Strongly stable surfaces in sub-Riemannian $3$-space forms

A surface of constant mean curvature (CMC) equal to $H$ in a sub-Riemannian $3$-manifold is strongly stable if it minimizes the functional $\text{area}+2H\,\text{volume}$ up to second order. In this paper we obtain some criteria ensuring strong stability of surfaces in Sasakian $3$-manifolds. We also produce new examples of $C^1$ complete CMC surfaces with empty singular set in the sub-Riemannian $3$-space forms by studying those ones containing a vertical line. As a consequence, we are able to find complete strongly stable non-vertical surfaces with empty singular set in the sub-Riemannian hyperbolic $3$-space $\mathbb{M}(-1)$. In relation to the Bernstein problem in $\mathbb{M}(-1)$ we discover strongly stable $C^\infty$ entire minimal graphs in $\mathbb{M}(-1)$ different from vertical planes. These examples are in clear contrast with the situation in the first Heisenberg group, where complete strongly stable surfaces with empty singular set are vertical planes. Finally, we analyze the strong stability of CMC surfaces of class $C^2$ and non-empty singular set in the sub-Riemannian $3$-space forms. When these surfaces have isolated singular points we deduce their strong stability even for variations moving the singular set.


Introduction
Let M be a Sasakian sub-Riemannian 3-manifold (to be defined in Section 2.1). From the first variation formulas, see for instance [25, Sect. 4.1], a surface Σ in M with ∂Σ = ∅ which is a critical point of the (sub-Riemannian) area A for any variation preserving the associated volume V has constant mean curvature H in the sense of (3.2). From here, it is easy to deduce that Σ satisfies (A + 2HV ) ′ (0) = 0 for any variation. Following standard terminology we will say that Σ is strongly stable if, furthermore, we have (A + 2HV ) ′′ (0) 0 for any variation. In particular, it is clear that A ′′ (0) 0 under volume-preserving variations. In the minimal case (H = 0) the strong stability is the classical condition that A ′′ (0) 0 for any variation.
In recent years constant mean curvature (CMC) surfaces, stability properties and Bernstein type problems have been extensively investigated in sub-Riemannian manifolds. The present paper aims to study these topics inside Sasakian 3-manifolds, focusing on the simplest and most symmetric ones: the space forms, defined in Section 2.2 as complete Sasakian sub-Riemannian 3-manifolds of constant Webster scalar curvature κ. In the simply connected case, a result of Tanno [34] establishes that the space form M(κ) is, up to isometries, the first Heisenberg group H 1 for κ = 0, the group of unit quaternions S 3 ⊂ R 4 for κ = 1, and the universal cover of the special linear group SL(2, R) for κ = −1. As the authors showed in [25, Sect. 2.2], standard arguments in Riemannian geometry produce 3-dimensional space forms with non-trivial topology.
The analysis of the stability condition requires an explicit expression for (A + 2HV ) ′′ (0). The second variation of the sub-Riemannian area has appeared in several contexts, see for instance [9], [3], [12], [26], [28], [23], [8], [22], [17], [27] and [20]. The computation of A ′′ (0) by differentiation under the integral sign in (2.4) involves a technical problem since the deformation could move the singular set Σ 0 , which consists of the points in Σ where the integrand |N h | vanishes. In [25] we solved this difficulty by assuming the employed deformations to be admissible, which allows to apply the Leibniz's rule for differentiating under the integral sign. For a CMC surface Σ inside a a Sasakian sub-Riemannian 3-manifold M , we were able to prove that for any admissible variation under suitable integrability conditions. In the previous formula, w denotes the normal component of the velocity vector field associated to the variation, div Σ G is the divergence relative to Σ of a certain tangent vector field G along Σ − Σ 0 , see [25, Eq. (7.1)], and Q is the (sub-Riemannian) index form of Σ defined in (3.4). For the particular case of variations supported on Σ − Σ 0 the divergence term vanishes, and we can use integration by parts to deduce where L is the second order differential operator given in (3.7). This operator plays in our setting the same role as the Jacobi operator introduced by Barbosa, do Carmo and Eschenburg [2] for CMC hypersurfaces in Riemannian manifolds. By analogy with the Riemannian situation we define a (sub-Riemannian) Jacobi function on Σ as a function ψ ∈ C 2 (Σ) for which L(ψ) = 0.
The second variation formula provides a bridge between the stability properties of Σ and the operator L. There is a vast literature exploring this connection in the Riemannian context. A relevant result in this line is a theorem of Fischer-Colbrie and Schoen [16] asserting that a CMC hypersurface having a positive Jacobi function is strongly stable. Recently, Montefalcone [27] has derived a sub-Riemannian counterpart of this theorem for minimal hypersurfaces with empty singular set in Carnot groups. In Section 3 of the present paper we provide a similar stability criterion for a CMC surface Σ with Σ 0 = ∅ in a Sasakian sub-Riemannian 3-manifold. Indeed, from the second variation formula (1.1) and the expression in equation (3.12) for the index form Q, we show in Theorem 3.5 strong stability of Σ provided there is a nowhere vanishing function ψ ∈ C 2 (Σ) such that ψ L(ψ) 0. Since the normal component of the Reeb vector field T in M is always a Jacobi function (Lemma 3.4) we deduce in Corollary 3.6 that, if Σ does not contain vertical points (those where T is tangent to Σ), then Σ is strongly stable. Moreover, an immediate application of Theorem 3.5 yields strong stability of Σ whenever the function |N h | satisfies L(|N h |) 0. It is worth mentioning that the relation between L(|N h |) and the stability properties of minimal surfaces has been investigated in several works, see [13], [23], [14], [33], [17], [18] and [20].
In Section 4 we study complete CMC surfaces with empty singular set in the model spaces M(κ) (though our construction and results can be extended to arbitrary sub-Riemannian 3-space forms). The existence of smooth stable examples in M(κ) is very restrictive due to some rigidity results that we know summarize. In the Heisenberg group M(0) any strongly stable minimal surface Σ with Σ 0 = ∅ is a vertical plane. By assuming C 2 regularity of Σ this was proved in [23] and [14] after some partial characterizations in [3] and [13]. In the C 1 case the statement has been recently obtained by Galli and Ritoré [20]. On the other hand, complete stable CMC surfaces of class C 2 and empty singular set in M(κ) were analyzed by the second author in [33]. More precisely, if Σ is such a surface, then either κ = 0 and Σ is a vertical plane, or κ = −1 and the mean curvature H of Σ satisfies H 2 1. Moreover, in the extremal case H 2 = 1, the surface Σ must be a vertical horocylinder. This result suggests that the stability condition is much less restrictive in the hyperbolic model M(−1) whenever 0 H 2 < 1. Motivated by this fact, the second author conjectured in [33,Re. 6.10] the existence of complete stable non-vertical CMC surfaces in M(−1) having empty singular set and mean curvature H ∈ [0, 1). In Section 4 we show existence of a continuum of such surfaces by means of a geometric construction which provides, at the same time, new examples of complete CMC surfaces with empty singular set in M(κ).
Let us motivate and explain our construction in more detail. Take a complete vertical surface Σ in M(κ) of constant mean curvature H. By the characterization result in [33,Prop. 4.5] the surface Σ is generated by some CC-geodesic γ of curvature H in M(κ) by means of vertical translations. Moreover, after some computations we get L(|N h |) = 4(H 2 + κ) along Σ. Hence, our stability criterion in Corollary 3.6 (iii) implies that Σ is strongly stable provided H 2 + κ 0. Now, the idea is to produce a deformation Σ ′ of Σ when H 2 + κ < 0, in such a way that Σ ′ has empty singular set (though it is possibly non vertical) and still satisfies L(|N h |) 0. Observe that a natural deformation of Σ arises when an arbitrary one-parameter family of vertical screw motions of M(κ) is acting on the generating CC-geodesic γ. In an equivalent way, we are led to introduce in (4.1) the sets Σ λ,σ obtained when we leave orthogonally from the vertical axis of M(κ) by CC-geodesics of the same curvature λ and initial velocity determined by a C n angle function σ with n 1. We must remark that the sets Σ 0,σ in M(0) were previously studied by Cheng and Hwang [7] in their classification of properly embedded minimal surfaces of helicoid type. Moreover, when cos σ = 0, the sets Σ 0,σ coincide with the entire graphical strips introduced by Danielli, Garofalo, Nhieu and Pauls [13] when they solved the Bernstein problem in M(0) for C 2 graphs with empty singular set.
In Theorem 4.1 we employ CC-Jacobi fields to prove some properties of Σ λ,σ . We analyze when these sets provide immersed surfaces with empty singular set. In such cases, we show that Σ λ,σ is complete, orientable, and has constant mean curvature λ. By the aforementioned rigidity results none of these surfaces is strongly stable in M(κ), κ 0, unless κ = 0, λ = 0 and σ is constant. Nevertheless, in the hyperbolic model M(−1) we are able to give a sufficient condition on the angle function σ ensuring that L(|N h |) 0, which ensures by Corollary 3.6 (iii) that Σ λ,σ is strongly stable. This allows us to deduce in Example 4.3 that, for any H ∈ [0, 1), there is a complete strongly stable non-vertical surface Σ in M(−1) with Σ 0 = ∅ and constant mean curvature H. Indeed, an embedded right handed helicoid is a minimal example in the previous conditions. We would like to emphasize that the existence of these examples is in clear contrast with the situation in M(0), where previous results guarantee that any complete strongly stable surface Σ with Σ 0 = ∅ is a vertical plane. It is also interesting to observe that the solution to the Bernstein problem in M(0) given in [13] has no direct counterpart in M(−1). We illustrate this fact in Example 4.4, since the surface y = xt is a strongly stable entire minimal graph with empty singular set and different from a vertical plane.
The family Σ λ,σ contains new examples of complete CMC surfaces with empty singular set in M(κ). We must point out that the classification of such surfaces is far from being established. In M(0) some partial results were obtained for minimal surfaces, [7], [3], and for CMC surfaces of revolution [31]. On the other hand, complete CMC vertical surfaces in Sasakian sub-Riemannian 3-manifolds were described in [33]. Following this line, in Theorem 4.1 (vi) we establish that any C 1 complete CMC surface in M(κ) having empty singular set and containing a vertical line must be congruent to some surface Σ λ,σ .
In the sub-Riemannian 3-sphere M(1) the study of compact CMC surfaces is particularly interesting. As a consequence of [9, Thm. E], we know that a compact CMC surface Σ with Σ 0 = ∅ is topologically a torus. As the authors proved in [24], if we further assume the mean curvature H to  [24]. In Example 4.5 of the present paper we find CMC tori in the family Σ λ,σ which are neither rotationally symmetric with respect to the vertical axis nor congruent to a vertical Clifford torus. This fact might suggest that the family of CMC tori with empty singular set in M(1) is very large. However, as we showed in [33] none of these tori is stable.
In Section 5 we discuss stability properties for the class of complete volume-preserving areastationary surfaces with non-empty singular set in sub-Riemannian 3-space forms. In the Heisenberg group M(0) it is possible to find many examples of area-minimizing surfaces with low analytical regularity, see for instance [29], [10], [28] and [30]. However, by assuming C 2 regularity the situation is considerably more rigid. Indeed, any C 2 surface Σ in the class above can be fully described in terms of the singular set Σ 0 and the mean curvature H = λ. From the work of Cheng, Hwang, Malchiodi and Yang [9], the set Σ 0 can contain isolated points and/or C 1 curves with non-vanishing tangent vector. We will analyze the two cases separately.
If there is an isolated point in Σ 0 then we have by [25] that Σ is congruent either to a planar surface P λ (p) as in (5.1), or to a spherical surface S λ (p) as in (5.2). The spheres S λ (p) are generalizations of the Pansu spheres in the Heisenberg group M(0). In [25] we proved that, though S λ (p) is not strongly stable, it is a second order minima of the area under volume-preserving admissible variations. In the present paper we complete the stability question in this setting by showing in Theorem 5.5 that the planes P λ (p) are strictly stable. This means that (A + 2λV ) ′′ (0) 0 for any admissible variation of P λ (p), and equality holds if and only if the associated velocity vector field is always tangent to P λ (p). We remark that employing admissible variations is necessary to apply the second variation formula (1.1) since we do not require the deformations to fix the singular point p. In fact, due to the integrability condition in Lemma 5.1 (i), the family of admissible variations of P λ (p) moving p is very large, see [25, App. B]. It is worth mentioning that the stability of the minimal plane P 0 (p) in M(0) is very well known; as a matter of fact P 0 (p) is area-minimizing by a calibration argument, see [32] and [3]. From a similar reasoning we may conclude the stability of the minimal planes P 0 (p) in the hyperbolic model M(−1). We stress that Theorem 5.5 is valid for all planes P λ (p) in sub-Riemannian 3-space forms of arbitrary topology.
The proof of Theorem 5.5 has two steps. In the first one, we employ the same arguments as for the spheres S λ (p) in [25] to infer that the divergence term in (1.1) vanishes. In the second step we see that Q(w, w) 0 for any w ∈ C 1 0 (P λ (p)), with equality if and only if w = 0. Observe that, since the planes P λ (p) have no vertical points, Corollary 3.6 (i) gives inequality Q(w, w) 0 provided the function w vanishes off of the pole p. The proof that Q(w, w) 0 for any w ∈ C 1 0 (P λ (p)) is more technical and relies on Lemma 5.4, where we use the analytical behaviour of P λ (p) around p to deduce an integration by parts formula involving the Jacobi function ψ = N, T . From Lemma 5.4 we can obtain Theorem 5.5 just by reproducing the proof of Theorem 3.5.
On the other hand, if the singular set Σ 0 contains a curve, then there is an ambient CC-geodesic Γ such that Σ is congruent to the surface C λ (Γ) defined in [25,Thm. 4.13], see also the references therein. Roughly speaking, the surfaces C λ (Γ) are produced by matching together some "fundamental pieces" in a suitable way. These pieces are surfaces Σ λ (Γ) as in (5.11), and they are obtained when one leaves from Γ by a family of orthogonal CC-geodesic segments of a given curvature and length.
In general, we cannot expect the surfaces C λ (Γ) to be strongly stable. Consider the simplest case of a CC-geodesic Γ of curvature µ in the Heisenberg group M(0). If µ = 0 then C 0 (Γ) is congruent to the hyperbolic paraboloid t = xy, and it is strongly stable (indeed area minimizing), see [32]. In the case µ = 0, Ritoré and the authors [23] found a variation of C 0 (Γ) moving the two singular curves of C 0 (Γ) while strictly decreasing the area. This leads us to study the stability of C λ (Γ) under variations supported off of the singular set. Surprisingly, in Theorem 5.8 we are able to deduce, as an immediate consequence of Corollary 3.6 (iii), that the regular set of any surface C λ (Γ) within a 3-dimensional space form is strongly stable. This implies in particular that, in order to show instability of C λ (Γ), one needs to use suitable variations moving the singular set.
The paper is organized into five sections. Section 2 contains background material about sub-Riemannian 3-manifolds. In Section 3 we introduce the variational setting and prove our stability criteria for CMC surfaces in Sasakian sub-Riemannian 3-manifolds. In Section 4 we construct and classify complete CMC surfaces in M(κ) containing a vertical line, focusing on strongly stable examples in M(−1). Finally, in Section 5 we establish our stability results for the surfaces P λ (p) and C λ (Γ).

Preliminaries
In this section we introduce the notation and gather some known results that will be used throughout the paper.

Sasakian sub-Riemannian 3-manifolds.
A contact sub-Riemannian manifold is a connected manifold M with ∂M = ∅ together with a Riemannian metric g h defined on an oriented contact distribution H, which we refer to as the horizontal distribution. A vector field U is horizontal if it coincides with its projection U h onto H.
The normalized form is the contact 1-form η on M such that Ker(η) = H and the restriction of the 2-form dη to H equals the area form on H. We will consider the orientation of M induced by η ∧ dη. The Reeb vector field is the vector field T transversal to H defined by η(T ) = 1 and dη(T, U ) = 0, for any U . If U is always proportional to T then we say that U is vertical.
We denote by J the orientation-preserving 90 degree rotation in (H, g h ). This is a contact structure on H since J 2 = −Id. We extend J to the tangent bundle of M by setting J(T ) := 0.
The canonical extension of g h is the Riemannian metric g = · , · on M extending g h , and such that T is a unit vector field orthogonal to H. The length of a vector field U is |U | := U, U 1/2 .
We say that M is complete if (M, g) is a complete Riemannian manifold.
By an isometry of M we mean a C ∞ diffeomorphism φ : M → M whose differential at any p ∈ M is an orientation-preserving linear isometry from H p to H φ(p) . We say that M is homogeneous if the group Iso(M ) of isometries of M acts transitively on M . In such a case M must be complete. Two subsets S 1 and By a Sasakian sub-Riemannian 3-manifold we mean a contact sub-Riemannian 3-manifold M such that g h is a Sasakian metric, i.e., any diffeomorphism of the one-parameter group of T is an isometry. This implies that (M, g) is a K-contact Riemannian manifold [5,Sect. 6.2]. It follows from [5, p. 67, Cor. 6.5, Thm. 6.3] that the Levi-Cività connection D associated to g satisfies In particular, the integral curves of T are geodesics in (M, g) parameterized by arc-length. We refer to these curves as vertical lines.
The Webster scalar curvature K of a contact sub-Riemannian 3-manifold M is the sectional curvature of H with respect to the Tanaka connection [5,Sect. 10.4]. If M is Sasakian then we have K = (1/4) (K h + 3), where K h denotes the sectional curvature of H in (M, g).
For κ = −1, 0 we denote M(κ) := N(κ) × R. Let (x, y, t) be the Euclidean coordinates in R 3 . We define in M(κ) the planar distribution H := Ker(η), where η := ρ (x dy − y dx) + dt and ρ(x, y, t) := ρ(x, y). A basis {X, Y, T } of vector fields on M(κ) such that X, Y are sections of H and T is the Reeb vector field associated to the contact 1-form η is given by For κ = 1 we consider M(κ) := S 3 with the planar distribution H := Ker(η), where η := Here (x 1 , y 1 , x 2 , y 2 ) are the Euclidean coordinates in R 4 . A basis {X, Y, T } of vector fields in the same conditions as above is defined by Some easy computations show the following bracket relations . We take the Riemannian metric g h on H for which {X p , Y p } is an orthonormal basis at any p ∈ M(κ). Hence, the associated orientation-preserving 90 degree rotation J satisfies J( In M(κ) there is a product * such that (M(κ), * ) is the Heisenberg group when κ = 0, the group of unit quaternions when κ = 1, and the universal covering of the special linear group SL(2, R) when κ = −1. The identity element for * is the point o := (0, 0, 0) when κ = −1, 0, or o := (1, 0, 0, 0) when κ = 1. The associated left translations (resp. right translations) are isometries of M(κ) when κ = −1, 0 (resp. κ = 1). Hence the isometry group of M(κ) acts transitively on M(κ). As a consequence M(κ) is homogeneous and, in particular, complete.
For any s ∈ R, we define the vertical translation φ s : It is easy to see that {φ s } s∈R is the one-parameter group of diffeomorphisms associated to T . Since any φ s is an isometry of M(κ) we conclude that M(κ) is a Sasakian sub-Riemannian 3-manifold. The vertical lines in M(κ) parameterize straight lines if κ = −1, 0 or great circles in S 3 if κ = 1. The vertical axis of M(κ) is the vertical line passing through o. By means of a left or right translation any vertical line is congruent to the vertical axis. We can check that a vertical rotation, i.e., a Euclidean rotation about the vertical axis is an isometry of M(κ). It follows that any vertical screw motion, defined as the composition of a vertical rotation and a vertical translation, is also an isometry of M(κ). We say that a set S ⊂ M(κ) is rotationally symmetric if it is invariant under vertical rotations.

Carnot-Carathéodory geodesics and Jacobi fields.
Let M be a Sasakian sub-Riemannian 3-manifold. A horizontal curve in M is a C 1 curve γ whose velocity vectorγ is horizontal. The length of γ over an interval [a, b] is b a |γ(s)| ds. Following the approach in [32, Sect. 3] and [33, Sect. 3], we say that a C 2 horizontal curve γ parameterized by arc-length is a Carnot-Carathéodory geodesic, or simply a CC-geodesic, if it is a critical point of length under C 2 variations by horizontal curves. As in [32, Prop. 3.1] this is equivalent to the existence of a constant λ ∈ R, called the curvature of γ, such that the second order ODE is satisfied. Here the prime ′ stands for the covariant derivative along γ in (M, g). It follows that any CC-geodesic is a C ∞ curve. If p ∈ M and v ∈ H p with |v| = 1, then the unique maximal solution γ to (2. . Then, the vector field V ε (s) := (∂F/∂ε)(ε, s) and the function v ε (s) := V ε (s), T satisfy these properties: Hence, if M has constant Webster scalar curvature K, then we have:

Horizontal geometry of surfaces.
Let Σ be a C 1 surface immersed in a Sasakian sub-Riemannian 3-manifold M . Unless explicitly stated we will assume that ∂Σ = ∅. The singular set Σ 0 of Σ consists of those points p ∈ Σ for which the tangent plane T p Σ equals the horizontal plane H p . Since H is a completely nonintegrable distribution, it follows by Frobenius theorem that Σ 0 is closed and has empty interior in Σ. Hence the regular set Σ − Σ 0 of Σ is open and dense in Σ. By using the arguments in [15,Lem. 1], see also [1, Thm. 1.2] and [22, App. A], we deduce that, for a C 2 surface Σ, the Hausdorff dimension of Σ 0 in (M, g) is less than or equal to 1. In particular, the Riemannian area of Σ 0 vanishes.
Suppose that Σ is oriented, and denote by N the unit normal along Σ in (M, g) which is compatible with the orientations of Σ and M . We define the (sub-Riemannian) area of Σ by where N h = N − N, T T and da is the area element in (M, g). In case Σ bounds a set Ω ⊂ M , then A(Σ) coincides with the sub-Riemannian perimeterá la De Giorgi of Ω, which can be introduced as in [6]. Note that Σ 0 = {p ∈ Σ ; N h (p) = 0}. In the regular set Σ − Σ 0 , we can define the horizontal Gauss map ν h and the characteristic vector field Z, by As Z is horizontal and orthogonal to ν h then Z is tangent to Σ. Hence Z p generates T p Σ ∩ H p for any p ∈ Σ − Σ 0 . The integral curves of Z in Σ − Σ 0 will be called (oriented ) characteristic curves of Σ. They are both tangent to Σ and to H. If we define where U ⊤ stands for the projection of a vector field U onto the tangent plane to Σ.
A vertical point is a point p ∈ Σ such that T p ∈ T p Σ. This is equivalent to that N p , T p = 0. We say that Σ is a vertical surface if any p ∈ Σ is a vertical point, i.e., N, T = 0 along Σ.
If Σ is an oriented C 2 surface immersed in M then, for any p ∈ Σ − Σ 0 and v ∈ T p M , we have these equalities, see [23,Lem. 3.5] and [33,Lem. 4 We denote by B the shape operator of Σ in (M, g). It is given by B(U ) := −D U N , for any vector U tangent to Σ. Finally, we will say that Σ is complete if it is complete with respect to the Riemannian metric induced by g.

Strongly stable surfaces
In this section we establish sufficient conditions ensuring that a surface inside a Sasakian sub-Riemannian 3-manifold is strongly stable (the precise definition is given in Section 3.2 below). We begin by recalling some known facts about critical points and second order minima of the area with or without a volume constraint.

Area-stationary surfaces.
Let M be a Sasakian sub-Riemannian 3-manifold and ϕ 0 : Σ → M an oriented C 1 surface immersed in M . By a variation of Σ we mean a C 1 map ϕ : I × Σ → M , where I ⊆ R is an open interval containing 0, and ϕ satisfies: (i) ϕ(0, p) = ϕ 0 (p) for any p ∈ Σ, (ii) the map ϕ s : Σ → M given by ϕ s (p) := ϕ(s, p) is an immersion for any s ∈ I, (iii) there is a compact set C ⊆ Σ such that ϕ s (p) = ϕ 0 (p) for any s ∈ I and any p ∈ Σ − C.
Observe that we do not assume neither a specific form for ϕ, nor that the velocity vector field U p := (∂ϕ/∂s)(0, p) is always normal to Σ or proportional to the horizontal Gauss map ν h in (2.5).
The area functional associated to ϕ is given by A(s) := A(Σ s ), see (2.4). We consider the volume functional V (s) defined in [2,Sect. 2] as the signed volume in (M, g) between Σ and Σ s . More precisely where dv denotes the volume element in (M, g). A variation ϕ is volume preserving if V (s) is constant for any s small enough. We say that Σ is area-stationary if A ′ (0) = 0 for any variation of Σ. We say that Σ is volume-preserving area-stationary or area-stationary under a volume constraint if A ′ (0) = 0 for any volume-preserving variation of Σ.
We denote by N the unit normal vector along Σ in (M, g) which is compatible with the orientations of Σ and M . If Σ is C 2 on Σ − Σ 0 , then the (sub-Riemannian) mean curvature of Σ is the function H on Σ − Σ 0 defined as in [31] and [33] by where {Z, S} is the orthonormal basis of the tangent plane to Σ − Σ 0 defined in (2.5) and (2.6). From (2.10) it follows that D S ν h is proportional to Z, and so D S ν h , S = 0. Since Z, ν h = 0 we deduce the identity Suppose now that Σ is C 1 and the vector field Z is C 1 along the characteristic curves. In such a case, we define the mean curvature of Σ by means of equality (3.2). This coincides, up to a factor, with the definition in [21, Eq. (5.1)]. We say that Σ has constant mean curvature (CMC) if H is constant on Σ − Σ 0 . When H = 0 on Σ − Σ 0 we say that Σ is a minimal surface. The next result characterizes C 1 volume-preserving area-stationary surfaces with empty singular set.
Proposition 3.1. Let Σ be an oriented C 1 surface immersed in a Sasakian sub-Riemannian 3manifold M . Suppose that Σ 0 = ∅ and the vector field Z is C 1 along the characteristic curves. Then, Σ is volume-preserving area-stationary (resp. area-stationary) if and only if H is constant (resp. H = 0) on Σ. In such a case, given any point p ∈ Σ, there is a unique characteristic curve γ through p, and γ is a CC-geodesic in M of curvature H. implies that γ is a C ∞ curve. Now, we can proceed as in the case where Σ is C 2 to conclude that γ is a CC-geodesic of curvature H.
The interested reader is referred to a recent work of Galli [19] for a generalization in contact sub-Riemannian 3-manifolds for domains with prescribed mean curvature and Lipschitz boundary which is locally a regular intrinsic graph.

Stability and second variation formula.
In this section we will assume more analytical regularity than in the previous one. For our purposes in the paper, it will suffice to consider C 2 surfaces Σ such that Σ − Σ 0 is C 3 , and C 2 variations ϕ of class C 3 off of Σ − Σ 0 . Let Σ be an oriented surface immersed in a Sasakian sub-Riemannian 3-manifold M . Suppose that Σ is area-stationary with or without a volume constraint, and let H be the constant mean curvature along Σ − Σ 0 . From the first variational formulas for area and volume, see for instance [25, Sect. 4.1], it follows that (A + 2HV ) ′ (0) = 0 for any variation of Σ. We say that Σ is For an area-stationary surface to be strongly stable agrees with the usual notion of stability for minimal surfaces in Riemannian geometry. Note that a strictly stable surface is in particular a strict minimum of the area under volume-preserving variations with non-tangent velocity. Obviously strict stability implies strong stability and strong stability implies stability.
In order to analyze the stability conditions we need to compute (A + 2HV ) ′′ (0 provided all the terms are locally integrable with respect to da. In the previous formula, w is the normal component of the velocity vector field U associated to the variation, div Σ G is the divergence relative to Σ of a certain tangent C 1 vector field G along Σ − Σ 0 (an explicit expression is found in [25, Eq. (7.1)]), and Q is the (sub-Riemannian) index form of Σ given by where K is the Webster scalar curvature of M and B is the shape operator of Σ in (M, g). Note that Q(u, v) is well defined if, for instance, u ∈ C 1 0 (Σ − Σ 0 ) and v ∈ C 1 (Σ − Σ 0 ). As was pointed out in [25, Ex. 8.2], if the variation ϕ : I × Σ → M is compactly supported on Σ − Σ 0 , then there is an interval I ′ ⊂⊂ I such that the restriction of ϕ to I ′ × Σ is admissible. Moreover, we can apply the Riemannian divergence theorem to get Σ div Σ G da = 0. From (3.3) we deduce that (3.5) (A + 2HV ) ′′ (0) = Q(w, w), provided ϕ is supported on Σ − Σ 0 .
In Riemannian geometry it is well known that the normal component along a CMC surface of an ambient Killing field is a Riemannian Jacobi function, see [2,Prop. (2.12)]. Here we prove in our sub-Riemannian context a similar result for the Reeb vector field. Proof. We compute L( N, T ) from (3.7). From (2.9) and the third equality in (2.7), we obtain On the other hand, note that D Z Z = (2H) ν h . This is a consequence of (3.2) together with identities D Z Z, Z = 0 and D Z Z, T = − Z, J(Z) = 0. Therefore, we deduce that From here, it is straightforward to check that The proof finishes by substituting the previous equalities into (3.7) and simplifying.
3.3. Criteria for strong stability.
Now, we have all the ingredients necessary to prove the following result ensuring strong stability of the regular set of a CMC surface.
As a consequence of Theorem 3.5 and Lemma 3.4 we can deduce the next result.

Strongly stable surfaces with empty singular set
In this section we obtain new examples of complete volume-preserving area-stationary surfaces with empty singular set in the model spaces M(κ). For that, we will produce suitable deformations of CMC vertical surfaces, see the Introduction for a motivation of our construction. Then, we will use Given a C n function σ : R → R with n ∈ N, we define a C n unit horizontal vector field along Γ by U (ε) := cos σ(ε) X(ε) + sin σ(ε) Y (ε). If κ = −1, 0, then it is easy to check that σ(ε) = θ(ε) + 2κε, where θ(ε) is the Euclidean angle between U (ε) and ∂ x in the plane generated by {∂ x , ∂ y }. In the case κ = 1, we have σ(ε) = θ(ε) + ε, where θ(ε) is the Euclidean angle between U (ε) and ∂ x2 in the plane generated by {∂ x2 , ∂ y2 }. Sometimes we will work with the angle function θ(ε) instead of σ(ε) since it provides a clearer geometric interpretation of our construction.
The following is the main result of this section. θ is a translation of the parameter along Γ. (iv) If Σ λ,σ is an immersed surface with empty singular set, then there is a Riemannian unit normal N such that any complete CC-geodesic γ ε : R → M(κ) is a characteristic curve of Σ λ,σ . Moreover, Σ λ,σ is a volume-preserving area-stationary surface with constant mean curvature λ with respect to N . (v) Suppose κ = −1 and λ 2 < 1. If the angle function θ is C 3 and satisfies 0 θ ′ 1 − λ 2 (resp. 0 θ ′ < 1 − λ 2 ) along Γ, then Σ λ,σ is a complete strongly stable (resp. strictly stable) surface in M(−1) with empty singular set. (vi) Any complete, connected, C 1 orientable surface with empty singular set and constant mean curvature which contains a vertical line is congruent to some surface Σ λ,σ .
Let v ε := V ε , T . From Lemma 2.1 (ii) we deduce that the expression of V ε with respect to the orthonormal frame {γ ε , J(γ ε ), T } is given by Note that where we have used (2.2). By substituting into the previous equation, we infer ). Now, if we compare the two expressions for V ′ ε (0) above, then we get v ′′ we can apply Lemma 2.1 (iii) to obtain an explicit expression for v ε . We denote τ := 4 (λ 2 + κ) and distinguish three cases.
since σ(ε) = θ(ε) + 2κε. We can proceed as in the previous case to get that Σ λ,σ is a C n surface immersed in M(κ), and that it has empty singular set if and only if θ ′ (ε) 0 for any ε ∈ R. This proves statement (i) of the theorem.
In order to prove (iv) we define the C n−1 vector field which clearly provides a unit normal along Σ λ,σ . Since we assume that the singular set is empty, then v ε (s) > 0 for any (ε, s) ∈ R 2 . From the definition of ν h and Z in (2.5), we get the equalities and so, any complete CC-geodesic γ ε is a characteristic curve of Σ λ,σ . By equations (3.1) and (2.3) we infer , from which Σ λ,σ has constant mean curvature λ with respect to N . We conclude that Σ λ,σ is volume-preserving area-stationary by Proposition 3.1.
We finish this section with some examples and remarks concerning the previous theorem. Example 4.2 (Surfaces Σ λ,σ in M(0)). It is well known that a CC-geodesic of vanishing curvature in the Heisenberg group M(0) is an affine parameterization of a horizontal straight line. Thus, for any angle function σ, the surface Σ 0,σ is the union of a family of horizontal lines contained in parallel planes. Since Γ(ε) = (0, 0, ε) and U (ε) = (cos σ(ε), sin σ(ε), 0), it follows that Observe that, for σ(ε) = Cε with C > 0, we get a right handed helicoid in R 3 . More generally, any Σ 0,σ is one of the embedded helicoid type minimal surfaces in M(0) described by Cheng and Hwang [7,Thm. B]. The fact that the singular set of Σ 0,σ is empty if and only if σ ′ 0 along Γ is contained in [7, Thm. C]. We also note that the entire graphical strips defined by Danielli, Garofalo, Nhieu and Pauls [13,Def. 1.3] in relation to the Bernstein problem in M(0) coincide with the surfaces Σ 0,σ having empty singular set and satisfying cos σ = 0 along Γ or sin σ = 0 along Γ.
To the best of our knowledge the surfaces Σ λ,σ such that λ = 0 and σ is a non-constant function with σ ′ 0 provide new examples of complete volume-preserving area-stationary surfaces in M(0) with empty singular set. These surfaces are never compact since they contain Γ. In general the surfaces Σ λ,σ are not embedded. From [33, Cor. 6.9] the unique stable surfaces Σ λ,σ of class C 2 with empty singular set are vertical planes. By a recent work of Galli and Ritoré [20] the vertical planes are also the only strongly stable surfaces Σ 0,σ of class C 1 in M(0) with empty singular set. Hence, for any angle function σ, we have From Theorem 4.1 (v), for any θ ∈ C 3 with 0 θ ′ 1 the surface Σ 0,σ is strongly stable and has empty singular set. This is a remarkable difference with respect to M(0), where the vertical planes are the only complete strongly stable minimal surfaces with empty singular set, see [23], [14], [20].
We can produce strongly stable non-minimal surfaces in M(−1) as follows. Choose numbers λ ∈ (−1, 1) and C ∈ (0, 1 − λ 2 ]. Consider the surface Σ λ,σ where σ(ε) = (C − 2)ε along Γ. This is an embedded right handed minimal helicoid when λ = 0. Note that θ(ε) = Cε, so that Theorem 4.1 implies that Σ λ,σ is a C ∞ complete, orientable, strongly stable, non-vertical surface in M(−1) with constant mean curvature λ and empty singular set.  1)). The classical Bernstein problem studies entire minimal graphs in Euclidean space. In R 3 it is well-known that the planes are the unique solutions to this problem. In the Heisenberg group M(0) it is possible to find entire minimal graphs (Euclidean or intrinsic) different from planes. Indeed, the best Bernstein type result in M(0) establishes that an entire strongly stable minimal graph with empty singular set must be a vertical plane, see [13, Thm. 1.8] and [3,Thm. 5.3]. To see that this rigidity result does not hold in M(−1), it suffices to consider a surface Σ 0,σ in M(−1) with angle function σ so that θ ∈ (−π/2, π/2) and 0 θ ′ 1 along Γ (there is a continuum of these functions). From (4.5) we have It follows from Theorem 4.1 that Σ 0,σ is a strongly stable entire minimal graph with empty singular set. In the particular case θ(ε) = arctan(ε) we get the Euclidean graph y = xt.
The vertical axis in M(1) is the great circle Γ(ε) = (cos ε, sin ε, 0, 0). Fix λ ∈ R such that λ/ √ 1 + λ 2 ∈ Q. This ensures by [24,Prop. 3.3] that all the CC-geodesics in M(1) of curvature λ are embedded circles of the same length. Take an angle function σ such that U := (cos σ) X + (sin σ) Y is 2π-periodic and θ ′ > −λ 2 along Γ. From Theorem 4.1 the surface Σ λ,σ is an immersed torus with constant mean curvature λ and empty singular set. Since Γ ⊂ Σ λ,σ then Σ λ,σ is not rotationally symmetric about Γ. Moreover, if θ is not a translation along Γ, then Σ λ,σ cannot be vertical and so, it is not congruent to T r . In general these surfaces Σ λ,σ are not embedded even if λ = 0. As a consequence of the stability result in [33,Cor. 6.9] none of these examples is stable. 2. It is natural to ask if other criteria different from Corollary 3.6 (iii) may be used to deduce strong/strict stability of Σ λ,σ when κ = −1, λ 2 < 1 and θ ′ 0. Note that the sufficient condition in Corollary 3.6 (ii) never holds on Σ λ,σ since any p ∈ Γ is vertical. As to the condition in Remark 3.2, we can follow the arguments in the proof of Theorem 4.1 to show that the function q := |B(Z) Now, it is easy to conclude that the inequality q 0 implies θ ′ 1 − λ 2 along Γ, thus recovering the stability condition in Theorem 4.1 (v).

Strong stability of surfaces with non-empty singular set
In this section we use our stability criteria to deduce stability properties of complete C 2 volumepreserving area-stationary surfaces with non-empty singular set in a 3-dimensional space form. In this setting there are rigidity results, see [25, Sect. 4] and the references therein, showing that these surfaces are uniquely determined by their singular set and their mean curvature. As was shown in [9], see also [17,Sect. 5], the singular set of a CMC surface of class C 2 consists of isolated points and/or curves with non-vanishing tangent vector. This leads us to divide this section in two parts, where we analyze the two cases separately.

Surfaces with isolated singular points.
Let M be a 3-dimensional space form of Webster scalar curvature κ. For any point p ∈ M and any number λ 0, we define The sets P λ (p) and S λ (p) were studied in [25,Sect. 3,Sect. 4.2]. We proved that they are the unique complete, connected, orientable, C 2 surfaces of constant mean curvature λ in M with at least one isolated singular point. The surfaces S λ (p) are generalizations of the well-known Pansu spheres in the first Heisenberg group M(0). The surfaces P λ (p) are volume-preserving areastationary immersed planes, whose singular set is the point p, called the pole of P λ (p). In M(0) any P λ (p) is a horizontal Euclidean plane P 0 (p). In M(1) there are no planes P λ (p). In M(−1), up to left translations, there is a unique plane P λ (p) (which is embedded) for any λ ∈ [0, 1]. All the planes P λ (p) are C ∞ surfaces off of the pole.
The stability properties of S λ (p) were analyzed in [25,Sect. 5], where it is shown that S λ (p) is stable but not strongly stable. In this section we will prove that all the planes P λ (p) are strictly stable. We first provide some basic facts and computations that will be helpful in the sequel.
Clearly v(s) > 0 for any s > 0. Since (∂F/∂s)(θ, s) =γ θ (s), which is a horizontal vector, then the map F : Let us see that It is clear that the right hand side above defines a unit normal along P λ (p) − {p}. By using (2.5) we have ν h = −J(γ θ ) and Z =γ θ . Hence any γ θ (s) with s > 0 is a characteristic curve of P λ (p) for this normal. By equations (3.1) and (2.3) we infer that the mean curvature H of P λ (p) with respect to this normal equals λ, which proves the claim. From (5.4) we deduce On the other hand, note that and so This proves statement (i). The inequality N, T > 0 on P λ (p) − {p} comes from the first identity in (5.5) since v ′ > 0 on R + . We also have N p = T p since p is a singular point. This proves (ii).
In the next step we simplify the second variation formula given in (3.3) for admissible variations of P λ (p). Indeed, the following result shows that (A + 2λV ) ′′ (0) = Q(w, w) even for variations moving the pole of P λ (p). We must remark that, since the function |N h | −1 is locally integrable on P λ (p), the family of admissible variations of P λ (p) moving the pole is very large, see [25, App. B].
Proposition 5.2. Let M be a 3-dimensional space form of Webster scalar curvature κ. Consider a plane P λ (p) immersed in M for some p ∈ M and λ 0 with λ 2 + κ 0. Let ϕ : I × P λ (p) → M be an admissible variation of class C 3 off of the pole. Denote w := U, N , where U is the velocity vector field and N is the unit normal with associated mean curvature H = λ. Then, the functional A + 2λV is twice differentiable at s = 0, and we have where Q is the index form defined in (3.4).
Note that, by statements (i) and (iv) in Lemma 5.1, all the terms in Q(u, v) are locally integrable whenever u, v ∈ C 1 (P λ (p)) and at least one of them has compact support on P λ (p).
Proof of Proposition 5.2. By equation (3.3) it suffices to check that div P λ (p) G is integrable with respect to da and has vanishing integral. This can be done by reproducing the arguments for the spherical surfaces S λ (p), see the proof of Thm. 5.2 in [25, App. A]. This requires Lemma 5.1 and the generalized divergence theorem for P λ (p) in Lemma 5.3 below. Lemma 5.3. Let U be a bounded and tangent C 1 vector field on P λ (p) − {p}, vanishing off of a compact set of P λ (p), and such that div P λ (p) U is integrable with respect to da. Then, we have Proof. The result follows from an approximation argument similar to the one in [25, Lem. 7.4]. Now, we begin to discuss the stability properties of P λ (p). By Proposition 5.2 and equation (3.4), it is natural to check first if q := |B(Z) + S| 2 + 4 (κ − 1) |N h | 2 is a non-positive function. Unfortunately Lemma 5.1 (iv) gives q 0 and q = 0 on P λ (p), so that we cannot obtain strong stability from this method. However, we can use Lemma 5.1 (ii) and Corollary 3.6 (ii) to deduce that P λ (p) − {p} is strongly stable. The technical difficulty in proving that P λ (p) is strongly stable arises when we deform P λ (p) by variations moving the pole. If we analyze the proof of Theorem 3.5 with ψ = N, T and w ∈ C 1 0 (P λ (p)), then we discover that this difficulty is solved provided equality (3.6) still holds for u = (w 2 /ψ 2 ) ψ and v = ψ (these functions has no compact support on P λ (p) − {p} if w(p) = 0). This is established in the next result.
Lemma 5.4. Let M be a 3-dimensional space form of Webster scalar curvature κ. Consider a plane P λ (p) immersed in M for some p ∈ M and λ 0 with λ 2 + κ 0. If Q is the index form of P λ (p) and we denote ψ := N, T , then Q(u, ψ) = 0 for any u ∈ C 1 0 (P λ (p)).
Proof. We follow the proof of [23,Prop. 3.14]. Let q := |B(Z) + S| 2 + 4 (κ − 1) |N h | 2 . We have The first equality comes from Lemma 3.4. The second one, which is valid for any ψ ∈ C 2 (P λ (p)), is a straightforward computation by using the definition of L(ψ) in (3.7) together with [33,Lem. 5.5] and equation (3.9). We define a tangent C 1 vector field on P λ (p) − {p} by where in the second equality we have taken into account (3.8). We know by Lemma 5.1 (iii) that Since u ∈ C 1 0 (P λ (p)) we deduce that U is bounded and vanishes off of a compact set of P λ (p). From (5.7), it follows that which is an integrable function with respect to da by statements (i) and (iv) in Lemma 5.1. Finally, we apply Lemma 5.3 to get and the lemma is proved. Now, we are ready to establish the main result of this section.
Theorem 5.5. Let M be a 3-dimensional space form of Webster scalar curvature κ. Consider a plane P λ (p) immersed in M for some p ∈ M and λ 0 with λ 2 + κ 0. Then, the index form satisfies Q(w, w) > 0, for any w ∈ C 1 0 (P λ (p)) with w = 0. Hence, P λ (p) is strictly stable under admissible variations.
Proof. Let N be the unit normal on P λ (p) with mean curvature H = λ. We denote q := |B(Z) + S| 2 + 4 (κ − 1) |N h | 2 and ψ := N, T . We know that ψ > 0 on P λ (p) by Lemma 5.1 (ii). Take w ∈ C 1 0 (P λ (p)) and define f := w/ψ, which is in C 1 0 (P λ (p)). As in the proof of Theorem 3.5 we get Now, we apply Lemma 5.4 with u := f 2 ψ to obtain where in the second equality we have used (5.8). We conclude that Suppose that Q(w, w) = 0. The fact that ψ > 0 on P λ (p) implies Z(f ) = 0, i.e., f is constant along the characteristic curves of P λ (p). Since these curves meet the pole and f ∈ C 0 (P λ (p)), then f = 0 and so w = 0. Finally, if ϕ is an admissible variation which is C 3 off of the pole, then Proposition 5.2 gives us (A + 2λV ) ′′ (0) = Q(w, w), where w := U, N and U is the velocity vector field. From here we conclude that P λ (p) is strictly stable and the proof finishes.
Remark 5.6. Montefalcone [27, Cor. 6.9] employed a criterion similar to Corollary 3.6 (ii) to discuss the strict stability of the regular set of minimal hyperplanes in Carnot groups of step 2.
In the next result we establish some properties of Σ λ (Γ). The lemma is known in M(κ) when κ 0, see [32, Prop. 6.3] and [24,Prop. 5.5]. Here we follow a general approach which does not use an explicit expression for the CC-geodesics in M .
Let Γ : R → M be a CC-geodesic of curvature µ in a 3-dimensional space form. In [25, Ex. 4.14] it is explained how to construct, for any number λ 0, a complete, immersed, orientable surface C λ (Γ) by matching together surfaces as in (5.11) in a suitable way. A precise description of C λ (Γ) in M(κ) for κ 0 is found in [32, Sect. 6] and [24,Sect. 5.2]. By Lemma 5.7 the surfaces C λ (Γ) are C ∞ off of the singular set, which can be parameterized by CC-geodesics of curvature µ. Moreover, by the characterization result in [25, Thm. 4.5] the surfaces C λ (Γ) are volume-preserving area-stationary with constant mean curvature λ. Indeed, in [25,Thm. 4.13] it is shown that these are the unique complete, connected, orientable, volume-preserving area-stationary C 2 surfaces in M whose singular set contains a curve.
Next, we discuss the stability properties of C λ (Γ). In general we cannot expect C λ (Γ) to be strongly stable. For instance, in [23,Thm. 5.4] it is proved that the minimal helicoids C 0 (Γ) with µ = 0 in M(0) are not strongly stable. We remark that the variation employed to obtain this result moves the two singular curves of C 0 (Γ) . This motivates the following question: is the regular set of C λ (Γ) a strongly stable surface? Surprisingly, this question has a positive answer.
Theorem 5.8. Let M be a 3-dimensional space form. Then, the regular set of any surface C λ (Γ) is strictly stable.