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

Abstract

Given a constant $k>1$, let Z be the family of round spheres of radius ${{\,\mathrm{artanh}\,}}(k^{-1})$ 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 $\mathcal{C}^1$-curve, parametrized by $\varepsilon \approx 0$, of embedded spheres in ${\mathbb {H}}^3$ having mean curvature $k +\varepsilon \phi$ at each point.

1 Introduction

Let K be a given function on the hyperbolic space ${\mathbb {H}}^3$. The K-bubble problem consists in finding a K-bubble, which is an immersed surface $u:{\mathbb {S}}^2\rightarrow {\mathbb {H}}^3$ 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 $\varGamma$, see for instance [1, 13]. In the Euclidean case, the impact of K-bubbles on nonexistence and lack of compactness phenomena in the Plateau problem has been investigated in [5, 8, 9].

To look for K-bubbles in the hyperbolic setting one can model ${\mathbb {H}}^3$ via the Poincaré upper half-space $({\mathbb {R}}^3_+, p_3^{-2}\delta _{h j})$ and consider the elliptic system

$$\begin{aligned} \varDelta u - 2 u_3^{-1} {{G}(\nabla u)} =2 u_3^{-1}K(u) ~\partial _x u\wedge \partial _y u \end{aligned}$$

(1.1)

for functions $u=(u_1,u_2,u_3)\in \mathcal{C}^2({\mathbb {S}}^2,{\mathbb {H}}^3)$. Here we used the stereographic projection to introduce local coordinates on ${\mathbb {S}}^2\equiv {\mathbb {R}}^2\cup \{\infty \}$ and put

$$\begin{aligned} {G}_{\ell }(\nabla u):=\nabla u_3\cdot \nabla u_\ell - \frac{1}{2}|\nabla u|^2 \delta _{\ell 3} =-\frac{1}{2}u_3\sum _{h,j=1}^3{\varGamma }^\ell _{hj}(u)\nabla u_h\cdot \nabla u_j,\quad \ell =1,2,3, \end{aligned}$$

(1.2)

where ${\varGamma }^\ell _{hj}$ are the Christoffel symbols. Any nonconstant solution u to (1.1) is a generalized K-bubble in ${\mathbb {H}}^3$ (see Lemma A.2 in the Appendix and [14, 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 [22], for surfaces in ${\mathbb {R}}^3$ (see for instance [7, 10, 12, 20] and references therein; see also the pioneering paper [23] by Ye and [3, 6, 19, 21] 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 ${\mathbb {R}}^3$ can be parameterized by an embedded k-bubble, which minimizes the energy functional

$$\begin{aligned} E_{\text {Eucl}}(u)=\frac{1}{2}\mathop {\int }\limits _{{\mathbb {R}}^{2}}|\nabla u|^2~ dz+\frac{2k}{3} \mathop {\int }\limits _{{\mathbb {R}}^{2}}u\cdot \partial _x u\wedge \partial _yu~dz \end{aligned}$$

on the Nehari manifold $\{\, u\ne \text {const.}\ |\ E_{\text {Eucl}}'(u)u=0\,\}$, see [7, Remark 2.6]. In contrast, no immersed hyperbolic k-bubble exists if $k\in (0,1]$, see for instance [16, Theorem 10.1.3]. If $k>1$, then any sphere in ${\mathbb {H}}^3$ of radius

$$\begin{aligned} \rho _{k}: ={{\,\mathrm{artanh}\,}}\frac{1}{k}=\frac{1}{2}\ln \frac{k+1}{k-1} \end{aligned}$$

can be parameterized by an embedding $U:{\mathbb {S}}^2\rightarrow {\mathbb {H}}^3$, which solves

and which is a critical point of the energy functional

$$\begin{aligned} E_{\text {hyp}}(u)=\frac{1}{2}\mathop {\int }\limits _{{\mathbb {R}}^{2}}u_3^{-2}|\nabla u|^2~ dz-k \mathop {\int }\limits _{{\mathbb {R}}^{2}}u_3^{-2}e_3\cdot \partial _x u\wedge \partial _yu~dz. \end{aligned}$$

(1.3)

As in the Euclidean case, the functional $E_{\text {hyp}}$ is unbounded from below (see Remark A.1). Therefore U does not minimize the energy $E_{\text {hyp}}$ on the Nehari manifold, which in fact fills $\{\,u\ne \text {const.}\,\}$.

Besides their invariance with respect to Möbius transformations, both system ($\mathcal {P}_{0}$) and the related energy $E_{\text {hyp}}$ 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 ($\mathcal {P}_{0}$). We explicitly describe the tangent space $T_ {U}Z$ at $U\in Z$ in formula (3.5).

As a further consequence of the invariances of problem ($\mathcal {P}_{0}$), any tangent direction $\varphi \in T_{U}Z$ solves the elliptic system

$$\begin{aligned} \varDelta {\varphi }-2U_3^{-1}{G}'(\nabla U)\nabla \varphi =-U_3^{-1}\varphi _3\varDelta U +2 {U}_3^{-1} k \big (\partial _x {\varphi }\wedge \partial _y {U}+\partial _x {U}\wedge \partial _y {\varphi }\big ), \end{aligned}$$

(1.4)

which is obtained by linearizing ($\mathcal {P}_{0}$) at U.

The next one is the main result of the present paper.

Theorem 1.1

(Nondegeneracy) Let $U\in Z$. If $\varphi \in \mathcal{C}^2({\mathbb {S}}^2,{\mathbb {R}}^3)$ solves the linear system (1.4), then $\varphi \in T_{U}Z$.

Different proofs of the nondegeneracy in the Euclidean case can be found in [11, 15, 17]. The proof of Theorem 1.1 (see Sect. 3), is considerably more involved. It requires the choice of a suitable orthogonal frame for functions in $\mathcal{C}^2({\mathbb {S}}^2,{\mathbb {R}}^3)$ and the complete classification of solutions of two systems of linear elliptic differential equations, which might have an independent geometrical interest (see Lemmata 3.3, 3.4).

As an application of Theorem 1.1, we provide sufficient conditions on a prescribed smooth function $\phi :{\mathbb {H}}^3\rightarrow {\mathbb {R}}$ that ensure the existence of embedded surfaces ${\mathbb {S}}^2\rightarrow {\mathbb {H}}^3$ having nonconstant mean curvature $k +\varepsilon \phi$. Our existence results involve the notion of stable critical point already used in [18] and inspired from [2, Chapter 2] (see Sect. 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 $\phi \in \mathcal{C}^1({\mathbb {H}}^3)$ be given. Assume that the function

$$\begin{aligned} F^{\phi }_k(q):=\int \limits _{B^{\mathbb {H}}_{\rho _{k}}(q)}\phi (p)~d{\mathbb {H}}^3_p,\quad F^{\phi }_k:{\mathbb {H}}^3\rightarrow {\mathbb {R}} \end{aligned}$$

(1.5)

has a stable critical point in an open set $A\Subset {\mathbb {H}}^3$. For every $\varepsilon \in {\mathbb {R}}$ close enough to 0 there exist a point $q^\varepsilon \in A$, a conformal parametrization ${\textsc {U} }_{q^\varepsilon }$ of a sphere of radius $\rho _k$ about $q^\varepsilon$, and a conformally embedded $(k+\varepsilon \phi )$-bubble $u^\varepsilon$, such that $\Vert u^\varepsilon -{\textsc {U} }_{q^\varepsilon }\Vert _{\mathcal{C}^2}=O(\varepsilon )$ as $\varepsilon \rightarrow 0$.

Moreover, any sequence $\varepsilon _h\rightarrow 0$ has a subsequence $\varepsilon _{h_j}$ such that $q^{\varepsilon _{h_j}}$ converges to a critical point for $F^{\phi }_k$. In particular, if $\hat{q}\in A$ is the unique critical point for $F^{\phi }_k$ in $\overline{A}$, then $u^\varepsilon \rightarrow {\textsc {U} }_{\hat{q}}$ in $\mathcal{C}^2({\mathbb {S}}^2,{\mathbb {H}}^3)$.

Theorem 1.3

Assume that $\phi \in \mathcal{C}^1({\mathbb {H}}^3)$ has a stable critical point in an open set $A\Subset {\mathbb {H}}^3$. Then there exists $k_0>1$ such that for any $k>k_0$ and for every $\varepsilon$ close enough to 0, there exists a conformally embedded $(k+\varepsilon \phi )$-bubble.

In Sect. 4 we first show that the existence of a critical point for $F^{\phi }_k(q)$ 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 weaken the regularity assumption on $\phi$ (from $\mathcal{C}^2$ to $\mathcal{C}^1$).

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 $\phi$ in Theorem 1.3.

2 Notation and preliminaries

The vector space ${\mathbb {R}}^n$ is endowed with the Euclidean scalar product $\xi \cdot \xi '$ and norm $|\xi |$. We denote by $\{e_1, e_2, e_3\}$ the canonical basis and by $\wedge$ the exterior product in ${\mathbb {R}}^3$.

We will often identify the complex number $z=x+iy$ with the vector $z=(x,y)\in {\mathbb {R}}^2$. Thus, $iz\equiv (-y,x)$, $z^2\equiv (x^2-y^2,2xy)$ and $z^{-1}\equiv |z|^{-2}(x,-y)$ if $z\ne 0$.

In local coordinates induced by the stereographic projections from the north and the south poles, the round metric on the sphere ${\mathbb {S}}^2$ is given by $g_{hj}=\mu ^2\delta _{hj}$, $d{\mathbb {S}}^2=\mu ^2dz$, where

$$\begin{aligned} \mu (z)=\frac{2}{1+|z|^2}. \end{aligned}$$

We identify the compactified plane ${\overline{{\mathbb {R}}}\,}^2= {{\mathbb {R}}^2 \cup \{\infty \}}$ with the sphere ${\mathbb {S}}^2$ through the inverse of the stereographic projection from the north pole, which is explicitly given by

$$\begin{aligned} \omega (x,y)=(\mu x,\mu y, 1-\mu ). \end{aligned}$$

(2.1)

The identity $|\omega |^2\equiv 1$ trivially gives $\omega \cdot \partial _x\omega \equiv 0$, $\omega \cdot \partial _y\omega \equiv 0$. We also notice that $\omega$ is a conformal (inward-pointing) parametrization of the unit sphere and satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} \varDelta \omega = 2 {\partial _{x}\omega } \wedge {\partial _{y}\omega },\quad -\varDelta \omega =2\mu ^2\omega \\ {\partial _{x}\omega } \cdot {\partial _{y}\omega }=0\,\\ |{\partial _{x}\omega }|^2 = |{\partial _{y}\omega }|^2 =\frac{1}{2}|\nabla \omega |^2 = \mu ^2. \\ \partial _{x}\omega \wedge \omega = \partial _{y} \omega ,\quad \omega \wedge \partial _{y} \omega = \partial _{x}\omega ,\quad \partial _{x}\omega \wedge \partial _y\omega = -\mu ^2 \omega . \end{array}\right. } \end{aligned}$$

(2.2)

2.1 The Poincaré half-space model

We adopt as model for the three dimensional hyperbolic space ${\mathbb {H}}^3$ the upper half-space ${\mathbb {R}}^3_+=\{ (p_1,p_2,p_3)\in {\mathbb {R}}^3\ | \ p_3 > 0 \}$ endowed with the Riemannian metric $g_{h j}= p_3^{-2}\delta _{h j}$ .

The hyperbolic distance $d_{\mathbb {H}}(p,q)$ in ${\mathbb {H}}^3$ is related to the Euclidean one by

$$\begin{aligned} \cosh d_{\mathbb {H}}(p,q) = 1 + \frac{|p-q|^2}{2 p_3 q_3}, \end{aligned}$$

and the hyperbolic ball $B^{\mathbb {H}}_\rho ({p})$ centered at $p=(p_1, p_2,p_3)$ is the Euclidean ball of center $(p_1, p_2, p_3 \cosh \rho )$ and radius $p_3 \sinh \rho$.

If $F:{\mathbb {H}}^3\rightarrow {\mathbb {R}}$ is a differentiable function, then $\nabla ^{{\mathbb {H}}} F(p)=p_3^2\nabla F(p)$, where $\nabla ^{{\mathbb {H}}}$, $\nabla$ are the hyperbolic and the Euclidean gradients, respectively. In particular, $\nabla ^{{\mathbb {H}}} F(p)=0$ if and only if $\nabla F(p)=0$. The hyperbolic volume form is related to the Euclidean one by $d{{\mathbb {H}}_p^3}=p_3^{-3}dp$.

2.2 Stable critical points

Let ${X}\in \mathcal{C}^1({\mathbb {H}}^3)$ and let $\varOmega \Subset {\mathbb {H}}^3$ be open. We say that X has a stable critical point in $\varOmega$ if there exists $r>0$ such that any function ${G}\in \mathcal{C}^1(\overline{\varOmega })$ satisfying $\displaystyle {\Vert {G}-{X}\Vert _{\mathcal{C}^1(\overline{\varOmega })}<r}$ has a critical point in $\varOmega$.

As noticed in [18], conditions to have the existence of a stable critical point $p\in \varOmega$ for X are easily given via elementary calculus. For instance, one can use Browder’s topological degree theory or can assume that

$$\begin{aligned} \displaystyle {{\min _{\partial \varOmega } {X}>\min _{\varOmega } {X}}}\quad \text {or}\quad \displaystyle {\max _{\partial \varOmega } {X}<\max _{\varOmega } {X}}. \end{aligned}$$

Finally, if X is of class $\mathcal{C}^2$ and $\varOmega$ contains a nondegenerate critical point $p_0$ (i.e. the Hessian matrix of X at $p_0$ is invertible), then $p_0$ is stable.

2.3 Function spaces

Any function f on ${\overline{{\mathbb {R}}}\,}^2$ is identified with $f\circ \omega ^{-1}$, which is a function on ${\mathbb {S}}^2$. If no confusion can arise, from now on we write f instead of $f\circ \omega ^{-1}$.

The Hilbertian norm on $L^2({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}}^n)\equiv L^2({\mathbb {S}}^2,{\mathbb {R}}^n)$ is given by

$$\begin{aligned} \Vert f\Vert _{2}^2=\int _{{{\mathbb {R}}}^{2}} |f|^2\,\mu ^2dz<\infty . \end{aligned}$$

Let $m\ge 0$. We endow

$$\begin{aligned} \mathcal{C}^m({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}}^n)=\{\,u\in \mathcal{C}^m({\mathbb {R}}^2,{\mathbb {R}}^n)~|~ u(z^{-1})\in \mathcal{C}^m({\mathbb {R}}^2,{\mathbb {R}}^n)\,\}\equiv \mathcal{C}^m({\mathbb {S}}^2,{\mathbb {R}}^n) \end{aligned}$$

with the standard Banach space structure (we agree that $\mathcal{C}^m({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}}^n)=\mathcal{C}^{\lfloor m\rfloor , m-\lfloor m\rfloor }({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}}^n)$ if m is not an integer). If m is an integer, a norm in $\mathcal{C}^m({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}}^n)$ is given by

$$\begin{aligned} \Vert u\Vert _{\mathcal{C}^m}= {\sum _{j=0}^m\Vert \mu ^{-j}\nabla ^j u\Vert _\infty ~}. \end{aligned}$$

(2.3)

Since we adopt the upper half-space model for ${\mathbb {H}}^3$, we are allowed to write

$$\begin{aligned} \mathcal{C}^m({\overline{{\mathbb {R}}}\,}^2,{\mathbb {H}}^3)=\mathcal{C}^m({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}}^3_+)=\{u\in \mathcal{C}^m({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}}^3)~|~ u_3>0\}, \end{aligned}$$

so that $\mathcal{C}^m({\overline{{\mathbb {R}}}\,}^2,{\mathbb {H}}^3)$ is an open subset of $\mathcal{C}^m({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}}^3)$.

If $\psi ,\varphi \in \mathcal{C}^1({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}}^3)$ and $\tau \in {\mathbb {R}}^2$, we put

$$\begin{aligned} \nabla \psi \cdot \nabla \varphi = \partial _{x} \psi \cdot \partial _{x} \varphi + \partial _{y} \psi \cdot \partial _{y}\varphi ,\quad \tau \nabla \varphi = \tau _1\partial _{x}\varphi +\tau _2\partial _{y}\varphi \end{aligned}$$

(notice that $\tau \nabla \varphi (z)=d\varphi (z)\tau$ for any $z\in {\mathbb {R}}^2$). For instance, we have

$$\begin{aligned} z^h\nabla \varphi ={\left\{ \begin{array}{ll} \partial _{x}\varphi &{}\text {if }h=0\\ x {\partial _{x}\varphi } + y {\partial _{y}\varphi }&{}\text {if }h=1 \end{array}\right. },\qquad iz^h\nabla \varphi ={\left\{ \begin{array}{ll} \partial _y\varphi &{}\text {if }h=0\\ -y {\partial _{x}\varphi } + x {\partial _{y}\varphi }&{}\text {if }h=1. \end{array}\right. } \end{aligned}$$

For future convenience we notice, without proof, that the next identities hold:

$$\begin{aligned} {\left\{ \begin{array}{ll} ~~~\partial _{x}\omega = e_1 - \omega _1 \omega - e_2 \wedge \omega \\ ~~z\nabla \omega = e_3 - \omega _3 \omega \\ z^2\nabla \omega = - (e_1 - \omega _1\omega + e_2 \wedge \omega ) \end{array}\right. } \quad \quad {\left\{ \begin{array}{ll} ~~~\partial _{y}\omega = e_2 - \omega _2 \omega + e_1 \wedge \omega \\ ~~iz\nabla \omega = e_3 \wedge \omega ,\\ iz^2\nabla \omega = e_2 -\omega _2 \omega - e_1 \wedge \omega . \end{array}\right. } \end{aligned}$$

(2.4)

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

$$\begin{aligned} H^1({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}}^n)= \{\, u\in H^1_{\mathrm{loc}}({\mathbb {R}}^2,{\mathbb {R}}^n)~|~|\nabla u|+|u|\,\mu \,\in L^2({\mathbb {R}}^2)\,\} \equiv H^1({\mathbb {S}}^2,{\mathbb {R}}^n). \end{aligned}$$

We simply write $L^2({\overline{{\mathbb {R}}}\,}^2)$, $\mathcal{C}^m({\overline{{\mathbb {R}}}\,}^2)$ and $H^1({\overline{{\mathbb {R}}}\,}^2)$ instead of $L^2({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}})$, $\mathcal{C}^m({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}})$ and $H^1({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}})$, respectively.

2.4 Möbius transformations and hyperbolic translations

Transformations in $PGL(2,{\mathbb {C}})$ are obtained by composing translations, dilations, rotations and complex inversion. Its Lie algebra admits as a basis the transforms

$$\begin{aligned} z\mapsto 1,~~z\mapsto i,~~ z\mapsto z,~~z\mapsto iz,~~z\mapsto z^2,~~z\mapsto iz^2\,. \end{aligned}$$

Therefore, for any $u\in \mathcal{C}^1({\overline{{\mathbb {R}}}\,}^2,{\mathbb {H}}^3)$, the functions

$$\begin{aligned} z^h\nabla u,\quad iz^h\nabla u,\qquad h=0,1,2, \end{aligned}$$

span the tangent space to the manifold $\{\, u\circ g~|~g\in PGL(2,{\mathbb {C}})\, \}$ at u.

Hyperbolic translations are obtained by composing a horizontal (Euclidean) translation $p\mapsto p+ae_1+be_2$, $a,b\in {\mathbb {R}}$ with an Euclidean homothety $p\mapsto tp$, $t>0$. Therefore, for any $u\in \mathcal{C}^m({\overline{{\mathbb {R}}}\,}^2,{\mathbb {H}}^3)$, the functions

$$\begin{aligned} e_1,\quad e_2,\quad u, \end{aligned}$$

span the tangent space to the manifold $\{\, u_q~|~q\in {\mathbb {H}}^3\, \}$ at u, where

$$\begin{aligned} u_q:=q_3 u+q-(q\cdot e_3)e_3\,. \end{aligned}$$

(2.5)

3 Nondegeneracy of hyperbolic k-bubbles

The proof of Theorem 1.1 needs some preliminary work. We put

$$\begin{aligned} {\textsc {U} } = r_{k}(\omega + ke_3),\quad r_{k}:=\sinh \rho _k=\frac{1}{k}\cosh \rho _{k}=\frac{1}{\sqrt{k^2-1}}, \end{aligned}$$

where $\omega$ is given by (2.1). Since ${\textsc {U} }$ 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 $\rho _k$ about $e_3$, then ${\textsc {U} }$ has curvature k and in fact it solves ($\mathcal {P}_{0}$). Accordingly with (2.5), we put

$$\begin{aligned} {\textsc {U} }_{q}:=q_3{\textsc {U} }+q-(q\cdot e_3)~e_3 \end{aligned}$$

(3.1)

(notice that ${\textsc {U} }_{e_3}={\textsc {U} }$), and introduce the 9-dimensional manifold

$$\begin{aligned} Z=\big \{\, {\textsc {U} }_{q}\circ g~|~g\in PGL(2,{\mathbb {C}}),~q\in {\mathbb {H}}^3\,\big \}. \end{aligned}$$

(3.2)

Remark 3.1

Any surface $U\in Z$ is an embedding and solves ($\mathcal {P}_{0}$). Conversely, let $U\in \mathcal{C}^2({\overline{{\mathbb {R}}}\,}^2,{\mathbb {H}}^3)$ be an embedding. If U solves ($\mathcal {P}_{0}$), then it is a k-bubble by Lemma A.2 and, thanks to an Alexandrov’ type argument (see for instance [16, Corollary 10.3.2]) it parametrizes a sphere of hyperbolic radius $\rho _k$ and Euclidean radius $r_k$. Since U is conformal, then $\varDelta U=2r_{k}^{-1}\partial _xU\wedge \partial _yU$. Therefore $U\in Z$ by the uniqueness result in [5].

By the remarks in Sect. 2.4 and since $\nabla {\textsc {U} }_{q}$ is proportional to $\nabla \omega$, we have that $T_{{\textsc {U} }_{q}}Z=T_{{\textsc {U} }}Z$ for any $q\in {\mathbb {H}}^3$, and

$$\begin{aligned} T_{{\textsc {U} }}Z=\big \langle \{z^h\nabla \omega , ~iz^h\nabla \omega ~|~h=0,1,2\}\big \rangle \oplus \big \langle e_1, e_2, {\textsc {U} }~\big \rangle . \end{aligned}$$

(3.3)

Moreover, any tangent direction $\tau \in T_{{\textsc {U} }}Z$ solves (1.4).

It is convenient to split $\mathcal{C}^m({\overline{{\mathbb {R}}}\,}^2, {\mathbb {R}}^3)$ in the direct sum of its closed subspaces

$$\begin{aligned} \langle \omega \rangle _{\mathcal{C}^m}^\perp : = \{\, \varphi \in \mathcal{C}^m({\overline{{\mathbb {R}}}\,}^2, {\mathbb {R}}^3) \ |\ \varphi \cdot \omega \equiv 0 \text { on }{\mathbb {R}}^2\,\},\quad \langle \omega \rangle _{\mathcal{C}^m}:=\{\,\eta \omega ~|~\eta \in \mathcal{C}^m({\overline{{\mathbb {R}}}\,}^2)\, \}. \end{aligned}$$

(3.4)

Since $T_{{\textsc {U} }}Z= \big (T_{{\textsc {U} }}Z\cap \langle \omega \rangle ^\perp _{{C}^2}\big )\oplus \big (T_{{\textsc {U} }}Z\cap \langle \omega \rangle _{{C}^2}\big )$, from (2.4) we infer another useful description of the tangent space, that is

$$\begin{aligned} T_{{\textsc {U} }}Z=\big \{\, s-(s\cdot \omega )\omega +t\wedge \omega ~|~s,t \in {\mathbb {R}}^3\,\big \}\oplus \big \{\,(\alpha \cdot (k\omega +e_3))~\omega ~|~\alpha \in {\mathbb {R}}^3\,\big \}\, . \end{aligned}$$

(3.5)

We now introduce the differential operator

$$\begin{aligned} J_0(u)= -\mathrm{div}(u_3^{-2}\nabla u)-u_3^{-3}|\nabla u|^2e_3+2ku_3^{-3}{\partial _{x}u}\wedge {\partial _{y}u}. \end{aligned}$$

Notice that $Z\subset \{J_0=0\}$. Further, let $I(z)=z^{-1}$. Since $J_0(u\circ I)=|z|^{-4}J_0(u)\circ I$ for any $u\in \mathcal{C}^{2+m}({\overline{{\mathbb {R}}}\,}^2,{\mathbb {H}}^3)$, $m\ge 0$, then $J_0$ is a $\mathcal{C}^1$ map

$$\begin{aligned} J_0: \mathcal{C}^{2+m}({\overline{{\mathbb {R}}}\,}^2,{\mathbb {H}}^3)\rightarrow \mathcal{C}^m({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}}^3). \end{aligned}$$

We denote by $J'_0(u) : \mathcal{C}^{2+m}({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}}^3)\rightarrow \mathcal{C}^m({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}}^3)$ its differential at u.

Finally, $J_0({\textsc {U} }_{q}\circ g)=0$ for any $g\in PGL(2,{\mathbb {C}})$, $q\in {\mathbb {H}}^3$, that implies $T_{{\textsc {U} }}Z\subseteq \ker J'_0({\textsc {U} })$. In order to prove Theorem 1.1 it suffices to show that

$$\begin{aligned} \ker {J}'_0({\textsc {U} })\subseteq T_{{\textsc {U} }}Z. \end{aligned}$$

Main computations

Recall that ${\textsc {U} }=r_{k}(\omega +ke_3)$ solves $J_0({\textsc {U} })=0$ to check that

$$\begin{aligned} J'_0({\textsc {U} })\varphi= & {} - \text {div}\big ({\textsc {U} }_3^{-2}{\nabla \varphi }\big )\\&+ 2{\textsc {U} }_3^{-3}\big [G'(\nabla {\textsc {U} })\nabla \varphi -\nabla U_3\nabla \varphi -\frac{1}{2}\varphi _3\varDelta {\textsc {U} } +k(\partial _x \varphi \wedge {\partial _{y}{\textsc {U} }} + {\partial _{x}{\textsc {U} }} \wedge \partial _y \varphi )\big ], \end{aligned}$$

where G is given by (1.2). Since $\nabla \omega _3=-\nabla \mu =\mu ^2z$, thanks to (2.2) we have

$$\begin{aligned}&r_{k}^2J'_0({\textsc {U} })\varphi = - \text {div}\big ((\omega _3+k)^{-2}{\nabla \varphi }\big )\nonumber \\&\quad + 2(\omega _3 + k)^{-3} \big [\big ({G}'(\nabla \omega )\nabla \varphi -\mu ^2z\nabla \varphi \big )+\mu ^2 \varphi _3\omega + {k} (\partial _x \varphi \wedge {\partial _{y}\omega } + {\partial _{x}\omega } \wedge \partial _y \varphi )\big ], \end{aligned}$$

(3.6)

$$\begin{aligned}&{G}'(\nabla \omega )\nabla \varphi -\mu ^2z\nabla \varphi =\nabla \varphi _3\nabla \omega -(\nabla \varphi \cdot \nabla \omega )e_3. \end{aligned}$$

(3.7)

To rewrite (3.6) in a less obscure form, we decompose any $\varphi \in \mathcal{C}^m({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}}^3)$, $m\ge 0$, as

$$\begin{aligned} \varphi ={{\mathcal {P}}\varphi } + (\varphi \cdot \omega ) \omega , \qquad {{\mathcal {P}}\varphi } := \varphi - (\varphi \cdot \omega )\omega =\mu ^{-2}\big ((\varphi \cdot \partial _x\omega )\partial _x\omega +(\varphi \cdot \partial _y\omega )\partial _y\omega \big ), \end{aligned}$$

(3.8)

compare with (3.4). Accordingly, for $\varphi \in \mathcal{C}^{2}({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}}^3)$ we have

$$\begin{aligned} J'_0({\textsc {U} })\varphi = \mathcal P\big (J_0'({\textsc {U} })\varphi \big )+(J_0'({\textsc {U} })\varphi \cdot \omega )\omega , \end{aligned}$$

so that we can reconstruct $J'_0({\textsc {U} })\varphi \in \mathcal{C}^0({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}}^3)$ by providing explicit expressions for $\mathcal P\big (J_0'({\textsc {U} })\varphi \big )$ and $J_0'({\textsc {U} })\varphi \cdot \omega$, separately. This will be done in the next Lemma.

Lemma 3.1

Let $\varphi \in \mathcal{C}^2({\overline{{\mathbb {R}}}\,}^2, {\mathbb {R}}^3)$. Then

$$\begin{aligned}&r_{k}^2\mathcal P\big (J_0'({\textsc {U} })\varphi \big ) = {{\mathcal {P}}} \Big (-\mathrm{div}\Big (\frac{\nabla {{\mathcal {P}}\varphi }}{(\omega _3 + k)^2} \Big )\Big )+ \frac{2\mu ^2}{(\omega _3 + k)^3}~(iz\nabla {{\mathcal {P}}\varphi }) \wedge \omega - \frac{2\mu ^2}{(\omega _3 + k)^2} ~{{\mathcal {P}}\varphi }, \end{aligned}$$

(3.9)

$$\begin{aligned}&r_{k}^2(J_0'({\textsc {U} })\varphi )\cdot \omega = -\mathrm{div}\Big (\frac{\nabla (\varphi \cdot \omega )}{(\omega _3 + k)^2}\Big ) - \frac{2k\, \mu ^2}{(\omega _3 + k)^3}~(\varphi \cdot \omega ). \end{aligned}$$

(3.10)

Proof

We introduce the differential operator $L=-\mathrm{div}\big ((\omega _3 + k)^{-2}\nabla ~\big )$ and start to prove (3.10) by noticing that

$$\begin{aligned} L\varphi \cdot \omega =L(\varphi \cdot \omega )+2(\omega _3+k)^{-3}\big [(\omega _3+k)\nabla \varphi \cdot \nabla \omega -\mu ^2\varphi \cdot (z\nabla \omega ) - \mu ^2(\omega _3 + k)(\varphi \cdot \omega )\big ]. \end{aligned}$$

(3.11)

Recalling that $\omega$ is pointwise orthogonal to $\partial _x\omega , \partial _y\omega$, from (3.7) we obtain

$$\begin{aligned} \big ({G}'(\nabla \omega )\nabla \varphi -\mu ^2z\nabla \varphi \big )\cdot \omega = -(\nabla \varphi \cdot \nabla \omega )\omega _3. \end{aligned}$$

Further, by (2.2) we have $(\partial _x \varphi \wedge {\partial _{y}\omega } + {\partial _{x}\omega } \wedge \partial _y \varphi )\cdot \omega = -\nabla \varphi \cdot \nabla \omega$. Finally, we obtain

$$\begin{aligned} r_{k}^2(J_0'({\textsc {U} })\varphi )\cdot \omega =L(\varphi \cdot \omega )-2(\omega _3+k)^{-3}\mu ^2 \big [\varphi \cdot (z\nabla \omega )-\varphi _3 + (\omega _3 + k) (\varphi \cdot \omega )\big ], \end{aligned}$$

and (3.10) follows, because $e_3=z\nabla \omega +\omega _3\omega$, see (2.4).

Next, using the equivalent formulation

$$\begin{aligned} \begin{aligned} {\textsc {U} }_3^2 J'_0({\textsc {U} })\varphi =-\varDelta \varphi + 2(\omega _3 + k)^{-1}\left[ G'(\nabla \omega )\nabla \varphi + \mu ^2 \omega \varphi _3+ k(\partial _{x}\varphi \wedge {\partial _{y}\omega } + {\partial _{x}\omega } \wedge \partial _{y}\varphi )\right] \end{aligned} \end{aligned}$$

we find that, for $\varphi = \eta \omega$, $\eta \in \mathcal{C}^2({\overline{{\mathbb {R}}}\,}^2)$, it holds

$$\begin{aligned} {\textsc {U} }_3^2 J'_0({\textsc {U} })(\eta \omega )\cdot \partial _x \omega = {\textsc {U} }_3^2 J'_0({\textsc {U} })(\eta \omega )\cdot \partial _y \omega = 0, \end{aligned}$$

whence we infer

$$\begin{aligned} {\mathcal {P}}\big (J_0'({\textsc {U} })(\varphi - {\mathcal {P}}\varphi )\big ) = 0, \quad \text {for every }\varphi \in \mathcal{C}^2({\overline{{\mathbb {R}}}\,}^2, {\mathbb {R}}^3)\,. \end{aligned}$$

(3.12)

Thanks to (3.10) and (3.12) we get ${\mathcal {P}}\big (J_0'({\textsc {U} })\varphi \big )=J_0'({\textsc {U} })({\mathcal {P}}\varphi )$, thus to conclude the proof we can assume that $\varphi ={\mathcal {P}}\varphi$. Since $\varphi$ is pointwise orthogonal to $\omega$, we trivially have

$$\begin{aligned} \partial _x\varphi \cdot \omega =-\varphi \cdot \partial _x\omega ,\quad \partial _y\varphi \cdot \omega =-\varphi \cdot \partial _y\omega . \end{aligned}$$

We start to handle (3.7). From $e_3=z\nabla \omega +\omega _3\omega$ we get

$$\begin{aligned}&(G'(\nabla \omega )\nabla \varphi -\mu ^2z\nabla \varphi )+\omega _3(\nabla \varphi \cdot \nabla \omega )\omega = \nabla \varphi _3\nabla \omega - (\nabla \varphi \cdot \nabla \omega )z\nabla \omega \\&\quad = \big (\partial _x\varphi _3-x (\nabla \varphi \cdot \nabla \omega )\big )\partial _x\omega +\big (\partial _y\varphi _3-y (\nabla \varphi \cdot \nabla \omega )\big )\partial _y\omega . \end{aligned}$$

Further,

$$\begin{aligned}&\partial _x\varphi _3-x (\nabla \varphi \cdot \nabla \omega )=\partial _x\varphi \cdot (z\nabla \omega +\omega _3\omega )-x(\nabla \varphi \cdot \nabla \omega )\\&\quad = \big (\partial _x\varphi \cdot (z\nabla \omega )-x (\nabla \varphi \cdot \nabla \omega )\big )-\omega _3\varphi \cdot \partial _x\omega =-(iz\nabla \varphi )\cdot \partial _y\omega -\omega _3\varphi \cdot \partial _x\omega . \end{aligned}$$

In a similar way one can check that $\partial _y\varphi _3-y(\nabla \varphi \cdot \nabla \omega )= (iz\nabla \varphi )\cdot \partial _x\omega -\omega _3\varphi \cdot \partial _y\omega$, thus

$$\begin{aligned} {G}'(\nabla \omega )\nabla \varphi -\mu ^2 z\nabla \varphi = \mu ^2 (iz \nabla \varphi ) \wedge \omega -\omega _3(\nabla \varphi \cdot \nabla \omega )\omega - \mu ^2\omega _3\varphi . \end{aligned}$$

Next, using (2.2) we can compute

$$\begin{aligned} \partial _x\varphi \wedge \partial _y\omega= & {} \partial _x\varphi \wedge (\partial _x\omega \wedge \omega )= -(\varphi \cdot \partial _x\omega )\partial _x\omega -(\partial _x\varphi \cdot \partial _x\omega )\omega \\ \partial _x\omega \wedge \partial _y\varphi= & {} (\omega \wedge \partial _y\omega )\wedge \partial _y\varphi = -(\varphi \cdot \partial _y\omega )\partial _y\omega -(\partial _y\varphi \cdot \partial _y\omega )\omega , \end{aligned}$$

which give the identity

$$\begin{aligned} \partial _x\varphi \wedge \partial _y\omega +\partial _x\omega \wedge \partial _y\varphi =-\mu ^2\varphi -(\nabla \varphi \cdot \nabla \omega )\omega , \end{aligned}$$

(3.13)

that holds for any $\varphi \in \langle \omega \rangle _{\mathcal{C}^m}^\perp$.

Putting together the above informations we arrive at

$$\begin{aligned} r_{k}^2J_0'({\textsc {U} })\varphi= & {} L\varphi + \frac{2\mu ^2}{(\omega _3 + k)^3}~(iz\nabla \varphi ) \wedge \omega - \frac{2\mu ^2}{(\omega _3 + k)^2} ~\varphi \\&+\frac{2}{(\omega _3+k)^3}\big [\mu ^2\varphi _3-(\omega _3+k)\nabla \varphi \cdot \nabla \omega \big ]\omega \, . \end{aligned}$$

Using (3.11) and $\varphi _3 = \varphi \cdot (z\nabla \omega )$, we conclude the proof. $\square$

Thanks to Lemma 3.1 we can study the system $J'_0({\textsc {U} })\varphi =0$ separately, on $\langle \omega \rangle _{\mathcal{C}^m}^\perp$ first, and on $\langle \omega \rangle _{\mathcal{C}^m}$ later. In fact, $\varphi \in \ker J'_0({\textsc {U} })$ if and only if the pair of functions

$$\begin{aligned} \psi := {\mathcal {P}}\varphi \in \langle \omega \rangle _{\mathcal{C}^2}^\perp \subset \mathcal{C}^2({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}}^3),\qquad \eta :=\varphi \cdot \omega \in \mathcal{C}^2({\overline{{\mathbb {R}}}\,}^2), \end{aligned}$$

solves

We begin by facing problem (3.14a). Firstly, we show that the quadratic form associated to the differential operator $J'_0({\textsc {U} })$ is nonnegative on $\langle \omega \rangle _{\mathcal{C}^{2}}^\perp$.

Lemma 3.2

Let $\psi \in \langle \omega \rangle _{\mathcal{C}^2}^\perp$. Then

Proof

Since $J_0'({\textsc {U} })\psi \cdot \psi ={\mathcal {P}}\big (J_0'({\textsc {U} })\psi \big )\cdot \psi$ and ${\mathcal {P}}\psi =\psi$, formula (3.9) gives

$$\begin{aligned} r_k^{2} \mathop {\int }\limits _{{\mathbb {R}}^{2}}J_0'({\textsc {U} })\psi \cdot \psi ~dz = \mathop {\int }\limits _{{\mathbb {R}}^{2}}\frac{|\nabla \psi |^2}{(\omega _3+k)^2}~dz+ 2\int \limits _{{{\mathbb {R}}}^{2}}\frac{\psi \cdot (iz \nabla \psi ) \wedge \omega }{(\omega _3 + k)^3}~\mu ^2d z -2\int \limits _{{{\mathbb {R}}}^{2}}\frac{|\psi |^2 }{(\omega _3 + k)^2}~\mu ^2d z. \end{aligned}$$

Now we prove the identity

$$\begin{aligned} B_\psi :=2\int \limits _{{{\mathbb {R}}}^{2}}\frac{\psi \cdot (iz \nabla \psi ) \wedge \omega }{(\omega _3 + k)^3}~\mu ^2d z = 2 \int \limits _{{{\mathbb {R}}}^{2}}\frac{\omega \cdot \partial _x \psi \wedge \partial _y \psi }{(\omega _3 + k)^2}\, d z + \int \limits _{{{\mathbb {R}}}^{2}}\frac{|\psi |^2 }{(\omega _3 + k)^2}~\mu ^2d z. \end{aligned}$$

(3.15)

We use polar coordinates $\rho ,\theta$ on ${\mathbb {R}}^2$ and notice that $\partial _\theta \psi = iz \nabla \psi$. From $\rho \mu ^2=\partial _\rho \omega _3$ we get

$$\begin{aligned} \begin{aligned} B_\psi =&- \int \limits _0^{2\pi }d\theta \int \limits _0^{\infty } ~(\psi \cdot \partial _\theta \psi \wedge \omega )\partial _\rho (\omega _3 + k)^{-2}\, d \rho \\ =&\int \limits _0^{\infty }d\rho \int \limits _0^{2\pi }\frac{\omega \cdot \partial _\rho \psi \wedge \partial _\theta \psi - \psi \cdot \partial _\rho \omega \wedge \partial _\theta \psi }{(\omega _3 + k)^2}d\theta + \int \limits _0^{\infty }d\rho \int \limits _0^{2\pi }\frac{\partial _{\rho \theta }\psi \cdot \omega \wedge \psi }{(\omega _3 + k)^2}d\theta \, \\ =&\int \limits _0^{\infty }d\rho \int \limits _0^{2\pi }\frac{\omega \cdot \partial _\rho \psi \wedge \partial _\theta \psi - \psi \cdot \partial _\rho \omega \wedge \partial _\theta \psi }{(\omega _3 + k)^2}d\theta \\&\quad + \int \limits _0^{\infty }d\rho \int \limits _0^{2\pi }\frac{ \omega \cdot \partial _\rho \psi \wedge \partial _{\theta } \psi - \psi \cdot \partial _\rho \psi \wedge \partial _\theta \omega }{(\omega _3 + k)^2}d\theta . \end{aligned} \end{aligned}$$

Using the elementary identity $\partial _\rho \alpha \wedge \partial _\theta \beta =\rho (\partial _x\alpha \wedge \partial _y\beta )$, we see that

$$\begin{aligned} B_\psi = 2\int \limits _{{{\mathbb {R}}}^{2}}\frac{\omega \cdot \partial _x \psi \wedge \partial _y \psi }{(\omega _3 + k)^2}\, d z - \int \limits _{{{\mathbb {R}}}^{2}}\frac{\psi \cdot (\partial _x \omega \wedge \partial _y\psi + \partial _x \psi \wedge \partial _y \omega )}{(\omega _3 + k)^2}\, d z, \end{aligned}$$

and (3.15) follows from (3.13) (with $\varphi$ replaced by $\psi$).

Thanks to (3.15), we have the identity

$$\begin{aligned} r_k^{2} \mathop {\int }\limits _{{\mathbb {R}}^{2}}J_0'({\textsc {U} })\psi \cdot \psi ~dz=\mathop {\int }\limits _{{\mathbb {R}}^{2}}\frac{|\nabla \psi |^2+2\omega \cdot \partial _x\psi \wedge \partial _y\psi -\mu ^2|\psi |^2}{ (\omega _3+k)^2}~dz, \end{aligned}$$

so that we only need to handle the function

$$\begin{aligned} b_\psi :=|\nabla \psi |^2+2\omega \cdot \partial _x\psi \wedge \partial _y\psi -\mu ^2|\psi |^2. \end{aligned}$$

We decompose $\partial _x\psi$ and $\partial _y\psi$ accordingly with (3.8), to obtain

$$\begin{aligned} \begin{aligned} \mu ^2\partial _x\psi&=(\partial _x \psi \cdot \partial _x\omega )\omega _x + (\partial _x\psi \cdot \partial _y\omega )\omega _y- \mu ^2(\psi \cdot \partial _x\omega )\omega ,\\ \mu ^2\partial _y\psi&=(\partial _y \psi \cdot \partial _x\omega )\omega _x + (\partial _y\psi \cdot \partial _y\omega )\omega _y- \mu ^2(\psi \cdot \partial _y\omega )\omega , \end{aligned} \end{aligned}$$

respectively. Since $|\nabla \psi |^2=|\partial _x\psi |^2+|\partial _y\psi |^2$, we infer

$$\begin{aligned} \mu ^2\big (|\nabla \psi |^2-\mu ^2|\psi |^2) =(\partial _x \psi \cdot \partial _x\omega )^2 + (\partial _x\psi \cdot \partial _y\omega )^2+ (\partial _y \psi \cdot \partial _x\omega )^2+ (\partial _y \psi \cdot \partial _y\omega )^2\,. \end{aligned}$$

Writing $\mu ^2\omega =-\partial _x\omega \wedge \partial _y\omega$, see (2.2), we get

$$\begin{aligned} \mu ^2\omega \cdot (\partial _x \psi \wedge \partial _y \psi ) =-(\partial _x\psi \cdot \partial _x \omega )(\partial _y \psi \cdot \partial _y\omega ) +(\partial _x \psi \cdot \partial _y\omega )( \partial _y\psi \cdot \partial _x\omega ), \end{aligned}$$

from which it readily follows that $\mu ^2 b_\psi =(\partial _x\psi \cdot \partial _x\omega - \partial _y\psi \cdot \partial _y\omega )^2 + (\partial _x\psi \cdot \partial _y \omega + \partial _y\psi \cdot \partial _x\omega )^2$. The proof is complete. $\square$

Lemma 3.3

Let $\psi \in \mathcal{C}^2({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}}^3)$ be a solution to (3.14a). There exist $s,t \in {\mathbb {R}}^3$ such that

$$\begin{aligned} \psi = s - (s\cdot \omega ) \omega + t \wedge \omega , \end{aligned}$$

and thus $\psi \in T_{{\textsc {U} }}Z\cap \langle \omega \rangle ^\perp _{{C}^2}=\{\, s - (s\cdot \omega ) \omega + t \wedge \omega \ |\ s, t \in {\mathbb {R}}^3\, \}$.

Proof

From (3.14a) it immediately follows that $\psi$ is pointwise orthogonal to $\omega$, which implies $\psi \in \langle \omega \rangle _{\mathcal{C}^2}^\perp$. Since $\mathcal P\psi =\psi$, then $J'_0({\textsc {U} })\psi =0$ by (3.9) and (3.10), hence

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _x \psi \cdot \partial _x\omega - \partial _y\psi \cdot \partial _y\omega = 0\\ \partial _x \psi \cdot \partial _y \omega + \partial _y\psi \cdot \partial _x \omega = 0 \end{array}\right. } \end{aligned}$$

(3.16)

by Lemma 3.2. Since $\psi \in \langle \partial _x\omega ,\partial _y\omega \rangle$ pointwise on ${\mathbb {R}}^2$, we can write

$$\begin{aligned} \psi =f\nabla \omega ,\quad \text {where}\quad f:=\mu ^{-2}(\psi \cdot \partial _x\omega , \psi \cdot \partial _y\omega )\in \mathcal{C}^2({\overline{{\mathbb {R}}}\,}^2, {\mathbb {R}}^2). \end{aligned}$$

We identify f with a complex valued function. A direct computation based on (2.2) shows that $\psi$ solves (3.16) if and only if f solves $\partial _x f + i \partial _y f = 0$ on ${\mathbb {R}}^2$. In polar coordinates we have that

$$\begin{aligned} \rho \partial _\rho f + i \partial _\theta f = 0. \end{aligned}$$

(3.17)

For every $\rho >0$ we expand the periodic function $f(\rho ,\cdot )$ in Fourier series,

$$\begin{aligned} \displaystyle f(\rho , \theta ) = \sum _{h\in {\mathbb {Z}}} \gamma _h(\rho ) e^{i h \theta }, \quad \gamma _h (\rho ) = \frac{1}{2\pi }\int \limits _0^{2\pi } f(\rho , \theta )e^{- ih\theta }d \theta . \end{aligned}$$

The coefficients $\gamma _h$ are complex-valued functions on the half-line ${\mathbb {R}}_+$ and solve

$$\begin{aligned} \gamma '_h - h \gamma _h = 0, \end{aligned}$$

because of (3.17). Thus for every $h \in {\mathbb {Z}}$ there exists $a_h\in {\mathbb {C}}$ such that $\gamma _h(\rho ) = a_h\rho ^h$. Now recall that $\mu \psi \in L^2({\mathbb {R}}^2,{\mathbb {R}}^3)$. Since

$$\begin{aligned} \int \limits _{{{\mathbb {R}}}^{2}}\mu ^2|\psi |^2\, dz = \int \limits _{{{\mathbb {R}}}^{2}}\mu ^4|f|^2\, d z \ge 2\pi \int _0^\infty \mu ^4\rho |\gamma _h|^2\, d\rho = a_h^2\mathop {\int }\limits _{{\mathbb {R}}^{2}}\mu ^4 |z|^{2h}\, dz , \quad \forall \ h \in {\mathbb {Z}}, \end{aligned}$$

we infer that $\gamma _h = 0$ for every $h \ne 0,1,2$. Thus $f(z) = \sum \limits _{h=0}^2 a_h z^h$, that is $\psi = \sum \limits _{h=0}^2 a_h z^h\nabla \omega$, and in particular the space of solutions of (3.14a) has (real) dimension 6. The conclusion of the proof follows from the relations (2.4). $\square$

Lemma 3.4

Let $\eta \in \mathcal{C}^2({\overline{{\mathbb {R}}}\,}^2)$ be a solution to (3.14b). There exists $\alpha \in {\mathbb {R}}^3$ such that

$$\begin{aligned} \eta = \alpha \cdot (k\omega +e_3), \end{aligned}$$

and thus $\eta \omega \in T_{{\textsc {U} }}Z\cap \langle \omega \rangle _{\mathcal {C}^2}=\{\, (\alpha \cdot (k\omega +e_3))~\omega ~|~\alpha \in {\mathbb {R}}^3\, \}$.

Proof

First of all, we notice that $\alpha \cdot (k\omega +e_3)$ solves (3.14b) for any $\alpha \in {\mathbb {R}}^3$.

By the Hilbert–Schmidt theorem, the eigenvalue problem

$$\begin{aligned} - \mathrm{div}\Big (\frac{\nabla \eta }{(\omega _3 + k)^2}\Big ) =\frac{ \lambda \, \mu ^2 }{(\omega _3 + k)^3}~\eta \quad \text {on }{\mathbb {R}}^2,\quad \eta \in \mathcal{C}^2({\mathbb {R}}^2), \end{aligned}$$

(3.18)

has a non decreasing, divergent sequence $(\lambda _h)_{h\ge 0}$ of eigenvalues which correspond to critical levels of the quotient

$$\begin{aligned} R(\eta ):=\frac{\displaystyle { \int _{{{\mathbb {R}}}^{2}} \frac{|\nabla \eta |^2}{(\omega _3 + k)^2}~dz }}{\displaystyle {\int _{{{\mathbb {R}}}^{2}}\frac{{|\eta |^2}}{(\omega _3 + k)^3}~\mu ^2dz}},\quad \eta \in H^1({\overline{{\mathbb {R}}}\,}^2)\setminus \{0\}. \end{aligned}$$

Clearly, $\lambda _0=0$ is simple, and its eigenfunctions are constant functions. We claim that the next eigenvalue is 2k, and that its eigenspace has dimension 3, which concludes the proof.

To this goal, we use the functional change

$$\begin{aligned} {\eta (z)= \frac{\mu (z)}{\mu (c_k z)}\varPhi (c_k z)},\quad c_k:=e^{\rho _k}=\sqrt{\frac{k+1}{k-1}}. \end{aligned}$$

By a direct computation involving the identity $(\omega _3(z) + k )\mu (c_k z) = (k-1)\mu (z)$ and integration by parts, one gets

$$\begin{aligned} \lambda _1=\mathop {\mathop {\inf }\limits _{\eta \in \mathcal{C}^2({\overline{{\mathbb {R}}}\,}^2)\setminus \{0\}}}\limits _{\scriptstyle \int _{{{\mathbb {R}}}^{2}}\frac{\eta ~ \mu ^2dz}{(\omega _3 + k)^3}=0} R(\eta )= 2k+ \mathop {\mathop {\inf }\limits _{\varPhi \in \mathcal{C}^2({\overline{{\mathbb {R}}}\,}^2)\setminus \{0\}}}\limits _{\scriptstyle \int _{{{\mathbb {R}}}^{2}}{\varPhi }\mu ^2dz=0} \frac{\displaystyle \int _{{{\mathbb {R}}}^{2}} |\nabla \varPhi |^2 dz- 2\displaystyle \int _{{{\mathbb {R}}}^{2}}{|\varPhi |^2}\mu ^2dz}{\displaystyle \int _{{{\mathbb {R}}}^{2}}\frac{|\varPhi |^2}{(k-\omega _3)}\, \mu ^2 dz}\,. \end{aligned}$$

On the other hand, it is well known that

$$\begin{aligned} \mathop {\mathop {\min }\limits _{\varPhi \in \mathcal{C}^2({\overline{{\mathbb {R}}}\,}^2)\setminus \{0\}}}\limits _{\int _{{{\mathbb {R}}}^{2}}{\varPhi }\mu ^2dz=0} \frac{\displaystyle \int _{{{\mathbb {R}}}^{2}} |\nabla \varPhi |^2 dz}{\displaystyle \int _{{{\mathbb {R}}}^{2}}{|\varPhi |^2}\mu ^2dz}=2 \end{aligned}$$

is the first nontrivial eigenvalue for the Laplace-Beltrami operator on the sphere and that its eigenspace has dimension 3, see for instance [4]. The proof is complete. $\square$

Remark 3.2

The third eigenvalue $\lambda _2$ of (3.18) verifies $\lambda _2 >2k$ by Lemma 3.4, and

$$\begin{aligned} \lambda _2=\min \Big \{\, R(\eta )~\Big |~~ \mathop {\int }\limits _{{\mathbb {R}}^{2}}\frac{{\eta }}{(\omega _3 + k)^3}~\mu ^2dz=\mathop {\int }\limits _{{\mathbb {R}}^{2}}\frac{{\eta }(k\omega _j+\delta _{j3})}{(\omega _3 + k)^3}~\mu ^2dz=0,~~j=1,2,3 \, \Big \}. \end{aligned}$$

Proof of Theorem 1.1

In fact, we only have to sum up the argument. Let $U \in Z$. Thanks to (3.2), $U = {\textsc {U} }_{q} \circ g$ for some $q \in {\mathbb {H}}^3$, $g \in PGL(2,{\mathbb {C}})$. Since

$$\begin{aligned} T_{{{\textsc {U} }_{q} \circ g}}Z = T_{{\textsc {U} }}Z\circ g, \quad \ker J'_0({\textsc {U} }_{q} \circ g) = \ker J_0'({\textsc {U} })\circ g, \quad \text {for every }q \in {\mathbb {H}}^3, \ g \in PGL(2,{\mathbb {C}}), \end{aligned}$$

it suffices to consider the case $U = {\textsc {U} }$.

If $\varphi \in \mathcal{C}^2({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}}^3)$ solves (1.4) then $J'_0({\textsc {U} })\varphi =0$, which means $\mathcal P\left( J_0'({\textsc {U} })\varphi \right) =0$ and $\left( J_0'({\textsc {U} })\varphi \right) \cdot \omega =0$. From Lemma 3.1 we infer that $\mathcal P\varphi$ solves (3.14a) and that $\varphi \cdot \omega$ solves (3.14b). Therefore, Lemmata 3.3, 3.4 give the existence of $s,t,\alpha \in {\mathbb {R}}^3$ such that

$$\begin{aligned} \mathcal P\varphi =s - (s\cdot \omega ) \omega + t \wedge \omega ,\quad \varphi \cdot \omega =\alpha \cdot (k\omega +e_3). \end{aligned}$$

Thus $\varphi =\mathcal P\varphi +(\varphi \cdot \omega )\omega \in T_{{\textsc {U} }}Z$ by (3.5), which concludes the proof. $\square$

3.1 Further results on the operator $J'_0({\textsc {U} })$

To shorten notation we put

$$\begin{aligned} H^1 = H^1({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}}^3). \end{aligned}$$

Since integration by parts gives

$$\begin{aligned} \mathop {\int }\limits _{{\mathbb {R}}^{2}}- \text {div}\Big (\frac{\nabla \varphi }{(\omega _3 + k)^{2}}\Big )\cdot \psi ~dz= \mathop {\int }\limits _{{\mathbb {R}}^{2}}\frac{\nabla \varphi \cdot \nabla \psi }{(\omega _3 + k)^{2}}~dz\ , \quad \varphi ,\psi \in \mathcal{C}^2({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}}^3), \end{aligned}$$

the quadratic form

$$\begin{aligned} (\varphi ,\psi )\mapsto \mathop {\int }\limits _{{\mathbb {R}}^{2}}J'_0({\textsc {U} })\varphi \cdot \psi ~dz \end{aligned}$$

(3.19)

can be extended to a continuous bilinear form $H^1\times H^1\rightarrow {\mathbb {R}}$ via a density argument. It can be checked by direct computation (see also Remark 4.1) that the quadratic form in (3.19) is self-adjoint on $H^1$, that is,

$$\begin{aligned} \mathop {\int }\limits _{{\mathbb {R}}^{2}}J'_0({\textsc {U} })\varphi \cdot \psi ~dz=\mathop {\int }\limits _{{\mathbb {R}}^{2}}J'_0({\textsc {U} })\psi \cdot \varphi ~dz\quad \text {for any }\varphi ,\psi \in H^1. \end{aligned}$$

(3.20)

Since $T_{{\textsc {U} }}Z$ is a subspace of $L^2({\overline{{\mathbb {R}}}\,}^2, {\mathbb {R}}^3)\equiv L^2({\mathbb {S}}^2,{\mathbb {R}}^3)$, we are allowed to put

$$\begin{aligned} T_{{\textsc {U} }}Z^\perp := \Big \{\, f \in L^2({\overline{{\mathbb {R}}}\,}^2, {\mathbb {R}}^3) \ \Big |\ \int \limits _{{{\mathbb {R}}}^{2}} f\cdot \tau ~\mu ^2dz= 0, \ \forall \ \tau \in T_{{\textsc {U} }}Z\, \Big \}. \end{aligned}$$

Moreover, we introduce on $L^2({\overline{{\mathbb {R}}}\,}^2, {\mathbb {R}}^3)$ the equivalent scalar product

$$\begin{aligned} (f, \psi )_* = \int \limits _{{{\mathbb {R}}}^{2}} \frac{{\mathcal {P}} f \cdot {\mathcal {P}}\psi }{(\omega _3 + k)^2}\, \mu ^2dz + \int \limits _{{{\mathbb {R}}}^{2}} \frac{(f \cdot \omega )(\psi \cdot \omega )}{(\omega _3 + k)^3}\, \mu ^2dz \end{aligned}$$

and the subspaces

$$\begin{aligned} T_{{\textsc {U} }}Z^\perp _* :=&\big \{\, f \in L^2({\overline{{\mathbb {R}}}\,}^2, {\mathbb {R}}^3) \ |\ (f, \tau )_* = 0, \ \forall \, \tau \in T_{{\textsc {U} }}Z\, \big \},\\ N_* :=&\langle \omega \rangle ^\perp _*= \big \{\, f \in L^2({\overline{{\mathbb {R}}}\,}^2, {\mathbb {R}}^3) \ |\ (f, \omega )_* = 0\, \big \}. \end{aligned}$$

We are in position to state the main result of this section.

Lemma 3.5

Let $q\in {\mathbb {H}}^3$. For any $v\in T_{{\textsc {U} }}Z^\perp$, there exists $\varphi _v\in H^1\cap T_{{\textsc {U} }}Z^\perp _*\cap N_*$ such that

$$\begin{aligned} J'_0({\textsc {U} }_{q})\varphi _v=v~\mu ^2\qquad \text {on }{\mathbb {R}}^2. \end{aligned}$$

(3.21)

If in addition $v\in \mathcal{C}^m({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}}^3)$ for some $m\in (0,1)$, then $\varphi _v\in \mathcal{C}^{2+m}({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}}^3)$.

In view of Lemma 3.1, we split the proof of Lemma 3.5 in few steps.

Lemma 3.6

Let $v \in T_{{\textsc {U} }}Z^\perp$ be such that $v \cdot \omega \equiv 0$ on ${\mathbb {R}}^2$. There exists $\varphi \in H^1\cap T_{{\textsc {U} }}Z^\perp _*$ such that $\varphi \cdot \omega \equiv 0$ on ${\mathbb {R}}^2$ and

$$\begin{aligned} J'_0({\textsc {U} })\varphi =v~\mu ^2\qquad \text {on }{\mathbb {R}}^2. \end{aligned}$$

(3.22)

Proof

We introduce

$$\begin{aligned} X:= \big \{\, \psi \in H^1~\big |~ \psi \cdot \omega \equiv 0~~\text {on }{\mathbb {R}}^2\, \big \}\cap T_{{\textsc {U} }}Z^\perp _*, \end{aligned}$$

which is a closed subspace of $H^1$. Notice that $\psi = {\mathcal {P}}\psi$ for any $\psi \in X$ and moreover

$$\begin{aligned} \mathop {\int }\limits _{{\mathbb {R}}^{2}}J'_0({\textsc {U} })\psi \cdot \psi ~dz= \int \limits _{{{\mathbb {R}}}^{2}} \frac{|\nabla {\psi }|^2}{(\omega _3 + k)^2}~dz + 2 \int \limits _{{{\mathbb {R}}}^{2}} \Big ( \frac{(\psi \cdot iz\nabla {\psi })\wedge \omega }{(\omega _3 + k)^3}- \frac{{|\psi |^2}}{(\omega _3 + k)^2}\Big )~\mu ^2dz, \end{aligned}$$

use (3.9) and a density argument. Next we put

$$\begin{aligned} \lambda := \inf _{\begin{array}{c} \psi \in X \\ \psi \ne 0 \end{array}}\frac{\displaystyle {\int _{{{\mathbb {R}}}^{2}} J'_0({\textsc {U} })\psi \cdot \psi ~dz}}{\displaystyle {\int _{{{\mathbb {R}}}^{2}}(\omega _3 + k)^{-2}{{|\psi |^2}}~\mu ^2dz}}, \end{aligned}$$

and notice that $\lambda \ge 0$ by Lemma 3.2. On the other hand, $\lambda$ is achieved by Rellich theorem. Thus $\lambda >0$, because of Lemma 3.3. It follows that the energy functional $I: X \rightarrow {\mathbb {R}}$,

$$\begin{aligned} I(\psi ) =\frac{1}{2}\mathop {\int }\limits _{{\mathbb {R}}^{2}}J'_0({\textsc {U} })\psi \cdot \psi ~dz - \int \limits _{{{\mathbb {R}}}^{2}} v \cdot \psi ~\mu ^2dz, \end{aligned}$$

is weakly lower semicontinuous and coercive. Thus its infimum is achieved by a function $\varphi \in X$ which satisfies

$$\begin{aligned} \int \limits _{{{\mathbb {R}}}^{2}}J'_0({\textsc {U} })\varphi \cdot \psi ~dz= \int \limits _{{{\mathbb {R}}}^{2}} v \cdot \psi ~\mu ^2dz, \quad \forall \ \psi \in X. \end{aligned}$$

(3.23)

If $\psi \in H^1$ we write

$$\begin{aligned} \psi = ({\mathcal {P}}\psi ^\top + {\mathcal {P}}\psi ^\perp ) + \eta \omega , \end{aligned}$$

where $\eta =\psi \cdot \omega$, ${\mathcal {P}}\psi ^\top \in T_{{\textsc {U} }}Z=\ker J'_0(U)$ is the orthogonal projection of ${\mathcal {P}}\psi =\psi -\eta \omega$ onto $T_{{\textsc {U} }}Z$ in the scalar product $(\cdot ,\cdot )_*$ and ${\mathcal {P}}\psi ^\perp := \psi -{\mathcal {P}}\psi ^\top - \eta \omega \in X$. We use (3.20) and (3.10) to compute

$$\begin{aligned} \int \limits _{{{\mathbb {R}}}^{2}}J'_0({\textsc {U} })\varphi \cdot {\mathcal {P}}\psi ^\top ~dz= & {} \int \limits _{{{\mathbb {R}}}^{2}}J'_0({\textsc {U} }){\mathcal {P}}\psi ^\top \cdot \psi ~dz=0,\\ \int \limits _{{{\mathbb {R}}}^{2}}J'_0({\textsc {U} })\varphi \cdot (\eta \omega ) ~dz= & {} \mathop {\int }\limits _{{\mathbb {R}}^{2}}\frac{\nabla (\varphi \cdot \omega )\cdot \nabla \eta }{(\omega _3 + k)^2} ~dz- 2k\mathop {\int }\limits _{{\mathbb {R}}^{2}}\frac{(\varphi \cdot \omega )\eta }{(\omega _3 + k)^3}~\mu ^2dz =0\, , \end{aligned}$$

because $\varphi \cdot \omega \equiv 0$. Therefore, (3.23) gives

$$\begin{aligned} \int \limits _{{{\mathbb {R}}}^{2}}J'_0({\textsc {U} })\varphi \cdot \psi ~dz =\int \limits _{{{\mathbb {R}}}^{2}}J'_0({\textsc {U} })\varphi \cdot {\mathcal {P}}\psi ^\perp ~dz= \mathop {\int }\limits _{{\mathbb {R}}^{2}}v \cdot {\mathcal {P}}\psi ^\perp ~\mu ^2dz= \mathop {\int }\limits _{{\mathbb {R}}^{2}}v \cdot \psi ~\mu ^2dz, \end{aligned}$$

as v is orthogonal to $T_{{\textsc {U} }}Z\ni {\mathcal {P}}\psi ^\top$ and to $\eta \omega$ in $L^2({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}}^3)$. We showed that $\varphi$ solves (3.22), and thus the proof is complete. $\square$

Lemma 3.7

Let $f \in H^1({\overline{{\mathbb {R}}}\,}^2)$ be such that $f \omega \in T_{{\textsc {U} }}Z^\perp$. There exists $\eta \in H^1({\overline{{\mathbb {R}}}\,}^2)$ such that $\eta \omega \in H^1\cap T_{{\textsc {U} }}Z^\perp _*\cap N_*$ and

$$\begin{aligned} J'_0({\textsc {U} })(\eta \omega )=f\omega ~\mu ^2\quad \text {on }{\mathbb {R}}^2. \end{aligned}$$

(3.24)

Proof

We introduce the space

$$\begin{aligned} Y:= \Big \{\,\eta \in H^1({\overline{{\mathbb {R}}}\,}^2)~\Big |~ \int \limits _{{{\mathbb {R}}}^{2}}\frac{\eta }{(\omega _3 + {k})^3} ~\mu ^2dz = \int \limits _{{{\mathbb {R}}}^{2}} \frac{\eta (\tau \cdot \omega )}{(\omega _3 + k)^3}~\mu ^2dz\, = 0,\ \forall \, \tau \in T_{{\textsc {U} }}Z\, \Big \}\, , \end{aligned}$$

so that $\eta \omega \in H^1\cap T_{{\textsc {U} }}Z^\perp _*\cap N_*$ for any $\eta \in Y$, and the energy functional $I:Y\rightarrow {\mathbb {R}}$,

$$\begin{aligned} \begin{aligned} I(\varphi ) =&~ \frac{1}{2}\int \limits _{{{\mathbb {R}}}^{2}}J'_0({\textsc {U} })(\eta \omega ) \cdot (\eta \omega )~dz- \int \limits _{{{\mathbb {R}}}^{2}} f\eta ~\mu ^2dz \\ =&~\frac{1}{2} \int \limits _{{{\mathbb {R}}}^{2}} \frac{|\nabla \eta |^2}{(\omega _3 + k)^2}~dz - k \int \limits _{{{\mathbb {R}}}^{2}}~\frac{|\eta |^2}{(\omega _3 + k)^3} \, \mu ^2dz - \int \limits _{{{\mathbb {R}}}^{2}} \eta f\, \mu ^2dz, \end{aligned} \end{aligned}$$

compare with (3.10). The functional I is weakly lower semicontinuous with respect to the $H^1({\overline{{\mathbb {R}}}\,}^2)$ topology and coercive by Remark 3.2. Thus its infimum is achieved by a function $\eta \in Y$. To conclude, argue as in the proof of Lemma 3.6 to show that $\eta$ solves (3.24). $\square$

Proof of Lemma 3.5

Since $J'_0({\textsc {U} }_{q})=q_3^{-2}J'_0({\textsc {U} })$, we can assume that $q=e_3$, that is, ${\textsc {U} }_{q}={\textsc {U} }$. We take any $v\in T_{{\textsc {U} }}Z^\perp$, and write

$$\begin{aligned} v = {\mathcal {P}} v + (v\cdot \omega )\omega , \end{aligned}$$

where ${\mathcal {P}} v= v-(v\cdot \omega )\omega$, as before. Since ${\mathcal {P}} v\in T_{{\textsc {U} }}Z^\perp$, by Lemma 3.6 there exists a unique $\hat{\varphi }\in H^1\cap T_{{\textsc {U} }}Z^\perp _*$ such that $\hat{\varphi }\cdot \omega \equiv 0$ on ${\mathbb {R}}^2$ and

$$\begin{aligned} \mathop {\int }\limits _{{\mathbb {R}}^{2}}J'_0({\textsc {U} })\hat{\varphi }\cdot \psi ~dz\, =\mathop {\int }\limits _{{\mathbb {R}}^{2}}{\mathcal {P}} v\cdot \psi ~\mu ^2dz, \quad \text {for any }\psi \in H^1. \end{aligned}$$

Next, notice that $(v\cdot \omega )\omega \in T_{{\textsc {U} }}Z^\perp$, so we can use Lemma 3.7 to find $\eta \in H^1({\overline{{\mathbb {R}}}\,}^2)$ such that $\eta \omega \in H^1\cap T_{{\textsc {U} }}Z^\perp _*\cap N_*$ solves

$$\begin{aligned} \mathop {\int }\limits _{{\mathbb {R}}^{2}}J'_0({\textsc {U} })(\eta \omega )\cdot \psi ~dz= \mathop {\int }\limits _{{\mathbb {R}}^{2}}(v\cdot \omega )(\psi \cdot \omega )~\mu ^2dz, \quad \text {for any }\psi \in H^1. \end{aligned}$$

The function $\varphi _v=\hat{\varphi }+\eta \omega$ solves (3.21).

To conclude the proof we have to show that if $v\in \mathcal{C}^{m}({\overline{{\mathbb {R}}}\,}^2, {\mathbb {R}}^3)$ then $\varphi _v \in \mathcal{C}^{2+m}({\overline{{\mathbb {R}}}\,}^2, {\mathbb {R}}^3)$. Since $\omega \in \mathcal{C}^{\infty }({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}}^3)$ and $\omega _3 + k$ is bounded and bounded away from zero, $\varphi _v$ solves a linear system of the form

$$\begin{aligned} -\varDelta \varphi _v=A(z)\varphi _v+ B(z)\nabla \varphi _v+\mu ^2(\omega _3+k)^2 v, \end{aligned}$$

for certain smooth matrices on ${\overline{{\mathbb {R}}}\,}^2$. A standard bootstrap argument and Schauder regularity theory plainly imply that $\varphi _v \in \mathcal{C}^{2+m}_{loc}({\mathbb {R}}^2, {\mathbb {R}}^3)$. The function $z\mapsto \varphi _v(z^{-1})$ satisfies a linear system of the same kind, hence $\varphi _v \in \mathcal{C}^{2+m}({\overline{{\mathbb {R}}}\,}^2, {\mathbb {R}}^3)$, as desired. $\square$

4 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 $\mathcal{C}^2$-functional $E_\varepsilon :\mathcal{C}^{2}({\overline{{\mathbb {R}}}\,}^2,{\mathbb {H}}^3)\rightarrow {\mathbb {R}}$,

$$\begin{aligned} E_\varepsilon (u):= \frac{1}{2}\mathop {\int }\limits _{{\mathbb {R}}^{2}}u_3^{-2}|\nabla u|^2~ dz- k\mathop {\int }\limits _{{\mathbb {R}}^{2}}u_3^{-2}e_3\cdot \partial _x u\wedge \partial _yu~dz +2\varepsilon ~V_\phi (u)=E_0(u)+2\varepsilon ~V_\phi (u) \end{aligned}$$

(notice that $E_0=E_{\text {hyp}}$, compare with (1.3)), solves

and has mean curvature $(k+\varepsilon \phi )$, apart from a finite set of branch points.

Due to the action of the Möbius transformations and of the hyperbolic translations, for any $u\in \mathcal{C}^{2}({\overline{{\mathbb {R}}}\,}^2,{\mathbb {H}}^3)$ we have the identities

$$\begin{aligned}&E_\varepsilon '(u)(z^h\nabla u)=0,~~ E_\varepsilon '(u)(iz^h\nabla u)=0,~~\text {for }h=0,1,2,\ \ \varepsilon \in {\mathbb {R}}, \end{aligned}$$

(4.1)

$$\begin{aligned}&E'_0(u) e_1 = 0, \quad E'_0(u)e_2=0,\quad E_0'(u) u = 0. \end{aligned}$$

(4.2)

Now we prove that

$$\begin{aligned} E_\varepsilon ({\textsc {U} }_{q})= E_0({\textsc {U} })- 2\varepsilon F^{\phi }_k(q), \end{aligned}$$

(4.3)

where $F^{\phi }_k$ is the Melnikov-type function in (1.5). The above mentioned invariances give $E_0({\textsc {U} }_{q})=E_0({\textsc {U} })$. Since the hyperbolic ball $B^{\mathbb {H}}_{\rho _k}(q)$ coincides with the Euclidean ball of radius $q_3r_{k}$ about the point $q^k:=(q_1,q_2,kr_{k}q_3)$, the divergence theorem gives

$$\begin{aligned} F^{\phi }_k(q)=\int \limits _{B^{\mathbb {H}}_{\rho _{k}}(q)}\phi (p)~d{\mathbb {H}}^3_p\, = \int \limits _{B_{q_3r_{k}}(q^k)}p_3^{-3}\phi (p)~dp\,= \int \limits _{\partial B_{q_3r_{k}}(q^k)}Q_\phi (p)\cdot \nu _p. \end{aligned}$$

Here $Q_\phi \in \mathcal{C}^1({\mathbb {R}}^3_+,{\mathbb {R}}^3)$ is any vectorfield such that $\text {div}Q_\phi (p)=p_3^{-3}\phi (p)$ and $\nu _p$ is the outer normal to $\partial B_{q_3r_{k}}(q^k)$ at p. The function ${\textsc {U} }_{q}$ in (3.1) parameterizes the Euclidean sphere $\partial B{q_3r_{k}(q^k)}$. Since $\partial _x {\textsc {U} }_{q}\wedge \partial _y{\textsc {U} }_{q}$ is inward-pointing, we have

$$\begin{aligned} F^{\phi }_k(q)=-\int \limits _{{{\mathbb {R}}}^{2}}Q_\phi (p)\cdot \partial _x {\textsc {U} }_{q}\wedge \partial _y{\textsc {U} }_{q}~dz=-V_\phi ({\textsc {U} }_{q}), \end{aligned}$$

(4.4)

and (4.3) is proved. Before going further, let us show that the existence of critical points for $F^{\phi }_k$ is a necessary condition for the conclusion in Theorem 1.2.

Theorem 4.1

Let $k>1$, $\phi \in \mathcal{C}^{1}({\mathbb {H}}^3)$. Assume that there exist sequences $\varepsilon _h\subset {\mathbb {R}}\setminus \{0\}$, $\varepsilon _h\rightarrow 0$, $u^h\in \mathcal{C}^2({\overline{{\mathbb {R}}}\,}^2,{\mathbb {H}}^3)$ and a point $q\in {\mathbb {H}}^3$ such that $u_h$ solves $(\mathcal P_{\varepsilon _h})$, and $u^h\rightarrow {\textsc {U} }_{q}$ in $\mathcal{C}^1({\overline{{\mathbb {R}}}\,}^2,{\mathbb {H}}^3)$. Then q is a stationary point for $F^{\phi }_k$.

Proof

The function $u^h$ is a stationary point for the energy functional $E_{\varepsilon _h}=E_0+2\varepsilon _h V_\phi$. From (4.2) we have $V'_\phi (u^h)e_j=0$ for $j=1,2$ and $V'_\phi (u^h)u^h=0$. We can plainly pass to the limit to obtain $V'_\phi ({\textsc {U} }_{q})e_j=0$ for $j=1,2$ and $V'_\phi ({\textsc {U} }_{q}){\textsc {U} }_{q}=0$. To conclude, use (4.4) and recall that $\partial _{q_j} {\textsc {U} }_{q} =e_j$ for $j=1,2$, and $\partial _{q_3} {\textsc {U} }_{q}={\textsc {U} }=q_3^{-1}({\textsc {U} }_{q}-q_1e_1-q_2e_2)$. $\square$

Now we fix $m\in (0,1)$. The operator $J_\varepsilon :\mathcal{C}^{2+m}({\overline{{\mathbb {R}}}\,}^2,{\mathbb {H}}^3)\rightarrow \mathcal{C}^m({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}}^3)$ defined by

$$\begin{aligned} J_\varepsilon (u)=-\mathrm{div}(u_3^{-2}\nabla u)-u_3^{-3}|\nabla u|^2e_3+2(k+ \varepsilon \phi )u_3^{-3}{\partial _{x}u}\wedge {\partial _{y}u}, \end{aligned}$$

is related to the differential of $E_\varepsilon$ via the identity

$$\begin{aligned} E_\varepsilon '(u)\varphi = \mathop {\int }\limits _{{\mathbb {R}}^{2}}J_\varepsilon (u)\cdot \varphi ~dz,\quad u\in \mathcal{C}^m({\overline{{\mathbb {R}}}\,}^2,{\mathbb {H}}^3),\ \varphi \in \mathcal{C}^m({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}}^3)\,. \end{aligned}$$

(4.5)

Remark 4.1

Since $E_\varepsilon$ is of class $\mathcal{C}^2$ and

$$\begin{aligned} E''_\varepsilon (u)[\varphi ,\psi ]=\displaystyle \int _{{{\mathbb {R}}}^{2}} J'_\varepsilon (u)\psi \cdot \varphi ~dz, \end{aligned}$$

then the quadratic form in the right hand side is a self-adjoint form on $H^1$.

We are in position to state and prove the next Lemma, which is the main step towards the proofs of Theorems 1.2, 1.3.

Lemma 4.1

(Dimension reduction) Let $\varOmega \Subset {\mathbb {H}}^3$ be an open set. There exist $\hat{\varepsilon } >0$ and a unique $\mathcal{C}^1$-map

$$\begin{aligned} {[}-\hat{\varepsilon }, \hat{\varepsilon }]\times \overline{\varOmega } \rightarrow \mathcal{C}^{2+m}({\overline{{\mathbb {R}}}\,}^2, {\mathbb {H}}^3), \quad (\varepsilon , q) \mapsto u^\varepsilon _q, \end{aligned}$$

such that the following facts hold:

• (i) $u^{\varepsilon }_q$ parameterizes an embedded ${\mathbb {S}}^2$-type surface, and $u^{0}_q = {\textsc {U} }_{q}$ ;

• (ii) $u^\varepsilon _q - {\textsc {U} }_{q} \in T_{{\textsc {U} }}Z^\perp \cap \mathcal {C}^{2+m}({\overline{{\mathbb {R}}}\,}^2, {\mathbb {R}}^3)$ and $E'_\varepsilon (u^\varepsilon _q)\varphi =0$ for any $\varphi \in T_{{\textsc {U} }}Z^\perp \cap \mathcal {C}^{0}({\overline{{\mathbb {R}}}\,}^2, {\mathbb {R}}^3)$ ;

• (iii) for any $\varepsilon \in [-\hat{\varepsilon }, \hat{\varepsilon }]$, the manifold $\{\,u^\varepsilon _q~|~ q\in \varOmega \,\}$ is a natural constraint for $E_\varepsilon$, that is, if $\nabla _{q} E_\varepsilon (u^\varepsilon _{q^\varepsilon }) = 0$ for some $q^\varepsilon \in \varOmega$, then $u^{\varepsilon }_{q^\varepsilon }$ is a $(k + \varepsilon \phi )$-bubble ;

• (iv) $\Vert E_\varepsilon (u^{\varepsilon }_q) - E_\varepsilon ({\textsc {U} }_{q})\Vert _{\mathcal{C}^1\left( \overline{\varOmega }\right) } = o (\varepsilon )$ as $\varepsilon \rightarrow 0$, uniformly on $\overline{\varOmega }$ .

Proof

To shorten the notation, we put $\mathcal {C}^m:=\mathcal {C}^m({\overline{{\mathbb {R}}}\,}^2, {\mathbb {R}}^3)$. For $s > 0$ and $\delta >0$ we write

$$\begin{aligned} \begin{aligned} \varOmega _s := \{\, p \in {\mathbb {H}}^3 \ |\ \text {dist}(p, \varOmega )< s\, \}\, ,\, \ \text {and}\ \ \mathcal {U}_\delta := \{\, \nu \in \mathcal{C}^{2+m} \ |\ |\nu (z)|<\delta \text { for every }z \in {\mathbb {R}}^2\, \}. \end{aligned} \end{aligned}$$

We fix s and $\delta =\delta (s)$ such that $\overline{\varOmega }_{2s} \subset {\mathbb {H}}^3$ and $({\textsc {U} }_{q}+ \nu )\cdot e_3 >0$ for $q \in \varOmega _{2s}$, $\nu \in \mathcal {U}_\delta$.

We define

$$\begin{aligned} \begin{array}{l@{\quad }l@{\quad }l} \tau _1:=c_0\partial _x\omega ,&{}\tau _3:= c_0\sqrt{2} z\nabla \omega ,&{}\tau _5:= c_0z^2\nabla \omega ,\\ \tau _2:=c_0\partial _y\omega ,&{}\tau _4:=c_0 \sqrt{2} iz\nabla \omega ,&{}\tau _6:=c_0iz^2\nabla \omega , \end{array} \qquad \gamma :=2c_0(k\omega +e_3), \end{aligned}$$

(4.6)

where $c_0:=\sqrt{\frac{3}{2^4\pi }}$ is a normalization constant. Thanks to (3.3), (3.5), we have

$$\begin{aligned} T_{{\textsc {U} }}Z=\langle \tau _1,\dots \tau _6\rangle \oplus ~\{\, (\alpha \cdot \gamma )~\omega ~|~\alpha \in {\mathbb {R}}^3\, \}. \end{aligned}$$

Trivially, $\tau _j\cdot \omega \equiv 0$ on ${\mathbb {R}}^2$. Elementary computations give

$$\begin{aligned} \int \limits _{{{\mathbb {R}}}^{2}}{\tau _i \cdot \tau _{j}}~\mu ^2dz = \delta _{ij},\quad \int \limits _{{{\mathbb {R}}}^{2}}{\gamma _h \gamma _\ell } ~\mu ^2dz = 0\quad \text {if }h\ne \ell , \end{aligned}$$

for $i,j \in \{1,\dots ,6\}$, $h,\ell \in \{1,2,3\}$, and moreover

$$\begin{aligned} \mathop {\int }\limits _{{\mathbb {R}}^{2}}\gamma _1^2~\mu ^2dz=\mathop {\int }\limits _{{\mathbb {R}}^{2}}\gamma _2^2~\mu ^2dz= k^2,\quad \mathop {\int }\limits _{{\mathbb {R}}^{2}}\gamma _3^2~\mu ^2dz=k^2+3. \end{aligned}$$

Construction of $\mathbf{u}^{\varepsilon }_{\mathbf{q}}$ satisfying i), ii). By our choices of s and $\delta$, the functions

$$\begin{aligned} \begin{aligned}&\mathcal {F}_1(\varepsilon ,q; \nu , \xi ,\alpha ) := \mu ^{-2}J_\varepsilon ({\textsc {U} }_{q} + \nu )- \sum _{j=1}^6 \xi _j\tau _j- (\alpha \cdot \gamma ) ~\omega ~\in \mathcal{C}^m , \\&\mathcal {F}_2(\varepsilon , q; \nu , \xi ,\alpha ) := \Big (\ \int \limits _{{{\mathbb {R}}}^{2}}{\nu \cdot \tau _1}\ \mu ^2d z\ ,\dots , \int \limits _{{{\mathbb {R}}}^{2}}{\nu \cdot \tau _6}\ \mu ^2d z~;\ \ \int \limits _{{{\mathbb {R}}}^{2}}\gamma ~(\nu \cdot \omega )\ \mu ^2d z\ \Big )~\in {\mathbb {R}}^6 \times {\mathbb {R}}^3 , \end{aligned} \end{aligned}$$

are well defined and continuously differentiable on ${\mathbb {R}} \times \varOmega _{2s}\times \mathcal {U}_\delta \times ({\mathbb {R}}^6 \times {\mathbb {R}}^3)$. Thus

$$\begin{aligned} \mathcal {F} := (\mathcal {F}_1, \mathcal {F}_2) : {\mathbb {R}} \times \varOmega _{2s}\times \mathcal {U}_\delta \times ({\mathbb {R}}^6 \times {\mathbb {R}}^3) \rightarrow \mathcal{C}^m\times ({\mathbb {R}}^6 \times {\mathbb {R}}^3) \end{aligned}$$

is of class $\mathcal{C}^1$ on its domain. Notice that ${\mathcal {F}}(0, q; 0, 0, 0) = 0$ for every $q \in \varOmega _{2s}$ because $J_0({\textsc {U} }_{q}) = 0$. Now we solve the equation ${\mathcal {F}}(\varepsilon , q; \nu , \xi ,\alpha ) =0$ in a neighborhood of (0, q; 0, 0, 0) via the implicit function theorem. Let

$$\begin{aligned} {\mathcal {L}} := ({\mathcal {L}}_1, {\mathcal {L}}_2) : \mathcal{C}^{2+m} \times ({\mathbb {R}}^6 \times {\mathbb {R}}^3) \rightarrow \mathcal{C}^m \times ({\mathbb {R}}^6 \times {\mathbb {R}}^3) \end{aligned}$$

given by

$$\begin{aligned} \begin{aligned}&\mathcal {L}_1(\varphi ; \zeta , \beta ) := \mu ^{-2}J'_0({\textsc {U} }_{q})\varphi - \sum _{j=1}^6 \zeta _j \tau _j - (\beta \cdot \gamma )~\omega ,\\&\mathcal {L}_2(\varphi ; \zeta , \beta ) := \mathcal {L}_2(\varphi )= \Big (\ \int \limits _{{{\mathbb {R}}}^{2}}\varphi \cdot \tau _1\ \mu ^2d z\ ,\dots , \int \limits _{{{\mathbb {R}}}^{2}}\varphi \cdot \tau _6\ \mu ^2d z\ ~;~ \int \limits _{{{\mathbb {R}}}^{2}}\gamma ~(\varphi \cdot \omega )\ \mu ^2d z\ \Big ), \end{aligned} \end{aligned}$$

so that ${\mathcal {L}}=({\mathcal {L}}_1,{\mathcal {L}}_2)$ is the differential of ${\mathcal {F}}(0,q; \cdot , \cdot , \cdot )$ evaluated in $(\nu , \xi ,\alpha ) = (0,0,0)$.

To prove that $\mathcal {L}$ is injective we assume that ${\mathcal {L}}(\varphi , \zeta , \beta ) = 0$ and put

$$\begin{aligned} v= \mu ^{-2}J'_0({\textsc {U} }_{q})\varphi \in T_{{\textsc {U} }}Z\,. \end{aligned}$$

From (3.20) we find

$$\begin{aligned} \int \limits _{{{\mathbb {R}}}^{2}}|v|^2\ \mu ^2d z = \int \limits _{{{\mathbb {R}}}^{2}} \big (\mu ^{-2}J'_0({\textsc {U} }_{q})\varphi )\cdot v\ \mu ^2 dz = \int \limits _{{{\mathbb {R}}}^{2}} J'_0({\textsc {U} }_{q})\varphi \cdot v \ dz=\int \limits _{{{\mathbb {R}}}^{2}} J'_0({\textsc {U} }_{q})v\cdot \varphi \ dz = 0, \end{aligned}$$

which implies $J'_0({\textsc {U} }_{q})\varphi =0$, that is, $\varphi \in T_{{\textsc {U} }}Z$. On the other hand, $\varphi \in T_{{\textsc {U} }}Z^\perp$ because ${\mathcal {L}}_2(\varphi ) = 0$. Thus $\varphi = 0$ and therefore also $\beta = \zeta = 0$.

To prove that ${\mathcal {L}}$ is surjective fix $v \in \mathcal{C}^m$ and $(\theta , b) \in {\mathbb {R}}^6 \times {\mathbb {R}}^3$. We have to find $\varphi \in \mathcal{C}^{2+m}$ and $(\zeta , \beta ) \in {\mathbb {R}}^6 \times {\mathbb {R}}^3$ such that ${\mathcal {L}}_1(\varphi ;\zeta ,\beta )=v$ and ${\mathcal {L}}_2(\varphi )=(\theta ,b)$. To this goal we introduce the minimal distance projection

$$\begin{aligned} P^\top : L^2({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}}^3)\rightarrow T_{{\textsc {U} }}Z, \quad w \mapsto P^{\top }w, \end{aligned}$$

so that ${\mathcal {L}}_2(w)$ is uniquely determined by $P^\top w$, and vice-versa. We find $\zeta _j$ and $\beta$ so that

$$\begin{aligned} \sum _{j=1}^6 \zeta _j \tau _j + (\beta \cdot \gamma )\omega =-P^{\top }v. \end{aligned}$$

Then, we use Lemma 3.5 to find $\widehat{\varphi }\in \mathcal{C}^{2+m}\cap T_{{\textsc {U} }}Z^\perp _*\cap N_*$ such that

$$\begin{aligned} J'_0({\textsc {U} }_{q})\widehat{\varphi }=(v-P^{\top }v)~\mu ^2. \end{aligned}$$

Finally, we take the unique tangent direction $\varphi ^\top \in T_{{\textsc {U} }}Z$ such that ${\mathcal {L}}_2(\varphi ^\top )=(\theta ,b)-{\mathcal {L}}_2(\widehat{\varphi })$. The triple $(\varphi ^\top +\widehat{\varphi };\zeta ,\beta )$ satisfies ${\mathcal {L}}(\varphi ^\top +\widehat{\varphi };\zeta ,\beta )=(v;\theta ,b)$ and surjectivity is proved. We are in the position to apply the implicit function theorem to ${\mathcal {F}}$, for any fixed $q \in \varOmega _{2s}$. In fact, thanks to a standard compactness argument, we get that there exist $\varepsilon ' >0$ and uniquely determined $\mathcal{C}^1$ functions

such that

$$\begin{aligned} \nu ^{0}_q \equiv 0, \qquad \alpha ^0(q) = 0, \qquad \xi ^{0}(q) = 0, \qquad \mathcal {F}(\varepsilon , q; \nu ^{\varepsilon }_q, \xi ^{\varepsilon }(q), \alpha ^{\varepsilon }(q)) = 0. \end{aligned}$$

(4.7)

By (4.7), the $\mathcal{C}^1$ function $(-\varepsilon ',\varepsilon ') \times \varOmega _s \rightarrow \mathcal{C}^{2+m}({\overline{{\mathbb {R}}}\,}^2, {\mathbb {H}}^3)$,

$$\begin{aligned} (\varepsilon , q) ~\mapsto ~u^\varepsilon _q := {\textsc {U} }_{q}+ \nu ^{\varepsilon }_q= ~\big (q_3{\textsc {U} }+q_1e_1+q_2e_2\big )+\nu ^{\varepsilon }_q, \end{aligned}$$

satisfies i), if $\varepsilon '$ is small enough. Further, using (4.5) (see also Lemma A.1) we rewrite the last identity in (4.7) as

$$\begin{aligned} \begin{aligned} E'_\varepsilon (u^\varepsilon _q)\varphi =&\mathop {\int }\limits _{{\mathbb {R}}^{2}}J'_\varepsilon ({\textsc {U} }_{q}+ \nu ^{\varepsilon }_q)\cdot \varphi ~dz\\ =&\sum _{j=1}^6 {\xi _j^{\varepsilon }(q)}\mathop {\int }\limits _{{\mathbb {R}}^{2}}\tau _j\cdot \varphi ~\mu ^2dz + \mathop {\int }\limits _{{\mathbb {R}}^{2}}(\alpha ^\varepsilon (q)\cdot \gamma )(\omega \cdot \varphi )~\mu ^2dz\quad \forall \,\varphi \in \mathcal{C}^0,\\ \int \limits _{{{\mathbb {R}}}^{2}}\nu ^{\varepsilon }_q \cdot \tau _j~ \mu ^2dz&= 0,\quad \forall \, j\in \{\, 1, \ldots , 6\,\}, \quad \ \int \limits _{{{\mathbb {R}}}^{2}}\gamma _\ell (\nu ^{\varepsilon }_q \cdot \omega )\ \mu ^2dz= 0,\quad \forall \, \ell \in \{\, 1,2,3\,\}. \end{aligned} \end{aligned}$$

(4.8)

In particular, claim ii) holds true.

Proof of $\mathbf{iii})$. As a straightforward consequence of (4.8) we have that

$$\begin{aligned} \int \limits _{{{\mathbb {R}}}^{2}}\partial _{q_i}\nu ^{\varepsilon }_q \cdot \tau _j\ \mu ^2dz = 0, \quad \quad \int \limits _{{{\mathbb {R}}}^{2}}\gamma _\ell ~(\partial _{q_i}\nu ^{\varepsilon }_q\cdot \omega )\ \mu ^2dz= 0, \end{aligned}$$

hence $E'_\varepsilon (u^\varepsilon _q)\partial _{q_i}\nu ^{\varepsilon }_q = 0$ for any $i=1,2,3$. We infer the identities

$$\begin{aligned} \begin{aligned}&\partial _{q_i}E_\varepsilon (u^\varepsilon _q) = E'_\varepsilon (u^\varepsilon _q)(e_i + \partial _{q_i}\nu ^{\varepsilon }_q) = E'_\varepsilon (u^\varepsilon _q)e_i, \quad i=1,2,\\&\partial _{q_3}E_\varepsilon (u^\varepsilon _q) = E'_\varepsilon (u^\varepsilon _q)({\textsc {U} } + \partial _{q_3}\nu ^{\varepsilon }_q) = E'_\varepsilon (u^\varepsilon _q){\textsc {U} }. \end{aligned} \end{aligned}$$

(4.9)

Now, from (2.4), (4.6) and (4.8) we find

$$\begin{aligned} 2c_0e_1= & {} \tau _1 - \tau _5 + k^{-1}\gamma _1 \omega , \quad 2c_0e_2 = \tau _2 + \tau _6 + k^{-1}\gamma _2 \omega ,\quad 2c_0{\textsc {U} } =k r_k(\sqrt{2}\tau _3 +k^{-1}\gamma _3 \omega ),\\ E'_\varepsilon (u^\varepsilon _q)\tau _j= & {} \xi ^{\varepsilon }_j(q),\qquad E'_\varepsilon (u^\varepsilon _q)(\gamma _\ell \omega ) = (k^2+3\delta _{\ell 3})\alpha _\ell ^{\varepsilon }(q), \end{aligned}$$

for any $j =1, \ldots , 6$, $\ell =1,2,3$. Thus by (4.9) we get

$$\begin{aligned} 2c_0\nabla _{q}E_\varepsilon (u^\varepsilon _q)=M_k\xi ^\varepsilon (q)+\varTheta _k\alpha ^\varepsilon (q), \end{aligned}$$

(4.10)

where $M_k$ and $\varTheta _k$ are constant matrices, namely

$$\begin{aligned} M_k=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 1&{}0&{}0&{}0&{}-1&{}0\\ 0&{}1&{}0&{}0&{}0&{}-1\\ 0&{}0&{}{\sqrt{2}kr_{k}}&{}0&{}0&{}0 \end{array}\right) ,\qquad \varTheta _k=\left( \begin{array}{c@{\quad }c@{\quad }c} k&{}0&{}0\\ 0&{}k&{}0\\ 0&{}0&{}(k^2+3)r_k \end{array}\right) . \end{aligned}$$

On the other hand, from (4.1) and using $\nabla {\textsc {U} }_{q} = r_kq_3 \nabla \omega$ we obtain

$$\begin{aligned} -q_3r_k~\xi ^{\varepsilon }_j(q)=E_\varepsilon '(u^{\varepsilon }_q)(\tau ^\varepsilon _j(q)), \end{aligned}$$

(4.11)

where, in the spirit of (4.6), we have put

$$\begin{aligned} \begin{array}{l@{\quad }l@{\quad }l} \tau ^\varepsilon _1(q):=c_0\partial _x\nu ^\varepsilon _q,&{}\tau ^\varepsilon _3(q):=c_0\sqrt{2}z\nabla \nu ^\varepsilon _q,&{}\tau ^\varepsilon _5(q):= c_0z^2\nabla \nu ^\varepsilon _q,\\ \tau ^\varepsilon _2(q):=c_0\partial _y\nu ^\varepsilon _q,&{}\tau ^\varepsilon _4(q):= c_0\sqrt{2} iz\nabla \nu ^\varepsilon _q,&{}\tau ^\varepsilon _6(q):=c_0iz^2\nabla \nu ^\varepsilon _q. \end{array} \end{aligned}$$

Notice that

$$\begin{aligned} \mathop {\int }\limits _{{\mathbb {R}}^{2}}|\tau ^\varepsilon _j(q)|^2~\mu ^2dz\le 2\mathop {\int }\limits _{{\mathbb {R}}^{2}}|\nabla _{z}\nu ^\varepsilon _q|^2~\mu ~dz\le 2~\Vert \nu ^\varepsilon _q\Vert _{\mathcal{C}^1}^2\mathop {\int }\limits _{{\mathbb {R}}^{2}}\mu ^3~dz=o(1), \end{aligned}$$

(4.12)

as $\varepsilon \rightarrow 0$, uniformly on $\overline{\varOmega }$, see (2.3).

For the sake of clarity, we make now some explicit computations. We denote by $\sigma _{\ell h}$ the entries of the $3\times 6$ constant matrix $\varTheta ^{-1}_kM_k$, and introduce the $6\times 6$ matrix $A^\varepsilon (q)=(a^\varepsilon _{jh}(q))_{j,h=1,\dots ,6}$, whose entries are given by

$$\begin{aligned} a^\varepsilon _{jh}(q)=\mathop {\int }\limits _{{\mathbb {R}}^{2}}\tau _h\cdot \tau ^\varepsilon _j(q)~\mu ^2dz- \sum _{\ell =1}^3 \sigma _{\ell h} \mathop {\int }\limits _{{\mathbb {R}}^{2}}\gamma _\ell (\omega \cdot \tau ^\varepsilon _j(q))~\mu ^2dz. \end{aligned}$$

Since $\tau ^\varepsilon _j\mu \rightarrow 0$ in $L^2({\mathbb {R}}^2,{\mathbb {R}}^3)$ by (4.12), then $A^\varepsilon \rightarrow 0$ uniformly on compact subsets of $(-\varepsilon ',\varepsilon ')\times \varOmega _s$. In particular, if $\hat{\varepsilon }\in (0,\varepsilon ')$ is small enough, then the determinant of the $6\times 6$ matrix $(A^\varepsilon (q)+q_3r_k\text {Id})$ is uniformly bounded away from 0 on $[-\hat{\varepsilon },\hat{\varepsilon }]\times \overline{\varOmega }$.

Assume that $\nabla _{q}E_\varepsilon (u^\varepsilon _{q^\varepsilon })=0$ for some $\varepsilon \in [-\hat{\varepsilon },\hat{\varepsilon }]$, $q^\varepsilon \in \varOmega$. From (4.10) we obtain $\alpha ^\varepsilon (q^\varepsilon )=-\varTheta _k^{-1}M_k\xi ^\varepsilon (q^\varepsilon )$. Thus (4.8) and (4.11) give

$$\begin{aligned} -q^\varepsilon _3 r_k~\xi ^\varepsilon (q^\varepsilon )= A^\varepsilon (q^\varepsilon )\xi ^\varepsilon (q^\varepsilon ), \end{aligned}$$

and hence $\xi ^\varepsilon (q^\varepsilon )=0$, because the matrix $(A^\varepsilon (q^\varepsilon )+q^\varepsilon _3r_k\text {Id})$ is invertible. But then (4.10) and $\nabla _{q}E_\varepsilon (u^\varepsilon _{q^\varepsilon })=0$ imply that $\alpha ^\varepsilon (q^\varepsilon )=0$ as well, hence $E'(u^\varepsilon _{q^\varepsilon })=0$ by (4.8).

Proof of $\mathbf{iv})$. The function $(\varepsilon , q) \mapsto \nu ^{\varepsilon }_q$ is of class $\mathcal{C}^1$, and in particular $\partial _\varepsilon \nu ^{\varepsilon }_q$ is uniformly bounded in $\mathcal{C}^2$ for $(\varepsilon ,q)\in [-\hat{\varepsilon },\hat{\varepsilon }]\times \overline{\varOmega }$. Thus Taylor expansion formula for

$$\begin{aligned} \varepsilon \mapsto E_\varepsilon (u^\varepsilon _q)-E_\varepsilon ({\textsc {U} }_{q})=E_0(u^\varepsilon _q)-E_0({\textsc {U} }_{q})+2\varepsilon \big (V_\phi (u^\varepsilon _q)-V_\phi ({\textsc {U} }_{q})) \end{aligned}$$

gives $E_\varepsilon (u^\varepsilon _q)-E_\varepsilon ({\textsc {U} }_{q})=o(\varepsilon )$ as $\varepsilon \rightarrow 0$, uniformly on $\overline{\varOmega }$.

Now we estimate $\nabla _{q}(E_\varepsilon (u^\varepsilon _q)-E_\varepsilon ({\textsc {U} }_{q}))$. We use (4.2), (4.9) to obtain, for $j=1,2$,

$$\begin{aligned} \begin{aligned} \partial _{q_j}(E_\varepsilon (u^\varepsilon _q)-E_\varepsilon ({\textsc {U} }_{q}))=&~\big (E'_0(u^\varepsilon _q)e_j-E'_0({\textsc {U} }_{q})e_j\big )+2\varepsilon \big (V'_\phi (u^\varepsilon _q)e_j-V'_\phi ({\textsc {U} }_{q})e_j\big )\\ =&~2\varepsilon \big (V'_\phi (u^\varepsilon _q)e_j-V'_\phi ({\textsc {U} }_{q})e_j\big )=o(\varepsilon ), \end{aligned} \end{aligned}$$

because $\Vert u^\varepsilon _q-{\textsc {U} }_{q}\Vert _{\mathcal{C}^{2+m}}=o(1)$ and $V_\phi$ is a $\mathcal{C}^1$-functional.

To handle the derivative with respect to $q_3$ we first argue as before to get

$$\begin{aligned} \begin{aligned} \partial _{q_3}(E_\varepsilon (u^\varepsilon _q)-E_\varepsilon ({\textsc {U} }_{q}))=&~\big (E'_0(u^\varepsilon _q){\textsc {U} }-E'_0({\textsc {U} }_{q}){\textsc {U} }\big )+2\varepsilon \big (V'_\phi (u^\varepsilon _q){\textsc {U} }-V'_\phi ({\textsc {U} }_{q}){\textsc {U} }\big )\\ =&~E'_0(u^\varepsilon _q){\textsc {U} }+ o(\varepsilon ), \end{aligned} \end{aligned}$$

uniformly on $\overline{\varOmega }$. Next, from $q_3{\textsc {U} }=u^\varepsilon _q-(q_1e_1+q_2e_2)-\nu ^\varepsilon _q$ and (4.2) we obtain

$$\begin{aligned} \begin{aligned} q_3E'_0(u^\varepsilon _q){\textsc {U} }=&E'_0(u^\varepsilon _q)(u^\varepsilon _q-(q_1e_1+q_2e_2)-\nu ^\varepsilon _q)\\ =&-E'_0(u^\varepsilon _q)\nu ^\varepsilon _q =-E'_\varepsilon (u^\varepsilon _q)\nu ^\varepsilon _q+2\varepsilon V'_\phi (u^\varepsilon _q)\nu ^\varepsilon _q=2\varepsilon V'_\phi (u^\varepsilon _q)\nu ^\varepsilon _q \end{aligned} \end{aligned}$$

because of (4.8). Since $\nu ^\varepsilon _q\rightarrow 0$ in $\mathcal{C}^{2+m}$ we infer that $E'_0(u^\varepsilon _q)u^\varepsilon _q=o(\varepsilon )$ uniformly on $\overline{\varOmega }$ as $\varepsilon \rightarrow 0$, which concludes the proof. $\square$

Proof of Theorem 1.2

Take an open set $\varOmega \Subset {\mathbb {R}}^3_+$ containing the closure of A, let $u^\varepsilon _q$ be the function given by Lemma 4.1 and notice that, by (4.4), $E_\varepsilon ({\textsc {U} }_{q})=E_0({\textsc {U} }_{q})- 2\varepsilon F^\phi _k(q)$. Thus for $\varepsilon \in [-\hat{\varepsilon },\hat{\varepsilon }], \varepsilon \ne 0$ we can estimate

$$\begin{aligned} \Big \Vert \frac{1}{2\varepsilon }\big (E_\varepsilon (u^\varepsilon _q)-E_0({\textsc {U} }_{q})\big )+F^\phi _k(q)\Big \Vert _{\mathcal{C}^1(\overline{A})}=\frac{1}{2|\varepsilon |}\Vert E_\varepsilon (u^\varepsilon _q)-E_\varepsilon ({\textsc {U} }_{q})\Vert _{\mathcal{C}^1(\overline{A})}=o(1), \end{aligned}$$

uniformly on $\overline{\varOmega }$ by iv) in Lemma 4.1. Recalling the definition of stable critical point presented in Sect. 2.2, we infer that for any $\varepsilon \approx 0$ the function $\frac{1}{2\varepsilon }\big (E_\varepsilon (u^\varepsilon _q)-E_0({\textsc {U} }_{q})\big )$ has a critical point $q^\varepsilon \in A$, to which corresponds the embedded $(k+\varepsilon \phi )$-bubble $u^\varepsilon :=u^\varepsilon _{q^\varepsilon }$ by iii) in Lemma 4.1. The continuity of $(\varepsilon ,q)\mapsto u^\varepsilon _q$ gives the continuity of $\varepsilon \mapsto u^\varepsilon$.

The last conclusion in Theorem 1.2 follows via a simple compactness argument and thanks to Theorem 4.1. $\square$

Proof of Theorem 1.3

Recalling that $q^k:=(q_1,q_2,kr_{k}q_3)$, we write

$$\begin{aligned} F^{\phi }_k(q)=\int \limits _{B_{r_{k}}(0)}(p_3+kr_{k})^{-3}\phi (q_3p+q^k)~dp. \end{aligned}$$

Since $r_k \rightarrow 0$ and $kr_k=k(k^2-1)^{-1/2}\rightarrow 1$ as $k\rightarrow \infty$, we infer that $q^k\rightarrow q$ uniformly on compact sets of ${\mathbb {R}}^3_+$ and

$$\begin{aligned} \frac{3}{4\pi r_{k}^3} F^{\phi }_k \rightarrow \phi \qquad \text {as }k\rightarrow \infty , \end{aligned}$$

uniformly on $\overline{\varOmega }$. Next, we easily compute

$$\begin{aligned} \begin{aligned} \partial _{q_j} F^{\phi }_k(q)&=\int \limits _{B_{r_{k}}(0)}(p_3+kr_{k})^{-3}\partial _{q_j}\phi (q_3p+q^k)~dp,\qquad j=1,2,\\ \partial _{q_3} F^{\phi }_k(q)&=\int \limits _{B_{r_{k}}(0)}(p_3+kr_{k})^{-3}\nabla \phi (q_3p+q^k)\cdot (p+kr_{k}e_3)~dp, \end{aligned} \end{aligned}$$

and thus we obtain, by the same argument,

$$\begin{aligned} \frac{3}{4\pi r_{k}^3} \nabla F^{\phi }_k \rightarrow \nabla \phi \qquad \text {as }k\rightarrow \infty , \end{aligned}$$

uniformly on $\overline{\varOmega }$. It follows that for k large enough, $F^{\phi }_k$ has a stable critical point in $\varOmega \Subset {\mathbb {H}}^3$, since having a stable critical point is a $\mathcal{C}^1$-open condition. Hence Theorem 1.1 applies and gives the conclusion of the proof. $\square$

Appendix

Let $K\in \mathcal{C}^0({\mathbb {H}}^3)$. Take any vectorfield $Q_K\in \mathcal{C}^1({\mathbb {R}}^3_+, {\mathbb {R}}^3)$ such that $\mathrm{div}Q_K(p)=p_3^{-3}K(p)$ for any $p\in {\mathbb {R}}^3_+$ (here $\mathrm{div}=\sum _j\partial _j$ is the Euclidean divergence). The functional

$$\begin{aligned} V_K(u):=\mathop {\int }\limits _{{\mathbb {R}}^{2}}Q_K(u)\cdot \partial _x u\wedge \partial _yu~dz,\qquad u\in \mathcal{C}^1({\overline{{\mathbb {R}}}\,}^2,{\mathbb {H}}^3), \end{aligned}$$

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 $\varOmega \Subset {\mathbb {R}}^3_+$ and if $\partial _xu\wedge \partial _y u$ is inward-pointing, then the divergence theorem gives

$$\begin{aligned} V_K(u)=-\int \limits _{\partial \varOmega } Q_K(u)\cdot \nu ~du=-\int \limits _\varOmega p_3^{-3}K~dp=-\int \limits _\varOmega K~d {\mathbb {H}}^3. \end{aligned}$$

Clearly, the functional $V_K$ does not depend on the choice of the vectorfield Q. Notice that if $K\equiv k$ is constant, then

$$\begin{aligned} V_k(u)=-\frac{k}{2} \mathop {\int }\limits _{{\mathbb {R}}^{2}}u_3^{-2}e_3\cdot \partial _x u\wedge \partial _yu~dz,\quad u\in \mathcal{C}^1({\overline{{\mathbb {R}}}\,}^2,{\mathbb {H}}^3). \end{aligned}$$

In the next Lemma we collect few simple remarks about the energy functional

$$\begin{aligned} {E}(u)=\frac{1}{2}\mathop {\int }\limits _{{\mathbb {R}}^{2}}u_3^{-2}|\nabla u|^2~ dz+2V_{K}(u). \end{aligned}$$

(A.1)

Lemma A.1

Let $K\in \mathcal{C}^0({\mathbb {H}}^3)$.

• (i) The functional $E:\mathcal{C}^1({\overline{{\mathbb {R}}}\,}^2,{\mathbb {H}}^3)\rightarrow {\mathbb {R}}$ is of class $\mathcal{C}^1$, and its differential is given by

  $$\begin{aligned} E'(u)\varphi =\mathop {\int }\limits _{{\mathbb {R}}^{2}}(u_3^{-2}\nabla u\cdot \nabla \varphi -u_3^{-3}|\nabla u|^2e_3\cdot \varphi )~dz+2\mathop {\int }\limits _{{\mathbb {R}}^{2}}u_3^{-3}K(u)\varphi \cdot \partial _x u\wedge \partial _yu~dz~; \end{aligned}$$

• (ii) If $u\in \mathcal{C}^2({\overline{{\mathbb {R}}}\,}^2,{\mathbb {H}}^3)$, then $E'(u)$ extends to a continuous form on $\mathcal{C}^0({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}}^3)$, namely

  $$\begin{aligned} E'(u)\varphi =\mathop {\int }\limits _{{\mathbb {R}}^{2}}(-\mathrm{div}(u_3^{-2}\nabla u)-u_3^{-3}|\nabla u|^2e_3+2u_3^{-3}K(u)\partial _x u\wedge \partial _yu) \cdot \varphi ~dz~; \end{aligned}$$

• (iii) If $K\in \mathcal{C}^1({\mathbb {H}}^3)$, then E is of class $\mathcal{C}^2$ on $\mathcal{C}^1({\overline{{\mathbb {R}}}\,}^2,{\mathbb {H}}^3)$.

In the next Lemma we show that critical points for E are in fact hyperbolic K-bubbles.

Lemma A.2

Let $K\in \mathcal{C}^0({\mathbb {H}}^3)$ and let $u\in \mathcal{C}^2({\overline{{\mathbb {R}}}\,}^2,{\mathbb {H}}^3)$ be a nonconstant critical point for E. Then u is conformal, that is,

$$\begin{aligned} |\partial _x u|=|\partial _y u|,~~ \partial _x u\cdot \partial _y u=0, \end{aligned}$$

hence it parameterizes an ${\mathbb {S}}^2$ type surface in ${\mathbb {H}}^3$, having mean curvature K, apart from a finite number of branch points.

Proof

Put $\alpha = \frac{1}{2}u_3^{-2}(|{\partial _{x}u}|^2 - |{\partial _{y}u}|^2)$, $\beta = - u_3^{-2} {\partial _{x}u} \cdot {\partial _{y}u}$, $\varphi = \alpha + i \beta$ and notice that $|\varphi |\le c_u|\nabla u|^2\in L^\infty ({\mathbb {R}}^2)$. By direct computation we find

$$\begin{aligned} \begin{aligned}&(\partial _x\alpha - \partial _y\beta ) u_3^3 =u_3 {\partial _{x}u} \cdot \varDelta u - (|{\partial _{x}u}|^2- |{\partial _{y}u}|^2)\partial _{x}u_3 - 2 ({\partial _{x}u} \cdot {\partial _{y}u})\partial _{y}u_3, \\&(\partial _y\alpha + \partial _x\beta ) u_3^3 = - u_3 {\partial _{y}u} \cdot \varDelta u - (|{\partial _{x}u}|^2- |{\partial _{y}u}|^2)\partial _{y}u_3 + 2 ({\partial _{x}u} \cdot {\partial _{y}u})\partial _{x}u_3. \end{aligned} \end{aligned}$$

(A.2)

Since u solves (1.1), it holds that

$$\begin{aligned} \begin{aligned}&u_3 {\partial _{x}u} \cdot \varDelta u = 2 {G}(\nabla u) \cdot {\partial _{x}u} = 2({\partial _{x}u} \cdot {\partial _{y}u}) \partial _{y}u_3 + (|{\partial _{x}u}|^2- |{\partial _{y}u}|^2)\partial _{x}u_3,\\&u_3 {\partial _{y}u} \cdot \varDelta u = 2 {G}(\nabla u) \cdot {\partial _{y}u} = 2({\partial _{x}u} \cdot {\partial _{y}u})\partial _{x}u_3 - (|{\partial _{x}u}|^2- |{\partial _{y}u}|^2)\partial _{y}u_3. \end{aligned} \end{aligned}$$

(A.3)

Putting together (A.2) and (A.3) we obtain $\partial _x\alpha - \partial _y\beta = \partial _y\alpha + \partial _x\beta = 0$, namely, $\varphi$ is an holomorphic function. Since $\varphi$ is bounded and vanishes at infinity then $\varphi \equiv 0$ on ${\mathbb {R}}^2$, hence u is conformal.

The last conclusion follows from Proposition 2.4 and Example 2.5(4) in [14]. $\square$

Remark A.1

Here we take $K\equiv k$ constant and point out two simple facts about the energy functional $E_{\text {hyp}}$ in (1.3).

By (4.2), the Nehari manifold contains any nonconstant function. Secondly, $E_{\text {hyp}}$ is unbounded from below. In fact, for $t>1$ we have

$$\begin{aligned} E_{\text {hyp}}(\omega +te_3)= & {} \frac{1}{2}\mathop {\int }\limits _{{\mathbb {R}}^{2}}(\omega _3+t)^{-2}~\mu ^2dz+k\mathop {\int }\limits _{{\mathbb {R}}^{2}}(\omega _3+t)^{-2}\omega _3~\mu ^2dz\\= & {} 4\pi \big (-\frac{kt-1}{t^2-1}+\frac{k}{2}\ln \frac{t+1}{t-1}\big ). \end{aligned}$$

Notice that $\omega +te_3$ approaches a horosphere as $t\rightarrow 1$, and that $\lim \limits _{t\rightarrow 1}E_{\text {hyp}}(\omega +te_3)=-\infty$.

Remark A.2

Differently from the Euclidean case, see for instance [5], the geometric and compactness properties of the energy functional E are far from being understood (also in the case of a constant curvature), and would deserve a careful analysis.

We conclude the paper by pointing out a necessary condition for the existence of embedded K bubbles.

Let $K\in \mathcal{C}^1({\mathbb {H}}^3)$ be given, and let $u\in \mathcal{C}^2({\overline{{\mathbb {R}}}\,}^2,{\mathbb {H}}^3)$ be an embedded solution to (1.1). By Lemma A.2, u is a conformal parametrization of the open set $\varOmega \subset {\mathbb {R}}^3_+$, which is the bounded connected component of ${\mathbb {R}}^3_+\setminus u({\mathbb {S}}^2)$. We can assume that the nowhere vanishing normal vector $\partial _x u\wedge \partial _y u$ is inward pointing. Since u is a critical point of the energy functional in (A.1), then for $j=1,2$ we have that

$$\begin{aligned} 0=E'(u)e_j=V'_K(u)e_j=\mathop {\int }\limits _{{\mathbb {R}}^{2}}u_3^{-3}K(u)e_j\cdot \partial _x u\wedge \partial _y u~dz= -\int \limits _{\varOmega }\mathrm{div}(p_3^{-3}K(p)e_j)~dp \end{aligned}$$

by the divergence theorem. Thus

$$\begin{aligned} \int \limits _{\varOmega }p_3^{-3}\partial _{p_j} K(p)~dp= 0. \end{aligned}$$

In a similar way, from $E'(u)u=0$ and since $\mathrm{div}(p_3^{-3}K(p)p)= p_3^{-3}\nabla K(p)\cdot p$, one gets

$$\begin{aligned} \int \limits _{\varOmega }p_3^{-3}\nabla K(p)\cdot p~dp= 0. \end{aligned}$$

In particular, $\partial _{p_1} K, \partial _{p_2} K$ and the radial derivative of K can not have constant sign in $\varOmega$. We infer the next nonexistence result (see [7, Proposition 4.1] for the Euclidean case).

Theorem A.3

Assume that $K\in \mathcal{C}^{1}({\mathbb {H}}^{3})$ satisfies one of the following conditions,

• (i) $K(p)=f(\nu \cdot p)$ for some direction $\nu$ orthogonal $e_3$, where f is strictly monotone;

• (ii) $K(p)=f(|p|)$, where f is strictly monotone.

Then (1.1) has no embedded solution $u\in \mathcal{C}^2({\overline{{\mathbb {R}}}\,}^2,{\mathbb {H}}^3)$.