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

We give new examples of entire area-minimizing t-graphs in the subriemannian Heisenberg group H^1. Most of the examples are locally lipschitz in Euclidean sense. Some regular examples have prescribed singular set consisting of either a horizontal line or a finite number of horizontal halflines extending from a given point. Amongst them, a large family of area-minimizing cones is obtained.


INTRODUCTION
Variational problems related to the subriemannian area in the Heisenberg group H 1 have received great attention recently. A major question in this theory is the regularity of minimizers. A related one is the construction of examples with low regularity properties. The study of minimal surfaces in subriemannian geometry was initiated in the paper by Garofalo and Nhieu [22]. Later Pauls [27] constructed minimal surfaces in H 1 as limits of minimal surfaces in Nil manifolds, the riemannian Heisenberg groups. Cheng, Hwang and Yang [9] have studied the weak solutions of the minimal surface equation for t-graphs and have proven existence and uniqueness results. Regularity of minimal surfaces, assuming that they are least C 1 , has been treated in the papers by Pauls [28] and Cheng, Hwang and Yang [10]. We would like also to mention the recently distributed notes by Bigolin and Serra Cassano [5], where they obtain regularity properties of an Hregular surface from regularity properties of its horizontal unit normal. Interesting examples 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 , for n > 1, has been recently obtained by Capogna, Citti and Manfredini [6].
Characterization in H 1 of solutions of the Bernstein problem for C 2 surfaces has been obtained by Cheng, Hwang, Malchiodi and Yang [8], and Ritoré and Rosales [29] for tgraphs, and by Barone Adessi, Serra Cassano and Vittone [4] and Garofalo and Pauls [23] for vertical graphs.
The aim of this paper is to provide new examples in H 1 of Euclidean locally lipschitz area-minimizing entire graphs over the xy-plane.
In section 3 we construct the basic examples. We start from a given horizontal line L, and a monotone angle function α : L → (0, π) over this line. For each p ∈ L, we consider the two horizontal halflines extending from p making an angle ±α(p) with L. We prove that in this way we always obtain an entire graph over the xy-plane which is Euclidean locally lipschitz and area-minimizing. The angle function α is only assumed to be continuous and monotone. Of course, further regularity on α yields more regularity on the graph. In case α is at least C 2 we get that the associated surface is C 1,1 .
The surfaces in section 3 are the building blocks for our next construction in section 4. We fix a point p ∈ H 1 , and a family of counter-clockwise oriented horizontal halflines R 1 , . . ., R n extending from p. We 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 between L i and R i . For every q ∈ L i , we consider the halflines extending from q with angles ±α i (q). In this way we also a family of area-minimizing t-graphs which are Euclidean locally lipschitz. In case the obtained surface is regular enough we have that the singular set is precisely 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 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 the graph obtained over a wedge of the xy-plane with the plane t = 0 along the halflines extending from 0 making an angle α(0) with L. When α is constant we get again an area-minimizing cone which is Euclidean locally lipschitz. These cones are a generalization of the one obtained by Cheng, Hwang and Yang [9,Ex. 7.2].
An interesting consequence of this construction is that we get a large number of Euclidean locally C 1,1 area-minimizing cones with prescribed singular set consisting on either a horizontal line or a finite number of horizontal halflines extending from a given point. It is an open question to decide if these examples are the only area-minimizing cones, together with vertical halfspaces and the example by Cheng, Hwang and Yang [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].

PRELIMINARIES
The Heisenberg group H 1 is the Lie group (R 3 , * ), where the product * is defined, for any pair of points [z, t] 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 translation) is given by The horizontal distribution 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 called horizontal if U = U H . A horizontal curve is a C 1 curve whose tangent vector lies in the horizontal distribution.
We denote by [U, V] the Lie bracket of two C 1 vector fields U, V on H 1 . Note that The last equality implies that H is a bracket generating distribution. Moreover, by Frobenius Theorem we have that H is nonintegrable. The vector fields X and Y generate the kernel of the (contact) 1-form ω := −y dx + x dy + dt.
We shall consider on H 1 the (left invariant) Riemannian metric g = · , · so that {X, Y, T} is an orthonormal basis at every point, and the associated Levi-Civitá connection D. The modulus of a vector field U will be denoted by |U|.
Let γ : I → H 1 be a piecewise C 1 curve defined on a compact interval I ⊂ R. The length of γ is the usual Riemannian length L(γ) := I |γ|, whereγ is the tangent vector 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. The Carnot-Carathédory distance d cc 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 curve which is a critical point of length under variations by horizontal curves. They satisfy the equation 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 are horizontal straight lines. The reader is referred to the section on geodesics in [29] for further details.
The volume |Ω| of a Borel set Ω ⊆ H 1 is the Riemannian volume of the left invariant metric g, which coincides with the Lebesgue measure in R 3 . We shall denote this volume element by dv g . The perimeter of E ⊂ H 1 in an open subset Ω ⊂ H 1 is defined as where N is the outer unit normal to ∂E, defined H 2 -almost everywhere. Here H 2 is the 2dimensional riemannian Hausdorff measure.
Let Ω ⊂ H 1 be an open set. We say that E ⊂ H 1 of locally finite perimeter is areaminimizing in Ω if, for any set F such that E = F outside Ω we have |∂E|(Ω) |∂F|(Ω).
The following extension of the divergence theorem will be needed to prove the areaminimizing property of sets of locally finite perimeter Theorem 2.1. Let E ⊂ H 1 be a set of locally finite perimeter, B ⊂ H 1 a set with piecewise smooth boundary, and U a C 1 horizontal vector field in int(B) that extends continuously to the boundary of B. Then Proof. The proof is modelled on [15, § 5.7]. Let s denote the riemannian distance function to Then h ε is a lipschitz function (in riemannian sense). For any smooth h with compact support in B we have div(hU) = h div(U) + ∇h, U . By applying the divergence theorem for sets of locally finite perimeter [17] we get By approximation, this formula is also valid for h ε . Taking limits when ε → 0 we have H ε → χ B . By the coarea formula for lipschitz functions and, taking again limits when ε → 0 and calling N B to the riemannian outer unit normal to ∂B (defined except on a small set), we have 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 and dense in Σ. It was proved in [14, Lemme 1], see also [3,Theorem 1.2], that, for a C 2 surface, the Hausdorff dimension with respect to the Riemannian distance 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 can describe the singular 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 , we can define the horizontal unit normal vector ν H , as in [12], [30] and [23] by Consider the characteristic vector field Z on Σ − Σ 0 given by 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 characteristic curves of Σ. They are both tangent to Σ and horizontal. Note that these curves depend on the unit normal N to Σ. If we define In the Heisenberg group H 1 there is a one-parameter group of dilations {ϕ s } s∈R generated by the vector field We may compute ϕ s in coordinates to obtain Conjugating with left translations we get the one-parameter family of dilations ϕ p,s := Any isometry of (H 1 , g) leaving invariant the horizontal distribution preserves the area of surfaces in H 1 . Examples of such isometries are left translations, which act transitively on H 1 . The Euclidean rotation of angle θ about the t-axis given by (x, y, t) → 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 transforms the orthonormal 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).

EXAMPLES WITH ONE SINGULAR LINE
Consider the x-axis in H 1 = R 3 parametrized by Γ(v) := (v, 0, 0). Take a non-increasing continuous function α : R → (0, π). For every v ∈ R, consider two horizontal halflines with angles α(v) and −α(v), respectively. The tangent vectors to these curves at Γ(v) are given by cos The parametric equations of this surface are given by One can eliminate the parameters v, w to get the implicit equation Letting β := cot(α), we get that β is a continuous non-decreasing function, and that the surface Σ β defined by the parametric equations (3.1) is given by the implicit equation Observe that, because of the monotonicity condition on α, the projection of relative interiors of the open horizontal halflines to the xy-plane together with the planar x-axis L x produce a partition of the plane. Since Σ β is the union of the horizonal lifting of these planar 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 For any (x, y) ∈ R 2 , denote by ξ β (x, y) the only value v ∈ R so that either Γ(v) = (x, y, 0), or (x, y, u β (x, y)) is contained in one of the two above described halflines leaving Γ(v).
Let us analyze first the properties of u β for regular β Lemma 3.1. Let β ∈ C k (R), k 2, be a non-decreasing function. Then The projection of the singular set of Σ β to the xy-plane is L x .
Proof. Along the proof we shall often drop 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 = 0 when β is C k .
To prove 2 we compute the partial derivatives of u β for y = 0. They are given by The limits, when y → 0, of (3.5) and (3.6) can be computed using (3.4). We conclude that the first derivatives of u β are continuous functions and so u β is a C 1 function on R 2 . To see that u β is merely lipschitz, we get from (3.6) and (3.4) Hence side derivatives exist, but they do not coincide unless β(x) = 0.
To prove 4 we use a calibration argument. We shall drop the subscript β to simplify the notation. Let F ⊂ H 1 such that F = E outside a Euclidean ball B centered at the origin. Let Vertical translations of the horizontal unit normal ν E , defined outside Π, provide two vector fields U 1 on H 1 , and U 2 on H 2 . They are C 2 in the interior of the halfspaces and extend continuously to the boundary plane Π. As in the proof of Theorem 5.3 in [29], we see that in the interior of the halfspaces. Here div U is the riemannian divergence of the vector field U. Observe that the vector field Y is the riemannian unit normal, and also the horizontal unit normal, to the plane Π. We may apply the divergence theorem to get Let D := Π ∩ B. Then, for every p ∈ D, we have ν int(H 1 )∩B = −Y, ν int(H 2 )∩B = Y, and Adding the above integrals we obtain We apply the same arguments to the set F and, since E = F on ∂B we conclude As E is a subgraph, |∂E|(Π) = 0 and so Cauchy-Schwarz inequality and the fact that |∂F| is a positive measure imply which implies 4.
To prove 5 simply take into account that the projection of the singular set of Σ β to the xy-plane is composed of those points ( i.e, when y = 0. In this case, from (3.6), we see that equation (u β ) y + x = 0 is trivially satisfied.
We now prove the general properties of Σ β from Lemma 3.1

Proof. Let
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 Let D ⊂ R 2 be a bounded subset. To check that u is lipschitz on D it is enough to prove that the first derivatives of u ε are uniformly bounded on D.
Since ρ −1 y (x) = ξ(x, y), we conclude that ρ −1 y is a continuous function that depends continuously on y.
A similar computation can be performed for z ε z. The consequence is that 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 compact subsets of R 2 .
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 [17] and proves 2.
To prove 3 we use approximation and the calibration argument. Let F ⊂ H 1 so that F = E outside a Euclidean ball B centered at the origin. For the functions β ε , consider the vector fields U i ε obtained by translating vertically the horizontal unit normal to the surface Σ ε . We repeat the arguments on the proof of 4 in Lemma 3.1 to conclude as in (3.7) that On the other 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) |∂F|(B), as desired.
Moreover, ν β admits continuous extensions to y = 0 from both sides of this line.
Proof. Since u β is lipschitz, it is differentiable almost everywhere on R 2 . On these points, The function −u β (x, y)/y is constant along the lines (x 0 , 0) Hence for almost every point on almost every line, we have Hence we have We conclude that the horizontal unit normal is proportional to (1, −β), which implies (3.9). The case y 0 is handled similarly. 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.
Take now In this case we obtain the graph which is simply locally Lipschitz.
This example was mentioned to me by Scott Pauls. Consider now a continuous nondecreasing 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 .

EXAMPLES WITH SEVERAL SINGULAR HALFLINES MEETING AT A POINT
Let α 0 1 , . . . , α 0 k , be a family of positive angles so that Let r β be the rotation of angle β around the origin in R 2 . Consider a family of closed halflines L i ⊂ R 2 , i ∈ Z k , extending from the origin, so that r α 0 (An alternative 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 .
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 , and define, as in the previous section, β i := cot(α i ). Let v i ∈ S 1 , i ∈ Z k , be such that L i = {sv i : s 0}. For every i ∈ Z k and s 0, we take the two closed halflines L ± s,i in R 2 extending from the point sv i with tangent vectors (cos α i (s), ± sin α i (s)). In this way we cover all of R 2 . We 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 horizontal halflines in H 1 extending from the unique point in L ′ i projecting onto sv 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 .
Obviously the angle functions α i (s) can be extended continuously and preserving the monotonicity, to an angle function α i : L i → (0, π), where L i is the straight line containing the halfline L i . The graph of u α restricted to W i coincides with the Euclidean locally lipschitz area-minimizing surface u β i , for β i := cot α i , constructed in the previous section.  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 contained in wedges and hence |u α (p) − u α (q)| |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 β i restricted to D. The general case is then obtained by approximating p and q by points in the condition of the assumption.
To prove that u α is area minimizing we first approximate α i by smooth angle functions (α i ) ε with (α i ) ε (0) = α i (0). In this way we obtain a calibrating vector field which is continuous 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 (Minimizing 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 . An 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 This vector field is invariant by rotations around the vertical axis. Hence in our construction, the normal vector field to Σ α is continuous. It is straightforward to show that it is locally lipschitz in Euclidean sense. . We consider a halfline L extending from the origin, and an angle function α : L → (0, π) continuous and nonincreasing as a function of the distance to the origin. We consider the union of the halflines L + α(q) , L − α(q) extending from q ∈ L with angles α(q), −α(q), respectively. We patch the area-minimizing surface defined by α in the wedge delimited by the halflines L + 0 , L − 0 , with the plane t = 0. In this way we get an entire area-minimizing t-graph, with lipschitz regularity. In case the angle function α is constant, we get an area-minimizing cone with center 0, which is defined by the equation u(x, y) := −xy + β 0 y|y|, −xy + β 0 y|y| 0, 0, −xy + β 0 y|y| 0.
This surface is composed of two smooth pieces patched together along the halflines x = β 0 |y|.