ESAIM: COCV 15 (2009) 839–862 DOI: 10.1051/cocv:2008051
ESAIM: Control, Optimisation and Calculus of Variations www.esaim-cocv.org
MINIMAL SURFACES IN SUB-RIEMANNIAN MANIFOLDS AND STRUCTURE OF THEIR SINGULAR SETS IN THE (2, 3) CASE
Nataliya Shcherbakova 1 Abstract. We study minimal surfaces in sub-Riemannian manifolds with sub-Riemannian structures of co-rank one. These surfaces can be defined as the critical points of the so-called horizontal area functional associated with the canonical horizontal area form. We derive the intrinsic equation in the general case and then consider in greater detail 2-dimensional surfaces in contact manifolds of dimension 3. We show that in this case minimal surfaces are projections of a special class of 2-dimensional surfaces in the horizontal spherical bundle over the base manifold. The singularities of minimal surfaces turn out to be the singularities of this projection, and we give a complete local classification of them. We illustrate our results by examples in the Heisenberg group and the group of roto-translations. Mathematics Subject Classification. 53C17, 32S25. Received August 29, 2007. Revised April 20, 2008. Published online July 19, 2008.
Introduction In the classical Riemannian geometry minimal surfaces realize the critical points of the area functional with respect to variations preserving the boundary of a given domain. In this paper we study the generalization of the notion of minimal surfaces in sub-Riemannian manifolds known also as the Carnot-Carath´eodory spaces. This problem was first introduced in the framework of Geometric Measure Theory for the Lie groups. Mainly the obtained results [6,7,10–12,15,16] concern the Heisenberg groups, in particular H1 ; in [9,13] the authors were studying the group E2 of roto-translations of the plane, in [7] there were also obtained some results for the case of S 3 . In [7], followed by just appeared paper [8], the authors considered the problem in a more general setting and introduced the notion of minimal surfaces associated with CR structures in pseudohermitian manifolds of any dimension. In this paper we develop a different approach using the methods of sub-Riemannian geometry. Though in particular cases of Lie groups Hm , E2 and S 3 the surfaces introduced in [7] are minimal also in the subRiemannian sense, in general it is not true. The sub-Riemannian point of view on the problem is based on the following construction. Keywords and phrases. Sub-Riemannian geometry, minimal surfaces, singular sets. 1
SISSA/ISAS, via Beirut 2-4, 34100, Trieste, Italy.
[email protected]
Article published by EDP Sciences
c EDP Sciences, SMAI 2008
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Consider an n-dimensional smooth manifold M and a co-rank 1 smooth vector distribution Δ in it (“horizontal” distribution). It is assumed that the sections of Δ are endowed with a Euclidean structure, which can be described by fixing an orthonormal basis of vector fields X1 , . . . , Xn−1 on Δ (see [5]). Then Δ defines a subRiemannian structure in M . In this case M is said to be a sub-Riemannian manifold. Given a sub-Riemannian structure there is a canonical way to define a volume form μ ∈ Λn M associated with it. In addition, for any hypersurface W ⊂ M the horizontal unite vector ν such that for any bounded domain Ω ⊂ W iν μ = sup iX μ, Ω
X∈Δ XΔ =1
Ω
plays the role of the Riemannian normal in the classical case, and the n − 1-form iν μ defines the horizontal area form on W . All these notions are direct generalizations of the classical ones in the Riemannian geometry (i.e., in the case Δ ≡ T M ). Going further in this direction, we define sub-Riemannian minimal surfaces in M as the critical points of the functional associated with the horizontal area form. It turns out that these surfaces satisfy the following intrinsic equation = 0, (0.1) (d ◦ iν μ) W \Σ
where Σ = {q ∈ W | Tq W ⊆ Δq } is the singular set of W , which along with the singular points of W contains also the so-called characteristic points, i.e., the points where W is tangent to Δ. The described construction does not require the existence of any additional global structure in M , and can be generalized for sub-Riemannian structures of greater co-rank. The existence of the singular set Σ is one of the main difficulties of the problem. In general, the set Σ can be quite large and have its own non-trivial intrinsic geometry. In Section 2 of this paper we show how this problem can be resolved in the case of 2-dimensional surfaces in 3-dimensional contact manifolds. It turns out that in the (2, 3) case, due to the relatively small dimension, there is an elegant way to extend the definition of a sub-Riemannian minimal surface over its singular set. Namely, in this case the intrinsic geometry of a surface W is encoded in its characteristic curves γ : [0, T ] → W such that γ(t) ˙ ∈ Tγ(t) W ∩ Δγ(t) for all t ∈ [0, T ]. The vector field η (the characteristic vector field) tangent to characteristic curves is Δ-orthogonal to the sub-Riemannian normal of W . We show that actually this vector field is a projection onto M of a special invariant vector field V in the horizontal spherical bundle SΔ M over M . In contrast with ν, the vector field V is well defined everywhere in SΔ M , and moreover, minimal surface equation (0.1) can be transformed into a quasilinear equation whose characteristics are exactly the integral curves of V . In particular, in this way one can define η also on Σ as the projection of V . These observations motivated the key idea of the present paper. By introducing an additional scalar parameter ϕ we show that the highly degenerate PDE (0.1) can be transformed into a system of ODEs on SΔ M , associated with the vector field V : q¯˙ = V (¯ q ), q¯ ∈ SΔ M . This allows us to consider the sub-Riemannian minimal surfaces in the (2, 3) case as the projections of a certain class of 2-dimensional surfaces in SΔ M foliated by the integral curves of V (the generating surfaces of the sub-Riemannian minimal surfaces). By varying initial conditions, one can provide a local characterization of all possible sub-Riemannian minimal surfaces, together with their singular sets. From the point of view that we develop in this paper the set Σ may contain • projections of the singular points of the generating surfaces; • singularities of the projection of the generating surface onto the base manifold M (singular characteristic points); • regular points of the projection of the generating surface onto M (regular characteristic points). In this work we focus out attention on the case of regular generating surfaces. Then their projections on M can have characteristic points of the last two types. We show that regular characteristic points form simple
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singular curves. Moreover, generically a small neighborhood of a sub-Riemannian minimal surface containing a singular characteristic point q has a structure of Whitney’s umbrella, and the point q gives rise to a curve of self-intersections and a pair of simple singular curves (see Thm. 2.9 in Sect. 2). Other types of singularities may appear in some particular cases. For instance, certain minimal surfaces may contain isolated characteristic points or entire curves of singular points (strongly singular curves). All described types of singularities are already present in the Heisenberg group H1 . In Section 3 we exhibit several examples to illustrate our results. There are still many natural questions concerning the structure of sub-Riemannian minimal surfaces, which are not touched in the present paper. First of all, we consider only surfaces generated by smooth curves in SΔ M . Then, we do not discuss the problem of gluing together smooth surfaces, as well as the existence of a smooth minimizing surface spanning a given contour. We leave these problems as the topics for the further studies.
1. Minimal surfaces in sub-Riemannian manifolds 1.1. Sub-Riemannian structures and associated objects Let M be an n-dimensional smooth manifold. Consider a co-rank 1 vector distribution Δ in M : Δq , Δq ⊂ Tq M, q ∈ M. Δ= q∈M
By definition, a sub-Riemannian structure in M is a pair (Δ, ·, · Δ ), where ·, · Δ denotes a smooth family of Euclidean inner products on Δ. In what follows we will call Δ the horizontal distribution and keep the same notation Δ both for the vector distribution and for the associated sub-Riemannian structure. Let Xi , i = 1, . . . , n − 1, be a horizontal orthonormal basis: Δq = span{X1 (q), . . . , Xn−1 (q)},
q ∈ M,
q ∈ M, i, j = 1, . . . , n − 1. Xi (q), Xj (q) Δ = δij , By Θ ∈ Λn−1 Δ we will denote the Euclidean volume form on Δ. Throughout this paper we assume that the fields Xi , i = 1, . . . , n − 1, are defined everywhere on M . In what follows we will also assume that Δ is bracket-generating in the sense that span{Xi (q), [Xi , Xj ](q), i, j = 1, . . . , n − 1} = Tq M,
q ∈ M.
Hereafter the square brackets denote the Lie brackets of vector fields. If Δ is bracket-generating, then by the Frobenius theorem it is completely non-holonomic, i.e., there is no invariant sub-manifold in M whose tangent space coincides with Δ at any point. We can also define the distribution Δ as the kernel of some differential 1-form. Let ω ∈ Λ1 M be such a form: Δq = Ker ωq = {v ∈ Tq M | ωq (v) = 0},
q ∈ M.
It is easy to check that Δ is bracket-generating at q ∈ M if and only if dq ω = 0. Though in general the form ω is defined up to a multiplication by a non-zero scalar function, there is a canonical way to choose it by using the Euclidean structure on Δ. Indeed, by standard construction the Euclidean structure on Δ can be extended to the spaces of forms Λk Δ, k ≤ n − 1. In particular, for any 2-form σ we set ⎞1 ⎛ ⎜
σq Δ = ⎝
n−1 i,j=1 i 2 and m2 > 1, the structure of the singular curves in the case φm 0 = 0, 2 = 0 will be analogous to one of the two situations (2.25) and (2.27) described above. The rigorous proof φm 1 in each singular case can be done in the same manner as above modulo a suitable modification, but this lies behind the scope of this paper. To conclude this discussion we have to consider the following two particularly degenerate cases. Namely, if φ0 = φ1 ≡ 0, the generating curve Γ∗ is everywhere tangent to the plane span{∂ϕ , V }, though (2.20) may be still verified. The projected curve γ∗ = π[Γ∗ ] consists of singular characteristic points (strongly singular curve), and any narrow enough stripe along γ∗ contains no other characteristic points of W . Indeed, the equation φ(t, s) = 0 has only trivial solution t = 0 and moreover, φ(t, s) = 0 for small enough t > 0 since φ2 (0, 0) = 0. The sub-Riemannian minimal surface generated by a purely “vertical curve” q¯(s) = (q0 , ϕ(s)) passing through q¯0 contains an isolated singular point q0 . In this case the whole strongly singular curve γ∗ collapses into a single point q0 = π[¯ q0 ]. The same argument as before, after an obvious modification, implies that q0 is an isolated singular point. Since, by assumption, q¯0 is a regular point of W, the component ξ4 (s) = ϕ (s) = 0. The characteristic vector η(q0 ) “rotates” monotonically in the plane Tq(0,0) W . Moreover, if we consider a closed curve γ(s) = π[eεV q¯(s)] for some fixed and sufficiently small ε, then to a complete revolution of the point along γ there corresponds one complete revolution of the vector η(γ(s)). In particular, it follows that the index of the isolated singular point is +1 [4]. We now conclude this section by the following classification of characteristic points of sub-Riemannian minimal surfaces: q0 ] is a Let W ∈ M be as in (2.8) and assume (2.20). Let q¯0 is be a regular point of W such that q0 = π[¯ characteristic point of the projected surface W = π[W]. Then in a small neighborhood of q0 ∈ W there realizes one of the following situations: • q0 is a regular point of W . In a small enough neighborhood of q the surface W contains a unique simple singular curve passing through q0 ; • q0 is a singular point of W . In addition, – it can be an isolated singular characteristic point; – in a small neighborhood of q0 in W there is a strongly singular curve passing through q0 ; – in a small neighborhood of q0 in W there is a smooth curve γ, which contains q0 and has a common tangent with the characteristic passing through this point. This curve has form (2.25), and its branches γ ± consist of regular characteristic points; – in a small enough neighborhood of q0 the surface W contains a pair of curves of form (2.25), which are both tangent to the characteristic passing through q0 and contain no other singular points besides q0 .
3. Example: singular sets of sub-Riemannian minimal surfaces in H1 We now illustrate the results of the previous section by examples of minimal surfaces associated with the Heisenberg distribution in H1 . Some of the facts that will be discussed below were already noticed in [6,7]. We limit ourselves to consider only the sub-Riemannian minimal surfaces generated by smooth generating curves. Due to the explicit parameterization (2.15) knowing the generating curve Γ(s), s ∈ R, is enough to reconstruct the whole projected surface together with its singular set.
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First of all observe that in the H1 case the Lie brackets of order greater than 2 of the generalized characteristic vector field V with the vector fields X1 , X2 and X3 vanish. This implies that for any vector field ξ ∈ Vec(M ) the right-hand sides of (2.18) contain at most quadratic terms w.r.t. t. More precisely, q ) − t ξ4 (¯ q ) sin ϕ(¯ q ), α1 (t) = ξ1 (¯
α2 (t) = ξ2 (¯ q ) + t ξ4 (¯ q ) cos ϕ(¯ q ),
α3 (t) = ξ3 (¯ q ) + t(ξ1 (¯ q ) sin ϕ(¯ q ) − ξ2 (¯ q ) cos ϕ(¯ q )) −
t2 ξ4 (¯ q ), 2
(3.1)
α4 (t) = ξ4 (¯ q ). Moreover, ζ ∧ V (¯ q ) = 0 for ζ ∈ Tq¯M if and only if ζ1 sin ϕ − ζ2 cos ϕ = 0,
ζ3 = ζ4 = 0.
Proposition 3.1. Let s → Γ(s) be an integral curve of a smooth vector field ξ ∈ Vec(M ) such that ξ ∧ V Γ = 0. Let W be a sub-Riemannian minimal surface generated by Γ. Then if for some sˆ = 0, and ξ4 (ˆ s) = 0, (3.2) 2ξ3 ξ4 + (ξ1 sin ϕ − ξ2 cos ϕ)2 Γ(ˆ s)
then the point q(tˆ, sˆ) is a singular point of W for ξ1 sin ϕ − ξ2 cos ϕ tˆ = · ξ4 Γ(ˆ s)
(3.3)
Moreover, the singular points of the type (tˆ, sˆ), where tˆ is defined by (3.3), are the only singular points of W . Proof. The point q∗ = π[¯ q (tˆ, sˆ)] is a singular point of W iff ˆ dim span{π∗ [ξ t ], η}
q∗
< 2,
i.e., s)α1 (tˆ; sˆ) = (cos ϕ(ˆ s)ξ2 (ˆ s) − sin ϕ(ˆ s)ξ1 (ˆ s)) + tξ4 (ˆ s) = 0, cos ϕ(ˆ s)α2 (tˆ; sˆ) − sin ϕ(ˆ ˆ ˆ cos ϕ(ˆ s)α3 (t; sˆ) = 0, sin ϕ(ˆ s)α3 (t; sˆ) = 0. Taking into account (3.1), one can easily see that (3.2) and (3.3) imply (3.4) and vice versa.
(3.4)
s) = 0 in (3.2) becomes clear from the following observation. If Remark 3.2. The role of the condition ξ4 (ˆ ξ4 (ˆ s) = 0 and cos ϕ(ˆ s)ξ2 (ˆ s) − sin ϕ(ˆ s)ξ1 (ˆ s) = 0 for some sˆ, then ξ3 (ˆ s) = 0 by the non-degeneracy assumption ξ ∧ V Γ = 0, and hence q(t, sˆ) is a regular point of W for any t. The left-hand side of (3.2) is the discriminant D(s) = (ξ1 (s) sin ϕ(s) − ξ2 (s) cos ϕ(s))2 + 2ξ3 (s)ξ4 (s)
(3.5)
t2 ξ4 (s) = 0, 2
(3.6)
of the quadratic equation ξ3 (s) + t(ξ1 (s) sin ϕ(s) − ξ2 (s) cos ϕ(s)) −
which describes the characteristic points of W . If ξ4 (s) = 0, then it has at most 2 real roots: ξ1 (s) sin ϕ(s) − ξ2 (s) cos ϕ(s) ± D(s) t± (s) = ∗ ξ4 (s)
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provided D(s) ≥ 0. According to Proposition 3.1, the singular points are characterized by the roots of the equation D(s) = 0. The two branches of simple singular curves described in Theorem 2.9 have the form ± x± ∗ (s) = t∗ (s) cos ϕ(s) + x0 (s),
y∗± (s) = t± ∗ (s) sin ϕ(s) + y0 (s),
(3.7)
t± ∗ (s) (x0 (s) sin ϕ(s) − y0 (s) cos ϕ(s)) + z0 (s) 2 and are defined for s such that D(s) > 0. z∗± (s) =
Example 3.3. The curve Γ(s) = (0, 0, s, s) for s ∈ R generates the counter-clockwise helicoid q(t, s) = (t cos s, t sin s, s), whose rulings are parallel to the (x, y)-plane. Since ξ1 (s) = ξ2 (s) = 0 and √ ξ3 (s) =√ξ4 (s) = 1, we found that D(s) = 2. Hence the singular set consists of two simple singular curves (± 2 cos s, ± 2 sin s, s) which never meet each other (Fig. 1). It is easy to see that all counter-clockwise helicoids have the same structure of singular set. Notice, that the clockwise oriented helicoids, for instance, the one generated by ˜ Γ(s) = (0, 0, −s, s), contain no singular curves neither singular points. All these facts obviously hold true for all helicoids with rulings parallel plane Δp and generated to any contact by the curve Γ(s) = (p + sw, ϕ0 ± s), where p = (x0 , y0 , z0 ) and w = y20 , − x20 , 1 . Indeed, these helicoids can be obtained just by shifting the origin to the point p in the previous example. In particular, it follows that the helicoids of the described class have no singular points. A similar result was also obtained in [6]. Example 3.4. Consider the curve Γ(s) = (0, 0, 16 s3 , s) for s ∈ [0, 2π]. This curve is the integral curve of the vector field ϕ2 ξ= X3 + X4 2 starting at the point q¯0 = (0, 0, 0, 0). In particular, for any s ∈ [0, 2π] we have ξ1 (s) = ξ2 (s) = 0, ξ3 (s) = 12 s2 and ξ4 (s) = 1, so that D(s) = s2 . Therefore q0 is the unique singular point of the resulting minimal surface. There are two pairs of simple singular curves g ± (s) = (±|s| cos s, ±|s| sin s, 16 s3 ), s = 0, whose concatenations (with the point q0 ) are smooth curves. In Figure 2 we show the general look of this surface (Fig. 2a) and the structure of its singular set (Fig. 2b). In worth to notice that for this example the genericity condition (2.21) fails and the singular point q0 does not give rise to a self-intersection. We will give a simple criterion for the existence of self-intersections for the minimal surfaces in H1 at the end of this section. Example 3.5. The surface generated by the curve s Γ(s) = sin s, − cos s, 1 + , s , 2 contains a strongly singular curve, actually is it the curve γ = π[Γ]. Indeed, Γ is the integral curve of the field ξ = cos ϕX1 + sin ϕX2 + X4 . Notice, that despite π∗ [ξ] = η condition (2.20) is still verified thanks to the fact that the forth component of the field ξ is different from zero. One can easily check that D ≡ 0, t∗ (s) ≡ 0, and the strongly singular curve is tangent to the characteristic field at every point (Fig. 3). As we have already seen, sub-Riemannian minimal surfaces generated by purely vertical lines (x0 , y0 , z0 , s) have isolated singular points. Indeed, it follows that in this case the resulting minimal surface is formed by a one-parametric family of ellipses (t cos ϕ + x0 , t sin ϕ + y0 , 2t (x0 sin ϕ − y0 cos ϕ) + z0 ), t ≥ 0, that fills the whole plane Δp , p = (x0 , y0 , z0 ). In particular, it follows that in H1 the only sub-Riemannian minimal surfaces having isolated characteristic points are planes. Moreover, they can contain at most one isolated singular point (this facts was first noticed in [7]), since they are formed by characteristics which are straight lines, which all intersect at the singular point. According to Theorem 2.9, any singular characteristic point of type (2.23) is a starting point of a germ of a curve of self-intersections of Whitney’s umbrella type. The next proposition describes all possible selfintersections for minimal surfaces in H1 case. This result is an immediate consequence of the explicit parameterization (2.15). Here we denote v(s) = η(γ(s)) ∈ R3 .
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Figure 2. An example of a surface with a pair of simple singular curves and one singular point. Proposition 3.6. Let W = {q(t, s) : s ∈ [−δ1 , δ2 ], t ∈ R} be a piece of sub-Riemannian minimal surface in H1 corresponding to the generating curve Γ(s) = (γ(s), ϕ(s)). If there exist a pair a, b ∈ [−δ1 , δ2 ] such that a = b and a pair of numbers τ1 , τ2 ∈ R such that γ(a) − γ(b) = τ1 v(a) + τ2 v(b),
(3.8)
then W contains a point of self-intersection q∗ = q(a, −τ1 ) = q(b, τ2 ). Example 3.7. Let Γ(s) = (−2 cos s, −2 sin s, cos s, s) be the generating curve and denote γ = π[Γ]. We have ξ1 (s) = 2 sin s, ξ2 (s) = −2 cos s, ξ3 (s) = −2−sin s and ξ4 (s) = 1. First we find the singular points. In the present case D(s) = −2 sin s, i.e., according to Proposition 3.1, the sub-Riemannian minimal surface generated by Γ contains two singular √ points q(2, 0) and q(2, π). At these points two simple singular curves q(t± (s), s) branch ± out. Here t (s) = 2± −2 sin s for s ∈ (π, 2π). The concatenation of these simple singular curves (with singular
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Figure 3. A piece of a surface containing a strongly singular curve.
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Figure 4. Illustration to Example 3.7.
points) is a unique closed smooth curve. Moreover, at both singular points the genericity assumption (2.21) is verified, so at these points we expect to have two germs of self-intersections. In order to complete the picture we apply the criterion of Proposition 3.6. Assume that a and b are two distinct numbers on [0, 2π]. Notice that v(s) = (cos s, sin s, 0) for all s. Therefore the pairs a and b, which may generate self-intersections, necessarily satisfy the relation cos a − cos b = 0. The non-trivial pairs of solutions of this equation are given by the pairs (a, b) where a = −b mod 2π. Substituting this condition into (3.8) and performing all necessary simplifications we find the following non-trivial pairs of solutions: a = −b mod 2π,
τ1 = −2, τ2 = 2.
Thus the surface in question intersects itself along the segment connecting the singular points q(2, 0) and q(2, π). Moreover, for a = π2 and b = 3π 2 any pair of numbers τi satisfying τ1 − τ2 + 4 = 0 is a solution of equation (3.8).
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Therefore the whole line passing throw the points γ( π2 ) = (0, −2, 0) and γ( 3π 2 ) = (0, 2, 0) is a line of selfintersection. In Figure 4a we show how the sub-Riemannian minimal surface described in this example looks like, and in Figure 4b its singular set (bold curves). Acknowledgements. The author is grateful to Prof. A. Agrachev whose vision of the problem inspired this work.
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