THE BERNSTEIN PROBLEM IN THE HEISENBERG GROUP

NICOLA GAROFALO AND SCOTT D. PAULS

Abstract. We establish a sub-Riemannian version for the first Heisenberg group ℍ^{1} of the classical theorem of Bernstein stating that the only complete minimal graphs in ℝ^{3} are planes.

1. Introduction

In recent years there has been considerable interest in the study of linear and nonlinear partial differential equations arising in Carnot-Carathéodory (CC) geometry. The common trend of such equations is a lack of ellipticity due to the non-abelian nature of the underlying algebraic structures which is reflected in non-commutativity of the relevant differential operators, and the absence of most of the symmetries which play an important role in the Riemannian setting. The basic geometric model in this context is represented by a sub-Riemannian, or CC manifold. This is a triple (M, HM, d), where (M, d_{R}) is a Riemannian manifold with metric d_{R}, HM is a given subbundle of the tangent bundle T M, and d is a distance, different from the Riemannian one d_{R}, which is obtained via a minimization procedure that selects only those paths γ : [a, b] → M whose tangent vector γ′(t) belongs to H_{γ(t)}M. The theorem of Chow-Rashevsky [Ch], [Ra] states that when M is connected and HM is bracket generating, there is a sufficiently large supply of such paths. Minimizing on their lengths one thus obtains the distance d, called the control, or Carnot-Carathéodory metric, generated by HM. The class of sub-Riemannian manifolds encompasses all Riemannian ones in the trivial case HM = T M, but also the Gromov-Hausdorff limit of some sequences of Riemannian manifolds.

A particularly interesting class of sub-Riemannian manifolds is that of Carnot groups, since they arise as the tangent spaces of sub-Riemannian spaces. A Carnot group is a connected, simply-connected Lie group G whose Lie algebra 𝔤 is graded, and r-nilpotent. This means that 𝔤 = V_{1} ⊕ V_{2} ⊕ ... ⊕ V_{r}, [V_{1}, V_{i}] = V_{i+1} for i = 1, ..., r − 1, and that [V_{1}, V_{r}] = {0}. In these hypothesis it is clear that if {e_{1}, ..., e_{m}} denotes a fixed orthonormal basis of the so-called horizontal layer V_{1} of 𝔤, defining X_{i}(g) = (L_{g})_{∗}(e_{i}), where (L_{g})_{∗} indicates the differential of the left-translation L_{g}(g′) = gg′, then the system of left-invariant vector fields X = {X_{1}, ..., X_{m}} fulfills the finite rank condition

rank Lie{X_{1}, ..., X_{m}} ≡ dim G = ∑_{i=1}^{r} dim V_{i} . (1.1)

Date: September 5, 2002.

The first author was supported by NSF grant No. DMS-0070492.

The second author was supported by NSF grant No. DMS-9971563.

Thanks to a fundamental theorem of Hörmander [H], this implies the hypoellipticity of the second-order partial differential operator on G

L = ∑_{i=1}^{m} X_{i}^{2} , (1.2)

which is known as the sub-Laplacian associated with the system X. It is worth nothing that, asides from the trivial (abelian) case when r = 1 and 𝔤 = V_{1} = ℝ^{n}, in which case (1.2) becomes the standard Laplacian in ℝ^{n}, the operator L fails to be elliptic at every point of G.

The simplest and most important non-abelian Carnot group is the so-called Heisenberg group ℍ^{n} [St], whose underlying manifold is ℂ^{n} × ℝ, with group law

g ∘ g′ = (z, t) ∘ (z′, t′) = (z + z′, t + t′ + \frac{1}{2}ℑ < z, \overline{z′} >) .

Here, we have let z = x + iy, z′ = x′ + iy′ ∈ ℂ^{n}, t, t′ ∈ ℝ, < z, \overline{z′} >= ∑_{j=1}^{n} z_{j}\overline{z′_{j}}. The generators of the (real) Heisenberg algebra 𝔥_{n} are the vector fields

X_{j}(g) = (L_{g})_{∗}(\frac{∂}{∂x_{j}}) = \frac{∂}{∂x_{j}} + \frac{y_{j}}{2} \frac{∂}{∂t} ,  X_{n+j}(g) = (L_{g})_{∗}(\frac{∂}{∂y_{j}}) = \frac{∂}{∂y_{j}} − \frac{x_{j}}{2} \frac{∂}{∂t} , (1.3)

j = 1, ..., n. The sub-Laplacian (1.2), given by L = ∑_{j=1}^{2n} X_{j}^{2}, is also known as the real part of the Kohn Laplacian in ℂ^{n+1}. One has [X_{i}, X_{n+j}] = −T δ_{ij}, where T = ∂/∂t represents the characteristic direction, all other commutators being trivial. Thereby, one has 𝔥_{n} = V_{1}⊕V_{2}, with V_{1} = ℂ^{n}×{0}, V_{2} = {0}×ℝ, and Hörmander’s finite rank condition (1.1) is fulfilled at step r = 2. The natural non-isotropic dilations on ℍ^{n} are δ_{λ}(z, t) = (λz, λ^{2}t), with corresponding homogeneous dimension Q = 2n + 2. Noting that Lebesgue measure dg = dzdt is a bi-invariant Haar measure on ℍ^{n}, one has in fact dδ_{λ}(g) = λ^{Q}dg.

To introduce the object of our study we recall the enormous influence that understanding minimal surfaces has had in the development of last century’s mathematics, and the central role played by the Bernstein problem. Formulated by S. Bernstein, the latter states that a smooth complete minimal graph in ℝ^{N+1}, i.e., a C^{2} entire solution u : ℝ^{N} → ℝ of the minimal surface equation

div (\frac{Du}{\sqrt{1 + |Du|^{2}}} ) = 0 , (1.4)

must necessarily be an affine hyperplane. Bernstein himself [Be] established this property when N = 2. Almost fifty years later Fleming [Fle] gave a completely different proof still for the case N = 2. His approach inspired a new major development in the subject. In [DG] De Giorgi was able to extend Bernstein’s theorem to the case N = 3. Almgren [Al] subsequently succeeded in treating the case N = 4, and Simons [Sim] finally obtained a solution for N ≤ 7. In their celebrated work [BDG] Bombieri, De Giorgi and Giusti established the surprising result that the Bernstein property fails if N ≥ 8, thus putting an end to the problem in Euclidean space. They proved in fact that: if N ≥ 8 there exist complete minimal graphs in ℝ^{N+1} which are not hyperplanes. For additional references and developments the reader should also consult [O], [N1], [G], [MM], [Bo], [N2], [Si], [GMS],

Minimal surfaces are surfaces of least perimeter, or area, among all those with the same boundary. We recall that in De Giorgi’s formulation of the Plateau problem one seeks to minimize the relative perimeter with respect to a given bounded open set Ω ⊂ ℝ^{n}, subjected to the constraint of coincidence with an assigned Caccioppoli set L outside Ω. In a CC space an appropriate sub-Riemannian version of perimeter generalizing the classical one was introduced in [CDG], and further developed in [GN], see also the papers [BM] and [FSS1] where two equivalent definitions were independently set forth. A basic result established in [GN] was the existence of a generalized solution of the Plateau problem in CC manifolds, see Theorems 1.23 and 1.25 in [GN]. Specialized to the setting of Carnot groups, this result states the following.

Theorem (Sub-Riemannian Plateau problem). Let G be a Carnot group, given a bounded open set Ω ⊂ G and a X-Caccioppoli set L ⊂ G, one can find an X-minimal surface in Ω, that is, an X-Caccioppoli set E ⊂ G such that

P_{X}(E; G) ≤ P_{X}(F; G)

for every F that coincides with L outside Ω.

We recall that in a sub-Riemannian space (M, HM, d) with a distribution of vector fields X = {X_{1}, ..., X_{m}} which (locally) generates the subbundle HM, a X-Caccioppoli set E ⊂ M is a measurable set of locally finite X-perimeter, i.e., P_{X}(E; Ω) < ∞, for every Ω ⊂⊂ M.

The existence of minimal surfaces naturally poses the problem of developing geometric measure theory in CC spaces. Citing from [GN]: “...It is only natural to predict other fascinating directions for research, such as the general structure of X-Caccioppoli sets and the connection between X-minimal surfaces and critical points of some natural area functionals”. Concerning the first point, in their recent paper [FSS2] Franchi, Serapioni and Serra Cassano have recently obtained the following remarkable structure theorem of De Giorgi type for sets with finite perimeter in the Heisenberg group ℍ^{n}.

Theorem (Partial regularity of X-Caccioppoli sets in ℍ^{n}). Let E ⊂ ℍ^{n} be a X-Caccioppoli set, then the reduced boundary of E, ∂_{X}^{∗}E, is X-rectifiable, i.e.,

∂_{X}^{∗}E = N ∪ ⋃_{j=1}^{∞} K_{j} ,

where H^{Q−1}(N) = 0, and K_{j} is a compact subset of a non-characteristic hypersurface S_{j} of class Γ^{1}. Moreover, one has for any g ∈ K_{j} and every ξ ∈ T_{H,g}S_{j}

< ν_{X}^{E}(g), ξ > = 0 ,

where ν_{X}^{E}(g) denotes the generalized horizontal outer normal to E in g, T_{H,g}S_{j} indicates the non-characteristic plane orthogonal to the horizontal normal to S_{j} in g, and H^{Q−1} is the (Q − 1)-dimensional Hausdorff measure in ℍ^{n} constructed with the CC distance.

We recall that, given a C^{1} hypersurface S, a point g_{o} ∈ S is called characteristic if the vector fields which generate the horizontal subbundle HG become tangent to S at g_{o}. Concerning the connection between minimal surfaces and a possible “area functional”, the most basic problem is that of an appropriate notion of mean curvature. Such question has been studied in the papers [Pau] and [DGN], where two (equivalent) notions of X-mean curvature were independently set forth. In [Pau], the second author used the approximation of (ℍ^{1}, d) by Riemannian manifolds (in the Gromov-Hausdorff topology) to characterize and investigate the solutions of the minimal surface and Plateau problems in (ℍ^{1}, d) via approximations by solutions to the analogous problems in the approximating Riemannian manifolds. With regard to the problem addressed in this paper, we recall results of D. Danielli, the first author, and D.M. Nhieu [DGN], where a notion of X-mean curvature is introduced and equations of mean curvature type are studied. In particular, the authors establish various properties of the isoperimetric sets in the Heisenberg group ℍ^{n}, and also propose a sub-Riemannian calculus on hypersurfaces. According to [DGN] a minimal surface is a C^{2} hypersurface with zero X-mean curvature H_{X}, see Definition 2.3. In the same paper the following problem was formulated.

Sub-Riemannian Bernstein problem : Let G be a Carnot group, and consider a complete X-minimal surface S ⊂ G. Under which assumptions on G and S is it true that S must be a vertical hyperplane, i.e., there exist (a_{1}, ..., a_{m}) ∈ ℝ^{m} \ {0}, and γ ∈ ℝ, such that

S = {g ∈ G | a_{1}x_{1}(g) + ... + a_{m}x_{m}(g) = γ} ?

Here, m = dim V_{1}, and ξ_{1}(g) = x_{1}(g)e_{1} + ... + x_{m}(g)e_{m} denotes the projection onto the horizontal layer V_{1} ⊂ 𝔤 of the exponential coordinates exp^{−1}(g) of g ∈ G.

It was conjectured in [DGN] that, at least in low dimension, the Bernstein property should hold provided that the surface S has no characteristic points. In this paper we provide a positive answer to this conjecture when G is the first Heisenberg group ℍ^{1}. Specifically, we prove the following result.

Theorem 1.1. Let S be a non-characteristic complete minimal surface in ℍ^{1}, then there exist a, b, γ ∈ ℝ, with a^{2} + b^{2} ≠ 0, such that

S = {g = (z, t) ∈ ℍ^{1} | ax + by = γ} .

Some comments are in order. The assumption that S be non-characteristic is crucial for the validity of Theorem 1.1. If we remove it, the theorem ceases to be true. For instance, the surface

S = \{(x, y, t) ∈ ℍ^{1} | t = \frac{xy}{2} \} (1.5)

is an example of a non-planar complete minimal graph. We note that its characteristic set is non-empty, and in fact it is given by the line Σ = {(x, y, t) ∈ ℍ^{1} | x = t = 0}. We also emphasize that, in addition to being a X-minimal surface according to Definition 2.4, the surface S also minimizes the X-perimeter. This was discovered in [Pau], where the second author showed that this example, all planes, and a family of surfaces given by quadratic polynomials are minimizers of X-perimeter as they are minimal surfaces in all of the Riemannian approximating spaces to (H, d_{cc}) mentioned above. In the same paper it was also recognized that the presence of characteristic points causes discontinuities in the horizontal Gauss map. This aspect has been separately investigated in [CGP], in the study of the connections between discontinuities of the Gauss map and regularity properties of the boundary of domains, such as non-tangential accessibility.

The motivation for the non-characteristic hypothesis in Theorem 1.1 is twofold. On one hand, there is the close relation between the Bernstein property and the classical Liouville theorem for harmonic functions. Such connection continues to hold in the sub-Riemannian setting. Now, the Liouville property for the sub-Laplacian (1.2) presents a striking new phenomenon with respect to the classical setting, namely that if u is a bounded entire solution of (1.2), then u depends only on the horizontal variables ξ_{1}(g) = x_{1}(g)e_{1} + ... + x_{m}(g)e_{m}. As a consequence, such a function must, in fact, be an ordinary harmonic function in the variables of the horizontal layer, and therefore by the classical Liouville theorem it is constant, see [DGN]. Secondly, when one adapts De Giorgi’s method of the blow-up to the sub-Riemannian setting, one is naturally led in the limit to blow-up sets which are vertical (non-characteristic) planes, see [FSS2] and [DGN], where this phenomenon was independently discovered. Additional evidence of the essential character of the assumption comes from the interesting recent paper [BL]. The latter contains an example proving that in the Heisenberg group ℍ^{n} the transposition of a famous conjecture of De Giorgi (also known as the ε-version of Bernstein problem [AAC]), concerning one-dimensional entire solutions of the Ginzburg-Landau equation

Δu = u^{3} − u ,

fails for the characteristic direction t.

2. The horizontal Gauss map and the minimal surface equation

In this section, we describe two notions of fundamental importance in the study of minimal surfaces in ℍ^{1}, the horizontal Gauss map, and the X-mean curvature. To set the stage, we will suppose ℍ^{1} endowed with a Riemannian metric which makes {X_{1}, X_{2}, T } an orthonormal basis, where the vector fields X_{1}, X_{2} are given by (1.3), and as before [X_{1}, X_{2}] = −T. We consider a surface S ⊂ ℍ^{1} defined by

S = {g ∈ ℍ^{1} | φ(g) = 0} , (2.1)

where φ : ℍ^{1} → ℝ is C^{2}, and there exists an open neighborhood O of S such that

|∇φ(g)| ≥ α > 0 g ∈ O .

We will think of S as the boundary of the open set Ω = {g ∈ ℍ^{1} | φ(g) < 0}, so that the Riemannian outer unit normal to S is given ν = ∇φ/|∇φ|. We denote by X = {X_{1}, X_{2}} the orthonormal basis of Hℍ^{1}, and indicate by

Xφ = X_{1}φ X_{1} + X_{2}φ X_{2}

the horizontal gradient of φ. We recall that the characteristic set of S is given by

Σ = {g ∈ S | X_{i}(g) ∈ T_{g}S, i = 1, 2} = {g ∈ S | |Xφ(g)| = 0} . (2.2)

Definition 2.1. We define the horizontal normal Y_{X} : O → Hℍ^{1}, relative to the basis X, by the formula

Y_{X} = < ν, X_{1} > X_{1} + < ν, X_{2} > X_{2} .

The horizontal Gauss map ν_{X} : O → 𝕊^{1} is defined by

ν_{X} = \frac{Y_{X}}{|Y_{X}|} = \frac{X_{1}φ}{|Xφ|} X_{1} + \frac{X_{2}φ}{|Xφ|} X_{2} ,

whenever

|Y_{X}(g)| = \frac{|Xφ(g)|}{|∇φ(g)|} ≠ 0 .

Throughout the following discussion we will denote by O_{X} the natural domain of ν_{X}, i.e., the open set

O_{X} = {g ∈ O | |Y_{X}(g)| ≠ 0} = {g ∈ O | |Xφ(g)| ≠ 0} .

We note explicitly that Y_{X} is the projection pr_{X}(ν) of the normal ν on the horizontal subbundle Hℍ^{1} ⊂ T ℍ^{1}, whereas ν_{X} is the normalized projection. An obvious, yet important consequence of the definition is

|ν_{X}|^{2} ≡ 1 , in O_{X} . (2.3)

We next consider a function u ∈ C^{1}(O). The following notion, introduced in [DGN], is central to the development of sub-Riemannian calculus on a hypersurface.

Definition 2.2. The tangential horizontal gradient of u on S with respect to X is defined as follows

δ_{X}u =^{def} Xu − < Xu, ν_{X} > ν_{X}

at every point g ∈ S \ Σ.

In the next definition we introduce a sub-Riemannian analog of the classical mean curvature H.

Definition 2.3. The X-mean curvature of S at points of S \ Σ is defined by

(Q − 1) H_{X} = ∑_{j=1}^{2} δ_{X,j} ν_{X,j} ,

where Q = 4 indicates the homogeneous dimension of ℍ^{1}. The X-mean curvature vector is given by

H_{X} = H_{X} ν_{X} .

If g_{o} ∈ Σ we let

H_{X}(g_{o}) = lim_{g→g_{o},g∈S∩O_{X}} H_{X}(g) ,

provided that such limit exists, finite or infinite. We do not define the X-mean curvature at those points g_{o} ∈ Σ at which the limit does not exist.

Definition 2.4. A C^{2} surface S is called X-minimal if its X-mean curvature with respect to any defining function φ of S vanishes everywhere.

It is worth noting that Definition 2.3, combined with (2.3), gives

(Q − 1) H_{X} = ∑_{j=1}^{2} X_{j} ν_{X,j} = ∑_{j=1}^{2} X_{j} \{\frac{X_{j}φ}{|Xφ|}\} . (2.4)

Thereby, the notion of X-minimal surface introduced in [DGN] coincides with that set forth in [Pau], and the corresponding minimal surface equation is given by

∑_{j=1}^{2} X_{j} \{\frac{X_{j}φ}{|Xφ|}\} = 0 . (2.5)

For a function u : ℍ^{1} → ℝ we define the symmetrized horizontal Hessian of u at g ∈ ℍ^{1} as the 2 × 2 matrix with entries

u_{,ij} =^{def} \frac{1}{2} \{X_{i}X_{j}u + X_{j}X_{i}u\} , i, j = 1, 2 . (2.6)

Setting Hess_{X}u = (u_{,ij}), we introduce the important matrix

H = (\begin{matrix}H_{11} & H_{12} \\ H_{21} & H_{22} \end{matrix}) =^{def} \frac{1}{|Xφ|^{3}} \{|Xφ|^{2}Hess_{X}φ − Hess_{X}φ(Xφ) ⊗ Xφ\} . (2.7)

The next proposition follows by a direct computation based on (2.4) and we leave the details to the reader.

Proposition 2.5. At every point of S \ Σ one has

(Q − 1) H_{X} = tr H .

In particular, the X-minimal surface equation for S can be re-written as

tr H = 0 . (2.8)

It is standard to verify that the eigenvalues of H are given by the equations

κ_{1} + κ_{2} = tr H , κ_{1} κ_{2} = det H . (2.9)

At non-characteristic points of S we have Xφ ≠ 0. If we consider the i-th entry of the vector H(Xφ), from the second equality in (2.7) we obtain

\frac{1}{|Xφ|^{3}} \{|Xφ|^{2} φ_{ij} X_{j}φ − φ_{ik} X_{k}φ X_{j}φ X_{j}φ\}

= \frac{1}{|Xφ|^{3}} \{|Xφ|^{2} φ_{ij} X_{j}φ − |Xφ|^{2} φ_{ik} X_{k}φ\} = 0 .

This proves that at points where Xφ ≠ 0, we have H(Xφ) = 0, so that Xφ is an eigenvector of H with corresponding eigenvalue κ_{1} = 0. From (2.9) we conclude that the second eigenvalue of H is given by κ_{2} = tr H. One easily finds that an eigenvector for κ_{2} is given by (H_{12}, H_{22})^{T}. Thanks to det H = 0, another eigenvector for κ_{2} is given by (H_{11}, H_{21})^{T}.

With an homage to classical notation, we now define

p = X_{1}φ ,  q = X_{2}φ ,  W = \sqrt{p^{2} + q^{2}} = |Xφ| .

Moreover, for convenience we let

\overline{p} = \frac{p}{W} , \overline{q} = \frac{q}{W} ,

so that the minimal surface equation (2.5) becomes X_{1}\overline{p} + X_{2}\overline{q} = 0. We also note the following useful equivalent form in terms of the functions p and q

X_{1}p̄ + X_{2}q̄ = \frac{1}{W^{3}} \{q^{2} X_{1}p + p^{2} X_{2}q − pq (X_{1}q + X_{2}p)\} = 0 . (2.10)

We stress that if (2.5) is satisfied, i.e., if S is X-minimal, then from (2.8) and (2.9), we conclude that also κ_{2} = 0, so that H has a double eigenvalue equal to zero with a single eigenvector Xφ = pX_{1} + qX_{2}.

The expression of H with respect to the variables p, q is

H = \frac{1}{W^{3}} (\begin{matrix}q^{2} X_{1}p − pqX_{1}q & p^{2}X_{1}q − pqX_{1}p \\ q^{2}X_{2}p − pqX_{2}q & p^{2}X_{2}q − pqX_{2}p \end{matrix}) . (2.11)

We now change coordinates to the new orthogonal basis {(p, q)^{T} , (q, −p)^{T} } and find that H transforms to the matrix H̃ = T H T^{−1}, where

T^{−1} = (\frac{q}{W^{2}} −\frac{p}{W^{2}}^{\frac{p}{W^{2}} \frac{q}{W^{2}}} )

is the inverse of the matrix for the change from the standard basis of ℝ^{2} to the new one. We obtain

H̃ = \frac{1}{W^{5}} (\begin{matrix}0 & pq(H_{11} − H_{22}) + q^{2}H_{21} − p^{2}H_{12} \\ 0 & q^{2}H_{11} + p^{2}H_{22} − pq(H_{12} + H_{21}) \end{matrix}) ,

where the two zero entries are justified by the equation H((p, q)^{T}) = (0, 0)^{T}. Using (2.11) we find

H̃ = \frac{1}{W^{3}} (\begin{matrix}0 & q^{2}X_{2}p − p^{2}X_{1}q + pq(X_{1}p − X_{2}q) \\ 0 & q^{2}X_{1}p + p^{2}X_{2}q − pq(X_{1}q + X_{2}p) \end{matrix}) . (2.12)

At this point we observe that, with respect to the variables p, q the quantity X_{2}p̄ − X_{1}q̄ can be expressed as

X_{2}p̄ − X_{1}q̄ = \frac{1}{W^{3}} \{q^{2}X_{2}p − p^{2}X_{1}q + pq(X_{1}p − X_{2}q)\} .

Using the latter equation and (2.10) in (2.12), we have proved the following lemma.

Lemma 2.6. At every point of S where Xφ ≠ 0 the matrix H, with respect to the similarity transformation H̃ = T H T^{−1}, can be written as

H̃ = (\begin{matrix}0 & X_{2}p̄ − X_{1}q̄ \\ 0 & X_{1}p̄ + X_{2}q̄ \end{matrix}) .

In particular, if S is X-minimal we obtain at such points

H̃ = (\begin{matrix}0 & X_{2}p̄ − X_{1}q̄ \\ 0 & 0 \end{matrix}) .

Returning to Definition 2.1 we see that with respect to the variables p̄ and q̄ the horizontal Gauss map is given by

ν_{X} = p̄ X_{1} + q̄ X_{2} .

The following result is a direct consequence of Lemma 2.6.

Lemma 2.7. Let S ⊂ ℍ^{1} be a X-minimal surface, then its horizontal Gauss map ν_{X} is constant if and only if X_{2}p̄ − X_{1}q̄ = 0.

3. Non-parametric minimal surfaces with no characteristic points

In this section we examine the possibilities for minimal surfaces with no characteristic points in two local non-parametric cases:

1. When a local patch can be written as a graph over a piece of the xy-plane.

2. When a local patch can only be written as a graph over one of the other coordinate planes.

By the implicit function theorem, we know that the first case corresponds to when φ_{t} ≠ 0 and the second to when φ_{t} = 0.

3.1. Type 1 : Graphs over the xy-plane. In this case, we can find an open set Ω ⊂ ℝ^{2}, and a function f : Ω → ℝ such that, with φ(x, y, t) = t − f(x, y), we have

S = ∂Ω̃ ∩ {g ∈ ℍ^{1} | φ(g) = 0} ,

where

Ω̃ = {g = (x, y, t) ∈ ℍ^{1} | (x, y) ∈ Ω , t < f(x, y)} .

We notice that the Riemannian unit normal to S pointing outward Ω̃ is given by

(3.1)  ν = − \frac{f_{x} − \frac{y}{2}}{\sqrt{1 + (f_{x} − \frac{y}{2})^{2} + (f_{y} + \frac{x}{2})^{2}}} X_{1} − \frac{f_{y} + \frac{x}{2}}{\sqrt{1 + (f_{x} − \frac{y}{2})^{2} + (f_{y} + \frac{x}{2})^{2}}} X_{2}

+ \frac{1}{\sqrt{1 + (f_{x} − \frac{y}{2})^{2} + (f_{y} + \frac{x}{2})^{2}}} T ,

where we have denoted by {X_{1}, X_{2}, T } the orthonormal basis of ℍ^{1} with respect to our fixed Riemannian metric.

Re-writing the minimal surface data under this assumption, we have

\begin{aligned}p & = f_{x} − \frac{y}{2} , q = f_{y} + \frac{x}{2} \\ W & = \sqrt{p^{2} + q^{2}} = \sqrt{|∇f|^{2} + \frac{|z|^{2}}{4} − < ∇f, z^{⊥} >} , \end{aligned} (3.2)

where we have denoted z^{⊥} = y − ix. Moreover, since from (3.2) we see that p̄ = p/W, q̄ = q/W do not depend on t, the minimal surface equation presently reads

div ν_{X} = 0 , (3.3)

where div is the usual divergence operator on ℝ^{2}. Thus, minimal surfaces of this nonparametric type correspond to divergence free unit vector fields on ℝ^{2}. We first use some elementary analysis to investigate these types of vector fields.

Lemma 3.1. If V (x, y) = (v(x, y), w(x, y)) is a C^{1}, unit, divergence free vector field on ℝ^{2}, then the integral curves of the perpendicular vector field V^{⊥} = (w, −v) are straight lines.

Proof: An integral curve γ(t) = (x(t), y(t)) of V^{⊥} satisfies the system

\begin{cases}x′ = w(x, y) ,  y′ = − v(x, y) , \\ x(0) = x ,  y(0) = y . \end{cases} (3.4)

Letting ψ(t) = V^{⊥}(γ(t)) = (w(x(t), y(t)), −v(x(t), y(t))), we find from (3.4)

ψ′(t) = ( < ∇w(γ(t)), V^{⊥}(γ(t)) >, − < ∇v(γ(t)), V^{⊥}(γ(t)) > ) . (3.5)

Since V is assumed to be unit, by differentiating the equation v^{2} + w^{2} = 1 with respect to x and y we obtain

v v_{x} + w w_{x} = 0 , v v_{y} + w w_{y} = 0 . (3.6)

Since V is also divergence free, we have

v_{x} + w_{y} = 0 . (3.7)

Combining (3.6) with (3.7) yields

< ∇w, V^{⊥} > ≡ 0 , < ∇v, V^{⊥} > ≡ 0 . (3.8)

If we use (3.8) in (3.5) we conclude that ψ′ ≡ 0, i.e., the vector field V^{⊥} is constant along its own integral curves. This is equivalent to saying that the integral curves are geodesics (straight lines) in ℝ^{2}. □

Remark 3.2. This proof uses two dimensions quite heavily. The equations (3.6), (3.7) do not have nearly as strong consequences in higher dimensions. This prevents the proof from extending to higher dimensional cases.

Corollary 3.3. If V is a C^{1}, unit, divergence free vector field defined on a connected open set Ω ⊂ ℝ^{2} containing at least two straight lines which are integral curves of V^{⊥}, then V is constant in Ω.

Proof: If we consider an integral curve γ(t) = (x(t), y(t)) of V^{⊥}, and define ψ(t) = V (γ(t)), then as in the proof of Lemma 3.1, using the chain rule and (3.4) we conclude that ψ′(t) ≡ 0, i.e., V is constant along the integral curves of V^{⊥} which are straight lines. Now, any two such lines in the plane either intersect or are parallel. If any two of the lines intersect at a single point P_{o}, then V cannot be defined at P_{o}, as V must be constant along both lines and perpendicular to both lines (we recall that, being unit, V never vanishes). Thus, all the integral curves of V^{⊥} must be parallel, and hence V is a constant vector field in Ω. □

Next, we investigate the size of the patches of the surface which are graphs over the xy-plane under the assumption that the surface has no characteristic points.

Lemma 3.4. Let Ω ⊂ ℝ^{2} be the largest open subset of ℝ^{2} where a non-characteristic Xminimal surface S such as (2.1) can be written as a graph of the form t = f(x, y), (x, y) ∈ Ω. Then, Ω is the union of infinitely many parallel straight lines which are integral curves of ν_{X}^{⊥} = q̄X_{1} − p̄X_{2}. Moreover, ν_{X} is constant on Ω.

Proof: Since we are assuming that the surface has no characteristic points, we know that ν_{X} is defined everywhere on Ω. Moreover, by the maximality hypothesis on Ω, we have

∂Ω = {(x, y) ∈ ℝ^{2}| φ(x, y, t_{o}) = 0, φ_{t}(x, y, t_{o}) = 0, for some t_{o} ∈ ℝ} .

Therefore, if we denote by Γ the lift of ∂Ω to the surface S, then at points of Γ we have

X_{1}φ = φ_{x} , X_{2}φ = φ_{y} ,

which imply, in particular, that Γ consists of points where the horizontal Gauss map ν_{X} coincides with the Riemannian unit normal ν to S.

We first consider the case that ∂Ω contains a segment γ which is transverse to the straight lines which are integral curves of ν_{X}^{⊥}. Denote by γ̃ the lift of γ to the minimal surface S. From what we have observed, ν_{X} restricted to γ̃ coincides with the Riemannian unit normal to S. Thus, if τ is the tangent field along Γ, it must be perpendicular to ν_{X}, hence τ is a linear combination of ν_{X}^{⊥} and T. However, the transversality assumption is violated if τ has no ν_{X} component. Thus, there cannot exist a piece of ∂Ω which is transverse to the integral curves of ν_{X}^{⊥}, and hence ∂Ω must be bounded by integral curves of ν_{X}^{⊥} which, by Lemma 3.1, are straight lines in the plane. Moreover, by Corollary 3.3, we have that these lines must be parallel. So, Ω is a region bounded by two parallel integral curves of ν_{X}^{⊥}. Moreover, again by Corollary 3.3, the fact that the integral curves are parallel forces ν_{X} to be constant on this region. □

To conclude this section, we will focus on how the characteristic locus Σ arises on surfaces of type 1. Consider the Riemannian normal vector,∇φ, to the surface S in question (recall that the normalized projection of this normal onto the span of X = {X_{1}, X_{2}} is ν_{X}). We wish to find all possible normal vector fields η corresponding to a smooth surface which have normalized projection onto the subbundle Hℍ^{1} equal to ν_{X}. This amounts to checking that the 2-plane field determined by η satisfies the Fröbenius condition. To set this up we look for a function α = α(x, y, t) so that

η = α \overline{p} X_{1} + α \overline{q} X_{2} + T . (3.9)

As S is a graph over a region of the xy-plane, α must be independent of t, i.e., α = α(x, y). Note that pr_{X}(η) = α\overline{p}X_{1} + α\overline{q}X_{2} which, when normalized, is ν_{X}. Moreover, any normal vector field can be realized via such an α by dividing by the coefficient of T. Now, the 2-plane field determined by this η is spanned by

V_{1} = X_{1} − α \overline{p} T , V_{2} = X_{2} − α \overline{q} T .

To verify the assumption in Fröbenius’ theorem, we must impose the condition that [V_{1}, V_{2}] = AV_{1} + BV_{2} for some {A, B}. Now,

[V_{1}, V_{2}] = − \{X_{1}(α\overline{q}) − X_{2}(α\overline{p}) + 1\} T . (3.10)

On the other hand, we have

A V_{1} + B V_{2} = AX_{1} + BX_{2} − (αp̄A + αq̄B) T . (3.11)

Equating (3.10) and (3.11) we find that we must have A = B = 0, and therefore,

X_{1}(α\overline{q}) − X_{2}(α\overline{p}) + 1 = 0 . (3.12)

Combining the fact that α = α(x, y) with the assumption that \overline{p} and \overline{q} are constant allows us to simplify equation (3.12) as

\overline{q} α_{x} − \overline{p} α_{y} + 1 = 0 , (3.13)

or,

< ∇α, ν_{X}^{⊥} > = − 1 . (3.14)

In other words, the directional derivative of α in the direction of ν_{X}^{⊥} is −1.

Remark 3.5. Note that if \overline{p} and \overline{q} are not constant, equation (3.12) takes the form

X_{1}α \overline{q} − X_{2}α \overline{p} = − 1 − α (X_{1}\overline{q} − X_{2}\overline{p}) .

Again, using the fact that α is independent of t, this reads

< ∇α, ν_{X}^{⊥} > = − 1 − α curl ν_{X} .

Lemma 3.6. If ν_{X} is constant on Ω, an open unbounded portion of the plane containing a straight line along the vector field ν_{X}^{⊥} then a X-minimal surface

S = {(x, y, t) ∈ ℍ^{1} | (x, y) ∈ Ω , t = f(x, y)} ,

must have characteristic points.

Proof: We will use the definitions and properties of α, η and ν_{X} described above. Consider the restriction ψ(s) = α(x(s), y(s)) of α to an integral curve (x(s), y(s)) of ν_{X}^{⊥}, which is a straight line inside Ω. By (3.14), we have

ψ′(s) = < ∇α(x(s), y(s)), ν_{X}^{⊥}(x(s), y(s), f(x(s), y(s)) > ≡ − 1 ,

so it must be ψ(s) = −s + c, for some constant c. In particular, ψ(c) = α(P_{o}) = 0, where P_{o} = (x(c), y(c)). However, thanks to (3.9), α(P_{o}) = 0 implies η = T. In view of the Definition 2.2, this is equivalent to saying that the point g_{o} = (P_{o}, t_{o}), where t_{o} = f(P_{o}), is a characteristic point of S. □

Remark 3.7. This lemma gives a great deal of information about the surfaces with constant horizontal Gauss map which are graphs over the xy-plane. In particular, we can show that a complete X-minimal graph with constant horizontal Gauss map ν_{X} is determined entirely by prescribing the characteristic set. This and other applications of this idea will be addressed in a future paper.

To summarize the results in this section, we end with a corollary.

Corollary 3.8. If S as in (2.1) is a maximal connected portion of a X-minimal surface, which can be written as a graph over the xy-plane, then S must have characteristic points.

Proof: Suppose that S had no characteristic points, then Lemma 3.4 would imply that S is defined over a region Ω ⊂ ℝ^{2} bounded by two parallel straight lines, which are integral curves of ν_{X}^{⊥}. Moreover, S has constant unit horizontal Gauss map over Ω. Lemma 3.6 implies that S must have characteristic points, thus leading to a contradiction. This proves the corollary. □

Corollary 3.9. If S as in (2.1) is a complete X-minimal surface with no characteristic points, then φ_{t} = 0.

Proof: We argue by contradiction and assume that there exist a point g_{o} ∈ S such that φ_{t}(g_{o}) ≠ 0. By continuity the same inequality is true in an open set U ⊂ ℍ^{1} containing g_{o}. The implicit function theorem would then yield a patch of surface of type 1. But this would contradict Corollary 3.8, since according to the latter no such patch can exist. □

3.2. Type 2 : Non-parametric surfaces with φ_{t} = 0. If φ_{t} = 0 on a neighborhood N, the φ(x, y, t) = φ_{o}(x, y) and S is a ruled surface whose generators are straight lines perpendicular (in ℝ^{3}) to a curve γ ⊂ ℝ^{2} × {0}. Using the implicit function theorem in ℝ^{2}, we can locally write γ as either y = f(x) or x = g(y). For y = f(x), we have the patch of surface defined by

f(x) − y = 0

In this case, p = f′(x), q = −1 and

ν_{X} = \frac{f′(x)}{\sqrt{1 + f′(x)^{2}}} X_{1} − \frac{1}{\sqrt{1 + f′(x)^{2}}} X_{2} .

Using the assumption that S is X-minimal, we take the divergence of ν_{X} to find that

0 = div ν_{X} = \frac{d}{dx} (\frac{f′}{\sqrt{1 + (f′)^{2}}} ) = \frac{f′′}{(1 + (f′)^{2})^{\frac{3}{2}}} ,

which gives f′′ = 0. This shows that the planar curve γ must be a straight line and ν_{X} is constant. A similar computation yields the same conclusion in the case x = g(y). Thus, surfaces of type 2 must be composed of pieces of vertical planes.

4. Proof of Theorem 1.1

At this point, we are ready to establish our main result. Let S ⊂ ℍ^{1} be a complete X-minimal surface with empty characteristic set Σ. Corollary 3.9 implies that φ_{t}(g) = 0 for all g ∈ ℍ^{1}. Therefore S is a surface of type 2. The argument above shows that S is then the union of pieces of vertical planes. Recalling the hypothesis that S is C^{2}, we see that the surface must be composed of a single vertical plane. In other words, it can be written as the translate of a line in the plane along the T axis. Moreover, if the line in the plane is given by the equation ax + by = 0, then S is given as ax + by = 0 as a subset of ℍ^{1}. □

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Purdue University, West Lafayette, IN 47906

   E-mail address: garofalo@math.purdue.edu

Dartmouth College, Hanover, NH 03755

   E-mail address: pauls@hilbert.dartmouth.edu