Bubbles with constant mean curvature, and almost constant mean curvature, in the hyperbolic space

Given a constant $k>1$, let $Z$ be the family of round spheres of radius $\textrm{artanh}(k^{-1})$ in the hyperbolic space $\mathbb{H}^3$, so that any sphere in $Z$ has mean curvature $k$. We prove a crucial nondegeneracy result involving the manifold $Z$. As an application, we provide sufficient conditions on a prescribed function $\phi$ on $\mathbb{H}^3$, which ensure the existence of a ${\cal C}^1$-curve, parametrized by $\varepsilon\approx 0$, of embedded spheres in $\mathbb{H}^3$ having mean curvature $k +\varepsilon\phi$ at each point.


Introduction
Let K be a given function on the hyperbolic space H 3 . The K-bubble problem consists in finding a K-bubble, which is an immersed surface u : S 2 → H 3 having mean curvature K at each point. Besides its independent interest, the significance of the K-bubble problem is due to its connection with the Plateau problem for disk-type parametric surfaces having prescribed mean curvature K and contour Γ, see for instance [1,12]. In the Euclidean case, the impact of K-bubbles on nonexistence and lack of compacteness phenomena in the Plateau problem has been investigated in [5,8,9].
To look for K-bubbles in the hyperbolic setting one can model H 3 via the upper halfspace (R 3 + , p −2 3 δ hj ) and consider the elliptic system introduce local coordinates on S 2 ≡ R 2 ∪ {∞} and put G ℓ (∇u) := ∇u 3 · ∇u ℓ − 1 2 Γ ℓ hj (u)∇u h · ∇u j , ℓ = 1, 2, 3 , (1. 2) where Γ ℓ hj are the Christoffel symbols. Any nonconstant solution u to (1.1) is a generalized K-bubble in H 3 (see Lemma A.2 in the Appendix and [13,Chapter 2]), that is, u is a conformal parametrization of a surface having mean curvature K(u), apart from a finite number of branch points. Once found a solution to (1.1), the next step should concern the study of the geometric regularity of the surface u, which might have self-intersections and branch points.
A remarkable feature of (1.1) is its variational structure, which means that its solutions are critical points of a certain energy functional E, see the Appendix for details. Because of their underlying geometrical meaning, both (1.1) and E are invariant with respect to the action of Möbius transformations. This produces some lack of compactness phenomena, similar to those observed in the largely studied K-bubble problem, raised by S.T. Yau [19], for surfaces in R 3 (see for instance [7,10,11,17] and references therein; see also [3,6,18] for related problems). However, the hyperbolic K-bubble problem is definitively more challenging, due to the homogeneity properties that characterize the hyperbolic-area and the hyperbolic-volume functionals.
The main differences between the Euclidean and the hyperbolic case are already evident when the prescribed curvature is a constant k > 0 (the case k < 0 is recovered by a change of orientation). Any round sphere of radius 1/k in R 3 can be parameterized by an embedded k-bubble, which minimizes the energy functional E Eucl (u) = 1 2R 2 |∇u| 2 dz + 2k 3R 2 u · ∂ x u ∧ ∂ y u dz on the Nehari manifold { u = const. | E ′ Eucl (u)u = 0 }, see [7,Remark 2.6]. In contrast, no immersed hyperbolic k-bubble exists if k ∈ (0, 1], see for instance [14,Theorem 10.1.3]. If k > 1, then any sphere in H 3 of radius can be parameterized by an embedding U : S 2 → H 3 , which solves and which is a critical point of the energy functional As in the Euclidean case, the functional E hyp is unbounded from below (see Remark A.3). Thus U does not minimize the energy E hyp on the Nehari manifold, which in fact fills { u = const. }.
Besides their invariance with respect to Möbius transformations, the system (P 0 ) and the related energy E hyp are invariant with respect to the 3-dimensional group of hyperbolic translations as well. Thus, any k-bubble generates a smooth 9-dimensional manifold Z of solutions to (P 0 ). We explicitly describe the tangent space T U Z at U ∈ Z in formula (3.4).
As a further consequence of the invariances of problem (P 0 ), any tangent direction ϕ ∈ T U Z solves the elliptic system which is obtained by linearizing (P 0 ) at U . The next one is the main result of the present paper.
In the Euclidean case the nondegeneracy of k-bubbles has been proved in [15, Proposition 3.1]. The proof of Theorem 1.1 (see Section 3), is considerably more involved. It requires the choice of a suitable orthogonal frame for functions in C 2 (S 2 , R 3 ) and the complete classification of solutions of two systems of linear elliptic differential equations, which might have an independent geometrical interest (see Lemmata 3.4,3.5).
As an application of Theorem 1.1, we provide sufficient conditions on a prescribed smooth function φ : H 3 → R that ensure the existence of embedded surfaces S 2 → H 3 having nonconstant mean curvature k + εφ. Our existence results involve the notion of stable critical point already used in [16] and inspired from [2, Chapter 2] (see Subsection 2.2). The main tool is a Lyapunov-Schmidt reduction technique combined with variational arguments, in the spirit of [2]. Theorem 1.2 Let k > 1 and φ ∈ C 1 (H 3 ) be given. Assume that the function has a stable critical point in an open set A ⋐ H 3 . For every ε ∈ R close enough to 0 there exist a point q ε ∈ A, a conformal parametrization U q ε of a sphere of radius ρ k about q ε , and a conformally embedded Moreover, any sequence ε h → 0 has a subsequence ε h j such that q ε h j converges to a critical point for F φ k . In particular, ifq ∈ Ω is the unique critical point for F φ k in Ω, then Then there exists k 0 > 1 such that for any k > k 0 and for every ε close enough to 0, there exists a conformally embedded (k + εφ)-bubble.
In Section 4 we first show that the existence of a critical point for F φ k (q) is a necessary condition in Theorem 1.2. Then we perform the dimension reduction and prove Theorems 1.2, 1.3. With respect to correspondent Euclidean results in [7], a different choice of the functional setting allows us to weaker the regularity assumption on φ (from C 2 to C 1 ).
We conclude the paper with an Appendix in which we collect some partially known results about the variational approach to (1.1) and prove a nonexistence result for (1.1) which, in particular, justifies the assumption on the existence of a critical point for φ in Theorem 1.3 .

Notation and preliminaries
The vector space R n is endowed with the Euclidean scalar product ξ · ξ ′ and norm |ξ|. We denote by {e 1 , e 2 , e 3 } the canonical basis and by ∧ the exterior product in R 3 .
We will often identify the complex number z = x + iy with the vector z = (x, y) ∈ R 2 .
In local coordinates induced by the stereographic projections from the north and the south poles, the round metric on the sphere S 2 is given by g hj = µ 2 δ hj , dS 2 = µ 2 dz, where We identify the compactified plane R 2 with the sphere S 2 through the inverse of the stereographic projection from the north pole, which is explicitly given by ω(x, y) = (µx, µy, 1 − µ) . (2.1) The identity |ω| 2 ≡ 1 trivially gives ω · ∂ x ω ≡ 0, ω · ∂ y ω ≡ 0. We also notice that ω is a conformal (inward-pointing) parametrization of the unit sphere and satisfies

The Poincaré half-space model
We adopt as model for the three dimensional hyperbolic space H 3 the upper half-space and the hyperbolic ball B H ρ (p) centered at p = (p 1 , p 2 , p 3 ) is the Euclidean ball of center (p 1 , p 2 , p 3 cosh ρ) and radius p 3 sinh ρ.
If F : H 3 → R is a differentiable function, then ∇ H F (p) = p 2 3 ∇F (p), where ∇ H , ∇ are the hyperbolic and the Euclidean gradients, respectively. In particular, ∇ H F (p) = 0 if and only if ∇F (p) = 0. The hyperbolic volume form is related to the Euclidean one by dH 3 p = p −3 3 dp.

Stable critical points
Let X ∈ C 1 (H 3 ) and let Ω ⋐ H 3 be open. We say that X has a stable critical point in Ω if there exists r > 0 such that any function G ∈ C 1 (Ω) satisfying G − X C 1 (Ω) < r has a critical point in Ω. As noticed in [16], conditions to have the existence of a stable critical point p ∈ Ω for X are easily given via elementary calculus. For instance, one can use Browder's topological degree theory or can assume that Finally, if X is of class C 2 and Ω contains a nondegenerate critical point p 0 (i.e. the Hessian matrix of X at p 0 is invertible), then p 0 is stable.

Function spaces
Any function f on R 2 is identified with f • ω −1 , which is a function on S 2 . If no confusion can arise, from now on we write f instead of f • ω −1 .
The Hilbertian norm on L 2 (R 2 , R n ) ≡ L 2 (S 2 , R n ) is given by Let m ≥ 0. We endow with the standard Banach space structure (we agree that C m (R 2 , R n ) = C ⌊m⌋,m−⌊m⌋ (R 2 , R n ) if m is not an integer). If m is an integer, a norm in C m (R 2 , R n ) is given by Since we adopt the upper space model for H 3 , we are allowed to write If ψ, ϕ ∈ C 1 (R 2 , R 3 ) and τ ∈ R 2 , we put (notice that τ ∇ϕ(z) = dϕ(z)τ for any z ∈ R 2 ). For instance, we have For future convenience we notice, without proof, that the next identities hold, The monograph [4] is our reference text for the theory of Sobolev spaces on Riemannian manifolds. In view of our purposes, it is important to notice that We simply write L 2 (R 2 ), C m (R 2 ) and H 1 (R 2 ) instead of L 2 (R 2 , R), C m (R 2 , R) and H 1 (R 2 , R), respectively.

Möbius transformations and hyperbolic translations
Transformations in P GL(2, C) are obtained by composing translations, dilations, rotations and complex inversion. Its Lie algebra admits as a basis the transforms Therefore, for any u ∈ C 1 (R 2 , H 3 ), the functions Hyperbolic translations are obtained by composing a horizontal (Euclidean) translation p → p + ae 1 + be 2 , a, b ∈ R with an Euclidean homothety p → tp, t > 0. Therefore, for any u ∈ C m (R 2 , H 3 ), the functions e 1 , e 2 , u , (2.5)

Nondegeneracy of hyperbolic k-bubbles
The proof of Theorem 1.1 needs some preliminary work. We put where ω is given by (2.1). Since U is a conformal parametrization of the Euclidean sphere of radius r k about kr k e 3 , which coincides with the hyperbolic sphere of radius ρ k about e 3 , then U has curvature k and in fact it solves (P 0 ). Accordingly with (2.5), we put (notice that U e 3 = U), and introduce the 9-dimensional manifold Remark 3.1 Any surface U ∈ Z is an embedding and solves (P 0 ). Conversely, let U ∈ C 2 (R 2 , H 3 ) be an embedding. If U solves (P 0 ), then it is a k-bubble by Lemma A.2 and, thanks to an Alexandrov' type argument (see for instance [14,Corollary 10.3.2]) it parametrizes a sphere of hyperbolic radius ρ k and Euclidean radius r k . Since U is conformal, then ∆U = 2r −1 k ∂ x U ∧ ∂ y U . Therefore U ∈ Z by the uniqueness result in [5].
By the remarks in Subsection 2.4 and since ∇U q is proportional to ∇ω, we have that T Uq Z = T U Z for any q ∈ H 3 , and we infer another useful description of the tangent space, that is We now introduce the differential operator We denote by . In order to prove Theorem 1.1 it suffices to show that
Next, using the equivalent formulation we find that, for ϕ = ηω, η ∈ C 2 (R 2 ), it holds Thanks to (3.9) and (3.11) we get P J ′ 0 (U)ϕ = J ′ 0 (U)(Pϕ), thus to conclude the proof we can assume that ϕ = Pϕ. Since ϕ is pointwise orthogonal to ω, we trivially have We start to handle (3.6). From e 3 = z∇ω + ω 3 ω we get In a similar way one can check that Next, using (2.2) we can compute that holds for any ϕ ∈ ω ⊥ C m . Putting together the above informations we arrive at Using (3.10) and ϕ 3 = ϕ · (z∇ω), we conclude the proof.
Thanks to Lemma 3.2 we can study the system J ′ 0 (U)ϕ = 0 separately, on ω ⊥ C m first, and on ω C m later. In fact, ϕ ∈ ker J ′ 0 (U) if and only if the pair of functions We begin by facing problem (3.13a). Firstly, we show that the quadratic form associated to the differential operator
We identify f with a complex valued function. A direct computation based on (2.2) shows that ψ solves (3.15) if and only if f solves ∂ x f + i∂ y f = 0 on R 2 . In polar coordinates we have that For every ρ > 0 we expand the periodic function f (ρ, ·) in Fourier series, The coefficients γ h are complex-valued functions on the half-line R + that solve because of (3.16). Thus for every h ∈ Z there exists a h ∈ C such that γ h (ρ) = a h ρ h . Now recall that µψ ∈ L 2 (R 2 , R 3 ). Sincê and in particular the space of solutions of (3.13a) has (real) dimension 6. The conclusion of the proof follows from the relations (2.4).
Lemma 3.5 Let η ∈ C 2 (R 2 ) be a solution to (3.13b). There exists α ∈ R 3 such that and thus ηω ∈ Proof. First of all, we notice that α · (kω + e 3 ) solves (3.13b) for any α ∈ R 3 . By the Hilbert-Schmidt theorem, the eigenvalue problem has a non decreasing, divergent sequence (λ h ) h≥0 of eigenvalues which correspond to critical levels of the quotient Clearly, λ 0 = 0 is simple, and its eigenfunctions are constant functions. We claim that the next eigenvalue is 2k, and that its eigenspace has dimension 3, that concludes the proof.
To this goal, we use the functional change By a direct computation involving the identity (ω 3 (z) + k)µ(c k z) = (k − 1)µ(z) and integration by parts, one gets On the other hand, it is well known that is the first nontrivial eigenvalue for the Laplace-Beltrami operator on the sphere and that its eigenspace has dimension 3, see for instance [4]. This concludes the proof.

Remark 3.6
The third eigenvalue λ 2 of (3.17) verifies λ 2 > 2k by Lemma 3.5, and Proof of Theorem 1.1 In fact, we only have to sum up the argument. Let U ∈ Z. Thanks to (3.2), U = U q • g for some q ∈ H 3 , g ∈ P GL(2, C). Since it suffices to consider the case U = U.

Further results on the operator J ′ 0 (U)
To shorten notation we put

Since integration by parts giveŝ
can be extended to a continuous bilinear form H 1 × H 1 → R via a density argument. It can be checked by direct computations (see also Remark 4.2), that the quadratic form in (3.18) is self-adjoint on H 1 , that is, Since T U Z is a subspace of L 2 (R 2 , R 3 ) ≡ L 2 (S 2 , R 3 ), we are allowed to put To shorten notation we introduce on L 2 (R 2 , R 3 ) the equivalent scalar product and the subspaces We are in position to state the main result of this section.
In view of Lemma 3.2, we split the proof of Lemma 3.7 in few steps.
Proof. We introduce which is a closed subspace of H 1 . Notice that ψ = Pψ for any ψ ∈ X and moreover R 2 use (3.8) and a density argument. Next we put and notice that λ ≥ 0 by Lemma 3.3. On the other hand, λ is achieved by Rellich theorem. Thus λ > 0, because of Lemma 3.4. It follows that the energy functional I : X → R, is weakly lower semicontinuous and coercive. Thus its infimum is achieved by a function ϕ ∈ X which satisfiesR is the orthogonal projection of Pψ = ψ − ηω onto T U Z in the scalar product (·, ·) * and Pψ ⊥ := ψ − Pψ ⊤ − ηω ∈ X. We use (3.19) and (3.9) as v is orthogonal to T U Z ∋ Pψ ⊤ and to ηω in L 2 (R 2 , R 3 ). We showed that ϕ solves (3.21), and thus the proof is complete.
Proof. We introduce the space so that ηω ∈ H 1 ∩ T U Z ⊥ * ∩ N * for any η ∈ Y , and the energy functional I : Y → R, compare with (3.9). The functional I is weakly lower semicontinuous with respect to the H 1 (R 2 ) topology and coercive by Remark 3.6. Thus its infimum is achieved by a function η ∈ Y . To conclude, argue as in the proof of Lemma 3.8 to show that η solves (3.23).
To conclude the proof we have to show Since ω ∈ C ∞ (R 2 , R 3 ) and ω 3 + k is bounded and bounded away from zero, ϕ v solves a linear system of the form for certain smooth matrices on R 2 . A standard bootstrap argument and Schauder regularity theory plainly imply that ϕ v ∈ C 2+m loc (R 2 , R 3 ). The function z → ϕ v (z −1 ) satisfies a linear system of the same kind, hence ϕ v ∈ C 2+m (R 2 , R 3 ), as desired.

The perturbed problem
In this Section we perform the finite dimensional reduction and prove Theorems 1.2, 1.3. By the results in the Appendix, any critical point of the and has mean curvature (k + εφ), apart from a finite set of branch points. Due to the action of the Möbius transformations and of the hyperbolic translations, for any u ∈ C 2 (R 2 , R 3 ) we have the identities where F φ k is the Melnikov type function in (1.5). The above mentioned invariances give E 0 (U q ) = E 0 (U). Since the hyperbolic ball B H ρ k (q) coincides with the Euclidean ball of radius q 3 r k about the point q k := (q 1 , q 2 , kr k q 3 ), the divergence theorem gives Here is any vectorfield such that divQ φ (p) = p −3 3 φ(p) and ν p is the outer normal to ∂B q 3 r k (q k ) at p. The function U q in (3.1) parameterizes the Euclidean sphere ∂Bq 3 r k (q k ). Since ∂ x U q ∧ ∂ y U q is inward-pointing, we have and (4.3) is proved. Before going further, let us show that the existence of critical points for F φ k is a necessary condition for the conclusion in Theorem 1.2. H 3 ) and a point q ∈ H 3 such that u h solves (P ε h ), and u h → U q in C 1 (R 2 , H 3 ). Then q is a stationary point for F φ k .
Proof. The function u h is a stationary point for the energy functional We can plainly pass to the limit to obtain V ′ φ (U q )e j = 0 for j = 1, 2 and V ′ φ (U q )U q = 0. To conclude, use (4.4) and recall that ∂ q j U q = e j for j = 1, 2, and ∂ q 3 Now we fix m ∈ (0, 1). The operator J ε : is related to the differential of E ε via the identity then the quadratic form in the right hand side is a self-adjoint form on H 1 .
We are in position to state and proof the next lemma, which is the main step towards the proofs of Theorems 1.2, 1.3.
To prove that L is surjective fix v ∈ C m and (θ, b) ∈ R 6 × R 3 . We have to find ϕ ∈ C 2+m and (ζ, β) ∈ R 6 ×R 3 such that L 1 (ϕ; ζ, β) = v and L 2 (ϕ) = (θ, b). To this goal we introduce the minimal distance projection so that L 2 (w) is uniquely determined by P ⊤ w, and vice-versa. We find ζ j and β so that Then, we use Lemma 3.7 to find ϕ ∈ C 2+m ∩ T U Z ⊥ * ∩ N * such that Finally, we take the unique tangent direction ϕ ⊤ ∈ T U Z such that L 2 (ϕ ⊤ ) = (θ, b) − L 2 ( ϕ).
The triple (ϕ ⊤ + ϕ, ζ, β) satisfies L(ϕ ⊤ + ϕ, ζ, β) = (v; θ, b) and surjectivity is proved. We are in the position to apply the implicit function theorem to F, for any fixed q ∈ Ω 2s . In fact, thanks to a standard compactness argument, we get that there exist ε ′ > 0 and uniquely determined C 1 functions satisfies i), if ε ′ is small enough. Further, using (4.5) (see also Lemma A.1) we rewrite the last identity in (4.7) as (4.8) In particular, claim ii) holds true.
Proof of Theorem 1.2. Take an open set Ω ⋐ R 3 + containing the closure of A, let u ε q be the function given by Lemma 4.3 and notice that, by (4.4), E ε (U q ) = E 0 (U q ) − 2εF φ k (q). Thus for ε ∈ [−ε,ε], ε = 0 we can estimate uniformly on Ω by iv) in Lemma 4.3. Recalling the definition of stable critical point presented in Subsection 2.2, we infer that for any ε ≈ 0 the function 1 2ε E ε (u ε q ) − E 0 (U q ) has a critical point q ε ∈ A, to which corresponds the embedded (k + εφ)-bubble u ε := u ε q ε by iii) in Lemma 4.3. The continuity of (ε, q) → u ε q gives the continuity of ε → u ε . The last conclusion in Theorem 1.2 follows via a simple compactness argument and thanks to Theorem 4.1.
Since r k → 0 and kr k = k(k 2 − 1) −1/2 → 1 as k → ∞, we infer that q k → q uniformly on compact sets of R 3 + and 3 4πr 3 uniformly on Ω. Next, we easily compute measures the signed (hyperbolic) volume enclosed by the surface u, with respect to the weight K. In fact, if u parameterizes the boundary of a smooth open set Ω ⋐ R 3 + and if ∂ x u ∧ ∂ y u is inward-pointing, then the divergence theorem gives Clearly, the functional V K does not depend on the choice of the vectorfield Q. Notice that if K ≡ k is constant, then In the next Lemma we collect few simple remarks about the energy functional i) The functional E : C 1 (R 2 , H 3 ) → R is of class C 1 , and its differential is given by ii) If u ∈ C 2 (R 2 , H 3 ), then E ′ (u) extends to a continuous form on C 0 (R 2 , R 3 ) , namely E ′ (u)ϕ =R 2 (−div(u −2 3 ∇u) − u −3 3 |∇u| 2 e 3 + 2u −3 3 K(u)∂ x u ∧ ∂ y u) · ϕ dz ; iii) If K ∈ C 1 (H 3 ), then E is of class C 2 on C 1 (R 2 , H 3 ).
In the next Lemma we show that critical points for E are in fact hyperbolic K-bubbles.
Lemma A.2 Let K ∈ C 0 (H 3 ) and let u ∈ C 2 (R 2 , H 3 ) be a nonconstant critical point for E. Then u is conformal, that is, |∂ x u| = |∂ y u| , ∂ x u · ∂ y u = 0 , hence it parameterizes an S 2 type surface in H 3 , having mean curvature K, apart from a finite number of branch points.
In particular, ∂ p 1 K, ∂ p 2 K and the radial derivative of K can not have constant sign in Ω. We infer the next nonexistence result (see [7,Proposition 4.1] for the Euclidean case).
Theorem A.5 Assume that K ∈ C 1 (H 3 ) satisfies one of the following conditions, i) K(p) = f (ν · p) for some direction ν orthogonal e 3 , where f is strictly monotone; ii) K(p) = f (|p|), where f is strictly monotone.