CMC spheres in the Heisenberg group

We study a family of spheres with constant mean curvature (CMC) in the Riemannian Heisenberg group $H^1$. These spheres are conjectured to be the isoperimetric sets of $H^1$. We prove several results supporting this conjecture. We also focus our attention on the sub-Riemannian limit.


Introduction
In this paper, we study a family of spheres with constant mean curvature (CMC) in the Riemannian Heisenberg group H 1 . We introduce in H 1 two real parameters that can be used to deform H 1 to the sub-Riemannian Heisenberg group, on the one hand, and to the Euclidean space, on the other hand. Even though we are not able to prove that these CMC spheres are in fact isoperimetric sets, we obtain several partial results in this direction. Our motivation comes from the sub-Riemannian Heisenberg group, where it is conjectured that the solution of the isoperimetric problem is obtained rotating a Carnot-Carathéodory geodesic around the center of the group, see [17]. This set is known as Pansu's sphere. The conjecture is proved only assuming some regularity (C 2 -regularity, convexity) or symmetry, see [4,7,15,16,18,19].
Given a real parameter τ ∈ R, let h = span{X, Y, T } be the three-dimensional real Lie algebra spanned by three elements X, Y, T satisfying the relations [X, Y ] = −2τ T and [X, T ] = [Y, T ] = 0. When τ = 0, this is the Heisenberg Lie algebra and we denote by H 1 the corresponding Lie group. We will omit reference to the parameter τ = 0 in our notation. In suitable coordinates, we can identify H 1 with C × R and assume that X, Y, T are left-invariant vector fields in H 1 of the form where (z, t) ∈ C × R and z = x + iy. The real parameters ε > 0 and σ = 0 are such that Let ·, · be the scalar product on h making X, Y, T orthonormal, that is extended to a left-invariant Riemannian metric g = ·, · in H 1 . The Riemannian volume of H 1 induced by this metric coincides with the Lebesgue measure L 3 on C×R and, in fact, it turns out to be independent of ε and σ (and hence of τ ). When ε = 1 and σ → 0, the Riemannian manifold (H 1 , g) converges to the Euclidean space. When σ = 0 and ε → 0 + , then H 1 endowed with the distance function induced by the rescaled metric ε −2 ·, · converges to the sub-Riemannian Heisenberg group. The boundary of an isoperimetric region is a surface with constant mean curvature. In this paper, we study a family of CMC spheres Σ R ⊂ H 1 , with R > 0, that foliate H 1 * = H 1 \ {0}, where 0 is the neutral element of H 1 . Each sphere Σ R is centered at 0 and can be described by an explicit formula that was first obtained by Tomter [20], see Theorem 2.1 below. We conjecture that, within its volume class and up to left translations, the sphere Σ R is the unique solution of the isoperimetric problem in H 1 . When ε = 1 and σ → 0, the spheres Σ R converge to the standard sphere of the Euclidean space. When σ = 0 is fixed and ε → 0 + , the spheres Σ R converge to the Pansu's sphere.
In Section 3, we study some preliminary properties of Σ R , its second fundamental form and principal curvatures. A central object in this setting is the left-invariant 1-form ϑ ∈ Γ(T * H 1 ) defined by ϑ(V ) = V, T for any V ∈ Γ(T H 1 ). (1. 3) The kernel of ϑ is the horizontal distribution. Let N be the north pole of Σ R and S = −N its south pole. In Σ * R = Σ R \ {±N } there is an orthonormal frame of vector fields X 1 , X 2 ∈ Γ(T Σ * R ) such that ϑ(X 1 ) = 0, i.e., X 1 is a linear combination of X and Y . In Theorem 3.1, we compute the second fundamental form of Σ R in this frame. We show that the principal directions of Σ R are given by a rotation of the frame X 1 , X 2 by a constant angle depending on the mean curvature of Σ R .
In Section 4, we link in a continuous fashion the foliation property of the Pansu's sphere with the foliation by meridians of the round sphere in the Euclidean space. The foliation H 1 * = R>0 Σ R determines a unit vector field N ∈ Γ(T H 1 * ) such that N (p) ⊥ T p Σ R for any p ∈ Σ R and R > 0. The covariant derivative ∇ N N , where ∇ denotes the Levi-Civita connection induced by the metric g, measures how far the integral lines of N are from being geodesics of H 1 (i.e., how far the CMC spheres Σ R are from being metric spheres). In space forms, we would have ∇ N N = 0, identically. Instead, in H 1 the normalized vector field is well-defined and smooth outside the center of H 1 . In Theorem 4.3, we prove that for any R > 0 we have where ∇ Σ R denotes the restriction of ∇ to Σ R . This means that the integral lines of M are Riemannian geodesics of Σ R . In the coordinates associated with the frame (1.1), when ε = 1 and τ = σ → 0 the integral lines of M converge to the meridians of the Euclidean sphere. When σ = 0 is fixed and ε → 0 + , the vector field M properly normalized converges to the line flow of the geodesic foliation of the Pansu's sphere, see Remark 4.5.
In Section 5, we give a proof of a known result that is announced in [1,Theorem 6] in the setting of three-dimensional homogeneous spaces (see also [13]). Namely, we show that any topological sphere with constant mean curvature in H 1 is isometric to a CMC sphere Σ R . The proof follows the scheme of the fundamental paper [2].
The surface Σ R is not totally umbilical and, for large enough R > 0, it even has negative Gauss curvature near the equator, see Remark 3.2. As a matter of fact, the distance from umbilicality is measured by a linear operator built up on the 1-form ϑ. We can restrict the tensor product ϑ ⊗ ϑ to any surface Σ in H 1 with constant mean curvature H and then define, at any point p ∈ Σ, a symmetric linear operator k ∈ Hom(T p Σ; T p Σ) by setting where h is the shape operator of Σ and q H is a rotation of each tangent plane T p Σ by an angle that depends only on H, see formula (5.1). In Theorem 5.7, we prove that for any topological sphere Σ ⊂ H 1 with constant mean curvature H, the linear operator k on Σ satisfies the equation k 0 = 0. This follows from the Codazzi's equations using Hopf's argument on holomorphic quadratic differentials, see [9]. The fact that Σ is a left translation of Σ R now follows from the analysis of the Gauss extension of the topological sphere, see Theorem 5.9.
In some respect, it is an interesting issue to link the results of Section 5 with the mass-transportation approach recently developed in [3].
In Section 6, we prove a stability result for the spheres Σ R . Let E R ⊂ H 1 be the region bounded by Σ R and let Σ ⊂ H 1 be the boundary of a smooth open set E ⊂ H 1 , Σ = ∂E, such that L 3 (E) = L 3 (E R ). Denoting by A (Σ) the Riemannian area of Σ, we conjecture that We also conjecture that a set E is isoperimetric (i.e., equality holds in (1.4)) if and only if it is a left translation of E R . We stress that if isoperimetric sets are topological spheres, this statement would follow from Theorem 5.9. It is well known that isoperimetric sets are stable for perturbations fixing the volume: in other words, the second variation of the area is nonnegative. On the other hand, using Jacobi fields arising from right-invariant vector fields of H 1 , it is possible to show that the spheres Σ R are stable with respect to variations supported in suitable hemispheres, see Section 6.
In the case of the northern and southern hemispheres, we can prove a stronger form of stability. Namely, using the coordinates associated with the frame (1.1), for R > 0 and 0 < δ < R we consider the cylinder where f (·; R) is the profile function of Σ R , see (2.2). Assume that the closure of E∆E R = E R \ E ∪ E \ E R is a compact subset of C δ,R . In Theorem 6.1, we prove that there exists a positive constant C Rτ ε > 0 such that the following quantitative isoperimetric inequality holds: The proof relies on a sub-calibration argument. This provides further evidence on the conjecture that isoperimetric sets are precisely left translations of Σ R . When ε = 1 and σ → 0, inequality (1.5) becomes a restricted form of the quantitative isoperimetric inequality in [8]. For fixed σ = 0 and ε → 0 + the rescaled area εA converges to the sub-Riemannian Heisenberg perimeter and εC Rτ ε converges to a positive constant, see Remark 6.2. Thus inequality (1.5) reduces to the isoperimetric inequality proved in [7].

2.
Foliation of H 1 * by concentric stationary spheres In this section, we compute the rotationally symmetric compact surfaces in H 1 that are area-stationary under a volume constraint. We show that, for any R > 0, there exists one such a sphere Σ R centered at 0. We will also show that H 1 * = H 1 \ {0} is foliated by the family of these spheres, i.e., Each Σ R is given by an explicit formula that is due to Tomter, see [20]. We work in the coordinates associated with the frame (1.1), where the parameters ε > 0 and σ ∈ R are related by (1.2). For any point (z, t) ∈ H 1 , we set r = |z| = x 2 + y 2 .
Theorem 2.1. For any R > 0 there exists a unique compact smooth embedded surface Σ R ⊂ H 1 that is area stationary under volume constraint and such that for a function f (·; R) ∈ C ∞ ([0, R)) continuous at r = R with f (R) = 0. Namely, for any 0 ≤ r ≤ R the function is given by where ω(r) = √ 1 + τ 2 ε 2 r 2 and p(r; R) = τ ε Proof. Let D R = {z ∈ C : |z| < R} and for a nonnegative radial function f ∈ C ∞ (D R ) consider the graph Σ = {(z, f (z)) ∈ H 1 : z ∈ D R }. A frame of tangent vector fields V 1 , V 2 ∈ Γ(T Σ) is given by Let g Σ = ·, · be the restriction of the metric g of H 1 to Σ. Using the entries of g Σ in the frame V 1 , V 2 , we compute the determinant where ∇f = (f x , f y ) is the standard gradient of f and |∇f | is its length. We clearly have xf y − yf x = 0 by the radial symmetry of f . Therefore, the area of Σ is given by where dz is the Lebesgue measure in the xy-plane. Thus, if Σ is area stationary under a volume constraint, then for any test function ϕ ∈ C ∞ c (D R ) that is radially symmetric and with vanishing mean (i.e., D R ϕ dz = 0) we have where div denotes the standard divergence in the xy-plane. It follows that there exists a constant H ∈ R such that With abuse of notation we let f (|z|) = f (z). Using the radial variable r = |z| and the short notation the above equation reads as follows: 1 r d dr r 2 g(r) = 1 r r 2 g r (r) + 2rg(r) = −εH.
Then there exists a constant K ∈ R such that r 2 g = −εr 2 H + K. Since g is bounded at r = 0, it must be K = 0 and thus g = −εH, and we get From this equation, we see that f r has a sign. Since Σ R is compact, it follows that H = 0. Since f ≥ 0 we have f r < 0 and therefore H > 0. The surface Σ R is smooth at the "equator" (i.e., where |z| = R and t = 0) and thus we have f r (R) = −∞. As we will see later, this is implied by the relation εHR = 1, (2.7) that will be assumed throughout the paper. Integrating the above equation we find Integrating this expression on the interval [r, R] and using f (R) = 0 we finally find After some computations, we obtain the explicit formula with ω(r) = √ 1 + τ 2 ε 2 r 2 . This is formula (2.2).
Remark 2.2. The function f (·; R) = f (·; R; τ ; ε) depends also on the parameters τ and ε, that are omitted in our notation. With ε = 1, we find When τ → 0, the spheres Σ R converge to Euclidean spheres with radius R > 0 in the three-dimensional space.
Remark 2.3. Starting from formula (2.2), we can compute the derivatives of f (·; R) in the variable R. The first order derivative is given by , (2.10) where : [0, ∞) → R is the function defined as . (2.11) The geometric meaning of will be clear in formula (4.1).
We now establish the foliation property (2.1).
Proof. Without loss of generality we can assume that t ≥ 0. After an integration by parts in (2.9), we obtain the formula Since ω r (r) > 0 for r > 0, we deduce that the function R → f (r; R) is strictly increasing for R ≥ r. Moreover, we have and hence for any r ≥ 0 there exists a unique R ≥ r such that f (r; R) = t.
Remark 2.5. By Proposition 2.4, we can define the function R : H 1 → [0, ∞) by letting R(0) = 0 and R(z, t) = R if and only if (z, t) ∈ Σ R for R > 0. The function R(z, t), in fact, depends on r = |z| and thus we may consider R(z, t) = R(r, t) as a function of r and t. This function is implicitly defined by the equation |t| = f (r; R(r, t)). Differentiating this equation, we find the derivatives of R, i.e., where f R is given by (2.10).

Second fundamental form of Σ R
In this section, we compute the second fundamental form of the spheres Σ R . In fact, we will see that H = 1/(εR) is the mean curvature of Σ R , as already clear from (2.6) and (2.7). Let N = (0, f (0; R)) ∈ Σ R be the north pole of Σ R and let where ϑ is the left-invariant 1-form introduced in (1.3). Explicit expressions for X 1 and X 2 are given in formula (3.9) below. This frame is unique up to the sign ±X 1 and ±X 2 . Here and in the rest of the paper, we denote by N the exterior unit normal to the spheres Σ R . The second fundamental form h of Σ R with respect to the frame X 1 , X 2 is given by Using the fact that the connection is torsion free and metric, it can be seen that ∇ is characterized by the following relations: Here and in the rest of the paper, we use the coordinates associated with the frame (1.1). For (z, t) ∈ H 1 , we set r = |z| and use the short notation = τ εr.
where R = 1/Hε and H is the mean curvature of Σ R . The principal curvatures of Σ R are given by (3.5) Outside the north and south poles, principal directions are given by Proof. Let a, b : Σ * R → R and c, p : Σ R → R be the following functions depending on the radial coordinate r = |z|: (3.8) In fact, b and p also depend on the sign of t. Namely, in b and p we choose the sign + in the northern hemisphere, that is for t ≥ 0, while we choose the sign − in the southern hemisphere, where t ≤ 0. Our computations are in the case t ≥ 0. The vector fields form an orthonormal frame for T Σ * R satisfying (3.1). This frame can be computed starting from (2.3). The outer unit normal to Σ R is given by Notice that this formula is well defined also at the poles. We compute the entries h 11 and h 12 . Using X 1 R = 0, we find where, by the fundamental relations (3.2), (3.12) Using the formulas we find the derivatives Inserting (3.13) and (3.12) into (3.11), we obtain (3.14) From this formula we get where p 2 + 1 = ω(R) 2 /ω(r) 2 and X 1 p can be computed starting from Namely, also using the formula for a and p in (3.8), we have

With (2.7) and (3.3), we finally find
¿From (3.14) we also deduce and using the formula for X 1 p and the formulas in (3.8) we obtain To compute the entry h 22 , we start from where, by (3.2) we have and thus Now X 2 p can be computed by using (3.15) and the formulas (3.8), and we obtain

By (2.7) and (3.3) we then conclude that
The principal curvatures κ 1 , κ 2 of Σ R are the solutions to the system They are given explicitly by the formulas (3.5). Now let K 1 , K 2 be tangent vectors as in (3.6). We identify h with the shape operator h ∈ Hom(T p Σ R ; T p Σ R ), h(K) = ∇ K N , at any point p ∈ Σ R and K ∈ T p Σ R . When = 0 (i.e., outside the north and south poles), the system of equations is satisfied if and only if the angle β = β H is chosen as in (3.7). The argument of arctan in (3.7) is in the interval (−1, 1) and thus β H ∈ (−π/4, π/4).
This means that, for large enough R, points in Σ R near the equator have strictly negative Gauss curvature.
Remark 3.3. The convergence of the Riemannian second fundamental form towards its sub-Riemannian counterpart is studied in [5], in the setting of Carnot groups.

Geodesic foliation of Σ R
We prove that each CMC sphere Σ R is foliated by a family of geodesics of Σ R joining the north to the south pole. In fact, we show that the foliation is governed by the normal N to the foliation H 1 * = R>0 Σ R . In the sub-Riemannian limit, we recover the foliation property of the Pansu's sphere. In the Euclidean limit, we find the foliation of the round sphere with meridians.
We need two preliminary lemmas. We define a function R : In fact, R(z, t) depends on r = |z| and t. The function p in (3.8) is of the form p = p(r, R(r, t)). Now, we compute the derivative of these functions in the normal direction N .
Proof. We start from the following expression for the unit normal (in the coordinates (x, y, t)): We just consider the case t ≥ 0. Using (2.12), we obtain Inserting into this formula the expression in (2.8) for f r we get and using formula (2.10) for f R , namely, we obtain formula (4.1).
To compute the derivatives of p in r and t, we have to consider p = p(r; R) and R = R(r, t). Using the formula in (3.8) for p and the expression (2.12) for R r yields and thus ∂ ∂r p(r, R(r, t)) = p r (r, R(r, t)) + p R (r, R(r, t))R r (r, t) Similarly, we compute The derivative of p along N is thus as in (4.2), when t ≥ 0. The case t < 0 is analogous.
In the next lemma, we compute the covariant derivative ∇ N N . The resulting vector field in H 1 * is tangent to each CMC sphere Σ R , for any R > 0.
Let N ∈ Γ(T H 1 * ) be the exterior unit normal to the family of CMC spheres Σ R centered at 0 ∈ H 1 . The vector field ∇ N N is tangent to Σ R for any R > 0, and for (z, t) ∈ Σ R we have ∇ N N (z, t) = 0 if and only if z = 0 or t = 0.
However, it can be checked that the normalized vector field is smoothly defined also at points (z, t) ∈ Σ R at the equator, where t = 0. We denote by ∇ Σ R the restriction of the Levi-Civita connection ∇ to Σ R . Proof. From (4.3) we obtain the following formula for M : where λ, µ : Σ * R → R are the functions with r = |z| and R = 1/(εH). The functions λ and µ are radially symmetric in z.
In defining λ we choose the sign +, when t ≥ 0, and the sign −, when t < 0. In the coordinates (x, y, t), the vector field M has the following expression where r∂ r = x∂ x + y∂ y , and so we have (4.12) Using (4.11), we compute and so we find (4.14) Now, inserting (4.13) and (4.14) into (4.12), we get The next computations are for the case t ≥ 0. Again from (4.11), we get (4.15) From (4.10) and (4.15) we have and so we finally obtain where we have set Comparing with (3.10), we deduce that The claim ∇ Σ R M M = 0 easily follows from the last formula. Remark 4.4. We compute the pointwise limit of M in (4.9) when σ → 0, for t ≥ 0.
In the southern hemisphere the situation is analogous. By (4.11), the vector field M is given by With ε = 1 we have Clearly, the vector field M is tangent to the round sphere of radius R > 0 in the three-dimensional Euclidean space and its integral lines turn out to be the meridians from the north to the south pole.

Now, it turns out that
The vector fieldM is horizontal and tangent to the Pansu's sphere. We denote by J the complex structure J(X) =Ȳ and J(Ȳ ) = −X. A computation similar to the one in the proof of Theorem 4.3 shows that Using (4.18) we can pass to the limit as ε → 0 in equation (4.8), properly scaled. An inspection of the right hand side in (4.16) shows that the right hand side of (4.8) is asymptotic to ε 4 . In fact, starting from (4.17) we get

Topological CMC spheres are left translations of Σ R
In this section, we prove that any topological sphere in H 1 having constant mean curvature is congruent to a sphere Σ R for some R > 0. This result was announced, in wider generality, in [1]. As in [2], our proof relies on the identification of a holomorphic quadratic differential for CMC surfaces in H 1 .
For an oriented surface Σ in H 1 with unit normal vector N , we denote by h ∈ Hom(T p Σ; T p Σ) the shape operator h(W ) = ∇ W N , at any point p ∈ Σ. The 1form ϑ in H 1 , defined by ϑ(W ) = W, T for W ∈ Γ(T H 1 ), can be restricted to the tangent bundle T Σ. The tensor product ϑ⊗ϑ ∈ Hom(T p Σ; T p Σ) is defined, as a linear operator, by the formula where X 1 , X 2 is any (local) orthonormal frame of T Σ. Finally, for any H ∈ R with H = 0, let α H ∈ (−π/4, π/4) be the angle and let q H ∈ Hom(T p Σ; T p Σ) be the (counterclockwise) rotation by the angle α H of each tangent plane T p Σ with p ∈ Σ.
Definition 5.1. Let Σ be an (immersed) surface in H 1 with constant mean curvature H = 0. At any point p ∈ Σ, we define the linear operator k ∈ Hom(T p Σ; T p Σ) by The operator k is symmetric, i.e., k(V ), W = V, k(W ) . The trace-free part of k is k 0 = k − 1 2 tr(k)Id. In fact, we have Formula (5.2) is analogous to the formula for the quadratic holomorphic differential discovered in [2]. In the following, we identify the linear operators h, k, ϑ ⊗ ϑ with the corresponding bilinear forms (V, W ) → h(V, W ) = h(V ), W , and so on.
The structure of k in (5.2) can be established in the following way. Let Σ R be the CMC sphere with R = 1/εH. From the formula (3.4), we deduce that, in the frame X 1 , X 2 in (3.1), the trace-free shape operator at the point (z, t) ∈ Σ R is given by where = τ ε|z|. On the other hand, from (3.9) and (3.8), we get ϑ(X 1 ) = 0 and ϑ(X 2 ) = √ τ 2 + H 2 τ 1 + 2 , and we therefore obtain the following formula for the trace-free tensor (ϑ ⊗ ϑ) 0 in the frame X 1 , X 2 : Now, in the unknowns c ∈ R and q (that is a rotation by an angle β), the system of equations h 0 + cq(ϑ ⊗ ϑ) 0 q −1 = 0 holds independently of if and only if c = 2τ 2 / √ H 2 + τ 2 and β is the angle in (5.1). We record this fact in the next: Proposition 5.2. The linear operator k on the sphere Σ R with mean curvature H, at the point (z, t) ∈ Σ R , is given by In particular, Σ R has vanishing k 0 (i.e., k 0 = 0).
In Theorem 5.7, we prove that any topological sphere in H 1 with constant mean curvature has vanishing k 0 . We need to work in a conformal frame of tangent vector fields to the surface.
Let z = x 1 + ix 2 be the complex variable. Let D ⊂ C be an open set and, for a given map F ∈ C ∞ (D; H 1 ), consider the immersed surface Σ = F (D) ⊂ H 1 . The parametrization F is conformal if there exists a positive function E ∈ C ∞ (D) such that, at any point in D, the vector fields V 1 = F * ∂ ∂x 1 and V 2 = F * ∂ ∂x 2 satisfy: We call V 1 , V 2 a conformal frame for Σ and we denote by N the normal vector field to Σ such that triple V 1 , V 2 , N forms a positively oriented frame, i.e., The second fundamental form of Σ in the frame V 1 , V 2 is denoted by where ∇ i = ∇ V i for i = 1, 2. This notation differs from (3.4), where the fixed frame is X 1 , X 2 , N . Finally, the mean curvature of Σ is By Hopf's technique on holomorphic quadratic differentials, the validity of the equation k 0 = 0 follows from the Codazzi's equations, which involve curvature terms. An interesting relation between the 1-form ϑ and the Riemann curvature operator, defined as for any U, V, W ∈ Γ(T H 1 ), is described in the following: Lemma 5.3. Let V 1 , V 2 be a conformal frame of an immersed surface Σ in H 1 with conformal factor E and unit normal N . Then, we have Proof. We use the notation From the fundamental relations (3.2), we obtain: . (12) Now, we have (9) + (10) + (11) + (12) = 0. In fact: For an immersed surface with conformal frame V 1 , V 2 , we use the notation Theorem 5.4 (Codazzi's Equations). Let Σ = F (D) be an immersed surface in H 1 with conformal frame V 1 , V 2 , conformal factor E and unit normal N . Then, we have where L, M, N, H are as in (5.6) and (5.7).
Proof. We start from the following well-known formulas for the derivatives of the mean curvature: Our claims (5.10) and (5.11) follow from these formulas and Lemma 5.3. For the reader's convenience, we give a short sketch of the proof of (5.12), see e.g. [12] for the flat case. For any i, j, k = 1, 2, we have (5.14) Setting i = j = 2 and k = 1 in (5.14), and using (5.7) we obtain Using the expression of ∇ i N in the conformal frame, we find 16) and from (5.15) and (5.16) we deduce that From (5.7), we have the further equation that, inserted into (5.17), gives claim (5.12).
Now we switch to the complex variable z = x 1 + ix 2 ∈ D and define the complex vector fields Equations (5.10)-(5.11) can be transformed into one single equation: The entries of b 0 as a quadratic form in the conformal frame V 1 , V 2 , with ϑ i = ϑ(V i ) and c H = 2τ 2 H 2 +τ 2 , are given by These entries can be computed starting from Proof. The complex equation (5.20) is equivalent to the system of real equations We check the first equation in (5.21). Since H is constant, we have For i, j = 1, 2, we have where, with the notation (5.9) and by the fundamental relations (3.2), By the definition (5.7) and (5.4), we have and thus, again from (5.22) and (5.23), we obtain From (5.25) and (5.24) we deduce that and finally In order to prove the second equation in (5.21), notice that By (5.26) we hence obtain Let Σ be an immersed surface in H 1 defined in terms of a conformal parametrization F ∈ C ∞ (D; H 1 ). Let f ∈ C ∞ (D; C) be the function of the complex variable z ∈ D given by that is equivalent to ∂zf = 0 in D.
Now, by a standard argument of Hopf, see [9] Chapter VI, for topological spheres the function f is identically zero. By Liouville's theorem, this follows from the estimate that can be obtained expressing the second fundamental forms in two different charts without the north and south pole, respectively. We skip the details of the proof of the next: Theorem 5.7. A topological sphere Σ immersed in H 1 with constant mean curvature has vanishing k 0 .
In the rest of this section, we show how to deduce from the equation k 0 = 0 that any topological sphere is congruent to a sphere Σ R . Differently from [2], we do not use the fact that the isometry group of H 1 is four-dimensional.
Let h be the Lie algebra of H 1 and let ·, · be the scalar product making X, Y, T orthonormal. We denote by S 2 = {ν ∈ h : |ν| = ν, ν = 1} the unit sphere in h. For any p ∈ H 1 , let τ p : H 1 → H 1 be the left-translation τ p (q) = p −1 · q by the inverse of p, where · is the group law of H 1 , and denote by τ p * ∈ Hom(T p H 1 ; h) its differential.
For any point (p, ν) ∈ H 1 ×S 2 there is a unique N ∈ T p H 1 such that ν = τ p * N and we define T ν p H 1 = {W ∈ T p H 1 : W, N = 0}. Depending on the point (p, ν) and on the parameters H, τ ∈ R, with H 2 + τ 2 = 0, below we define the linear operator L H ∈ Hom(T ν p H 1 ; T ν S 2 ). The definition is motivated by the proof of Proposition 5.8. For any W ∈ T ν p M , we let where ∇ W τ p * ∈ Hom(T p H 1 ; h) is the covariant derivative of τ p * in the direction W and the trace-free operator (ϑ ⊗ ϑ) 0 ∈ Hom(T ν p H 1 ; T ν p H 1 ) is The operator q H ∈ Hom(T ν p H 1 ; T ν p H 1 ) is the rotation by the angle α H in (5.1). The operator L H is well-defined, i.e., L H W ∈ h and L H W, ν = 0 for any W ∈ T ν p H 1 .
This can be checked using the identity |N | = 1 and working with the formula where Y 1 , Y 2 , Y 3 is any frame of orthonormal left-invariant vector fields. Finally, for any point (p, ν) ∈ H 1 × S 2 , define The distribution E H origins from CMC surfaces with mean curvature H and vanishing k 0 . Let Σ be a smooth oriented surface immersed in H 1 given by a parameterization Proof. Let N be the unit normal to Σ. For any tangent section W ∈ Γ(T Σ), we have where h(W ) = ∇ W N is the shape operator. Therefore, the set of all sections of the tangent bundle of Σ is and thus the sections of Σ are of the form This concludes the proof. Proof. Let H > 0 be the mean curvature of Σ, let R = 1/Hε, and recall that the sphere Σ R has mean curvature H. Let T Σ (p) ∈ T p Σ be the orthogonal projection of the vertical vector field T onto T p Σ. Since Σ is a topological sphere, there exists a point p ∈ Σ such that T Σ (p) = 0. This implies that either T = N or T = −N at the point p, where N is the outer normal to Σ at p. Assume that T = N .
Let ι be the left translation such that ι(p) = N , where N is the north pole of Σ R . At the point N the vector T is the outer normal to Σ R . Since ι * T = T (this holds for any isometry), we deduce that Σ R and ι(Σ) are two surfaces such that:

Quantitative stability of Σ R in vertical cylinders
In this section, we prove a quantitative isoperimetric inequality for the CMC spheres Σ R with respect to compact perturbations in vertical cylinders, see Theorem 6.1. This is a strong form of stability of Σ R in the northern and southern hemispheres.
A CMC surface Σ in H 1 with normal N is stable in an open region A ⊂ Σ if for any function g ∈ C ∞ c (A) with Σ gdA = 0, where A is the Riemannian area measure of Σ, we have The functional S (g) is the second variation, with fixed volume, of the area of Σ with respect to the infinitesimal deformation of Σ in the direction gN . Above, |∇g| is the length of the tangential gradient of g, |h| 2 is the squared norm of the second fundamental form of Σ and Ric(N ) is the Ricci curvature of H 1 in the direction N .
The Jacobi operator associated with the second variation functional S is L g = ∆g + (|h| 2 + Ric(N ))g, where ∆ is the Laplace-Beltrami operator of Σ. As a consequence of Theorem 1 in [6], if there exists a strictly positive solution g ∈ C ∞ (A) to equation L g = 0 on A, then Σ is stable in A (even without the restriction A gdA = 0). Now consider in H 1 the right-invariant vector fields These are generators of left-translations in H 1 , and the functions are solutions to L g = 0. By the previous discussion, the CMC sphere Σ R is stable in the hemispheres In particular, Σ R is stable in the northern hemisphere A T = {(z, t) ∈ Σ R : t > 0}.
In fact, we believe that the whole Σ R is stable. Actually, this would follow from the isoperimetric property for Σ R . The proof of the stability of Σ R requires a deeper analysis and it is not yet clear. However, in the case of the northern (or southern) hemisphere we can prove a strong form of stability in terms of a quantitative isoperimetric inequality. Some stability results in various sub-Riemannian settings have been recently obtained in [14,10,11].
For R > 0, let E R ⊂ H 1 be the open domain bounded by the CMC sphere Σ R , where f (·; R) is the profile function of Σ R in (2.2). For 0 ≤ δ < R, we define the half-cylinder where t R,δ = f (r R,δ ; R) and r R,δ = R − δ. In the following, we use the short notation .
We denote by A the Riemannian surface-area measure in H 1 .
(i) If E∆E R ⊂⊂ C R,δ with 0 < δ < R then we have Remark 6.2. When Σ ⊂ H 1 is a t-graph, Σ = {(z, f (z)) ∈ H 1 : z ∈ D} for some f ∈ C 1 (D), from (2.4) and (2.5) we see that the Riemannian area of Σ is The integral in the right-hand side is the sub-Riemannian area of Σ.
The proof of Theorem 6.1 is based on the foliation of the cylinder C R,δ by a family of CMC surfaces with quantitative estimates on the mean curvature. Theorem 6.3. For any R > 0 and 0 ≤ δ < R, there exists a continuous function u : C R,δ → R with level sets S λ = (z, t) ∈ C R,δ : u(z, t) = λ , λ ∈ R, such that the following claims hold: (i) u ∈ C 1 (C R,δ ∩ E R ) ∩ C 1 (C R,δ \ E R ) and the normalized Riemannian gradient ∇u/|∇u| is continuously defined on C R,δ . (ii) λ>R S λ = C R,δ ∩ E R and λ≤R S λ = C R,δ \ E R . (iii) Each S λ is a smooth surface with constant mean curvature H λ = 1/(ελ) for λ > R and H λ = 1/(εR) for λ ≤ R. (iv) For any point (z, f (|z|; R) − t) ∈ S λ with λ > R we have and , when 0 < δ < R. (6.5) Proof of Theorem 6.3. For points (z, t) ∈ C R,δ \ E R we let Then u satisfies u(z, t) ≤ R for t ≥ f (|z|; R) and u(z, t) = R if t = f (|z|; R). In order to define u in the set C R,δ ∩ E R , for 0 ≤ r < r R,δ , t R,δ < t < f (r; R), and λ > R we consider the function The function F also depends on δ. We claim that for any point (z, t) ∈ C R,δ ∩ E R there exists a unique λ > R such that F (|z|, t, λ) = 0. In this case, we can define u(z, t) = λ if and only if F (|z|, t, λ) = 0. (6.7) We prove the previous claim. Let (z, t) ∈ C R,δ ∩ E R and use the notation r = |z|.
First of all, we have We claim that we also have To prove this, we let f (r; Using the asymptotic approximation (1), and thus f (r; λ) − f (r R,δ ; λ) = o(1), where o(1) → 0 as λ → ∞. Since λ → F (r, t, λ) is continuous, (6.8) and (6.9) imply the existence of a solution λ of F (r, t, λ) = 0. The uniqueness follows from ∂ λ F (r, t, λ) < 0. This inequality can be proved starting from (2.10) and we skip the details. This finishes the proof of our initial claim.
Claims (i) and (ii) can be checked from the construction of u. Claim (iii) follows, by Theorem 3.1, from the fact that S λ for λ > R is a vertical translation (this is an isometry of H 1 ) of the t-graph of z → f (z; λ).
We can now prove Theorem 6.1, the last result of the paper. The proof follows the lines of [7].