Examples of area-minimizing surfaces in the subriemannian Heisenberg group H^1 with low regularity

EXAMPLES OF AREA-MINIMIZING SURF ACES IN THE SUBRIEMANNIAN HEISENBERG GROUP H 1 WITH LOW REGULARITY MANUEL RITOR ´ E A B S T R AC T . W e give new examples of entire ar ea-minimizing t -graphs in the subrieman- nian Hei senberg group H 1 . Most of the examples are locally lipschitz in Euclidean sen se . Some regular examples hav e pr escribed singular set consisting of either a horizontal line or a finite number of h o rizontal halflines extending from a given point. Amongst t hem, a lar ge family of area-minimizing cones is obtained. 1. I N T R O D U C T I O N V ariational problems related to the subriemannian area in the Heisenberg gr oup H 1 have received great attention r ecently . A major question in th is theory is the regularity of minimizers. A related one is the constr uction of examples with low regularity pr op- erties. The study of minimal surfa c es in subriemannian geometry was initiated in the paper by Garofalo and Nhieu [22]. Later Pauls [27] constr ucted minimal surfaces in H 1 as limits of minimal surfaces in Nil manifolds, the riemannian Heisenberg groups. Cheng, Hwang and Y ang [9] hav e studied the weak solutions of the minimal surface equation for t -graphs and have proven existence a nd uniqueness results . Regularity of minim al surfaces, assuming that they are least C 1 , has been treated in the papers by Pauls [2 8] and Cheng, Hwang and Y ang [10]. W e would li ke also to mention the recently distributed notes by Bigolin and Serra Cassano [5], where they obtain regularity properties of a n H - regular surface from regularity properties of its horizontal unit normal. Interesting exa m- ples of minimal surfaces which are not area-minimizing are obtained in [ 11]. See also [13]. Smoothness of lipschitz minimal intrinsic graphs in Heisenberg groups H n , f or n > 1 , has been recently obtained by Capogna, Citti and M anfredini [6]. Characteriza tion in H 1 of solutions of the Bernstein problem for C 2 surfaces has been obtained by Cheng, Hwang, Malchiodi and Y ang [ 8], and Ritor ´ e and Rosales [29] for t - graphs, and by Barone Adessi, Serra Cassano and V itto ne [ 4] and Garofalo and Pauls [23] for vertica l graphs. Additional contributions concerning variational problems related to the subrieman- nian a rea in the Heisenberg groups include [26], [2], [8 ], [9], [10], [ 2 1], [21], [20], [ 19], [18], [17], [16], [25], [2 9]. The recent monograph by Capogna, Danielli, Pauls and T yson [ 7] gives a recent overv ie w of the subject with an exhaustive list of references. W e would like Date : October 26, 2018. 2000 M athematics Subject Classificatio n. 53C17, 49Q20. Key words and phrases. S ub-Riemannian g e ometry , Heisenbe rg group, minimal surfaces, minimal cones. Researc h supported by ME C-Feder grant MTM2007-61919. 2 M. RITOR ´ E to stress that, in H 1 , the condition H ≡ 0 is not enough to guara ntee that a given surf a ce of class C 2 is even a stationary point for the a rea functional, see Ritor ´ e and Rosales [29], and Cheng, Hwang a nd Y ang [9] f or minimizing t - graphs. The aim of this pa per is to provide new examples in H 1 of Euclidean locally lipschitz area-minimizing entire graphs over the x y -plane. In section 3 we construct the basic exa mples. W e start from a given horizontal line L , and a monotone a ngle function α : L → ( 0 , π ) over this line. For each p ∈ L , we consider the two horizontal ha lflines e xtending f rom p making an angle ± α ( p ) with L . W e prove that in this way we a lways obtain a n entire graph over the xy -plane which is Euclidean locally lipschitz and area-minimizing. The angle function α is only a ssumed to be con- tinuous and monoto ne. Of course, f urther regularity on α yields more regularity on the graph. In case α is at least C 2 we get that the a ssociated surface is C 1,1 . The surfaces in sec tion 3 are the building blocks for our next construction in section 4. W e fix a point p ∈ H 1 , and a family of counter-clockwise oriented horizontal halflines R 1 , . . . , R n extending from p . W e choose the bisector L i of the wedge determined by R i − 1 and R i , and we consider angle functions α i : L i → ( 0, π ) which are continuous, nonincreasing as a function of the distance to p , and such that α ( p ) is equal to the angle b e tween L i and R i . For every q ∈ L i , we consider the halflines exte nding from q with angles ± α i ( q ) . In this way we also a f a mily of area-minimizing t -graphs which are Euclide an locally lipschitz. In case the obta ined surfa ce is regular enough we have that the singular set is precisely S n i = 1 L i . If the angle functions α i are constant, then we obtain area-minimizing cones (the original motivation of this paper ) , which are Euclidean locally C 1,1 minimizers, and C ∞ outside the singular set S n i = 1 L i . For a single halfline L extending from the origin and an angle function α : L → ( 0, π ) , continuous and nonincreasing as a function of the distance to 0, we patch th e graph obtained over a wedge of the x y -plane with the plane t = 0 along the ha lflines extending from 0 making an angle α ( 0 ) with L . When α is constant we get aga in an area-minimizing cone which is E uc lide an locally lipschitz. These cones a re a generalization of the one obta ined by Cheng, Hwang and Y a ng [9, Ex. 7.2 ]. An interesting consequence of this construction is that we get a large number of Eu- clidean locally C 1,1 area-minimizing cones with prescribed singular set consisting on ei- ther a horizontal line or a finite number o f horizontal halflines extending fr om a given point. It is an open question to decid e if these examples are the only a rea-minimizing cones, together with vertical halfspa ces and the e xample by Cheng, Hwang and Y ang [9, Ex. 7.2 ] with a singular halfline and its generalizations in the last section. The importance of tangent cones has been recently stressed in [1]. 2. P R E L I M I N A R I E S The Heisenber g group H 1 is the Lie group ( R 3 , ∗ ) , wher e the product ∗ is defined, for any pair of points [ z , t ] , [ z ′ , t ′ ] ∈ R 3 ≡ C × R , as [ z , t ] ∗ [ z ′ , t ′ ] : = [ z + z ′ , t + t ′ + Im ( z z ′ ) ] , ( z = x + i y ) . EXAMPLES OF ARE A-MINIMIZING S URF ACES 3 For p ∈ H 1 , the left translation by p is the diffeomorphism L p ( q ) = p ∗ q . A basis of left invariant vector fields (i.e. , invariant by any left tr a nslation) is given by X : = ∂ ∂ x + y ∂ ∂ t , Y : = ∂ ∂ y − x ∂ ∂ t , T : = ∂ ∂ t . The horizontal d istribution H in H 1 is the smooth planar one generated by X and Y . The horizontal projection of a vector U onto H will be denoted by U H . A vector field U is ca lled horizontal if U = U H . A horizontal curve is a C 1 curve whose tangent vector lies in the horizontal distribution. W e denote by [ U , V ] the Lie bracket of two C 1 vector fields U , V on H 1 . Note that [ X , T ] = [ Y , T ] = 0, while [ X , Y ] = − 2 T . The last equality implies that H is a bracket gen- erating distribution. Moreover , by Frobenius Theorem we have that H is nonintegrable. The vector fields X and Y generate the kernel of the (contact) 1-f orm ω : = − y d x + x d y + d t . W e shall consider on H 1 the (lef t invariant) Riemannian metric g =  · , ·  so that { X , Y , T } is an orthonormal basis a t every point, a nd the associated Levi-Civit ´ a connec- tion D . The modulus of a vector field U will b e denoted by | U | . Let γ : I → H 1 be a piecewise C 1 curve defined on a compact in terval I ⊂ R . The length of γ is the usual Riemannian length L ( γ ) : = R I | ˙ γ | , where ˙ γ is the tangent vec tor of γ . For two given points in H 1 we can find, by Chow’s connectivity Theorem [24, p. 95 ], a horizontal curve joining these points. T he Carnot-Carath ´ edory distance d c c between two points in H 1 is defined as the infimum of the length of horizontal curves joining the given points. A geodesic γ : H 1 → R is a horizontal curv e which is a critical point of length under va riations by horizontal curves. They satisfy the equation (2.1) D ˙ γ ˙ γ + 2 λ J ( ˙ γ ) = 0, where λ ∈ R is the curvature of the geodesic, and J is the π /2 -degrees oriented rotation in the horizontal distribution. Geodesics in H 1 with λ = 0 a re horizontal straight lines. The reader is referred to the section on geodesics in [29] f or further d etails. The v o lum e | Ω | of a Borel set Ω ⊆ H 1 is the Riemannian volume of the left invaria nt metric g , which coincides with the Lebesgue measure in R 3 . W e shall de note this volume element by d v g . The perimeter of E ⊂ H 1 in an open subset Ω ⊂ H 1 is d efined as (2.2) | ∂ E | ( Ω ) : = sup  Z Ω div U d v g : U horizontal and C 1 , | U | 6 1, supp ( U ) ⊂ Ω  , where supp ( U ) is the support of U . A set E ⊂ H 1 is of locally finite perimeter if P ( E , Ω ) < + ∞ for any bounded open set Ω ⊂ H 1 . A set of locally finite perimeter has a measur- able horizonta l unit normal ν E , that satisfies the following divergence theorem [17, Corol- lary 7 . 6]: if U is a horizontal vector field with compact support, then Z E div U d v g = Z H 1  U , ν E  d | ∂ E | . If E ⊂ H 1 has Euclidean lipschitz boundary , then [17, Corollary 7.7 ] (2.3) | ∂ E | ( Ω ) = Z ∂ E ∩ Ω | N H | d H 2 , 4 M. RITOR ´ E where N is the outer unit normal to ∂ E , de fined H 2 -almost everywhere. Here H 2 is the 2- dimensional riemannian Hausdorff measure. Let Ω ⊂ H 1 be an open set. W e say that E ⊂ H 1 of locally finite perimeter is area- minimizing in Ω if , for a ny set F such that E = F outside Ω we have | ∂ E | ( Ω ) 6 | ∂ F | ( Ω ) . The following extension of the divergence theorem will be needed to prove the area- minimizing property of sets of locally finite perimeter Theorem 2.1 . Let E ⊂ H 1 be a set of locally finite p erimeter , B ⊂ H 1 a set with piecewise smooth boundary , and U a C 1 horizontal vector field in int ( B ) t h at extends cont inuously t o the boundary of B. Th en (2.4) Z E ∩ B div U d v g = Z B  U , ν E  d | ∂ E | + Z E  U , ν B  d | ∂ B | . Proof. The proof is modelled on [15, § 5 .7]. Let s denote the riemannian distance f unction to H 1 − B . For ε > 0, de fine h ε ( p ) : = ( 1, ε 6 s ( p ) , s ( p ) / ε , 0 6 s ( p ) 6 ε , Then h ε is a lipschitz function (in rie ma nnian sense). For a ny smoot h h with compact sup- port in B we have div ( h U ) = h div ( U ) +  ∇ h , U  . By applying the d ivergence theorem for sets of locally finite perimeter [17] we get Z H 1 h  U , ν E  d | ∂ E | = Z E h div ( U ) + Z E  ∇ h , U  . By approximation, this formula is also valid for h ε . T aking limits when ε → 0 we have H ε → χ B . B y the coarea formula for lipschitz functions 1 ε Z { 0 6 s 6 ε } χ E  ∇ s , U  = 1 ε Z ε 0  Z { s = r } χ E  ∇ s , U  d H 2  dr , and, ta king again limits when ε → 0 a nd calling N B to the riema nnian outer unit normal to ∂ B (d efined except on a small set), we have lim ε → 0 Z E  ∇ h ε , U  = Z ∂ B χ E  N B , U  d H 2 = Z E  ν B , U  d | ∂ B | . Hence (2 .4) is proved.  For a C 1 surface Σ ⊂ H 1 the singular set Σ 0 consists of those points p ∈ Σ for which the tangent plane T p Σ coincides with the horizontal distribution. As Σ 0 is closed and has empty interior in Σ , the regular set Σ − Σ 0 of Σ is open a nd dense in Σ . It was proved in [14, Lemme 1] , see also [3, Theorem 1.2], that, for a C 2 surface, the Ha usdorff d imension with respect to the Riemannian dista nce on H 1 of Σ 0 is less than two. If Σ is a C 1 oriented surface with unit normal vector N , then we ca n describe the singu- lar set Σ 0 ⊂ Σ , in terms of N H , as Σ 0 = { p ∈ Σ : N H ( p ) = 0 } . In the regular part Σ − Σ 0 , EXAMPLES OF ARE A-MINIMIZING S URF ACES 5 we can define the horizontal unit normal vector ν H , as in [12], [ 30] and [2 3] by (2.5) ν H : = N H | N H | . Consider the characteristic vector fi eld Z on Σ − Σ 0 given by (2.6) Z : = J ( ν H ) . As Z is horizontal and orthogonal to ν H , we conclude that Z is tangent to Σ . Hence Z p generates the intersection of T p Σ with the horizontal distribution . The integral curves of Z in Σ − Σ 0 will be called cha ract eristic curves of Σ . They are both tangent to Σ and horizontal. Note that these curves depe nd on the unit normal N to Σ . If we define (2.7) S : =  N , T  ν H − | N H | T , then { Z p , S p } is an orthonormal basis of T p Σ whenever p ∈ Σ − Σ 0 . In the Heisenberg group H 1 there is a one-parameter group of dilations { ϕ s } s ∈ R gener- ated by the vector field (2.8) W : = x X + yY + 2 t T . W e may compute ϕ s in coordinates to obta in (2.9) ϕ s ( x 0 , y 0 , t 0 ) = ( e s x 0 , e s y 0 , e 2 s t 0 ) . Conjugating with le f t translations we get the one-par a meter family of dilations ϕ p , s : = L p ◦ ϕ s ◦ L − 1 p with center at a ny point p ∈ H 1 . A set E ⊂ H 1 is a cone of center p if ϕ p , s ( E ) ⊂ E for a ll s ∈ R . Any isometry of ( H 1 , g ) leaving invaria nt the horizontal d istribution preserves the area of surfa c e s in H 1 . Examples of such isometries are left translations, which act transitively on H 1 . The Euclidean rotation of a ngle θ about the t -axis given by ( x , y , t ) 7→ r θ ( x , y , t ) = ( cos θ x − sin θ y , sin θ x + cos θ y , t ) , is also an area-preserving isometry in ( H 1 , g ) since it tra nsforms the ortho normal basis { X , Y , T } at the point p into the orthonormal basis { cos θ X + sin θ Y , − sin θ X + cos θ Y , T } at the point r θ ( p ) . 3. E X A M P L E S W I T H O N E S I N G U L A R L I N E Consider the x -axis in H 1 = R 3 parametrized by Γ ( v ) : = ( v , 0 , 0 ) . T ake a non-increasing continuous function α : R → ( 0, π ) . For e v e ry v ∈ R , consider two horizontal halflines L + v , L − v extending from Γ ( v ) with a ngles α ( v ) and − α ( v ) , respectively . The ta ngent ve c- tors to these curv e s at Γ ( v ) are given by cos α ( v ) X Γ ( v ) + sin α ( v ) Y Γ ( v ) and cos α ( v ) X Γ ( v ) − sin α ( v ) Y Γ ( v ) , respectively . The par a metric equations of this surface are given by (3.1) ( v , w ) 7 → ( ( v + w cos α ( v ) , w sin α ( v ) , − v w sin α ( v ) ) , w > 0 , ( v + | w | c os α ( v ) , − | w | sin α ( v ) , v | w | sin α ( v ) ) , w 6 0 , 6 M. RITOR ´ E One can eliminate the parameters v , w to ge t the implicit equation t + x y − y | y | cot α  − t y  = 0 . Letting β : = cot ( α ) , we get that β is a continuous non-decreasing function, a nd that the surface Σ β defined by the para metric equations (3.1) is given by the implicit equation (3.2) 0 = f β ( x , y , t ) : = t + x y − y | y | β  − t y  . Observe that, because of the monotonicity condition on α , the projection of relative inte- riors of the open horizontal halflines to the x y - plane together with th e planar x -axis L x produce a partition of the plane. Since Σ β is the union of the horizonal lifting of these pla- nar halflines and the x -axis to H 1 , it is the graph of a continuous function u β : R 2 → R . For ( x , y ) ∈ R 2 , the only point in the intersection of Σ β with the vertical line passing through ( x , y ) is precisely ( x , y , u β ( x , y ) ) . Obviously (3.3) f β ( x , y , u β ( x , y ) ) = 0 . For any ( x , y ) ∈ R 2 , denote by ξ β ( x , y ) the only va lue v ∈ R so that either Γ ( v ) = ( x , y , 0 ) , or ( x , y , u β ( x , y ) ) is c ontained in one of the two above described halflines leav ing Γ ( v ) . T rivially ξ β ( x , 0 ) = x . Using (3.1) one checks that (3.4) ξ β ( x , y ) = − u β ( x , y ) y , y 6 = 0. Recalling that α = c ot − 1 ( β ) , we see that the map p ing ( v , w ) 7 → ( ( v + w cos α ( v ) , w sin α ( v ) ) , w > 0 , ( v + | w | cos α ( v ) , − | w | sin α ( v ) ) , w 6 0 , is a n homeomorphism of R 2 whose inverse is given by ( x , y ) 7 → ( ξ β ( x , y ) , sgn ( y ) | ( x − ξ β ( x , y ) , y ) | ) , where sgn ( y ) : = y / | y | for y 6 = 0. Hence ξ β : R 2 → R is a c ontinuous f unction. By ( 3.4), the function u β ( x , y ) / y admits a continuous extension to R 2 . Let us a nalyze first the properties of u β for regular β Lemma 3.1 . Let β ∈ C k ( R ) , k > 2 , be a non-decreasing function. T h en (i) u β is a C k function in R 2 − L x , (ii) u β is merely C 1,1 near the x -a x is when β 6 = 0 , (iii) u β is C ∞ in ξ − 1 ( I ) when β ≡ 0 on any open set I ⊂ R , a nd (iv) Σ β is area-minimizing. (v) The projection of the singular set of Σ β to the x y - plane is L x . Proof. Along the proof we shall often d rop the subscript β for f β , u β , ξ β and Σ β . The proof of 1 is just an application of the Implicit Function Theorem since f β is a C k function for y 6 = 0 when β is C k . EXAMPLES OF ARE A-MINIMIZING S URF ACES 7 T o prove 2 we c ompute the pa rtial derivative s of u β for y 6 = 0. They are given by ( u β ) x ( x , y ) = − y 1 + | y | β ′  ξ β ( x , y )  , (3.5) ( u β ) y ( x , y ) = − x + | y |  2 β  ξ β ( x , y )  − β ′  ξ β ( x , y )  ξ β ( x , y )  1 + | y | β ′  ξ β ( x , y )  . (3.6) Since u β ( x , 0 ) = 0 for a ll x ∈ R we get ( u β ) x ( x , 0 ) = 0 . On the other hand ( u β ) y ( x , 0 ) = li m y → 0 u β ( x , y ) y = − lim y → 0 ξ β ( x , y ) = − ξ β ( x , 0 ) = − x . The limits, when y → 0 , of (3.5) and ( 3 .6) can be computed using (3.4). W e conclude that the first de rivatives of u β are continuous f unctions and so u β is a C 1 function on R 2 . T o see that u β is merely lipschitz, we get f rom (3.6) and (3.4) ( u β ) y y ( x , 0 ) = lim y → 0 ± ( u β ) y ( x , y ) + x y = lim y → 0 ± | y |  2 β  ξ β ( x , y )  − β ′  ξ β ( x , y )  ξ β ( x , y ) + x β ′  ξ β ( x , y )  y  1 + | y | β ′  ξ β ( x , y )  = ± 2 β ( x ) . Hence side derivatives exist, but they do not coincide unless β ( x ) = 0. As u β   ξ − 1 ( I ) = − x y , 3 f ollows easily . T o prove 4 we use a calibration argument . W e shall drop the subscript β to simplify the notation. Let F ⊂ H 1 such that F = E outside a Euclidean ball B ce ntered a t the origin. Let H 1 : = { ( x , y , t ) : y > 0 } , H 2 : = { ( x , y , t ) : y 6 0 } , Π : = { ( x , y , t ) : y = 0 } . V erti- cal tr anslations of the horizontal unit normal ν E , defined outside Π , provide two vector fields U 1 on H 1 , a nd U 2 on H 2 . They are C 2 in the interior of the halfspaces and extend continuously to the boundary plane Π . A s in the proof of T heorem 5.3 in [2 9], we see that div U i = 0, i = 1, 2, in the interior of the ha lf spaces. He re d iv U is the riemannian d ivergence of the vector field U . Observe that the vector field Y is the riemannian unit normal, and also the hori- zontal unit normal, to the plane Π . W e may a pply the divergence theorem to get 0 = Z E ∩ int ( H i ) ∩ B div U i = Z E  U i , ν int ( H i ) ∩ B  d | ∂ ( int ( H i ) ∩ B ) | + Z int ( H i ) ∩ B  U i , ν E  d | ∂ E | . Let D : = Π ∩ B . Then, for every p ∈ D , we have ν int ( H 1 ) ∩ B = − Y , ν int ( H 2 ) ∩ B = Y , and U 1 = J ( v ) , U 2 = J ( w ) , where v − w is proportional to Y , by the construction of Σ β . Hence  U 1 , ν int ( H 1 ) ∩ B  +  U 2 , ν int ( H 2 ) ∩ B  =  v − w , J ( Y )  = 0 , p ∈ D . 8 M. RITOR ´ E Adding the above integrals we obtain 0 = ∑ i = 1,2 Z E  U i , ν B  d | ∂ B | + ∑ i = 1,2 Z B ∩ int ( H i )  U i , ν E  d | ∂ E | . W e apply the same a rguments to the set F and, since E = F on ∂ B we conclude (3.7) ∑ i = 1,2 Z B ∩ int ( H i )  U i , ν E  d | ∂ E | = ∑ i = 1,2 Z B ∩ int ( H i )  U i , ν F  d | ∂ F | . As E is a subgraph, | ∂ E | ( Π ) = 0 and so | ∂ E | ( B ) = ∑ i = 1,2 Z B ∩ int ( H i )  U i , ν B  d | ∂ E | . Cauchy-Schwarz inequality and the fact that | ∂ F | is a positive measure imply ∑ i = 1,2 Z B ∩ int ( H i )  U i , ν F  d | ∂ F | 6 | ∂ F | ( B ) , which implies 4 . T o prove 5 simply take into account that the projection of the singular set of Σ β to the x y -plane is composed of those points ( x , y ) such that ( u β ) − x − y = ( u β ) y + x = 0. From (3.5) we get that ( u β ) x − y = 0 if and only if y  2 + | y | β ′ ( ξ β ( x , y ) )  = 0 , i.e, when y = 0. In this case, from (3.6), we see that equation ( u β ) y + x = 0 is trivially satisfied.  W e now prove the general properties of Σ β from Lemma 3.1 Proposition 3.2. Let β : R → R be a c ontinuous non-decreasing function. Let u β be the o nly solution of equation (3. 3) , Σ β the graph of u β , and E β the subgraph of u β . Then (i) u β is locally lip schitz in Euclidean sense, (ii) E β is a set of locally finite perimeter in H 1 , and (iii) Σ β is area-minimizing in H 1 . Proof. Let β ε ( x ) : = Z R β ( y ) η ε ( x − y ) d y the usual convolution, where η is a Dirac function and η ε ( x ) : = η ( x / ε ) , see [15]. Then β ε is a C ∞ non-decreasing function, and β ε converges uniformly , on compact subsets of R , to β . Let u = u β , u ε = u β ε , f = f β , f ε = f β ε . Let D ⊂ R 2 be a bounded subset. T o c heck that u is lipschitz on D it is enough to prove that the first deriva tives of u ε are uniformly bounded on D . From (3.3) we get ξ ( x , y ) + | y | β  ξ ( x , y )  = x , y 6 = 0. For y fixed, define the continuous strictly increasing function ρ y ( x ) : = x + | y | β ( x ) . EXAMPLES OF ARE A-MINIMIZING S URF ACES 9 Hence we get (3.8) ξ ( x , y ) = ρ − 1 y ( x ) . W e can also define ( ρ ε ) y ( x ) : = x + | y | β ε ( x ) . Equation (3. 8) holds replacing u , β by u ε , β ε . Since ρ − 1 y ( x ) = ξ ( x , y ) , we conclude that ρ − 1 y is a continuo us function that d epends continuously on y . Let us estimate | ( ρ ε ) − 1 y ( x ) − ρ − 1 y ( x ) | . Let z ε : = ( ρ ε ) − 1 y ( x ) , z = ρ − 1 y ( x ) . Then x = ( ρ ε ) y ( z ε ) = ρ y ( z ) and we have, assuming z ε > z . 0 = ( ρ ε ) y ( z ε ) − ρ y ( z ) = z ε + | y | β ε ( z ε ) −  z + | y | β ( z )  = ( z ε − z ) + | y |  β ε ( z ε ) − β ε ( z )  + | y |  β ε ( z ) − β ( z )  > ( z ε − z ) + | y |  β ε ( z ) − β ( z )  . A similar computation can be performed for z ε > z . The consequence is that | z ε − z | 6 | y | | β ε ( z ) − β ( z ) | , or , equivalently , | ( ρ ε ) − 1 y ( x ) − ρ − 1 y ( x ) | 6 | y | | β ε ( ρ − 1 y ( x ) ) − β ( ρ − 1 y ( x ) ) | . As β ε → β uniformly on compact subsets of R , we have uniform convergence of ( ρ ε ) − 1 y ( x ) to ρ − 1 y ( x ) on compact subsets of R 2 . This also implies the uniform convergence of ξ ε ( x , y ) to ξ ( x , y ) on compact subsets. Hence also u ε ( x , y ) converges uniformly to u ( x , y ) on com- pact subsets of R 2 . From (3.5) and (3 .6) we hav e | ( u ε ) x ( x , y ) | 6 | y | , | ( u ε ) y ( x , y ) | 6 | x | + 2 | y |   β ε  ξ ε ( x , y )    +   ξ ε ( x , y )   . As β ε → β and ξ ε ( x , y ) → ξ ( x , y ) uniformly on compact subsets, we have that the first derivatives of u ε are uniformly bounded on compact subsets. Hence u is locally lipschitz. The subgraph of u β is a set of locally finite perimeter in H 1 since its boundary is locally lipschitz by 1. This follows from [1 7] and proves 2. T o prove 3 we use approximation and the calibration a rgument. Let F ⊂ H 1 so tha t F = E outside a Euclidea n ball B centered at the origin. Fo r the functions β ε , consider the vector fields U i ε obtained by translating vertically the ho rizontal unit no rmal to the surface Σ ε . W e repeat the arguments on the proof of 4 in Lemma 3.1 to conclude a s in (3. 7) that ∑ i = 1,2 Z B ∩ int ( H i )  U i ε , ν E  d | ∂ E | = ∑ i = 1,2 Z B ∩ int ( H i )  U i ε , ν F  d | ∂ F | . T rivially we have ∑ i = 1,2 Z B ∩ int ( H i )  U i ε , ν F  d | ∂ F | 6 | ∂ F | ( B ) . 10 M. RITOR ´ E On the oth er hand, U i ε converges uniformly , on compact subsets, to U i by Lemma 3. 3. Passing to the limit when ε → 0 and taking into account that U i = ν E we conclude | ∂ E | ( B ) 6 | ∂ F | ( B ) , as desired.  Lemma 3 .3. Let β be a continuous non-d ecreasing function. Th en the h orizontal unit normal of Σ β is given, in { X , Y } -coordinates, by (3.9) ν β ( x , y ) =  1 ( 1 + β 2 ) 1/ 2 , − sgn ( y ) β ( 1 + β 2 ) 1/ 2   ξ β ( x , y )  , y 6 = 0. Moreover , ν β admits continuous extensions to y = 0 from both sides of this line. Proof. Since u β is lipschitz, it is d ifferentiable almost eve r ywhere on R 2 . On these points, ν β ( x , y ) = (( u β ) x − y , ( u β ) y + x ) . The function − u β ( x , y ) / y is constant along the lines ( x 0 , 0 ) + λ ( 1 + β 2 ) − 1 /2 ( β , ± 1 ) ( x 0 ) , for λ > 0 . Let y > 0. From (3.2) we have 0 = − x 0 + x − y β ( x 0 ) . Let v : = ( 1 + β 2 ) − 1 /2 ( β , 1 ) ( x 0 ) . Then v ( − u β ( x , y ) / y ) = 0. Hence for almost every point on a lmost every line, we have β ( x 0 ) ( u β ) x + ( u β ) y = − x 0 . Hence we have ( u β ) y + x = − x 0 − β ( x 0 ) ( u β ) x + x 0 + y β ( x 0 ) = β ( x 0 ) ( − ( u β ) x + y ) . W e conclude that the horizontal unit normal is proportio nal to ( 1, − β ) , which implies (3.9). The case y 6 0 is handled similarly .  Example 3.4. T aking β ( x ) : = x we get u β ( x , y ) = − x y 1 + | y | , which is a Euclidean C 1,1 graph. Another family of interesting exa mples a re the minimal cones obtained by taking the constant function β ( x ) : = β 0 . In this case we get u β ( x , y ) = − x y + β 0 y | y | . In this case Σ β is a C 1,1 surface which is invariant by the dilations centered at any point of the singular line. T ake now β ( x ) : = ( 0, x 6 0 , x , x > 0 . In this case we obtain the graph u β ( x , y ) : = ( − x y , x 6 0 , − x y 1 + | y | , x > 0 , EXAMPLES OF ARE A-MINIMIZING S URF ACES 11 which is simply locally Lipschitz. This example was mentioned to me by Scott Pauls. Consider now a continuous non- decreasing function β : R → R , constant outside the Cantor set C ⊂ [ 0, 1 ] with β ( 0 ) = 0, β ( 1 ) = 1. Then the associated surface Σ β is an area-minimizing surface in H 1 . 4. E X A M P L E S W I T H S E V E R A L S I N G U L A R H A L FL I N E S M E E T I N G AT A P O I N T Let α 0 1 , . . . , α 0 k , be a family of positive angles so that k ∑ i = 1 α 0 i = π . Let r β be the rotation of a ngle β around the origin in R 2 . Consider a family of closed halflines L i ⊂ R 2 , i ∈ Z k , extending f rom the origin, so that r α 0 i + α 0 i + 1 ( L i ) = L i + 1 . Finally , define R i : = r α 0 i ( L i ) . (An a lternative way of defining this configuration is to start from a family of counter-clockwise oriented halflines R i ⊂ R 2 , i ∈ Z k , choosing L i , i ∈ Z k , as the bisector of the angle determined by R i − 1 and R i , and defining α 0 i as the angle between L i and R i ). Define W i as the closed wedge, containing L i , bordered by R i − 1 and R i . W 2 L 1 L 3 R 3 α 0 3 R 1 α 0 1 L 2 R 2 α 0 3 α 0 1 α 0 2 W 1 W 3 α 0 2 F I G U R E 1 . The initial configuration with three halflines L 1 , L 2 , L 3 . For every i ∈ Z k , let α i : [ 0, ∞ ) → ( 0, π ) be a continuous nonincreasing function so that α i ( 0 ) = α 0 i , a nd define, as in the previous section, β i : = cot ( α i ) . L e t v i ∈ S 1 , i ∈ Z k , be such that L i = { s v i : s > 0 } . Fo r every i ∈ Z k and s > 0, we take the two closed halflines L ± s , i in R 2 extending from the point s v i with tangent vectors ( cos α i ( s ) , ± sin α i ( s ) ) . In this way we cover all of R 2 . W e shall define α : = ( α 1 , . . . , α k ) . Lift L 1 , . . . , L k to horizontal halflines L ′ 1 , . . . , L ′ k in H 1 from the origin, and L ± s , i to hor - izontal halflines in H 1 extending from the unique point in L ′ i projecting onto s v i . In this way we obtain a continuous function u α : R 2 → R . The graph Σ α of u α is a topological surface in H 1 . 12 M. RITOR ´ E Obviously the angle functions α i ( s ) ca n be extended continuously and preserving the monotoni city , to an angle function e α i : e L i → ( 0, π ) , where e L i is the straight line contain- ing the halfline L i . The graph of u α restricted to W i coincides with the Euclidean locally lipschitz area-minimizing surfa ce u e β i , for e β i : = c ot e α i , constructed in the previous section. So the examples in this section can be seen a s pieces of the exa mples of the previous one patched together . Theorem 4.1. Under th e above cond itions (i) Th e function u α is locally lip schitz in the Euclidean sense. (ii) The surface Σ α is area-minimizing. Proof. It is immediate that u α is a graph which is locally lipschitz in Euclidean sense: choose a disk D ⊂ R 2 . Let p , q ∈ D . Assume first that ( p , q ) intersects the halflines R 1 , . . . , R k transversally at the points x 1 , . . . , x n . Then [ p , x 1 ] , [ x 1 , x 2 ] , . . . , [ x n , p ] are c on- tained in wedges a nd hence | u α ( p ) − u α ( q ) | 6 | u α ( p ) − u α ( x 1 ) | + · · · + | u α ( x n ) − u α ( q ) | C  | p − x 1 | + · · · | x n − q |  = C | p − q | , where C is the supremum of the Lipschitz constants of u e β i restricted to D . The general case is then obtained by approximating p and q by p oints in the c ondition of the assumption. T o prove that u α is area minimizing we first approximate α i by smooth a ngle functions ( α i ) ε with ( α i ) ε ( 0 ) = α i ( 0 ) . In this wa y we obtain a calibrating vector field which is con- tinuous along the vertical planes passing through R i by Lemma 3.3. This allows us to apply the calibration argument to prove the area-minimizing property of Σ α .  Example 4.2 (M inimizing cones) . Let α i ( s ) = α 0 i be a constant for all i . Then the subgraph of Σ α is a minimizing cone with center at 0. Restricted to the interior of the wedges W i , the surface Σ α is C 1,1 . A n easy computation shows that, taking β ( s ) : = β 0 in the construction of the first section, the Riemannian normal to Σ β along the halflines β 0 | y | = x , x > 0 (that make angle ± cot − 1 ( β 0 ) with the positive x - axis) is given by N = − 2 y X + 2 β 0 | y | Y − T q 1 + 4 y 2 + 4 β 2 0 y 2 = − 2 y X + 2 x Y − T p 1 + 4 x 2 + 4 y 2 . This vec tor field is invariant by rotations around the ve rtical axis. Hence in our construc- tion, the normal vec tor field to Σ α is continuo us. It is straightforward to show that it is locally lipschitz in Euclide a n sense. Example 4.3 (Area- minimizing surfaces with a singular halfline) . These ex amples are in- spired b y [9, Exa mple 7. 2]. W e consider a halfline L extending from the orig in, and an angle function α : L → ( 0, π ) c ontinuous and nonincreasing a s a function of the distance to the origin. W e consider the union of the halflines L + α ( q ) , L − α ( q ) extending from q ∈ L with angles α ( q ) , − α ( q ) , respectively . W e patch the a rea-minimizing surface defined by α in the wedge delimited by the halflines L + 0 , L − 0 , with the pla ne t = 0. In this way we get an entire area-minimizing t -graph, with lipschitz regularity . In case the angle f unction α is EXAMPLES OF ARE A-MINIMIZING S URF ACES 13 constant, we get an a rea-minimizing cone with center 0, which is defined by the equation u ( x , y ) : = ( − x y + β 0 y | y | , − x y + β 0 y | y | > 0, 0, − x y + β 0 y | y | 6 0. 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