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.
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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
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
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
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
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
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
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
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
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
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
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
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
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
Let \(m\ge 0\). We endow
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
Since we adopt the upper half-space model for \({\mathbb {H}}^3\), we are allowed to write
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
(notice that \(\tau \nabla \varphi (z)=d\varphi (z)\tau \) for any \(z\in {\mathbb {R}}^2\)). 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 \(L^2({\overline{{\mathbb {R}}}\,}^2)\), \(\mathcal{C}^m({\overline{{\mathbb {R}}}\,}^2)\) 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
Therefore, for any \(u\in \mathcal{C}^1({\overline{{\mathbb {R}}}\,}^2,{\mathbb {H}}^3)\), the functions
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
span the tangent space to the manifold \(\{\, u_q~|~q\in {\mathbb {H}}^3\, \}\) at u, where
3 Nondegeneracy of hyperbolic k-bubbles
The proof of Theorem 1.1 needs some preliminary work. We put
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
(notice that \({\textsc {U} }_{e_3}={\textsc {U} }\)), and introduce the 9-dimensional manifold
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
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
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
We now introduce the differential operator
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
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
Main computations
Recall that \({\textsc {U} }=r_{k}(\omega +ke_3)\) solves \(J_0({\textsc {U} })=0\) to check that
where G is given by (1.2). Since \(\nabla \omega _3=-\nabla \mu =\mu ^2z\), thanks to (2.2) we have
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
compare with (3.4). Accordingly, for \(\varphi \in \mathcal{C}^{2}({\overline{{\mathbb {R}}}\,}^2,{\mathbb {R}}^3)\) we have
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
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
Recalling that \(\omega \) is pointwise orthogonal to \(\partial _x\omega , \partial _y\omega \), from (3.7) we obtain
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
and (3.10) follows, because \(e_3=z\nabla \omega +\omega _3\omega \), see (2.4).
Next, using the equivalent formulation
we find that, for \(\varphi = \eta \omega \), \(\eta \in \mathcal{C}^2({\overline{{\mathbb {R}}}\,}^2)\), it holds
whence we infer
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
We start to handle (3.7). From \(e_3=z\nabla \omega +\omega _3\omega \) we get
Further,
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
Next, using (2.2) we can compute
which give the identity
that holds for any \(\varphi \in \langle \omega \rangle _{\mathcal{C}^m}^\perp \).
Putting together the above informations we arrive at
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
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
Now we prove the identity
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
Using the elementary identity \(\partial _\rho \alpha \wedge \partial _\theta \beta =\rho (\partial _x\alpha \wedge \partial _y\beta )\), we see that
and (3.15) follows from (3.13) (with \(\varphi \) replaced by \(\psi \)).
Thanks to (3.15), we have the identity
so that we only need to handle the function
We decompose \(\partial _x\psi \) and \(\partial _y\psi \) accordingly with (3.8), to obtain
respectively. Since \(|\nabla \psi |^2=|\partial _x\psi |^2+|\partial _y\psi |^2\), we infer
Writing \(\mu ^2\omega =-\partial _x\omega \wedge \partial _y\omega \), see (2.2), we get
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
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
by Lemma 3.2. Since \(\psi \in \langle \partial _x\omega ,\partial _y\omega \rangle \) pointwise on \({\mathbb {R}}^2\), we can write
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
For every \(\rho >0\) we expand the periodic function \(f(\rho ,\cdot )\) in Fourier series,
The coefficients \(\gamma _h\) are complex-valued functions on the half-line \({\mathbb {R}}_+\) and solve
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
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
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
has a non decreasing, divergent sequence \((\lambda _h)_{h\ge 0}\) of eigenvalues which correspond to critical levels of the quotient
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
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
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]. 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
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
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
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
Since integration by parts gives
the quadratic form
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,
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
Moreover, we introduce on \(L^2({\overline{{\mathbb {R}}}\,}^2, {\mathbb {R}}^3)\) the equivalent scalar product
and the subspaces
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
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
Proof
We introduce
which is a closed subspace of \(H^1\). Notice that \(\psi = {\mathcal {P}}\psi \) for any \(\psi \in X\) and moreover
use (3.9) and a density argument. Next we put
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 \(I: X \rightarrow {\mathbb {R}}\),
is weakly lower semicontinuous and coercive. Thus its infimum is achieved by a function \(\varphi \in X\) which satisfies
If \(\psi \in H^1\) we write
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
because \(\varphi \cdot \omega \equiv 0\). Therefore, (3.23) gives
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
Proof
We introduce the space
so that \(\eta \omega \in H^1\cap T_{{\textsc {U} }}Z^\perp _*\cap N_*\) for any \(\eta \in Y\), and the energy functional \(I:Y\rightarrow {\mathbb {R}}\),
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
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
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
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
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 \(E_\varepsilon :\mathcal{C}^{2}({\overline{{\mathbb {R}}}\,}^2,{\mathbb {H}}^3)\rightarrow {\mathbb {R}}\),
(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
Now we prove that
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
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
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
is related to the differential of \(E_\varepsilon \) via the identity
Remark 4.1
Since \(E_\varepsilon \) is of class \(\mathcal{C}^2\) and
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
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
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
where \(c_0:=\sqrt{\frac{3}{2^4\pi }}\) is a normalization constant. Thanks to (3.3), (3.5), we have
Trivially, \(\tau _j\cdot \omega \equiv 0\) on \({\mathbb {R}}^2\). Elementary computations give
for \(i,j \in \{1,\dots ,6\}\), \(h,\ell \in \{1,2,3\}\), and moreover
Construction of \(\mathbf{u}^{\varepsilon }_{\mathbf{q}}\) satisfying i), ii). By our choices of s and \(\delta \), the functions
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
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
given by
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
From (3.20) we find
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
so that \({\mathcal {L}}_2(w)\) is uniquely determined by \(P^\top w\), and vice-versa. We find \(\zeta _j\) and \(\beta \) so that
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
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
By (4.7), the \(\mathcal{C}^1\) function \((-\varepsilon ',\varepsilon ') \times \varOmega _s \rightarrow \mathcal{C}^{2+m}({\overline{{\mathbb {R}}}\,}^2, {\mathbb {H}}^3)\),
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
In particular, claim ii) holds true.
Proof of \(\mathbf{iii})\). As a straightforward consequence of (4.8) we have that
hence \(E'_\varepsilon (u^\varepsilon _q)\partial _{q_i}\nu ^{\varepsilon }_q = 0\) for any \( i=1,2,3\). We infer the identities
Now, from (2.4), (4.6) and (4.8) we find
for any \(j =1, \ldots , 6\), \(\ell =1,2,3\). Thus by (4.9) we get
where \(M_k\) and \(\varTheta _k\) are constant matrices, namely
On the other hand, from (4.1) and using \(\nabla {\textsc {U} }_{q} = r_kq_3 \nabla \omega \) we obtain
where, in the spirit of (4.6), we have put
Notice that
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
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
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
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 \(j=1,2\),
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
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
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
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
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
uniformly on \(\overline{\varOmega }\). Next, we easily compute
and thus we obtain, by the same argument,
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. \(\square \)
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Acknowledgements
This work is partially supported by PRID Projects PRIDEN and VAPROGE, Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Università di Udine.
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Appendix
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
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
Clearly, the functional \(V_K\) does not depend on the choice of the vectorfield Q. Notice that if \(K\equiv k\) is constant, then
In the next Lemma we collect few simple remarks about the energy functional
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,
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
Since u solves (1.1), it holds that
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
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
by the divergence theorem. Thus
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
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 \(u\in \mathcal{C}^2({\overline{{\mathbb {R}}}\,}^2,{\mathbb {H}}^3)\).
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Cora, G., Musina, R. Bubbles with constant mean curvature, and almost constant mean curvature, in the hyperbolic space. Calc. Var. 60, 222 (2021). https://doi.org/10.1007/s00526-021-01932-8
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DOI: https://doi.org/10.1007/s00526-021-01932-8