In Prop osition 3.1 1 v ariations in the direction of w T of a C 2 area-s tationary sur face Σ with singular curves of cla ss C 3 will be tak en. Here w has compact supp ort near the singular curves, a nd it is consta nt along the characteristic curves of Σ. B oth types of v ariations will b e combined to pro duce g lobal ones in Pr op osition 5.2. Second v ariation formulas of the area for v ariations su pp orte d in the r e gular set hav e app eared in several co ntexts. In [10], such a for mula was obtained for C 3 surfaces inside a 3-dimensio nal pseudo-hermitian manifold. In [4], a second v ariation formula w as pr ov ed for v ariations by intrinsic graphs o f clas s C 2 . In [12], it is computed the second der iv a tive of the area asso ciated to a C 2 v a riation of a C 2 surface a long E uclidean straight lines. Once we hav e the second v a riation formula w e pro ceed int o tw o steps. First we pr ove in Theorem 4.7 that a C 2 complete oriented stable area -stationar y surface with empt y singular set mu st b e a vertical pla ne. In fact, for such a surface Σ, the se cond deriv a tive o f the ar ea for a compactly s uppo rted v ariation as in Theorem 3.7 is given by I ( u, u ) = − Z Σ u L ( u ) , where u is the no rmal c o mp o nent of the v ariation, and L is the hypo elliptic o pe r ator o n Σ given in ( 3.44). By analo gy with the Riemannia n situation [3 ] w e refer to I a s the index form asso cia ted to Σ and to L as the stability op er ator of Σ. In Propo sition 3.12 we see that the stabilit y condition for Σ implies that I ( u, u ) > 0 for any u ∈ C 0 (Σ) which is als o C 1 along the c hara cteristic lines. Then w e c ho ose the function u := | N h | , where N is the Riemannian unit normal to Σ for the left inv ariant Riemannian metric g on H 1 deﬁned in Section 2, N h is the ho rizontal pr o jection of N , and the mo dulus is computed with resp ect to the metric g . W e see in Pro po sition 4.6 that this function u sa tisﬁes L ( u ) > 0 , and the inequality is s tr ict in piece s of Σ which are not contained inside Euclidean vertical planes. In suc h a ca s e w e pro duce a compactly supp orted non- negative function v in Σ so that inequality I ( v , v ) < 0 still holds. T o construc t the function v we use the Jacobi vector ﬁeld o n Σ asso cia ted to the family of horizo nt al straight lines ruling Σ and which is s tudied in Le mma 4 .5. Observe that the function | N h | is asso cia ted to the v a riational vector ﬁeld induced by the surfaces equidistant to Σ in the Car no t-Carath´ eo dory distance, see [1]. H ence, our construction of the test function v is, in spirit, similar to that in the Euclidean c a se, where the equiv a lent test function is u ≡ 1. Using Fischer-Colbrie’s results [20], a stable minimal surface in R 3 is conformally a compa ct Riema nn surface min us a ﬁnite nu mber of p o ints, so that a lo garithmic cut-oﬀ function v o f u ≡ 1 has compa ct supp ort and yields instability unless the sur face is a pla ne. W e remark that the function | N h | was already used as a test function in [4], [1 3] and [15]. In the seco nd step of the pr o of of Theorem 6.1 we consider a complete ar ea-stationa ry surface Σ with non- empt y singular set. F rom the c lassiﬁcation in [33], w e conclude that Σ m ust b e a no n-vertical plane, congr uent to the hype r b olic parab o loid t = xy , or cong ruent to a left-handed helicoid, see Prop osition 5.1 for a pre c is e statemen t. The ﬁrst tw o types o f surfaces are t -graphs and then they are a rea-minimizing by a ca libration arg ument [33]. F or 4 A. HUR T ADO, M. RITOR ´ E, AND C. ROSALES the third type we will co mbine o ur second v ariation formulas in Theorem 3.7 and Pr o p o- sition 3.11 to pro duce the stability inequality Q ( u ) > 0, where Q is the quadra tic form deﬁned in (5.8). The co nstruction of appropr iate test functions with Q ( u ) < 0 will prov e the instability of the helicoids. It is interesting to observe that Q ( u ) > 0 for functions u with supp ort in the regula r part of the helicoids. In the Heisenberg gro ups H n , with n > 5 , ther e is no counterpart to Theor em 4.7, as some examples hav e b een constructed in [4] of complete ar ea-minimizing intrinsic graphs diﬀerent from Euclidean vertical hyperpla nes. F or n = 2 , 3 , 4 it is still unkno wn if similar examples can b e obtained. W e would like to mention that ex amples of ar ea-minimizing sur faces in H 1 with low Euclidean regula rity have b een o btained in [11], [31], [3 4] and [30]. Hence our results a re optimal in the c lass of C 2 area-s tationary surfaces. Finally , the techniques in this pap er can b e employ ed to prov e c lassiﬁcation results for complete stable a r ea-statio nary surfaces under a volume constr aint in the ﬁrst Heisenberg group [36], and inside the sub-Riema nnian three-sphere [2 7]. W e hav e org anized this paper as follo ws: the next section contains some background ma- terial in several s ubsections. In the third one we r ecall known facts about area - stationary surfaces and w e co mpute second v ariation formulas for the area. The fourth and ﬁfth sec- tions treat co mplete s ta ble s urfaces without a nd with singula r p oints, resp ectively . In the sixth section we state and pr ove the main result. After the distribution o f this pap er we were informed b y Pro f. Nicola Garofalo that The - orem 4.7 was prov en, for the case of e mbedded sur faces, by Danielli, Gar ofalo, Nhieu and Pauls in late 200 6, [14]. 2. Preliminaries In this section we gather some previous res ults that will b e used throug hout the pa p er . W e hav e organized it in s everal pa rts. 2.1. The Heisenberg group. The Heisenb er g gr oup H 1 is the Lie gr o up ( R 3 , ∗ ), where the pro duct ∗ is deﬁned, for any pair of p oints [ z , t ], [ z ′ , t ′ ] ∈ R 3 ≡ C × R , by [ z , t ] ∗ [ z ′ , t ′ ] := [ z + z ′ , t + t ′ + Im( z z ′ )] , ( z = x + iy ) . F o r p ∈ H 1 , the left tr anslation by p is the diﬀeomor phis m L p ( q ) = p ∗ q . A basis of left inv a riant vector ﬁelds (i.e., inv ar iant by any le ft translatio n) is given by X := ∂ ∂ x + y ∂ ∂ t , Y := ∂ ∂ y − x ∂ ∂ t , T := ∂ ∂ t . The horizontal distribution H in H 1 is the smo oth planar distr ibution g enerated by X and Y . The hori zontal pr oje ct ion of a tangent vector U onto H will be denoted by U h . A vector ﬁeld U is horizontal if U = U h . W e denote by [ U, V ] the Lie bracket of tw o C 1 vector ﬁelds U and V on H 1 . Note that [ X , T ] = [ Y , T ] = 0, while [ X , Y ] = − 2 T , so that H is a bracket -gener ating distribution. Moreov er, b y F rob enius theorem we ha ve that H is no nin tegra ble. The vector ﬁelds X and Y generate the kernel of the (contact) 1-form ω := − y dx + x dy + dt . 2.2. The l eft i n v ariant metri c. W e shall consider on H 1 the Riema nnia n metr ic g =  · , ·  so that { X , Y , T } is a n o r thonormal bas is at every po int. The r estriction of g to H coin- cides with the us ual sub-Riemannia n metr ic in H 1 . Let D b e the Levi-Civ ita connection COMPLETE ST ABLE SURF ACES IN THE HE ISENBERG GROUP H 1 5 asso ciated to g . F rom Koszul for mula and the Lie bra cket r elations we g et D X X = 0 , D Y Y = 0 , D T T = 0 , D X Y = − T , D X T = Y , D Y T = − X , (2.1) D Y X = T , D T X = Y , D T Y = − X . F o r any ta ngent vector U on H 1 we deﬁne J ( U ) := D U T . Then we hav e J ( X ) = Y , J ( Y ) = − X and J ( T ) = 0, so that J 2 = − Id when res tricted to H . It is a lso clear that (2.2)  J ( U ) , V  +  U, J ( V )  = 0 , for a ny pair o f ta ngent vectors U and V . The inv olution J : H → H together with the 1-form ω = − y dx + x dy + dt , provides a pseudo-hermitian structur e on H 1 , see [7, Sect. 6.4]. Let R b e the Riemannian curv ature tensor of g deﬁned for tang ent vectors U, V , W by R ( U, V ) W = D V D U W − D U D V W + D [ U,V ] W . F r om (2.1) and the Lie bracket relations we ca n obtain the following identit ies R ( X , Y ) X = − 3 Y , R ( X , Y ) Y = 3 X , R ( X , Y ) T = 0 , R ( X , T ) X = T , R ( X , T ) Y = 0 , R ( X , T ) T = − X, (2.3) R ( Y , T ) X = 0 , R ( Y , T ) Y = T , R ( Y , T ) T = − Y . W e denote by Ric the Ricci curv ature in ( H 1 , g ) deﬁned, for any pair of tangent vectors U and V , as the trac e of the map W 7→ R ( U, W ) V . These eq ualities can b e chec ked b y taking int o acco unt (2.3) Ric( X , Y ) = 0 , Ric( X, T ) = 0 , Ric( Y , T ) = 0 , (2.4) Ric( X, X ) = − 2 , Ric( Y , Y ) = − 2 , Ric( T , T ) = 2 . 2.3. Horizon tal curv es and Carnot-Carath´ eo dory distance. Let γ : I → H 1 be a piecewise C 1 curve deﬁned on a compa c t interv a l I ⊂ R . The length of γ is the usual Rie- mannian length L ( γ ) := R I | ˙ γ ( ε ) | dε , where ˙ γ is the tangent vector of γ . A horizontal curve γ in H 1 is a C 1 curve whose tangent vector always lies in the horizontal distributio n. F or tw o given p oints in H 1 we can ﬁnd, by Chow’s connectivity theorem [2 4, Sect. 1 .2.B], a horizontal curve joining these p o int s. The Carnot-Car ath´ eo dory distanc e d cc betw een t wo points in H 1 is deﬁned as the inﬁmum of the length of horizontal curves joining the given p oints. The top ology asso c ia ted to d cc coincides with the usual to p o logy in R 3 , see [5, Co r. 2.6]. 2.4. Geo desi cs and Jacobi ﬁelds in ( H 1 , g ) . A ge o desic in ( H 1 , g ) is a C 2 curve γ such that the cov a riant deriv ative of the tange nt vector ﬁeld ˙ γ v anishes along γ . Let γ ( s ) = ( x ( s ) , y ( s ) , t ( s )). Dots will indicate deriv atives with res pe ct to s . W e write ˙ γ = ˙ x X + ˙ y Y + ( ˙ t − ˙ xy + x ˙ y ) T . Then γ is a geo des ic in ( H 1 , g ) if and only if ¨ x = 2  ˙ γ , T  ˙ y , ¨ y = − 2  ˙ γ , T  ˙ x, d ds  ˙ γ , T  = 0 . 6 A. HUR T ADO, M. RITOR ´ E, AND C. ROSALES Let λ b e the constant ˙ t − ˙ xy + x ˙ y =  ˙ γ , T  . An easy integration sho ws that the geo desic with initial conditions ( x (0) , y (0) , t (0)) = ( x 0 , y 0 , t 0 ) and ( ˙ x (0) , ˙ y (0) , ˙ t (0)) = ( A, B , C ) is given by x ( s ) = x 0 + As f (2 λs ) + B s g (2 λs ) , y ( s ) = y 0 − As g (2 λs ) + B s f (2 λs ) , (2.5) t ( s ) = t 0 + λs + ( A 2 + B 2 ) s 2 h (2 λs ) + ( Ax 0 + B y 0 ) s g (2 λs ) + ( Ay 0 − B x 0 ) s f (2 λs ) , where f , g and h a re the r eal analytic functions f ( x ) :=    sin( x ) x , x 6 = 0 1 , x = 0 , g ( x ) :=    1 − cos( x ) x , x 6 = 0 0 , x = 0 , h ( x ) :=    x − sin( x ) x 2 , x 6 = 0 0 , x = 0 . In particula r , w e hav e (2.6) exp p ( sv ) = p + sv , for p ∈ H 1 and v ∈ H p or v || T p , which is a ho rizontal or vertical straig ht line. Here exp p denotes the ex po nential map of ( H 1 , g ) at p . In the nex t res ult we construct Riemannia n Jaco bi ﬁelds asso ciated to C 1 families of Riemannian geo desics . Lemma 2.1. L et α : I → H 1 b e a C 1 curve deﬁne d on some op en interval I ⊆ R . F or any C 1 ve ctor ﬁeld U along α we c onsider the map F : I × R → H 1 given by F ( ε, s ) := exp α ( ε ) ( s U α ( ε ) ) . Then, the variational ve ctor ﬁeld V ε ( s ) := ( ∂ F /∂ ε )( ε, s ) is C ∞ along the ge o desic γ ε ( s ) := F ( ε, s ) . As a c onse quenc e, [ ˙ γ ε , V ε ] = 0 and V ε satisﬁes the Jac obi e qu ation (2.7) V ′′ ε + R ( ˙ γ ε , V ε ) ˙ γ ε = 0 , wher e the prime ′ denotes the c ovariant derivative along the ge o desic γ ε . Mor e over, if γ ε is a horizontal str aight line, then (2.8) V ′′ ε − 3  V ε , J ( ˙ γ ε )  J ( ˙ γ ε ) + | ˙ γ ε | 2  V ε , T  T = 0 . Remark 2.2. The classica l pro ofs in Riemannian geo metry of [ ˙ γ ε , V ε ] = 0 and the fact that V ε satisﬁes the Jaco bi equation do not a pply directly in our s e tting since we only suppos e that F is a C 1 map. Pr o of of Le mma 2.1 . Le t ( x 0 ( ε ) , y 0 ( ε ) , t 0 ( ε )) a nd ( A ( ε ) , B ( ε ) , C ( ε )) be the E uclidean co or- dinates of α ( ε ) and U α ( ε ) , resp ectively . By using the expressio n of the Riemannian geo desics in (2.5), we see that the map F ( ε, s ) ca n b e written as x ( ε, s ) = x 0 ( ε ) + A ( ε ) s f (2 λ ( ε ) s ) + B ( ε ) s g (2 λ ( ε ) s ) , y ( ε, s ) = y 0 ( ε ) − A ( ε ) s g (2 λ ( ε ) s ) + B ( ε ) s f (2 λ ( ε ) s ) , t ( ε, s ) = t 0 ( ε ) + λ ( ε ) s + ( A 2 + B 2 )( ε ) s 2 h (2 λ ( ε ) s ) + ( A ( ε ) x 0 ( ε ) + B ( ε ) y 0 ( ε )) s g (2 λ ( ε ) s ) + ( A ( ε ) y 0 ( ε ) − B ( ε ) x 0 ( ε )) s f (2 λ ( ε ) s ) , where λ ( ε ) := C ( ε ) − A ( ε ) y 0 ( ε ) + B ( ε ) x 0 ( ε ). Observe that the functions x 0 ( ε ), y 0 ( ε ), t 0 ( ε ), A ( ε ), B ( ε ), C ( ε ) a nd λ ( ε ) are C 1 . A dir ect computation of ( ∂ F /∂ ε )( ε, s ) shows that V ε ( s ) is C ∞ along the geo des ic γ ε ( s ). On the other hand, w e ca n chec k that for all k ∈ N and a ny of the Euclidean co mp o nents φ ( ε, s ) o f F ( ε, s ), the partia l der iv a tives ∂ k +1 φ/∂ ε ∂ k s exist and are contin uous functions. In par ticular, it follows from the cla ssical Sch warz’s theorem tha t ∂ 2 φ/∂ ε∂ s = ∂ 2 φ/∂ s∂ ε and ∂ 3 φ/∂ ε∂ s 2 = ∂ 3 φ/∂ s∂ ε∂ s . N ow, the classical pro ofs in [17, p. 68 a nd p. 111] can be traced to pr ov e that [ ˙ γ ε , V ε ] = 0 and that V ε satisﬁes the Jacobi eq ua tion. Finally , to get COMPLETE ST ABLE SURF ACES IN THE HE ISENBERG GROUP H 1 7 (2.8) from (2 .7) it suﬃces to use (2.3) to obtain R ( w , v ) w = − 3  v , J ( w )  J ( w ) + | w | 2  v , T  T provided w is a horizo ntal v ector.  2.5. Geometry of surfaces in H 1 . Unless explicitly stated we shall consider surfaces with empt y bo undary . Let Σ be a C 1 surface immer sed in H 1 . The singular set Σ 0 consists o f those points p ∈ Σ for which the ta ngent pla ne T p Σ coincides with H p . As Σ 0 is closed and has empty in terior in Σ, the r e gular set Σ − Σ 0 of Σ is op en and dense in Σ. It was pr ov ed in [16, Lem. 1], see a ls o [2, Thm. 1.2], that, for a C 2 surface, the Hausdo rﬀ dimensio n of Σ 0 with resp ect to the Riema nnian distance o n H 1 is less than o r equal to one. In pa rticular, the Riema nnia n a rea of Σ 0 v a nishes. If N is a unit nor mal vector to Σ in ( H 1 , g ), then we can describ e the singular set as Σ 0 = { p ∈ Σ; N h ( p ) = 0 } , where N h = N −  N , T  T . In the regula r part Σ − Σ 0 , we can deﬁne the horizontal Gauss map ν h and the char acteristic ve ctor ﬁeld Z , by (2.9) ν h := N h | N h | , Z = J ( ν h ) . As Z is horizo ntal and or thogonal to ν h , we co nc lude that Z is tangent to Σ. Hence Z p generates T p Σ ∩ H p . The in tegral cur ves of Z in Σ − Σ 0 will b e called ( ori ente d ) char acteristic curves o f Σ. They a r e b oth tange nt to Σ and horizontal. If we deﬁne (2.10) S :=  N , T  ν h − | N h | T , then { Z p , S p } is a n o rthonorma l basis of T p Σ whenever p ∈ Σ − Σ 0 . Mo reov er, for any p ∈ Σ − Σ 0 we ha ve the o rthonorma l basis of T p H 1 given b y { Z p , ( ν h ) p , T p } . F rom here we deduce the following ident ities o n Σ − Σ 0 (2.11) | N h | 2 +  N , T  2 = 1 , ( ν h ) ⊤ =  N , T  S, T ⊤ = −| N h | S, where U ⊤ stands for the pro jection o f a vector ﬁeld U onto the tangent plane to Σ. Given a C 1 immersed surface Σ with a unit norma l vector N , w e deﬁne the ar e a of Σ b y (2.12) A (Σ) := Z Σ | N h | d Σ , where d Σ is the Riemannian ar ea element on Σ. If Σ is a C 2 surface b ounding a s e t Ω, then A (Σ) c oincides with a ll the notions of p erimeter of Ω and area of Σ intro duce d by other authors, see [2 2, Prop. 2.14 ], [29, Thm. 5.1] a nd [22, Cor. 7 .7]. Finally , for a C 2 immersed surface Σ with a unit normal vector N , w e denote by B the Riemannian shap e op era tor of Σ with r esp ect to N . It is deﬁned for any v ector W tangent to Σ b y B ( W ) = − D W N . The Riemannian mean curv a ture of Σ is − 2 H R = div Σ N , where div Σ denotes the Riemannia n divergence relative to Σ. 2.6. Isometries and di l ations. By a horizontal isometry of H 1 we mean an iso metr y of ( H 1 , g ) leaving in v ariant the hor izontal distributio n. These isometries prese r ve the a rea de- ﬁned in (2 .12). E xamples of such isometries are the left transla tions and the Euclidean rotations ab out the t -axis. W e say that tw o surfaces Σ 1 and Σ 2 are c ongruent if ther e is a horizontal isometry φ such that φ (Σ 1 ) = Σ 2 . In the Heisenberg gr oup H 1 there is a one-par ameter group of C ∞ dilations { δ λ } λ ∈ R given in co or dinates ( x, y, t ) by (2.13) δ λ ( x, y, t ) = ( e λ x, e λ y , e 2 λ t ) . F r om (2.13) it is easy to chec k that an y δ λ preserves the horizo ntal a nd the v ertical dis tr i- butions. The b ehaviour of the ar ea with resp ect to δ λ is contained in the for mu la (2.14) A ( δ λ (Σ)) = e 3 λ A (Σ) . 8 A. HUR T ADO, M. RITOR ´ E, AND C. ROSALES F o r a pro o f o f (2.1 4) see [33, P r o of of Thm. 4 .1 2]. 2.7. A weak Riemannian divergence theorem. Let Σ b e a C 2 Riemannian surface. F or any in teger r > 1 we de no te by C r 0 (Σ) and C r (Σ) the spaces of functions of class C r with or without compact suppo r t in Σ. F or r > 1 let L r (Σ) be the corresp onding space of in tegrable functions w ith resp ect to the Riema nnian measur e d Σ. Let U b e a C 1 tangent vector ﬁeld on Σ. Given a co ntin uous function f on Σ, a contin uous vector ﬁeld V on Σ, and a p oint p ∈ Σ, w e deﬁne U p ( f ) = ( f ◦ α ) ′ (0) and ( D U V )( p ) = V ′ α ( s ) (0). Here α is the in tegra l curve of U with α (0) = p , while the primes denote deriv ativ es of functions dep ending on s and cov ariant deriv atives alo ng α ( s ). W e say that f and V are C 1 in the U -direction if U ( f ) and D U V are well deﬁned and they a r e contin uous on Σ. W e a ls o set (2.15) div Σ ( f U ) := f div Σ U + U ( f ) , where div Σ U stands for the Riema nnian divergence of U . Note that these deﬁnitions coin- cide with the classical o nes when f ∈ C 1 (Σ) and V is a C 1 vector ﬁeld on Σ. In the same wa y w e can intro duce deriv ativ es of higher order in the U - direction. Now we extend the cla ssical Riemannia n div ergence theorem in Σ to certain vector ﬁelds with compact supp or t which are not C 1 on Σ. Fir st we need an approximation result. Lemma 2.3 . L et Σ b e a C 2 Rie mannian surfac e. Consider a C 1 tangent ve ctor ﬁeld U on Σ such that U p 6 = 0 for any p ∈ Σ . Then, for any function f ∈ C 0 (Σ) which is also C 1 in the U -dir e ction, ther e is a c omp act set K ⊆ Σ and a se quenc e of functions { f ε } ε> 0 in C 1 0 (Σ) such that the supp orts of f and f ε ar e c ontaine d in K for any ε > 0 , and (i) { f ε } → f in L r (Σ) for any int e ger r > 1 , (ii) { U ( f ε ) } → U ( f ) in L r (Σ) for any int e ger r > 1 . Pr o of. Let p ∈ Σ. By using the lo c a l ﬂow of U in Σ and tha t U p 6 = 0, we can ﬁnd a loc a l C 1 chart ( D , φ = ( x, y )) of Σ around p such that K = D is compact and the restric tio n of U to D coincides with the bas ic vector ﬁeld ∂ y . This means that U ( h ) = ( ∂ ( h ◦ φ − 1 ) /∂ y ) ◦ φ for any function h which is C 1 in the U -direction. T o ﬁnish the pr o of it suﬃces, by a sta ndard partition of unity argument , to prov e the claim when the supp ort of f is contained in D . Let D ′ = φ ( D ) and g = f ◦ φ − 1 . W e hav e g ∈ C 0 ( D ′ ) and ∂ g /∂ y = U ( f ) ◦ φ − 1 ∈ C 0 ( D ′ ). F r om the sta ndard r egulariza tion by conv olution in R 2 , see for instance [1 9, Sect. 4.2.1], w e ca n ﬁnd a s equence { g ε } ε> 0 in C ∞ 0 ( R 2 ) such that { g ε } → g and { ∂ g ε /∂ y } → ∂ g /∂ y uniformly in R 2 , while the suppo rts of g ε are contained in D ′ for any ε > 0. It follows that the family { f ε } ε> 0 with f ε = g ε ◦ φ sa tisﬁes { f ε } → f and { U ( f ε ) } → U ( f ) uniformly in D , while the suppo rt of f ε is contained in D ⊂ K for any ε > 0 . Clea rly { f ε } ε> 0 prov es the lemma.  Lemma 2.4. L et Σ b e a C 2 Rie mannian surfac e. Consider a C 1 tangent ve ctor ﬁeld U on Σ such that U p 6 = 0 for any p ∈ Σ . Then, for any f ∈ C 0 (Σ) which is also C 1 in the U - dir e ction, we have Z Σ div Σ ( f U ) d Σ = 0 . Pr o of. By deﬁnition (2.1 5) it fo llows that div Σ ( f U ) ∈ L 1 (Σ) since f has compact suppor t and U ( f ) is contin uous. B y Lemma 2.3 we can ﬁnd a sequence { f ε } ε> 0 in C 1 0 (Σ) such that { f ε } → f and { U ( f ε ) } → U ( f ) in L 1 (Σ), while the supports of f ε and f a re co ntained in the same compact set K ⊆ Σ for any ε > 0. In particular, we deduce { f ε div Σ U } → f div Σ U in L 1 (Σ) since div Σ U is cont inuous. By using the Riemannian divergence theor em for C 1 vector ﬁelds with co mpact supp o rt, we obtain 0 = Z Σ div Σ ( f ε U ) d Σ = Z Σ f ε div Σ U d Σ + Z Σ U ( f ε ) d Σ , ε > 0 . COMPLETE ST ABLE SURF ACES IN THE HE ISENBERG GROUP H 1 9 Letting ε → 0 in the prev io us equality the claim is pr oven.  3. St able surf ace s. Second v aria tion f ormulas of the area In this section we deﬁne s table surfa c es and w e show that they satisfy an analytical in- equality by means o f a s e c ond v ariation formula for the ar ea functional deﬁned in (2.12). W e ﬁrst introduce the appropr iate v ar iational background. Let Σ b e a C 2 oriented surface immer sed in H 1 with singular s et Σ 0 . By a variation of Σ we mean a C 1 map ϕ : I × Σ → H 1 , where I is a n open interv al containing the orig in, satisfying the following prop erties: (i) ϕ (0 , p ) = p for any p ∈ Σ, (ii) The set Σ s = { ϕ ( s, p ); p ∈ Σ } is a C 1 surface immersed in H 1 for any s ∈ I , (iii) The map ϕ s : Σ → Σ s given b y ϕ s ( p ) = ϕ ( s, p ) is a diﬀeomorphism for a ny s ∈ I . W e say that the v a riation is c omp actly su pp orte d if there is a compact set K ⊆ Σ such that ϕ s ( p ) = p for any s ∈ I and p ∈ Σ − K . If, in addition, the set K is contained inside Σ − Σ 0 then the v ar iation is nonsingular . The area functional as so ciated to the v aria tion is A ( s ) := A (Σ s ). No te that only the deformatio n ov er the compact set K contributes to the change of ar e a. W e say that Σ is ar e a-stationary if A ′ (0) = 0 for a ny co mpactly supp or ted v a riation. W e say that Σ is stable (resp. stable under non-singu lar variations ) if it is ar ea- stationary a nd A ′′ (0) > 0 for any compactly supp orted (r esp. non-sing ula r) v ar iation o f Σ. Finally by an ar e a-minimizing surface in H 1 we mean a C 2 orientable sur fa ce Σ such that any compact reg ion M ⊂ Σ satisﬁes A ( M ) 6 A ( M ′ ) for any o ther C 1 compact surface M ′ in H 1 with ∂ M = ∂ M ′ . Clearly any area-minimizing surface is s table. Remark 3 .1. Consider a C 1 vector ﬁeld U with compact supp ort on Σ. F o r any s ∈ R we denote ϕ s ( p ) = exp p ( sU p ), wher e exp p is the ex po nential map o f ( H 1 , g ) at p . It is easy to see that, for s small enough, { ϕ s } s deﬁnes a compactly supp orted v ariation of Σ. In case the supp or t of U is c ontained in Σ − Σ 0 then the induced v ariation is nonsingular. This was the p oint of view used in [33] to deﬁne v ariations of a C 2 surface. In par ticular, our notion of area- s tationary surface implies the o ne int ro duced in [33, Sect. 4]. It is clear that stability is preserved under left tra nslations and vertical ro tations since they a re horizontal isometries in H 1 . In the next res ult we prov e that any dilation δ λ as deﬁned in (2.1 3) satisﬁes the sa me prop erty . Lemma 3. 2. L et Σ b e a C 2 immerse d oriente d surfac e in H 1 . Then Σ is stable ( r esp. stable under non-singular variations ) if and only if the same hold s for δ λ (Σ) . Pr o of. Let Σ λ = δ λ (Σ). T ake a compactly supp or ted v ariation { ϕ s } s ∈ I of Σ λ . By using that the family of dilations is a one-par ameter gro up of diﬀeomorphisms we can see that { ψ s } s ∈ I with ψ s = δ − λ ◦ ϕ s ◦ δ λ provides a compa c tly suppo rted v ariation o f Σ. Moreov er, the v ar iation { ψ s } s ∈ I is nonsingular if and only if { ϕ s } s ∈ I is nonsingular . By (2 .14) we get A (Σ s ) = A ( ψ s (Σ)) = A ( δ − λ ((Σ λ ) s )) = e − 3 λ A ((Σ λ ) s ) . F r om here it is easy to deduce that if Σ is stable (resp. sta ble under non-singular v a riations) then the same ho lds for Σ λ . T o prov e the reverse statement it suﬃces to c hange the roles of Σ and Σ λ .  10 A. HUR T ADO, M. RITOR ´ E, AND C. ROSALES 3.1. Area-stationary surfaces. In this pa rt of the sectio n we ga ther some facts ab out area-s tationary surfaces in H 1 that will b e useful in the sequel. Let Σ b e a C 2 immersed surface in H 1 with a unit nor mal v ector N . W e deﬁne the me an curvatur e of Σ as in [32] a nd [33], by the equality (3.1) − 2 H ( p ) = (div Σ ν h )( p ) , p ∈ Σ − Σ 0 , where ν h is the horizontal Gauss map deﬁned in (2.9) and div Σ U stands for the divergence relative to Σ of a C 1 vector ﬁeld U . W e say that Σ is a minimal surfac e if the mea n curv ature v a nishes on Σ − Σ 0 . In the follo wing propo sition we r ecall so me fea tures abo ut a rea-sta tionary and minimal surfaces in H 1 inv olving the structure of the regular and the s ingular set, see [1 0, Sect. 3], [33, Sect. 4] and the re fer ences there in. Similar results also hold in other s ub- Riemannian spaces, see [2 6] and [28]. Prop ositi o n 3.3. L et Σ b e a C 2 immerse d oriente d minimal surfac e in H 1 with singular set Σ 0 . Then we have (i) Any char acteristic curve of Σ is a se gment of a horizontal str aight line. (ii) Σ 0 c onsists of isolate d p oints and C 1 curves with non-vanishing tangent ve ctor ( singular curves ) . (iii) If Γ is a singular curve and p ∈ Γ , then ther e is a neighb orho o d B of p in Σ such that B − Γ is the u nion of t wo disjoint domains B + and B − c ontaine d in Σ − Σ 0 . Mor e over, the ve ctor ﬁelds Z and ν h extend c ontinu ously t o p fr om B + and B − in such a way that Z + p = − Z − p and ( ν h ) + p = − ( ν h ) − p . (iv) If Σ is any C 2 immerse d oriente d surfac e, then Σ is ar e a-stationary if and only if Σ is minimal and the char acteristic curves me et ortho gonal ly the singular curves. Now w e prov e a regularity result for minimal s urfaces in H 1 . Given a C 2 surface Σ in H 1 with unit normal vector N , it is clea r that the vector ﬁeld D Z N is well deﬁned on Σ − Σ 0 and it is con tinuous. By using the ruling pr op erty of minimal surfaces in Prop ositio n 3.3 (i) we ca n obtain more reg ularity for N in the Z -direction. Lemma 3.4. L et Σ b e a C 2 immerse d oriente d surfac e in H 1 . I f Σ is m inimal t hen, in Σ − Σ 0 , the normal ve ctor N is C ∞ in the dir e ction of the char acteristic ﬁeld Z . Pr o of. T ake p ∈ Σ − Σ 0 . Let γ b e the characteristic cur ve thr o ugh p . Co nsider a C 1 curve α : ( − ε 0 , ε 0 ) → Σ − Σ 0 transverse to γ with α (0) = p . Deﬁne F ( ε, s ) := α ( ε ) + s Z α ( ε ) . By using (2.6) and Lemma 2.1 we get that V ( s ) := ( ∂ F /∂ ε )(0 , s ) is a C ∞ Jacobi ﬁeld a long γ . Since b oth ˙ γ ( s ) and V ( s ) are C ∞ and linear ly indep endent for s sma ll e no ugh, the unit normal N to Σ along γ is given by N = ± ˙ γ × V   ˙ γ × V   , where × denotes the cross pro duct in ( H 1 , g ). W e conclude that N is a C ∞ vector ﬁeld along γ .  3.2. Second v ariation of the area. In this part of the section we pro vide some formulas for the s e c ond de r iv ativ e of the area functional asso cia ted to some v ariations of a n area - stationary surface . W e ﬁr st give some preliminary computations. Lemma 3.5. L et Σ ⊂ H 1 b e a C 2 immerse d surfac e with unit normal ve ctor N and singular set Σ 0 . Consider a p oint p ∈ Σ − Σ 0 , the horizontal Gauss m ap ν h and the char acteristic COMPLETE ST ABLE SURF ACES IN THE HE ISENBERG GROUP H 1 11 ﬁeld Z deﬁne d in (2.9) . F or any v ∈ T p H 1 we have D v N h = ( D v N ) h −  N , T  J ( v ) −  N , J ( v )  T , (3.2) v ( | N h | ) =  D v N , ν h  −  N , T   J ( v ) , ν h  , (3.3) v (  N , T  ) =  D v N , T  +  N , J ( v )  , (3.4) D v ν h = | N h | − 1  D v N , Z  −  N , T   J ( v ) , Z  Z +  Z, v  T . (3.5) Pr o of. E q ualities (3.2) a nd (3.3) ar e eas ily obtained sinc e N h = N −  N , T  T . The pro of of (3.4) is immedia te. Let us sho w that (3.5) holds. As | ν h | = 1 and { Z p , ( ν h ) p , T p } is an orthonor mal ba s is of T p H 1 , we get D v ν h =  D v ν h , Z  Z +  D v ν h , T  T . Note that  D v ν h , T  = −  ν h , J ( v )  =  Z, v  by (2.2). On the other hand, by using (3.2) and the fact tha t Z is tangent and horizontal, we deduce  D v ν h , Z  = | N h | − 1  D v N h , Z  = | N h | − 1  D v N , Z  −  N , T  J ( v ) , Z  , and the pro o f follows.  Remark 3.6. In Σ − Σ 0 we can consider the orthono rmal bas is { Z , S } deﬁned in (2.9) and (2.10). By using the deﬁnition of mean curv ature in (3.1) we hav e − 2 H = div Σ ν h =  D Z ν h , Z  +  D S ν h , S  . By (3.5) we get D Z ν h = T − | N h | − 1  B ( Z ) , Z  Z , and that D S ν h is prop ortiona l to Z . It follows that, in Σ − Σ 0 2 H = | N h | − 1  B ( Z ) , Z  , (3.6) D Z ν h = T − (2 H ) Z, (3.7) where B is the Riemannian sha p e op erato r of Σ. On the other ha nd, the vector D Z Z is orthogo nal to Z and T since | Z | = 1 and  J ( Z ) , Z  = 0. It follows that D Z Z is prop ortiona l to ν h . F rom (3.7) we o bta in (3.8) D Z Z =  D Z Z, ν h  ν h = 2 H ν h . The second deriv ative of the a rea for no n-singular v ariatio ns of a minimal sur face in H 1 has app eared in se veral contexts, see [10, Pr op. 6.1], [4, Sect. 3.2], [12, Sect. 14 ], [30, P ro of of Thm. 3.5] and [2 5, Thm. E]. In the next theorem we compute the second deriv ative o f the area functional for so me non-singular varia tions by Riemannian ge o desics of a C 2 minimal surface (maybe with non-empty boundary) in H 1 . Theorem 3.7. Le t Σ ⊂ H 1 b e a C 2 immerse d minimal surfac e with b oundary ∂ Σ and sin- gular set Σ 0 . Consider the C 1 ve ctor ﬁeld U = v N + w T , wher e N is a u nit normal ve ctor to Σ and v , w ∈ C 1 0 (Σ − Σ 0 ) . If u =  U, N  , then the se c ond derivative of t he ar e a for t he variation induc e d by U is given by A ′′ (0) = Z Σ | N h | − 1  Z ( u ) 2 −  | B ( Z ) + S | 2 − 4 | N h | 2  u 2  d Σ (3.9) + Z Σ div Σ ( ξ Z ) d Σ + Z Σ div Σ ( µZ ) d Σ . Her e { Z , S } is the orthonormal b asis in (2.9 ) and (2.1 0) , B is the Riemannian shap e op er- ator of Σ , t he functions ξ and µ ar e deﬁne d by ξ =  N , T  (1 −  B ( Z ) , S  ) u 2 , (3.10) µ = | N h | 2  N , T  (1 −  B ( Z ) , S  ) w 2 − 2  B ( Z ) , S  v w  , (3.11) 12 A. HUR T ADO, M. RITOR ´ E, AND C. ROSALES and the diver genc e terms ar e u n dersto o d in the sen s e of (2.15) . In p articular, if ∂ Σ is empty, then (3.12) A ′′ (0) = Z Σ | N h | − 1  Z ( u ) 2 −  | B ( Z ) + S | 2 − 4 | N h | 2  u 2  d Σ . Pr o of. W e will follow closely the ar guments in [3 7, § 9]. Let ϕ s ( p ) = exp p ( sU p ), for s small, be the v ariation induced by U . Then any Σ s = ϕ s (Σ) is a C 1 immersed oriented surface. W e extend the vector U along the v ariation by s etting U ( ϕ s ( p )) = ( d/dt ) | t = s ϕ t ( p ). Let N be a co ntin uous vector ﬁeld along the v ariation whose res tr iction to an y Σ s is a unit normal vector. By using (2.1 2), the coa rea formula, and that the Riema nnia n a rea of Σ 0 v a nishes, we have (3.13) A ( s ) = A (Σ s ) = Z Σ s | N h | d Σ s = Z Σ − Σ 0 ( | N h | ◦ ϕ s ) | Jac ϕ s | d Σ , where Jac ϕ s is the J acobian determinant of the diﬀeomorphism ϕ s : Σ → Σ s . W e can suppo se that | N h | ( ϕ s ( p )) > 0 whenever p ∈ Σ − Σ 0 and | s | < s 0 . T ake a po int p ∈ Σ − Σ 0 and co nsider the orthonor mal basis { e 1 , e 2 } o f T p Σ given by e 1 = Z p and e 2 = S p . Let γ be the Riema nnian geo des ic deﬁned by γ ( s ) = ϕ s ( p ) = exp p ( sU p ). De- note by N ( s ) the unit no rmal to Σ s at γ ( s ). Let α i : ( − ε 0 , ε 0 ) → Σ − Σ 0 be a C 1 curve such tha t α i (0) = p a nd ˙ α i (0) = e i . W e deﬁne the C 1 map F i : ( − ε 0 , ε 0 ) × R → H 1 given by F i ( ε, s ) = ϕ s ( α i ( ε )) = exp α i ( ε ) ( sU α i ( ε ) ). By using Lemma 2.1 we deduce that E i ( s ) = ( ∂ F i /∂ ε )(0 , s ) = e i ( ϕ s ) is a C ∞ Jacobi vector ﬁeld along γ with [ ˙ γ , E i ] = 0 and E i (0) = e i . Therefore , w e hav e the following iden tities along γ D U D U E i = − R ( U, E i ) U, (3.14) D U E i = D E i U. (3.15) On the other hand, it is clear that { E 1 ( s ) , E 2 ( s ) } provide a basis of the tangent spa ce to Σ s at γ ( s ). In particula r | Ja c ϕ s | = ( | E 1 | 2 | E 2 | 2 −  E 1 , E 2  2 ) 1 / 2 ( s ), and so | Jac ϕ s | is C ∞ along γ . Moreov er, we have N ( s ) = ± | E 1 × E 2 | − 1 ( E 1 × E 2 )( s ), which is C ∞ on γ . Here × is the cr oss pro duct in ( H 1 , g ). W e conclude that | N h | ( s ) is C ∞ along γ a s well. Thus we can apply the classical r esult of diﬀerent iation under the int egr a l sign to deduce, fro m (3.13), that (3.16) A ′′ (0) = Z Σ − Σ 0 {| N h | ′′ (0) + 2 | N h | ′ (0) | Jac ϕ s | ′ (0) + | N h | | Jac ϕ s | ′′ (0) } d Σ , where we ha ve used that ϕ 0 ( p ) = p for any p ∈ Σ, and so | Ja c ϕ 0 | = 1 . Now we compute the diﬀerent terms in (3.16). The ca lculus o f | Ja c ϕ s | ′ (0) and | Jac ϕ s | ′′ (0) is found in [3 7, § 9] for C 2 v a riations o f a C 1 surface in Euclidean space. The arguments can be gener alized to an y Riemannian manifold for a C 1 v a riation obtained when we leav e from a C 2 surface b y geo desics. As U = v N + wT o n Σ, w e deduce, by using div Σ T = 0 a nd the third equality in (2.11), that (3.17) | Jac ϕ s | ′ (0) = div Σ U = ( − 2 H R ) v − | N h | S ( w ) = −  B ( S ) , S  v − | N h | S ( w ) . T o get the seco nd equality w e hav e taken into account (3.6) to obtain 2 H R = − div Σ N =  B ( Z ) , Z  +  B ( S ) , S  =  B ( S ) , S  . COMPLETE ST ABLE SURF ACES IN THE HE ISENBERG GROUP H 1 13 On the other ha nd, it is k nown that | Jac ϕ s | ′′ (0) =(div Σ U ) 2 + 2 X i =1 | ( D e i U ) ⊥ | 2 − 2 X i =1  R ( U, e i ) U, e i  − 2 X i,j =1  D e i U, e j   D e j U, e i  . Hence from (3.1 7), equality (3.18) D e U = e ( v ) N − v B ( e ) + e ( w ) T + w J ( e ) , and equation (2.3), we get | Jac ϕ s | ′′ (0) = |∇ Σ v | 2 +  N , T  2 |∇ Σ w | 2 − 2 | N h | Z ( v ) w − 2 | N h |  B ( Z ) , S  Z ( w ) v (3.19) + 2  N , T  Z ( v ) Z ( w ) + 2  N , T  S ( v ) S ( w ) − (Ric( N , N ) + | B | 2 −  B ( S ) , S  2 ) v 2 − 4  N , T  v w , where ∇ Σ is the gr adient relative to Σ, Ric is the Ricci tensor in ( H 1 , g ), and | B | 2 is the squared norm of the Riemannian sha pe op era tor of Σ. Let us compute | N h | ′ (0) and | N h | ′′ (0). F rom (3.3) and (2.2) it follows that | N h | ′ ( s ) = U ( | N h | ) =  D U N , ν h  −  N , T   J ( U ) , ν h  =  D U N , ν h  +  N , T   U, Z  . Note that U = u N − ( | N h | w ) S . Then  U, Z  = 0 and D U N = −∇ Σ u + ( | N h | w ) B ( S ) on Σ − Σ 0 . By the second equa lit y in (2.11) we obtain (3.20) | N h | ′ (0) =  D U N , ν h  = −  N , T  S ( u ) + | N h |  N , T   B ( S ) , S  w. W e also deduce the following | N h | ′′ (0) =  D U D U N , ν h  +  D U N , D U ν h  (3.21) + U (  N , T  )  U, Z  +  N , T  U (  U, Z  ) =  D U D U N , ν h  +  D U N , D U ν h  +  N , T   U, D U Z  , since  U, Z  = 0 and D U U = 0 o n Σ − Σ 0 . W e can compute D U ν h from (3.5). By using that D U N = −∇ Σ u + ( | N h | w ) B ( S ) and J ( U ) = ( | N h | v ) Z on Σ − Σ 0 , we get D U ν h = −| N h | − 1 ( Z ( u ) − | N h |  B ( Z ) , S  w + | N h |  N , T  v ) Z, and so  D U N , D U ν h  = | N h | − 1 Z ( u ) 2 +  N , T  Z ( u ) v − 2  B ( Z ) , S  Z ( u ) w (3.22) − | N h |  N , T   B ( Z ) , S  v w + | N h |  B ( Z ) , S  2 w 2 . Now we c ompute D U Z . The coor dinates o f this vector with respect to the orthono r mal basis { Z, ν h , T } are g iven by  D U Z, Z  = 0 ,  D U Z, ν h  = | N h | − 1 Z ( u ) −  B ( Z ) , S  w +  N , T  v ,  D U Z, T  = −| N h | v . The previous equa lities and the fact that U = v N + w T on Σ − Σ 0 imply that (3.23)  U, D U Z  = Z ( u ) v − | N h | (1 +  B ( Z ) , S  ) vw . It remains to co mpute D U D U N . Note that { E 1 , E 2 , N } provides an orthonorma l basis of T p H 1 . As a co nsequence D U D U N = 2 X i =1  D U D U N , E i  E i +  D U D U N , N  N . 14 A. HUR T ADO, M. RITOR ´ E, AND C. ROSALES As  N , E i  = 0 along γ we get  D U D U N , E i  = − 2  D U N , D U E i  −  N , D U D U E i  = − 2  D U N , D E i U  +  N , R ( U, E i ) U  . The second equa lity follows from (3.15) and (3.1 4). Now recall that e 1 = Z p and e 2 = S p . It follows that  D U D U N , ν h  = − 2  N , T   D U N , D S U  +  N , T   N , R ( U, S ) U  (3.24) + | N h |  D U D U N , N  . By taking into account (3.1 8) and that D U N = −∇ Σ u + ( | N h | w ) B ( S ), we obtain  D U N , D S U  = − | N h | 2  B ( S ) , S  S ( w ) w −  N , T  Z ( u ) w (3.25) + | N h | S ( u ) S ( w ) + B ( S )( u ) v + | N h |  N , T   B ( Z ) , S  w 2 − | N h | | B ( S ) | 2 v w . On the other ha nd, we use (2.3) so that, after a stra ightforw ard computation, we co nc lude (3.26)  R ( U, S ) U, N  = | N h | ( v +  N , T  w ) w . Moreov er, since | N | 2 = 1 on Σ − Σ 0 we have (3.27)  D U D U N , N  = −| D U N | 2 = −|∇ Σ u | 2 + 2 | N h | B ( S )( u ) w − | N h | 2 | B ( S ) | 2 w 2 . By substituting (3.25), (3.26) and (3.27) into (3.24) we get  D U D U N , ν h  . F r om (3.2 4), (3.22) and (3.23), after simplifying, e q uality (3.21) b ecomes | N h | ′′ (0) = | N h | − 1 Z ( u ) 2 − | N h | |∇ Σ u | 2 + 2  N , T  ( Z ( u ) v − B ( S )( u ) v ) (3.28) + 2  N , T  2 + | N h | 2  B ( Z ) , S  −  B ( Z ) , S  Z ( u ) w + 2 | N h | 2  B ( S ) , S   S ( u ) w +  N , T  S ( w ) w  − 2 | N h |  N , T  S ( u ) S ( w ) + 2 | N h |  N , T   | B ( S ) | 2 −  B ( Z ) , S  v w +  | N h |  N , T  2 (1 −  B ( Z ) , S  ) 2 − | N h | 3  B ( S ) , S  2  w 2 . Now, since u = v +  N , T  w , we have ∇ Σ u = ∇ Σ v + w ∇ Σ (  N , T  ) +  N , T  ∇ Σ w . By (3.4) and (2.1 1) it is eas y to see that Z (  N , T  ) = | N h | (  B ( Z ) , S  − 1) , (3.29) S (  N , T  ) = | N h |  B ( S ) , S  . This allows us to compute the term | N h | |∇ Σ u | 2 in (3.28). At this moment, we use (3.28), (3.20), (3.17) a nd (3.19) so that, after simplifying, we g et that | N h | ′′ (0) + 2 | N h | ′ (0) | Jac ϕ s | ′ (0) + | N h | | Jac ϕ s | ′′ (0) is equal to | N h | − 1 Z ( u ) 2 + 2  N , T  (1 −  B ( Z ) , S  ) ( Z ( v ) v + Z ( w ) w ) (3.30) + 2(  N , T  2 −  B ( Z ) , S  ) ( Z ( w ) v + Z ( v ) w ) + q 1 v 2 + 2 | N h |  N , T  (  B ( Z ) , S  − 3) v w − | N h | (1 −  B ( Z ) , S  ) 2 w 2 , where q 1 is the function g iven by q 1 = | N h |  B ( S ) , S  2 − Ric( N , N ) − | B | 2  . COMPLETE ST ABLE SURF ACES IN THE HE ISENBERG GROUP H 1 15 In order to obtain (3.9) from (3.16) and (3.3 0), we apply Lemma 3.9 b elow. W e deduce the following | N h | ′′ (0) + 2 | N h | ′ (0) | Jac ϕ s | ′ (0) + | N h | | Jac ϕ s | ′′ (0) = | N h | − 1 Z ( u ) 2 + div Σ ( ρZ ) +  q 1 +  B ( Z ) , S  − 1  N , T  q 2 + Z (  N , T  )  +  N , T  Z (  B ( Z ) , S  )  v 2 +  B ( Z ) , S  − 1  N , T  q 2 + Z (  N , T  )  +  N , T  Z (  B ( Z ) , S  ) −| N h | (1 −  B ( Z ) , S  ) 2  w 2 + 2 n | N h |  N , T  (  B ( Z ) , S  − 3) − Z (  N , T  2 ) −  N , T  2 q 2 +  B ( Z ) , S  q 2 + Z (  B ( Z ) , S  ) o v w , where ρ is the function  N , T  (1 −  B ( Z ) , S  ) ( v 2 + w 2 ) + 2 (  N , T  2 −  B ( Z ) , S  ) vw . A straig ht forward computation using (3.29), (3.34), the identities Ric( N , N ) = 2 − 4 | N h | 2 (it follows from (2.4)) , | B | 2 =  B ( Z ) , Z  2 +  B ( S ) , S  2 + 2  B ( Z ) , S  2 =  B ( S ) , S  2 + 2  B ( Z ) , S  2 , B ( Z ) =  B ( Z ) , Z  Z +  B ( Z ) , S  S =  B ( Z ) , S  S, and that u = v +  N , T  w , gives us | N h | ′′ (0) + 2 | N h | ′ (0) | Jac ϕ s | ′ (0) + | N h | | Jac ϕ s | ′′ (0) = | N h | − 1 Z ( u ) 2 − | N h | − 1 ( | B ( Z ) + S | 2 − 4 | N h | 2 ) u 2 + div Σ ( ξ Z ) + div Σ ( µZ ) , where ξ and µ ar e the functions given in (3.10) a nd (3 .1 1). Finally , suppo se that ∂ Σ is empty . Then ξ and µ ar e co nt inuous functions w ith co mpact suppo rt in Σ − Σ 0 and they are also C 1 in the Z -dir ection b y Lemma 3.10. Hence the in te- grals o f div Σ ( ξ Z ) a nd div Σ ( µZ ) v anish by virtue o f the divergence theo rem in Lemma 2.4. This prov es (3.12).  Remark 3. 8 . The divergence terms in (3.9) need not v anish if ∂ Σ is nonempty . In the pro of of Pro p osition 5.2 we will show that these terms play an impo rtant role. Lemma 3.9. L et Σ b e a C 2 immerse d oriente d surfac e in H 1 and φ ∈ C 1 (Σ) . Then, in the r e gular set Σ − Σ 0 , we have div Σ ( φ Z ) = Z ( φ ) + q 2 φ, wher e q 2 is the function given by q 2 = | N h | − 1  N , T  (1 +  B ( Z ) , S  ) . Pr o of. Clea rly w e have (3.31) div Σ ( φ Z ) = (div Σ Z ) φ + Z ( φ ) . Note that div Σ Z =  D Z Z, Z  +  D S Z, S  =  D S Z, S  , since | Z | 2 = 1. W e compute the comp onents of D S Z in the orthonormal bas is { Z , ν h , T } . Observe that D S Z is orthogo nal to Z . By using (3.5 ) and that J ( S ) =  N , T  Z , we get  D S Z, ν h  = −  Z, D S ν h  = | N h | − 1 (  B ( Z ) , S  +  N , T  2 ) ,  D S Z, T  = −  Z, J ( S )  = −  N , T  . 16 A. HUR T ADO, M. RITOR ´ E, AND C. ROSALES F r om here we deduce (3.32) D S Z = | N h | − 1 (  B ( Z ) , S  + 1 − | N h | 2 ) ν h −  N , T  T . As a cons equence, w e o btain (3.33) div Σ Z =  D S Z, S  = | N h | − 1  N , T  (1 +  B ( Z ) , S  ) . The pro of ﬁnishes by subs tituting (3.33) into (3.31).  Lemma 3.10. L et Σ b e a C 2 immerse d oriente d minimal surfac e in H 1 . Th en, in the r e gular set Σ − Σ 0 , we have (i) The functions  N , T  and | N h | ar e C ∞ in the Z -dir e ction. (ii) The ve ctor ﬁelds ν h and S ar e C ∞ in the Z -dir e ction. (iii) The function  B ( Z ) , S  is C ∞ in the Z -dir e ction, and (3.34) Z (  B ( Z ) , S  ) = 4 | N h |  N , T  − 2 | N h | − 1  N , T   B ( Z ) , S  (1 +  B ( Z ) , S  ) . Pr o of. Reca ll that N is C ∞ in the Z -direction by Lemma 3 .4. This implies (i). Assertions (ii) and (iii) follow from (i) by the deﬁnition o f ν h and S in (2.9) a nd (2.10). T o compute Z (  B ( Z ) , S  ) note that Z (  B ( Z ) , S  ) = Z ( −  D Z N , S  ) = −  D Z D Z N , S  −  D Z N , D Z S  . It is cle a r that D Z N is tangent to Σ. On the other hand, D Z S is pr op ortiona l to N . This comes from the fact that  D Z S, Z  = −  S, D Z Z  = 0 by (3.8), whe r eas  D Z S, S  = 0. Therefore we ha ve  D Z N , D Z S  = 0 , (3.35) Z (  B ( Z ) , S  ) = −  D Z D Z N , S  =  N , D Z D Z S  . (3.36) It remains to compute D Z D Z S . F r om (3.32) we see that D S Z is C ∞ in the Z -dir ection. As a consequence [ Z , S ] = D Z S − D S Z is a lso C ∞ in the Z direction, a nd D Z [ Z, S ] = D Z D Z S − D Z D S Z . Thus equa tion (3.36) b ecomes Z (  B ( Z ) , S  ) =  N , D Z [ Z, S ]  +  N , D Z D S Z  (3.37) =  N , D Z [ Z, S ]  +  N , D S D Z Z  −  N , R ( Z, S ) Z  +  N , D [ Z,S ] Z  =  N , D Z [ Z, S ]  −  N , R ( Z, S ) Z  +  N , D [ Z,S ] Z  , where R is the Riema nnia n curv ature tensor and we ha ve used (3 .8) to get D S D Z Z = 0. Now, o bserve that  [ Z, S ] , N  =  D Z S, N  −  D S Z, N  = −  S, D Z N  +  Z, D S N  = 0 , which implies that [ Z , S ] is tang ent to Σ. Therefore, we deduce  N , D Z [ Z, S ]  =  B ( Z ) , [ Z, S ]  =  B ( Z ) , D Z S  −  B ( Z ) , D S Z  = −  B ( Z ) , D S Z  ,  N , D [ Z,S ] Z  = −  D [ Z,S ] N , Z  =  B ( Z ) , [ Z, S ]  = −  B ( Z ) , D S Z  , where we ha ve used (3.35). If we put this information int o (3.37), we obtain (3.38) Z (  B ( Z ) , S  ) = − 2  B ( Z ) , D S Z  −  N , R ( Z, S ) Z  . T o compute the ﬁrst term a bove we tak e into account (3.3 2). After simplifying, we get (3.39)  B ( Z ) , D S Z  = | N h | − 1  N , T   B ( Z ) , S  (1 +  B ( Z ) , S  ) . F o r the seco nd term, we apply (2.3) so tha t, after a straig htforward calculus, we conclude (3.40)  N , R ( Z, S ) Z  = − 4 | N h |  N , T  . The pro of ﬁnishes by subs tituting (3.39) and (3.4 0) into (3.38).  COMPLETE ST ABLE SURF ACES IN THE HE ISENBERG GROUP H 1 17 In the next result we co mpute the seco nd deriv ative of the area for some vertic al varia- tions of an ar ea-stationa ry surface Σ whose singular cuves (Σ 0 ) c are C 3 (in [33, Pro p. 4.20] we prov ed that they are alwa ys C 2 ). W e supp ose that the v a r iation is co nstant along the characteristic curves of a tubular neighbor ho o d around (Σ 0 ) c . By a tubular neigh b orho o d of radius ε > 0 we mean the union of all the characteristic se gments of length 2 ε cent ered at (Σ 0 ) c . Prop ositi o n 3.11. L et Σ b e a C 2 immerse d oriente d ar e a-stationary surfac e in H 1 such that the singular curves (Σ 0 ) c of Σ ar e of class C 3 . L et ϕ r ( p ) := exp p ( rw ( p ) T p ) , for r smal l, b e the vertic al variation of Σ induc e d by a function w ∈ C 2 0 (Σ) . Supp ose that ther e is a tubular neighb orho o d E 0 of s upp( w ) ∩ (Σ 0 ) c wher e Z ( w ) = 0 . Then, ther e is a tubular neighb orho o d E of supp ( w ) ∩ (Σ 0 ) c such that d 2 dr 2     r =0 A ( ϕ r ( E )) = Z (Σ 0 ) c S ( w ) 2 dl , wher e S is any c ontinuous ext ension of the ve ctor ﬁeld S deﬁne d in (2.10) to (Σ 0 ) c and dl denotes the Riemannian lengt h element. Pr o of. W e ca n restric t our selves to a neighborho o d o f a single singular curve Γ. W e co n- sider a para meterization Γ( ε ) = ( x ( ε ) , y ( ε ) , t ( ε )) b y ar c-length. B y Prop o sition 3.3 the area-s tationary surface Σ can b e parameteriz e d in a neighborho o d of s upp( w ) ∩ Γ by ( ε, s ) 7→ Γ( ε ) + sJ ( ˙ Γ( ε )) , so that the curv es with ε constant are the c haracter istic curves o f Σ. In E uclidean co or di- nates we ha ve x ( ε, s ) = x ( ε ) − s ˙ y ( ε ) , y ( ε, s ) = y ( ε ) + s ˙ x ( ε ) , t ( ε, s ) = t ( ε ) − s ( x ˙ x + y ˙ y )( ε ) . As Z ( w ) = 0 w e get that w is a function o f ε alo ne. The deformation ϕ r ( p ) = exp p ( rw ( p ) T p ) consists on changing the t -co or dinate of the ab ove para meterization by t ( ε, s ) + rw ( ε ) . A simple co mputation shows that the tangent space to the surface Σ r := ϕ r (Σ) is g enerated by the vectors (3.41) − ˙ y X + ˙ xY , ( ˙ x − s ¨ y ) X + ( ˙ y + s ¨ x ) Y +  s ( − 2 + sh ) + r ˙ w  T , where x , y and t ar e the co or dinates of Γ, dots represent deriv a tives with resp ect to ε , and h = h ( ε ) = ( ˙ x ¨ y − ˙ y ¨ x )( ε ) is the Euclidean geo de s ic curv atur e of the xy -pro jection o f Γ. Hence the singula r p oints of Σ r corres p o nds to the zero set of F ( ε, s, r ) := s ( − 2 + sh ( ε )) + r ˙ w ( ε ). Observe that F is a C 1 function since the singular c urves are assumed to be o f class C 3 and w ∈ C 2 . As ( ∂ F /∂ s )( ε, 0 , 0) = − 2, we can apply the Implicit F unction Theorem and a compactness argument to show that there a re p ositive v alues ε 0 , s 0 , r 0 , and a C 1 function s : ( − ε 0 , ε 0 ) × ( − r 0 , r 0 ) → ( − s 0 , s 0 ) with s ( ε , 0) = 0 sa tisfying F ( ε, s ( ε, r ) , r ) = 0 . Here ε 0 > 0 is taken so that supp( w ) ∩ Γ ⊂ [ − ε 0 , ε 0 ]. W e deﬁne E := F (( − ε 0 , ε 0 ) × ( − s 0 , s 0 )). On the other hand, a computation using (3.41) shows that | ( N h ) r | d Σ r = | s ( − 2 + sh ( ε )) + r ˙ w ( ε ) | dε ds. Hence we ha ve A ( ϕ r ( E )) = Z ε 0 − ε 0  Z s 0 − s 0 | s ( − 2 + sh ( ε )) + r ˙ w ( ε ) | ds  dε. 18 A. HUR T ADO, M. RITOR ´ E, AND C. ROSALES Denote by f ε ( r ) the integral be tw een brack ets. As ( ∂ F /∂ s )( ε, 0 , 0) < 0 we deduce f ε ( r ) = Z s ( ε,r ) − s 0  s ( − 2 + sh ( ε )) + r ˙ w ( ε )  ds + Z s 0 s ( ε,r )  s (2 − sh ( ε )) − r ˙ w ( ε )  ds. T a king deriv atives with r esp ect to r we obtain f ′ ε ( r ) = Z s ( ε,r ) − s 0 ˙ w ( ε ) ds − Z s 0 s ( ε,r ) ˙ w ( ε ) ds = 2 ˙ w ( ε ) s ( ε, r ) . T a king deriv atives ag a in w e have f ′′ ε ( r ) = 2 ˙ w ( ε ) ∂ s ∂ r ( ε, r ) . Since ( ∂ s/∂ r )( ε, 0) = ˙ w ( ε ) / 2 we conclude f ′′ ε (0) = ˙ w ( ε ) 2 , and so d 2 dr 2     r =0 A ( ϕ r ( E )) = Z ε 0 − ε 0 ˙ w ( ε ) 2 dε. By Pr op osition 3.3 we know that the vector ﬁeld S deﬁned in (2 .10) ex tends contin uously to Γ as a unit tangent v ector to Γ. Then ˙ w ( ε ) 2 = S ( w ) 2 and the c laim follows.  3.3. A stabil it y criterion for s table surfaces in H 1 . Here we obtain a useful criterion to chec k if a g iven area-statio nary sur face is unsta ble. Firs t we need a deﬁnition. Let Σ b e a C 2 oriented minimal surface immersed in H 1 . F or tw o functions u, v ∈ C 0 (Σ − Σ 0 ) which are also C 1 in the Z -direction, we denote (3.42) I ( u, v ) := Z Σ | N h | − 1  Z ( u ) Z ( v ) −  | B ( Z ) + S | 2 − 4 | N h | 2  uv  d Σ , where { Z , S } is the orthonormal basis in (2.9) and (2.1 0), and B is the Riemannian sha pe op erator o f Σ. The e x pression (3.4 2) deﬁnes a symmetric bilinear form, which we call the index form asso ciated to Σ by a nalogy with the Riemannian situation, see [3]. Prop ositi o n 3. 12. L et Σ b e a C 2 immerse d oriente d ar e a-stationary su rfac e in H 1 with singular set Σ 0 . If Σ is stable under non-singular variations then t he index form deﬁne d in (3.42) satisﬁes I ( u , u ) > 0 for any fu n ction u ∈ C 0 (Σ − Σ 0 ) whi ch is also C 1 in the dir e ction of the char acteristic ﬁeld Z . Pr o of. Let N b e the unit normal vector to Σ. T ake u ∈ C 1 0 (Σ − Σ 0 ) and consider the vector ﬁeld U = uN . Note that Σ is a minimal sur face since it is ar e a-stationa r y . H ence Theorem 3.7 implies that the seco nd deriv a tive of the area for the v aria tion induced b y U is A ′′ (0) = I ( u, u ). As Σ is stable under non-singular v ariations we deduce that (3.43) I ( u, u ) > 0 , for any u ∈ C 1 0 (Σ − Σ 0 ) . Now ﬁx a function u ∈ C 0 (Σ − Σ 0 ) which is also C 1 in the Z -dire ction. By using Lemma 2.3 and that Σ 0 has v anishing Riemannian area , we can ﬁnd a compact set K ⊆ Σ − Σ 0 and a sequence of functions { u ε } ε> 0 in C 1 0 (Σ − Σ 0 ) such that { u ε } → u in L 2 (Σ), { Z ( u ε ) } → Z ( u ) in L 2 (Σ), while the supp orts o f u ε and u are co ntained in K for any ε > 0. F rom her e it is not diﬃcult to chec k that {| N h | − 1 / 2 Z ( u ε ) } → | N h | − 1 / 2 Z ( u ), { ( | N h | − 1 f 1 ) 1 / 2 u ε } → ( | N h | − 1 f 1 ) 1 / 2 u and { ( | N h | − 1 f 2 ) 1 / 2 u ε } → ( | N h | − 1 f 2 ) 1 / 2 u in L 2 (Σ), where f 1 = | B ( Z ) + S | 2 and f 2 = 4 | N h | 2 . It follows that lim ε → 0 I ( u ε , u ε ) = I ( u, u ), so that inequality (3.43) pr ov es the claim.  Remark 3 . 13. As in [1 2, Thm. 15.2 ] and [30, Thm. 3.5, Cor. 3.7] the previo us result can be seen as a Poincar´ e t yp e inequality for stable surfac e s in H 1 . COMPLETE ST ABLE SURF ACES IN THE HE ISENBERG GROUP H 1 19 3.4. In tegration b y parts. The stabilit y op erator in H 1 . In Riemannian geometry the index form of a minimal surfac e ca n b e expressed in terms of a second order elliptic op erator deﬁned on the surface, see [3]. In this par t o f the section we prove a similar prop erty fo r the index form (3 .4 2) of a minimal sur face in H 1 which inv olves a h yp o elliptic second order diﬀerential o p erator on the s urface. Prop ositi o n 3. 14 (Integration by parts I) . L et Σ ⊂ H 1 b e a C 2 immerse d surfac e with unit normal ve ctor N and singular set Σ 0 . Consider two funct ions u ∈ C 0 (Σ − Σ 0 ) and v ∈ C (Σ − Σ 0 ) which ar e C 1 and C 2 in the Z -dir e ction, r esp e ctively. Then we have I ( u, v ) = − Z Σ u L ( v ) d Σ , wher e I is the index form deﬁne d in (3.4 2) , and L is the se c ond or der diﬀer ential op er ator L ( v ) := | N h | − 1  Z ( Z ( v )) + 2 | N h | − 1  N , T   B ( Z ) , S  Z ( v ) (3.44) + ( | B ( Z ) + S | 2 − 4 | N h | 2 ) v  . Pr o of. Along this pro of we shall denote q = | B ( Z ) + S | 2 − 4 | N h | 2 . First no te that in Σ − Σ 0 the hypotheses abo ut u and v ensure that | N h | − 1 Z ( v ) a nd | N h | − 1 Z ( v ) u ar e C 1 in the Z -direction. Supp ose proved that (3.45) L ( v ) = div Σ ( | N h | − 1 Z ( v ) Z ) + | N h | − 1 q v . In such a case, we would apply the divergence theor e m in Lemma 2.4 in o rder to get 0 = Z Σ div Σ ( | N h | − 1 Z ( v ) u Z ) d Σ = Z Σ u div Σ ( | N h | − 1 Z ( v ) Z ) d Σ + Z Σ | N h | − 1 Z ( u ) Z ( v ) d Σ . = Z Σ u L ( v ) d Σ + I ( u, v ) , and this would ﬁnish the pro o f. T o obtain (3.4 5) observe that (3.46) div Σ ( | N h | − 1 Z ( v ) Z ) = | N h | − 1 Z ( v ) div Σ Z + Z ( | N h | − 1 Z ( v )) . The computation o f div Σ Z is given in (3.33). On the other hand, we hav e Z ( | N h | − 1 Z ( v )) = | N h | − 1 Z ( Z ( v )) + Z ( | N h | − 1 ) Z ( v ) (3.47) = | N h | − 1 Z ( Z ( v )) − | N h | − 2 Z ( | N h | ) Z ( v ) = | N h | − 1 Z ( Z ( v )) + | N h | − 2  N , T  (  B ( Z ) , S  − 1) Z ( v ) , where we ha ve used (3.3) to compute Z ( | N h | ). T o deduce (3.45) it suﬃces to simplify in (3.46) after substituting the informa tion of (3 .33) and (3.47).  Remark 3 .15. If Σ is a minimal surface then the functional L in (3.44) provides a Sturm- Liouville diﬀerential oper ator along a ny of the characteristic segmen ts of Σ. As a direc t consequence of Prop os itio n 3.14 and Pro po sition 3.12 we deduce Corollary 3 .16. L et Σ b e a C 2 immerse d oriente d ar e a-stationary surfac e in H 1 . If Σ is stable under non- s ingular variations then we have − Z Σ u L ( u ) d Σ > 0 , for any function u ∈ C 0 (Σ − Σ 0 ) which is also C 2 in the Z -dir e ction. Finally , with the same technique as in Prop osition 3.14 w e c an prove the following lemma. 20 A. HUR T ADO, M. RITOR ´ E, AND C. ROSALES Lemma 3.17 (Integration by parts I I) . L et Σ ⊂ H 1 b e a C 2 immerse d surfac e with unit nor- mal ve ctor N and singular set Σ 0 . Consider two fu n ctions u ∈ C 0 (Σ − Σ 0 ) and v ∈ C (Σ − Σ 0 ) which ar e C 1 and C 2 in the Z -dir e ction, r esp e ctively. Then we have Z Σ | N h |  Z ( u ) Z ( v ) + u Z ( Z ( v )) + 2 | N h | − 1  N , T  u Z ( v )  d Σ = 0 . Pr o of. Obs erve that in Σ − Σ 0 div Σ ( | N h | u Z ( v ) Z ) = u Z ( v )  Z ( | N h | ) + | N h | div Σ Z  + | N h | Z ( u ) Z ( v ) + | N h | u Z ( Z ( v )) , and that the function in the left-hand side has v anishing integral b y Lemma 2.4. On the other hand, (3.3) gives us (3.48) Z ( | N h | ) =  N , T  −  N , T   B ( Z ) , S  , which together with (3.33) implies Z ( | N h | ) + | N h | div Σ Z = 2  N , T  . The result follo ws.  Remark 3.18. Some other integration by parts fo r mulas in H 1 can be found in [12, Sect. 10]. 4. Complete st able surf aces with empty singular set In this section we provide the classiﬁca tion of C 2 complete stable surfaces in H 1 with empt y singular set. Reca ll that if Σ 0 = ∅ then Σ is area -stationar y if and only if Σ is min- imal by Pr op osition 3.3 (iv). W e say that a n immer sed surface Σ in H 1 is c omplete if it is complete in the Riemannian manifold ( H 1 , g ). F or a C 2 complete area- stationary sur face Σ with Σ 0 = ∅ the characteristic cur ves ar e straight lines by Prop osition 3.3 (i). In particular Σ cannot b e compact. Some classiﬁcatio n results for area -stationar y surfa c e s with empty singular set can b e found in [32, Thm. 5.4], [9] and [33, P rop. 6.16]. Note also that for such surfaces to b e stable is e quiv a lent to b e stable under non-singular v ariations. In Euclidean three -space the descriptio n o f complete stable area-stationa ry surfaces can be o btained b y means of a logarithmic cut-oﬀ o f the function u = 1 asso cia ted to the v a ria- tion by lev el surfaces of the dis tance function, see [18]. In H 1 the vector ﬁeld induced by the family o f equidistants for the Carnot-Carath´ eo dory dista nc e d cc to a C 2 surface with empty singular s et coincides, up to a sign, with the hor izontal Gauss map ν h , see [1, Thm. 1.1 and 1.2]. This leads us to use the sta bility condition in Pr op osition 3.12 with a test function of the form f = u | N h | , wher e f is contin uous with compact supp ort on the sur fa ce and C 1 in the dir e ction of the characteris tic ﬁeld Z . W e ﬁrst compute the index fo rm for these t yp e of functions. Lemma 4. 1 . L et Σ ⊂ H 1 b e a C 2 immerse d minimal s urfac e in H 1 with unit normal ve ctor N and singular set Σ 0 . Then, for any function f ∈ C 0 (Σ − Σ 0 ) which is also C 1 in the Z - dir e ction, we have (4.1) I ( f | N h | , f | N h | ) = Z Σ | N h |  Z ( f ) 2 − L ( | N h | ) f 2  d Σ , wher e I is the index form in (3.42) , and L is the diﬀer ential op er ator in (3.44) . Pr o of. Along this pro of we shall denote w = f | N h | a nd q = | B ( Z ) + S | 2 − 4 | N h | 2 . Note that w is C 1 in the Z -dir ection and Z ( w ) = f Z ( | N h | ) + | N h | Z ( f ). If we introduce w in the index form we obtain I ( w , w ) = Z Σ  | N h | Z ( f ) 2 + | N h | − 1 Z ( | N h | ) 2 f 2 + Z ( f 2 ) Z ( | N h | ) − | N h | q f 2  d Σ . (4.2) COMPLETE ST ABLE SURF ACES IN THE HE ISENBERG GROUP H 1 21 On the o ther hand, we know from Lemma 3.10 (i) that | N h | is C ∞ in the Z -direction. Therefore, we can apply Pro po sition 3.14 with u = f 2 | N h | and v = | N h | , so that we g et − Z Σ | N h | L ( | N h | ) f 2 d Σ = Z Σ  | N h | − 1 Z ( | N h | ) 2 f 2 + Z ( f 2 ) Z ( | N h | ) − | N h | q f 2  d Σ = I ( w, w ) − Z Σ | N h | Z ( f ) 2 d Σ , where in the second equality w e hav e used (4.2). This prov es the claim.  Remark 4.2. So me other versions of (4.1) for v aria tions o f a C 2 surface Σ with asso cia ted vector ﬁeld f ν h , f ∈ C 2 0 (Σ − Σ 0 ), can b e found in [13, Lem. 3.9] and [1 5, Thm. 3.4]. See also [4, Sect. 3.2] and [3 0, Thm. 3.5] for the case o f an intrinsic gr a ph ass o ciated to a function with less reg ularity than C 2 . In the next lemma we particular iz e (4 .1) for f = u v − 1 . This type of test functions will be used to prov e Theorem 4.7. Lemma 4. 3 . L et Σ ⊂ H 1 b e a C 2 immerse d minimal s urfac e in H 1 with unit normal ve ctor N and singular set Σ 0 . Consider two functions u ∈ C 0 (Σ − Σ 0 ) and v ∈ C (Σ − Σ 0 ) which ar e C 1 and C 2 in t he Z -dir e ction, r esp e ctively. If v never vanishes, then I ( uv − 1 | N h | ,uv − 1 | N h | ) = Z Σ | N h | v − 2 Z ( u ) 2 d Σ (4.3) + Z Σ | N h | u 2  Z ( v − 1 ) 2 − 1 2 Z ( Z ( v − 2 )) − | N h | − 1  N , T  Z ( v − 2 )  d Σ − Z Σ | N h | L ( | N h | ) ( uv − 1 ) 2 d Σ , wher e I is the index form in (3.42) , and L is the diﬀer ential op er ator in (3.44) . Pr o of. F rom (4.1) we only hav e to compute Z Σ | N h | Z ( uv − 1 ) 2 d Σ . Since Z ( u v − 1 ) 2 = v − 2 Z ( u ) 2 + u 2 Z ( v − 1 ) 2 + 1 2 Z ( u 2 ) Z ( v − 2 ) , and Lemma 3 .1 7 implies Z Σ 1 2 | N h | Z ( u 2 ) Z ( v − 2 ) d Σ = − Z Σ | N h | u 2  1 2 Z ( Z ( v − 2 )) + | N h | − 1  N , T  Z ( v − 2 )  d Σ , we see that (4.3) holds.  The previo us lemmas sugg est that, for a function u = f | N h | , the stability condition in Prop os itio n 3.12 is mo re restrictive if L ( | N h | ) > 0. Thus it is interesting to compute L ( | N h | ) and to study its sign. Lemma 4.4. L et Σ b e a C 2 immerse d minimal surfac e in H 1 with unit normal ve ctor N . Consider t he b asis { Z, S } deﬁne d in (2.9) and (2.10) . L et B b e the Riemannian shap e op er ator of Σ . Then, in the r e gular set Σ − Σ 0 , we have (4.4) L ( | N h | ) = 4  | N h | − 2  B ( Z ) , S  − 1  , wher e L is the se c ond or der op er ator in (3.44) . 22 A. HUR T ADO, M. RITOR ´ E, AND C. ROSALES Pr o of. F rom Lemma 3.10 (i) we know that | N h | is C ∞ in the Z -direction. W e m ust compute Z ( | N h | ) and Z ( Z ( | N h | )). By (3.4 8) w e have Z ( | N h | ) =  N , T  −  N , T   B ( Z ) , S  , and so Z ( Z ( | N h | )) = Z (  N , T  ) − Z (  N , T  )  B ( Z ) , S  −  N , T  Z (  B ( Z ) , S  ) . Now we use (3.29) and (3.34), so tha t we get Z ( Z ( | N h | )) = − 5 | N h | + 4 | N h | 3 + 2 | N h | − 1  B ( Z ) , S  (4.5) + 2 | N h | − 1  B ( Z ) , S  2 − 3 | N h |  B ( Z ) , S  2 . By substituting (3.48) and (4.5) into (3.44), we o bta in L ( | N h | ) = − 5 −  B ( Z ) , S  2 + 4 | N h | − 2  B ( Z ) , S  − 2  B ( Z ) , S )  + | B ( Z ) + S | 2 = 4( | N h | − 2  B ( Z ) , S  − 1) , where in the se cond equality w e hav e applied that Σ is minimal, a nd so B ( Z ) =  B ( Z ) , S  S by (3.6). This prov es (4.4).  In the next result we sho w some pr op erties of the Ja cobi ﬁeld asso ciated to the family of characteristic segments o f a minimal s urface in H 1 . This will a llows us to study the sign of L ( | N h | ) and to construct suitable test functions to intro duce in (4.3) when Σ is a c omplete minimal surface with empty singular set. Lemma 4.5. L et Σ ⊂ H 1 b e a C 2 immerse d minimal surfac e with unit n ormal N and sin- gular set Σ 0 . Consider an inte gr al curve Γ : I → Σ − Σ 0 of the ve ctor ﬁeld S in (2.10) . We deﬁne the map F : I × I ′ → Σ − Σ 0 by F ( ε, s ) := Γ( ε ) + s Z Γ( ε ) . L et V ε ( s ) := ( ∂ F /∂ ε )( ε, s ) . Then V ε is a C ∞ Jac obi ve ctor ﬁeld along γ ε ( s ) := F ( ε, s ) . Mor e over, we have (i) The vertic al c omp onent of V ε is given by  V ε , T  ( s ) = a ε s 2 + b ε s + c ε , with b 2 ε − 4 a ε c ε = −| N h | 2 (Γ( ε )) L ( | N h | )(Γ( ε )) . (ii) V ε is always ortho gonal to γ ε and never vanishes along γ ε . (iii) The function v ε ( s ) := |  V ε , T  ( s ) | 1 / 2 satisﬁes Z ( v − 1 ε ) 2 − 1 2 Z ( Z ( v − 2 ε )) − | N h | − 1  N , T  Z ( v − 2 ε ) = 1 4 | V ε | | N h | L ( | N h | ) , along any se gment γ ε ( s ) wher e  V ε , T  ( s ) never vanishes. Pr o of. T o simplify the no tation we will a void the subscript ε a long the pro of. W e will use primes for b oth the der iv a tive of functions depe nding on s and the cov ar iant deriv ative along γ ( s ). By Pr op osition 3.3 (i) the curve γ is a characteristic curve of Σ. It follows from (2.6) and Lemma 2 .1 that V is a C ∞ Jacobi ﬁeld alo ng γ with [ ˙ γ , V ] = 0. Note that (4.6)  V , T  ′ =  V ′ , T  +  V , T ′  = − 2  V , ν h  , since T ′ = J ( Z ) = − ν h , and (4.7)  V ′ , T  =  D Z V , T  =  D V Z, T  = −  Z, J ( V )  =  J ( Z ) , V  = −  V , ν h  . If we deriv e aga in in (4.6) then we obtain (4.8)  V , T  ′′ = − 2  V ′ , ν h  − 2  V , ν ′ h  = − 2  V ′ , ν h  +  V , T  , since ν ′ h = D Z ν h = T by (3.7) and the fact that Σ is minimal. Hence (4.9) ( − 1 / 2)  V , T  ′′′ =  V ′ , ν h  +  V , T  ′ =  V ′′ , ν h  + 2  V ′ , T  +  V , T ′  = 0 , COMPLETE ST ABLE SURF ACES IN THE HE ISENBERG GROUP H 1 23 where we have used the J acobi equatio n (2.8), equalit y (4.7), and that T ′ = − ν h . T o simplify (4.8) we compute  V ′ , ν h  . By (3 .5) and the fact that V is tangent to Σ, we deduce  V ′ , ν h  =  D V Z, ν h  = −  Z, D V ν h  = −| N h | − 1  D V N , Z  −  N , T   J ( V ) , Z  = | N h | − 1  B ( Z ) , V  +  N , T   V , ν h  , and so, after substituting into (4.8), we get (4.10)  V , T  ′′ = − 2 | N h | − 1  B ( Z ) , V  +  N , T   V , ν h  + | N h |  V , T  . F r om (4.9) we conclude that  V , T  ( s ) is a p o lynomial of degr ee at most tw o. W rite (4.11)  V , T  ( s ) = as 2 + bs + c. Denote p = Γ( ε ). As V (0) = S p , it is ea sy to chec k by (4.6) and (4.1 0), that c =  V , T  (0) = − | N h | ( p ) , b =  V , T  ′ (0) = − 2  V , ν h  ( p ) = − 2  N , T  ( p ) , a = (1 / 2)  V , T  ′′ (0) = −| N h | − 1  B ( Z ) , S  +  N , T  2 − | N h | 2  ( p ) . In particula r , it follows from (4.4) that b 2 − 4 ac = − 4  B ( Z ) , S  − | N h | 2  ( p ) = −| N h | 2 ( p ) L ( | N h | )( p ) , which proves as sertion (i) in the statement. T o prove asse r tion (ii), observe that  V , ˙ γ  ′ =  V ′ , ˙ γ  +  V , ˙ γ ′  =  D V Z, Z  +  V , D Z Z  = 0 , by (3.8). This implies that  V , ˙ γ  = 0 along γ since V (0) = S p . Hence there is a C 1 function f : I ′ → R s uch that V = f S alo ng γ . Clearly | f | = | V | , and so  V , T  = ± | V | | N h | . By (4.11) the vector V v anishes a t mo st tw o times a lo ng γ . Suppose that s 0 ∈ I ′ is the ﬁrst p ositive v alue wher e V ( s 0 ) = 0 . Note that the sign o f f / | V | is c o nstant along a small int erv al ( s 0 − δ, s 0 ). By (4.6) and (4.10) we g et  V , T  ( s 0 ) =  V , T  ′ ( s 0 ) =  V , T  ′′ ( s 0 ) = 0. By using L’Hˆ opital’s rule twice, w e deduce ±| N h | ( γ ( s 0 )) = lim s ↑ s 0  V , T  | V | ( s ) = lim s ↑ s 0 | V |  V , T  ′  V , V ′  ( s ) = lim s ↑ s 0 | V | ′  V , T  ′ + | V |  V , T  ′′ | V ′ | 2 +  V , V ′′  ( s ) . The numerator tends to zero since | V | ′ =  V / | V | , V ′  6 | V ′ | 6 M on ( s 0 − δ, s 0 ). The denom- inator g o es to | V ′ ( s 0 ) | 2 , which is positive; other wise, the Jacobi ﬁeld V would b e identically zero along γ . It follows that | N h | ( γ ( s 0 )) = 0, a contradiction since γ ( s 0 ) ∈ Σ − Σ 0 . T o pr ov e (iii) let us supp os e tha t  V , T  never v anis he s along γ . The n it is cle ar that v = |  V , T  | 1 / 2 = ( −  V , T  ) 1 / 2 since V (0) = S p . In pa rticular, we get f = | V | > 0 a long γ . Now we der ive v = ( −  V , T  ) 1 / 2 = ( f | N h | ) 1 / 2 with r esp ect to s . By taking into account 24 A. HUR T ADO, M. RITOR ´ E, AND C. ROSALES (4.6) and (4.1 0), w e o btain Z ( v − 1 ) = 1 2 ( −  V , T  − 3 / 2 )  V , T  ′ = − v − 3  V , ν h  = −  N , T  f 1 / 2 | N h | 3 / 2 , Z ( v − 2 ) =  V , T  − 2  V , T  ′ = − 2 v − 4  V , ν h  = − 2  N , T  f | N h | 2 , Z ( Z ( v − 2 )) =  V , T  − 2  V , T  ′  ′ = − 2  V , T  − 3 (  V , T  ′ ) 2 +  V , T  − 2  V , T  ′′ = 8  N , T  2 f | N h | 3 − 2 f 2 | N h | 3  B ( Z ) , V  +  N , T   V , ν h  + | N h |  V , T  = 8  N , T  2 f | N h | 3 − 2 f | N h | 3  B ( Z ) , S  +  N , T  2 − | N h | 2  . After simplifying, we co nclude by (4.4) that Z ( v − 1 ) 2 − 1 2 Z ( Z ( v − 2 )) − | N h | − 1  N , T  Z ( v − 2 ) = 1 f | N h |  | N h | − 2  B ( Z ) , S  − 1  = 1 4 f | N h | L ( | N h | ) , which proves the claim.  Prop ositi o n 4.6. L et Σ b e a C 2 c omplete, oriente d, ar e a-stationary surfac e immerse d in H 1 with empty singu lar set. Then the op er ator L deﬁne d in (3.44) satisﬁes L ( | N h | ) > 0 on Σ . Mor e over, L ( | N h | )( p ) = 0 for a p oint p ∈ Σ if and only if  N , T  = 0 and  B ( Z ) , S  = 1 along t he char acteristic line of Σ p assing thr ough p . As a c onse quenc e, L ( | N h | ) ≡ 0 on Σ if and only if any c onne cte d c omp onent of Σ is a Euclide an vertic al plane. Pr o of. T ake a p oint p ∈ Σ. Let Γ : I → Σ b e the integral curve through p of the vector ﬁeld S in (2.10). W e de ﬁne the map F : I × R → H 1 by F ( ε, s ) := Γ( ε ) + s Z Γ( ε ) . By the completeness o f Σ and Pro po sition 3.3 (i), a n y γ ε ( s ) := F ( ε, s ) is a c haracter istic cur ve of Σ. In particular , F ( I × R ) ⊆ Σ. Let V ( s ) := ( ∂ F /∂ ε )(0 , s ). B y using Lemma 4.5 we deduce that, along the complete line γ ( s ) := γ 0 ( s ), the v ectors V ( s ) and ˙ γ ( s ) gener ate the tangent plane to Σ at γ ( s ). Since Σ has empty singular s et, it follows that the function  V , T  ( s ) = a s 2 + bs + c never v anishe s along γ ( s ). In case a = 0 we must hav e b = 0 (otherwise w e would ﬁnd a ro ot of as 2 + bs + c ). In case a 6 = 0 we must ha ve b 2 − 4 ac < 0. Anyw ay , w e get b 2 − 4 ac 6 0 and so L ( | N h | )( p ) > 0 by Lemma 4 .5 (i). Observe that L ( | N h | )( p ) = 0 if and o nly if a = b = 0. This is equiv alent to that  V , T  is constant along γ . It follows from (4.6) and (4.10) that  N , T  = 0 a nd  B ( Z ) , S  = 1 a long γ . Conv ersely , if  N , T  = 0 a nd  B ( Z ) , S  = 1 a long γ then (4.4) implies that L ( | N h | ) = 0 along γ . Finally , if L ( | N h | ) ≡ 0 o n Σ then  N , T  ≡ 0 on Σ. By [33, Pro p. 6.16 ] w e conclude that any connected comp onent of Σ must b e a Euclidean vertical plane. Conv ersely , it is not diﬃcult to se e that L ( | N h | ) ≡ 0 holds for a ny Euclidean vertical plane.  Now w e are ready to pr ov e the main res ult of this sec tion. Theorem 4.7. L et Σ b e a C 2 c omplete, oriente d, c onne cte d, ar e a-stationary surfac e im- merse d in H 1 with empty singular set. If Σ is not a Eu clide an vertic al plane then Σ is unstable. COMPLETE ST ABLE SURF ACES IN THE HE ISENBERG GROUP H 1 25 Pr o of. Let N be the unit norma l vector to Σ. W e can ﬁnd p ∈ Σ such that  N , T  ( p ) 6 = 0. Otherwise Σ would b e a Euclidea n vertical pla ne by [33, Pr op. 6.16]. By us ing Pr o p osi- tion 3.3 (i) and the completeness o f Σ, we can para meterize Σ, around the c hara cteristic line containing p , by the ma p F : I × R → Σ given by F ( ε, s ) = Γ( ε ) + s Z Γ( ε ) , wher e Γ( ε ) is a piece o f the in tegra l curve through p of the vector ﬁeld S in (2.1 0). Let γ ε ( s ) := F ( ε, s ). By Lemma 4.5 w e know that V ε ( s ) := ( ∂ F /∂ ε )( ε, s ) is a no n- v a nishing J a cobi ﬁeld or thogonal to γ ε ( s ). Mor eov er, the function  V ε ( s ) , T  is strictly negative since Σ has empt y singula r set and V ε (0) = S Γ( ε ) . W e consider the function v ( ε, s ) := |  V ε ( s ) , T  | 1 / 2 = ( | N h | | V ε ( s ) | ) 1 / 2 , which is contin uous and C ∞ along any γ ε ( s ). Now we use the co area formula to compute the index form (4.3) in terms of the co ordinates ( ε, s ). The Riema nnian ar ea element can be expressed as d Σ = | V ε | dε ds. Hence by using the deﬁnition of v together with Lemma 4.5 (iii), equation (4.3) r eads (4.12) I ( uv − 1 | N h | , uv − 1 | N h | ) = Z I × R  ∂ u ∂ s  2 dε ds − 3 4 Z I × R L ( | N h | ) u 2 dε ds, for any u ∈ C 0 ( I × R ) which is also C 1 with resp ect to s . T a ke a non- negative C ∞ function φ : I → R with φ (0) > 0 and compact suppo rt con- tained inside a bo unded interv a l I ′ ⊆ I . Denote ℓ := length( I ′ ). Let M b e a p ositive constant so that | φ ′ ( ε ) | 6 M , ε ∈ I . F or any k ∈ N we deﬁne the function u k ( ε, s ) := φ ( ε ) φ ( s/k ) . It is clear that u k ∈ C 0 ( I ′ × k I ′ ), and that u k is C ∞ with res p ect to s . By F ubini’s theorem Z I × R  ∂ u k ∂ s  2 dε ds = 1 k 2  Z I ′ φ ( ε ) 2 dε   Z kI ′ φ ′ ( s/k ) 2 ds  6 ℓM 2 k Z I ′ φ ( ε ) 2 dε, which go es to 0 when k → ∞ . Note also that { u k } k ∈ N po int wise con verges when k → ∞ to u ( ε, s ) = φ (0) φ ( ε ). By Prop osition 4.6 we hav e L ( | N h | ) > 0 on Σ. Thus we ca n a pply F a tou’s lemma to o btain lim inf k →∞ Z I × R L ( | N h | ) u 2 k dε ds > Z I × R L ( | N h | ) u 2 dε ds. W e conclude from (4.1 2) that lim sup k →∞ I ( u k v − 1 | N h | , u k v − 1 | N h | ) = − 3 4 lim inf k →∞ Z I × R L ( | N h | ) u 2 k dε ds 6 − 3 4 Z I × R L ( | N h | ) u 2 dε ds, which is strictly negative b y Pr op osition 4.6 since  N , T  6 = 0 inside an o pe n neighbor ho o d around p . Hence Σ is unstable.  Corollary 4.8. L et Σ b e a C 2 c omplete, oriente d, c onne cte d, ar e a-stationary surfac e im- merse d in H 1 with empty singular set. Then Σ is st able if and only if Σ is a Euclide an vertic al plane. Pr o of. The neces s ary conditio n follows from Theor em 4.7. Conv ersely , supp ose that Σ is a vertical plane. W e can prov e that Σ is an area-minimizing surface in H 1 by using a calibra- tion argument similar to the one in [33, Thm. 5.3 ], see a ls o [4, Ex. 2.2]. In particular, Σ is stable.  26 A. HUR T ADO, M. RITOR ´ E, AND C. ROSALES Remark 4.9. Previous results related to Coro llary 4.8 were obtained in [4] a nd [1 5]. P re- cisely , in [4, Thm. 5.1] it is proved that the Euclide a n v ertical planes ar e the only complete stable intrinsic g raphs in H 1 asso ciated to a C 2 function. In [15, Thm. 1.8] vertical planes are characterized as the unique complete stable C 2 Euclidean g raphs with empt y sing ular set. As we p ointed out in the introduction of the pa p e r, Corolla ry 4.8 do e s not follow from the aforementioned results. F or ex a mple, they do not apply for the family of sub-Riemannian catenoids t 2 = λ 2 ( x 2 + y 2 − λ 2 ), λ 6 = 0. 5. Complete st able surf aces with non-empty singular set In this section we give the classiﬁcation of C 2 complete stable surfaces in H 1 with non- empt y singular set. By Pr op osition 3 .3 the singula r set of a C 2 area-s tationary surface consists of isolated p o ints and c ur ves of class C 1 . Moreover, the characteristic cur ves in the regula r set meet the singular cur ves orthogonally . By using these facts we were able to obtain the following result in [3 3, Thm. 6.1 5]. Prop ositi o n 5 .1. L et Σ b e a C 2 c omplete, oriente d, c onne cte d, ar e a-st ationary surfac e immerse d in H 1 with singular set Σ 0 . (i) If Σ 0 c ontains an isolate d p oint then Σ c oincides with a Euclid e an non-vertic al plane. (ii) If Σ 0 c ontains a singular curve then Σ is either c ongruent to t he hyp erb olic p ar ab oloid t = xy or to one of the helic oidal su rfac es H R deﬁne d b elow. In [3 3, Ex. 6.14] we describ ed the helicoid H R as the union of all the horizo ntal straight lines orthog onal to the sub-Riemannian ge o desic in H 1 obtained by the ho rizontal lift of the circle in the xy -plane of ra dius 1 /R centered at the origin. W e can para meterize H R by means of the C ∞ diﬀeomorphism F : R 2 → H R deﬁned by (5.1) F ( ε, s ) = ( s sin( Rε ) , s cos( Rε ) , ε/R ) . The singular set of H R consists of the helices s = ± 1 / R . Note that the family {H R } R> 0 is inv a riant under the dilatio ns δ λ deﬁned in (2.13). In fact, it can b e chec ked from (5.1) that δ λ ( H R ) = H R ′ with R ′ = e − λ R . T he surfaces H R coincide with the class ical left- handed minimal helicoids in R 3 . In particula r, they are embedded surfaces con taining the vertical axis. W e remark that the classica l right-handed minimal helicoids in R 3 are com- plete area -stationar y surfaces in H 1 with empty singular set, and so they are unstable by Theorem 4.7. Prop os itio n 5.1 indicates us that the study of s table surfaces in H 1 with non-empt y singu- lar set can be reduced to three cases: Euclidean non-vertical planes, the hyperb o loid t = xy and the helico ids H R . In [3 3, Thm. 5.3] w e show ed that any complete C 2 area-s tationary graph over the xy -pla ne is an area -minimizing surface. This gives us the stability of any plane t = ax + by and an y surface congruent to t = xy . So it remains to analyze the stability of the helicoidal s urfaces H R . W e ﬁrs t compute some geometric terms o f a helicoid H R with resp ect to the sy s tem o f co ordinates ( ε, s ) in (5.1). Note that ∂ F ∂ ε = Rs cos( Rε ) X − Rs sin( R ε ) Y + f ( s ) T , ∂ F ∂ s = sin( Rε ) X + cos( Rε ) Y , where f : R → R is deﬁned by f ( s ) = 1 R − R s 2 . COMPLETE ST ABLE SURF ACES IN THE HE ISENBERG GROUP H 1 27 As a cons equence, the Riemannian a rea element is given by (5.2) d Σ = p f ( s ) 2 + R 2 s 2 dε ds. On the other hand, the cross pr o duct of ∂ F /∂ s and ∂ F /∂ ε in ( H 1 , g ) pr ovides the following unit normal vector to H R (5.3) N = f ( s ) cos( Rε ) X − f ( s ) sin( R ε ) Y − R s T p f ( s ) 2 + R 2 s 2 , and so (5.4) | N h | = | f ( s ) | p f ( s ) 2 + R 2 s 2 ,  N , T  = − Rs p f ( s ) 2 + R 2 s 2 . It follows that the straight lines γ ε ( s ) = F ( ε, s ), s ∈ R , satisfy (5.5) ˙ γ ε ( s ) = sign(1 /R − | s | ) Z, | s | 6 = 1 /R. By taking into account (3.2 9) and (5.4) we g et, for | s | 6 = 1 /R , that (5.6)  B ( Z ) , S  = 1 + | N h | − 1 Z (  N , T  ) = 2 f ( s ) 2 − R f ( s ) f ( s ) 2 + R 2 s 2 − 1 , which in particular implies (5.7) | B ( Z ) + S | 2 − 4 | N h | 2 = ( R 2 − 4) f ( s ) 2 ( f ( s ) 2 + R 2 s 2 ) 2 . Now we are ready to deduce fr om Theorem 3.7 and Prop osition 3.11 a stabilit y criter ion for helicoidal sur faces that plays the same ro le as P rop osition 3.12. Prop ositi o n 5. 2. L et Σ b e the helic oid H R . If Σ is stable t hen, for any function u ∈ C 2 0 (Σ) such that Z ( u ) = 0 inside a smal l tubular n eighb orho o d of Σ 0 , we have Q ( u ) > 0 , wher e Q ( u ) := Z Σ | N h | − 1  Z ( u ) 2 −  | B ( Z ) + S | 2 − 4 | N h | 2  u 2  d Σ (5.8) − 4 Z Σ 0 u 2 d Σ 0 + Z Σ 0 S ( u ) 2 d Σ 0 . Her e { Z , S } is the orthonormal b asis in (2.9 ) and (2.1 0) , B is the Riemannian shap e op er- ator of Σ , and d Σ 0 is the Riemannian lengt h me asur e on Σ 0 . Pr o of. W e suppo se that the unit no rmal N to Σ is the o ne in (5.3). F or simplicity we de- note q = | B ( Z ) + S | 2 − 4 | N h | 2 . By (5.7), (5.4) and (5 .2) it fo llows that | N h | − 1 q u 2 ∈ L 1 (Σ) provided u ∈ C 0 (Σ). In par ticular, Q ( u ) is w ell deﬁned for an y u ∈ C 0 (Σ) whic h is piecewise C 1 in the Z -direction, satisﬁes | N h | − 1 Z ( u ) 2 ∈ L 1 (Σ), and who se restrictio n to Σ 0 is C 1 . Let us show, in a ﬁrs t step, the following statemen t Q ( v ) > 0 , for any v ∈ C 2 0 (Σ) such that Z ( v /  N , T  ) = 0 (5.9) in a small tubular neighborho o d E of Σ 0 . Note tha t a function v as a b ov e satisﬁes Z ( v ) 2 = ( Z (  N , T  ) 2 /  N , T  2 ) v 2 in E . It follows from (5.4) and (5.2) that | N h | − 1 Z ( v ) 2 ∈ L 1 (Σ), and s o Q ( v ) < ∞ . Let σ 0 be the ra dius of E and K the suppor t of v . F or a ny σ ∈ (0 , σ 0 / 2) let E σ be the tubular neighbo rho o d of Σ 0 of r adius σ . W e consider functions h σ , g σ ∈ C ∞ 0 (Σ) such that g σ = 1 on K ∩ E σ , supp( g σ ) ⊂ E 2 σ and h σ + g σ = 1 on K . W e deﬁne the C 2 vector ﬁeld U σ := ( h σ v ) N + g σ v  N , T  T , 28 A. HUR T ADO, M. RITOR ´ E, AND C. ROSALES whose suppo rt is contained in K . Note that  U σ , N  = v on K . Let ϕ σ r ( p ) := exp p ( r ( U σ ) p ) be the v ariation asso cia ted to U σ and A σ ( r ) := A ( ϕ σ r (Σ)) the corresp onding area functional. The v ariation ϕ σ r is vertical when res tricted to E σ . Hence we can supp ose, by applying Prop os itio n 3.11 to w = v /  N , T  , that the second deriv ativ e o f A 1 σ ( r ) := A ( ϕ σ r ( E σ )) is given b y A ′′ 1 σ (0) = Z Σ 0 S ( v ) 2 d Σ 0 . In the previous equality w e have used that  N , T  = ± 1 on Σ 0 . On the other hand, the second deriv ative of A 2 σ ( r ) := A ( ϕ σ r (Σ − E σ )) can b e computed fro m Theorem 3.7. W e obtain the following expressio n A ′′ 2 σ (0) = Z Σ − E σ | N h | − 1  Z ( v ) 2 − q v 2  d Σ + Z Σ − E σ div Σ ( ξ Z ) d Σ + Z Σ − E σ div Σ ( µZ ) d Σ , where ξ =  N , T  (1 −  B ( Z ) , S  ) v 2 , µ = | N h | 2 (  N , T  (1 −  B ( Z ) , S  ) g 2 σ v 2  N , T  2 − 2  B ( Z ) , S  h σ g σ v 2  N , T  ) . If Σ is stable then A ′′ σ (0) > 0. As A σ ( r ) = A 1 σ ( r ) + A 2 σ ( r ) we deduce, by using the classical Riemannian divergence theorem that, for any σ ∈ (0 , σ 0 / 2), we hav e the inequality (5.10) Z Σ − E σ | N h | − 1  Z ( v ) 2 − q v 2  d Σ − Z ∂ E σ ( ξ + µ )  Z, η  dl + Z Σ 0 S ( v ) 2 d Σ 0 > 0 , where η is the unit normal to ∂ E σ po intin g into Σ − E σ and dl denotes the Riemannia n length element. Let us compute the b oundar y ter m ab ov e. Fix k ∈ { 1 , 2 } . Let Λ b e one o f the tw o comp onents of ∂ E σ at distance σ of the singular curv e where  N , T  = ( − 1) k +1 . By ta king int o account (5.5) it follows that η = ( − 1 ) k +1 Z a long Λ. Mo reov er, the functions ξ and µ are constant along Λ. Since g σ = 1 and h σ = 0 on Λ we hav e Z Λ ξ  Z, η  d Λ = ( − 1) k +1  N , T  (1 −  B ( Z ) , S  ) Z Λ v 2 d Λ , (5.11) Z Λ µ  Z, η  d Λ = ( − 1) k +1 | N h | 2  N , T  − 1 (1 −  B ( Z ) , S  ) Z Λ v 2 d Λ . (5.12) Now w e let σ → 0 in (5.10). F rom the dominated convergence theorem we get lim σ → 0 Z Σ − E σ | N h | − 1  Z ( v ) 2 − q v 2  d Σ = Z Σ | N h | − 1  Z ( v ) 2 − q v 2  d Σ , lim σ → 0 Z ∂ E σ v 2 dl = 2 Z Σ 0 v 2 d Σ 0 . On the o ther hand, equation (5.6) yie lds  B ( Z ) , S  → − 1 when we approach Σ 0 . Mor eov er, we kno w that | N h | → 0 and  N , T  → ± 1 when σ → 0. This fac ts , together with (5.11) and (5.12) imply tha t lim σ → 0 Z ∂ E σ ( ξ + µ )  Z, η  dl = 4 Z Σ 0 v 2 d Σ 0 . Hence we obtain Q ( v ) > 0 from (5.10). This pr ov es (5.9). Now w e take u ∈ C 2 0 (Σ) with Z ( u ) = 0 inside a small tubular neighborho o d E of Σ 0 . F or any σ ∈ (0 , 1) let D σ be the open neig hborho o d of Σ 0 such that |  N , T  | = 1 − σ on ∂ D σ . COMPLETE ST ABLE SURF ACES IN THE HE ISENBERG GROUP H 1 29 W e can ﬁnd σ 0 > 0 such that D σ ⊂ E for σ ∈ (0 , σ 0 ). F o r suc h v alues of σ we deﬁne the function φ σ : Σ → [0 , 1] given by φ σ = ( |  N , T  | , in D σ , 1 − σ , in Σ − D σ . Clearly φ σ is contin uous and piecewise C 1 in the Z -direction. Moreov er, the sequence { φ σ } σ ∈ (0 , σ 0 ) po int wise conv erges to 1 when σ → 0. By using (5.4) and (5.2) we can see that | N h | − 1 Z (  N , T  ) 2 extends to a contin uous function on Σ, a nd so lim σ → 0 Z Σ | N h | − 1 Z ( φ σ ) 2 d Σ = 0 . By a standar d a pproximation ar gument we can slightly mo dify φ σ around ∂ D σ in or der to c o nstruct a seq ue nc e o f C 2 functions { ψ σ } σ ∈ (0 , σ 0 ) satisfying the same pr o p erties. De- ﬁne v σ := ψ σ u . This provides a sequence of functions in C 2 0 (Σ) such that v σ = u in Σ 0 and Z ( v σ /  N , T  ) = 0 inside a small tubular neighborho o d of Σ 0 . As a consequence of (5.9) we have Q ( v σ ) > 0 for a ny σ ∈ (0 , σ 0 ). Finally , it is stra ightforw ard to c heck by us- ing the dominated co nv ergence theorem and the Cauch y-Sch wartz inequalit y in L 2 (Σ) that {Q ( v σ ) } → Q ( u ) when σ → 0. The prop osition is pr ov ed.  Remark 5.3. By using Prop o sition 3.11, inequalit y Q ( u ) > 0 can be g e ne r alized for any C 2 stable s olution Σ of the Plateau problem whose singular curv es a re of cla ss C 3 whenever supp( u ) is co ntained in the interior o f Σ and Z ( u ) = 0 inside a tubular neighborho o d of the singular curves. Now w e are ready to pr ov e the main res ult of this sec tion. Theorem 5.4. The helic oidal surfac es H R ar e al l u nstable. Pr o of. T o prov e the claim it suﬃces to show that H 2 is unstable. In fact, for any R > 0 we hav e H R = δ λ ( H 2 ), wher e δ λ is the dila tion deﬁned in (2.13) with λ = log(2 / R ). By vir tue of Lemma 3.2 we deduce that H R is stable if and only if H 2 is stable. Let Σ := H 2 . Cons ider the diﬀeomo rphism F : R 2 → Σ in (5.1). W e deno te γ ε ( s ) = F ( ε, s ), s ∈ R . The sing ula r set Σ 0 consists o f the singular cur ves F ( ε, − 1 / 2) and F ( ε, 1 / 2), ε ∈ R . W e s upp os e that the no rmal N to Σ is the o ne in (5.3). By equation (5.7) we get | B ( Z ) + S | 2 − 4 | N h | 2 = 0 , on Σ − Σ 0 . In particula r , the quadratic form Q in (5.8) is given b y (5.13) Q ( u ) = Z Σ | N h | − 1 Z ( u ) 2 d Σ − 4 Z Σ 0 u 2 d Σ 0 + Z Σ 0 S ( u ) 2 d Σ 0 , for any u ∈ C 0 (Σ) which is piecew is e C 1 in the Z -direction, satisﬁes | N h | − 1 Z ( u ) 2 ∈ L 1 (Σ), and whose restr iction to Σ 0 is C 1 . W e a pply in (5.13) the coarea formula. By using (5.4), (5.5) and (5.2), we deduce that Q ( u ) = Z R 2 f ( s ) 2 + 4 s 2 | f ( s ) |  ∂ u ∂ s  2 dε ds − 4 Z R u ( ε, − 1 / 2) 2 dε − 4 Z R u ( ε, 1 / 2) 2 dε (5.14) + Z R  d dε u ( ε, − 1 / 2)  2 dε + Z R  d dε u ( ε, 1 / 2)  2 dε. Let φ : R → R b e any C ∞ function with compact support [ − ε 0 , ε 0 ]. F or any k > 1 / 2 a nd δ > 0, let φ kδ : R → [0 , 1] b e the symmetric function with res pe ct to the origin given, for 30 A. HUR T ADO, M. RITOR ´ E, AND C. ROSALES s > 0 , by φ kδ ( s ) =    1 , 0 6 s 6 k , δ − 1 ( − s + δ + k ) , k 6 s 6 k + δ, 0 , s > k + δ. Now we deﬁne the function u kδ on Σ whose expressio n in co ordinates ( ε, s ) is u kδ ( ε, s ) = φ ( ε ) φ kδ ( s ) . Clearly u kδ is a function in C 0 (Σ) which is also C ∞ with resp ect to ε a nd piecewise C ∞ in the Z -direction. Note also that (5.15) u kδ ( ε, − 1 / 2) = u kδ ( ε, 1 / 2) = φ ( ε ) , ε ∈ R . Moreov er ( ∂ u kδ /∂ s )( ε, s ) = φ ( ε ) φ ′ kδ ( s ), which v anishes if | s | < k or | s | > k + δ , and equa ls ± φ ( ε ) /δ if k < | s | < k + δ . This implies that Z ( u kδ ) = 0 inside a tubular neig hborho o d of Σ 0 . By using F ubini’s theo r em and that | f ( s ) | − 1 ( f ( s ) 2 + 4 s 2 ) is symmetric with resp ect to the orig in, w e hav e (5.16) Z R 2 f ( s ) 2 + 4 s 2 | f ( s ) |  ∂ u kδ ∂ s  2 dε ds =  Z ε 0 − ε 0 φ ( ε ) 2 dε  2 δ 2 Z k + δ k f ( s ) 2 + 4 s 2 | f ( s ) | ds ! . The second integral in the r ight-hand side can b e ea sily computed. W e o btain 2 Z k + δ k f ( s ) 2 + 4 s 2 | f ( s ) | ds = Z k + δ k 16 s 4 + 8 s 2 + 1 4 s 2 − 1 ds = 4 s 3 3 + 3 s + log  2 s − 1 2 s + 1   k + δ k . By an elementary analysis we can ﬁnd a v alue k > 1 / 2 and δ = 2 k + 1 s uch that the in tegral ab ov e times 1 /δ 2 is strictly less than 8 . By substituting this information into (5.1 6), and using (5.1 4) together with (5.15), we conclude for v := u kδ Q ( v ) < M Z ε 0 − ε 0 φ ( ε ) 2 dε + 2 Z ε 0 − ε 0 φ ′ ( ε ) 2 dε, for so me c onstant M < 0 whic h do es not dep end on the function φ . If ε 0 is large enough, then we ca n cho ose φ with compact supp or t [ − ε 0 , ε 0 ] s uch that the right-hand side of the previous equation is strictly nega tive. This can b e done since inf (  Z R φ ′ ( ε ) 2 dε   Z R φ ( ε ) 2 dε  − 1 ; φ ∈ C ∞ 0 ( R ) ) = 0 . Denote ¯ φ := φ kδ for the par ticular v alues of k and δ found ab ov e. W e mollify ¯ φ in or der to obtain a sequence of functions v σ ( ε, s ) = φ ( ε ) ¯ φ σ ( s ) in C ∞ 0 (Σ) with v σ = v on Σ 0 and lim σ → 0 Z Σ | N h | − 1 Z ( v σ ) 2 d Σ = Z Σ | N h | − 1 Z ( v ) 2 d Σ . Hence we ha ve lim σ → 0 Q ( v σ ) = Q ( v ) < 0 . By Prop o sition 5.2 we co nclude that Σ is unstable.  Remark 5.5. Though the helicoids H R are unstable, it is p ossible to obtain b y mea ns of a calibratio n argument similar to the one used for the hyperb oloid t = xy in [3 3, Thm. 5.3 ] that the surface o btained by removing the vertical axis from H R is are a -minimizing. On the other ha nd, the seco nd deriv ative of the area in Theor em 3.7 indicates us that any non- singular v ariation induced by a vector ﬁeld U = vN + w T such that v and w are C 1 functions whose supp or t is contained in the r egular set of H 2 satisﬁes A ′′ (0) > 0. This means that H 2 is also stable under the variations use d in Theo rem 3.7. The pro of of Theorem 5.4 shows that, to get that H 2 is unstable, we need to consider a function whose supp ort intersects a large piece o f H 2 containing the vertical a x is and the singula r set. COMPLETE ST ABLE SURF ACES IN THE HE ISENBERG GROUP H 1 31 6. Main resul t As a cons equence of our pre v ious stability results we ca n prove the following. Theorem 6.1. L et Σ b e a C 2 c omplete, oriente d, c onne cte d, ar e a-stationary surfac e im- merse d in H 1 . Then Σ is stable if and only if Σ is a Euclide an plane or Σ is c ongruent to the hyp erb olic p ar ab oloid t = xy . In p articular, Σ is ar e a-minimizing. Pr o of. If Σ is stable and the sing ular s et Σ 0 is empt y then Σ m ust be a vertical pla ne by Theorem 4.7. If Σ is stable a nd Σ 0 6 = ∅ then Prop o sition 5.1 and Theo rem 5.4 imply that Σ coincides with a non- vertical E uclidean pla ne, or it is congr uent to the hyperb olic parab o loid t = xy . 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The classification of complete stable area-stationary surfaces in the Heisenberg group $mathbb{H}^1$

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