Higher genus capillary surfaces in the unit ball of ${\mathbb{R}}^{3}$
 Filippo Morabito^{1, 2}Email author
DOI: 10.1186/168727702014130
© Morabito; licensee Springer. 2014
Received: 16 December 2013
Accepted: 30 April 2014
Published: 22 May 2014
Abstract
We construct the first examples of capillary surfaces of positive genus, embedded in the unit ball of ${\mathbb{R}}^{3}$ with vanishing mean curvature and locally constant contact angles along their three boundary curves. These surfaces come in families depending on one parameter and they converge to the triple equatorial disk. Such surfaces are obtained by deforming the CostaHoffmanMeeks minimal surfaces.
MSC:53A10, 35R35, 53C21.
Keywords
minimal surface perturbation method nonlinear pde’s1 Introduction
The study of capillarity started in the beginning of the 19th century by the work of PS de Laplace and T Young. They considered a liquid contained in a vertical tube of small radius dipped in a reservoir and studied the shape of the free surface interface between the liquid and the air. Such a surface is called capillary surface. More generally a capillary surface is the surface interface between a liquid situated adjacent to another immiscible liquid or gas.
where H is the mean curvature, λ is a constant to be determined by physical condition (volume of the fluid and boundary conditions) and k is positive (resp. negative) when denser fluid lies below (resp. above) the interface.
Here ν is the unit normal vector to the tube along the boundary of the surface. It says that the capillary surface meets the tube in a constant contact angle (equal to α). See Finn [1], for a survey on more recent discoveries about capillarity.
Existence and uniqueness for the solution of capillarity problem for graphs over domains of ${\mathbb{R}}^{n}$ $n\ge 2$ (also in the more general form where $H=f$, for an assigned function f), has been extensively studied in the past, see e.g. Gerhardt [2], Lieberman [3], Simon and Spruck [4], Spruck [5], Uraltseva [6].
A more recent series of works (see e.g. [7–9]) deals with the existence and regularity of capillary graphs with constant mean curvature in vertical cylinders containing corners or cusps. Huff and McCuan [10] showed the existence of Scherktype capillary minimal graphs.
Very recently, Calle and Shahriyari in [11] have solved the prescribed mean curvature equation with a boundary contact angle condition. They show the existence of graphs over domains in ${\mathbb{M}}^{n}\times \mathbb{R}$, where ${\mathbb{M}}^{n}$ is a ndimensional Riemannian submanifold of ${\mathbb{R}}^{n+1}$. In [12] Lira and Wanderley show the existence of Killing graphs with prescribed mean curvature and prescribed contact angle along their boundary in a wide class of Riemannian manifolds endowed with a Killing vector field.
Fall and Mercuri in [13] constructed by a perturbation method disktype minimal surfaces embedded in an infinite cylinder in ${\mathbb{R}}^{3}$ and which intersect its boundary orthogonally. In [14] they extended this result to Riemannian manifolds.
In [15] Fall showed that, given a bounded domain of ${\mathbb{R}}^{3}$ there exist embedded constant mean curvature (cmc) surfaces contained in Ω and whose boundary intersects ∂ Ω orthogonally. Also he showed that, given a stable stationary point p for the mean curvature of ∂ Ω, there exists near p a family of embedded surfaces with cmc equal to ${\epsilon}^{1}$, which, after scaling and translation, converges to a hemisphere of radius 1 as $\epsilon \to 0$.
In [16] Fall and Mahmoudi showed that if Ω is a domain of ${\mathbb{R}}^{m+1}$ and K a kdimensional nondegenerate minimal submanifold, then there exists a family of embedded constant mean curvature hypersurfaces which, as their mean curvature tends to infinity, concentrate along K and intersect ∂ Ω orthogonally.
where $H(p)$ denotes the mean curvature at the point p; ${N}_{i}(p)$ and ${\nu}_{i}(p)$ denote, respectively, the unit normal vector to the surface ${S}_{\tau}$ and to $\partial {B}^{3}$ at $p\in {\lambda}_{i}$. The functions $({\psi}_{t}(\tau ),{\psi}_{m}(\tau ),{\psi}_{b}(\tau ))=(\psi (\tau ),0,\psi (\tau ))$ are decreasing smooth and nonzero for $\tau \in (0,{\tau}_{0})$. We will describe them below.
The solution of the previous system is based on the deformation of a compact piece of a scaled CostaHoffmanMeeks minimal surface contained in the unit ball. More precisely we consider the image by a homothety of ratio τ. Such a surface is denoted by ${M}_{k,\tau}$. As we will explain in Section 2.1, ${M}_{k,\tau}$ is asymptotic to a top half catenoid, to a bottom half catenoid and to a horizontal plane. The functions $({\psi}_{t}(\tau ),{\psi}_{m}(\tau ),{\psi}_{b}(\tau ))$ are defined to be the values of the scalar product ${N}_{i}(p)\cdot {\nu}_{i}(p)$ we obtain if we replace ${S}_{\tau}$ by the two halves catenoid and the plane. In particular ${\psi}_{m}=0$.
We provide the first examples of capillary type surfaces with nontrivial topology, having vanishing mean curvature and locally constant contact angles with the sphere. They are equal to the contact angles made by the asymptotic catenoids and the plane described above with the sphere. Such surfaces are obtained by deformation of minimal surfaces by a function in the space described by Definition 2.1.
Here is the statement of the result we get. The cartesian coordinates in ${\mathbb{R}}^{3}$ are denoted by $({x}_{1},{x}_{2},{x}_{3})$.
Such surfaces are invariant under the action of the rotation of angle $\frac{2\pi}{k+1}$ about the ${x}_{3}$axis, under the action of the reflection in the ${x}_{2}=0$ plane and under the action of the composition of a rotation of angle $\frac{\pi}{k+1}$ about the ${x}_{3}$axis and the reflection in the ${x}_{3}=0$ plane.
We observe that for values of τ in the range of validity of our theorem ${\psi}_{t}(\tau ),{\psi}_{b}(\tau )\ne 0$. In other terms the surface cannot make a constant angle equal to $\pi /2$ with $\partial {B}^{3}$ along ${\lambda}_{t}$, ${\lambda}_{b}$. We point out that ${lim}_{\tau \to 0}{\psi}_{i}(\tau )=0$. As τ is the homothety ratio, this says that, as τ tends to 0 the limit of ${S}_{\tau}$ consists in the triple equatorial disk.
The proof can easily be modified in order to handle the case of capillary surfaces with boundary on a vertical cylinder.
Among the works dealing with capillary surfaces in a ball we cite [17] by Ros and Souam. They showed that a stable minimal capillary surface (that is, stationary surfaces with nonnegative second variation of the area) in a ball of ${\mathbb{R}}^{3}$ is a totally geodesic disk or a surfaces of genus 1 with boundary having at most 3 connected components. Consequently, at least for $k>1$, the surfaces described by Theorem 1.1 are unstable.
The interest in capillary surfaces in the unit ball has been rekindled by the recent works of Fraser and Schoen [18, 19]. They considered free boundary minimal surfaces embedded in the unit ball of ${\mathbb{R}}^{n}$, i.e. surfaces which meet orthogonally the boundary of the ball.
Free boundary minimal submanifolds are critical for the problem of extremizing the volume among deformations which preserve the ball. Such solutions arise from variational min/max constructions, and examples include equatorial disks, the (critical) catenoid, as well as the cone over any minimal submanifold of the sphere. If Σ is a compact Riemannian surface with $\partial \mathrm{\Sigma}\ne \mathrm{\varnothing}$ then the DirichlettoNeumann operator maps a function u on ∂ Σ to the normal derivative of the harmonic extension of u to the interior. A submanifold properly immersed in the unit ball is a free boundary submanifold if and only if its coordinate functions are Steklov eigenfunctions with eigenvalue 1. Using this characterization they prove the existence of free boundary minimal surfaces in the unit ball of ${\mathbb{R}}^{3}$ of genus 0 with boundary having k connected components, for any finite $k\ge 1$. The authors conjecture the existence of higher genus examples of free boundary embedded minimal surfaces which have three boundary components and converge to the union of the critical vertical catenoid and the equatorial disk.
The minimal surfaces described in Theorem 1.1 come in 1parameter families, they have finite genus ≥1, they meet orthogonally the boundary of the ball only along the middle boundary curve. Furthermore, for any value of the genus, the limit for values of the parameter close to zero consists in the triple equatorial disk.
2 Preliminaries
The proof of the existence of solutions of the capillarity type problem is based on the deformation of a compact piece of the minimal surfaces ${M}_{k,\tau}$. We describe this family of surfaces in Section 2.1.
We will show that it is possible to deform a surface Σ in this family in order to get a surface satisfying (3). More precisely we will prove the existence of a function u defined on Σ and of small norm such that its normal graph ${S}_{u}$ over Σ has vanishing mean curvature and the scalar product of the unit normal vectors, ${({N}_{{S}_{u}})}_{i}\cdot {\nu}_{i}$, equals ${\psi}_{i}$ at each point of the i th component of $\partial {S}_{u}$, with $i\in \{1,2,3\}$.
We will adapt to our setting some arguments used in [20, 21].
2.1 The scaled CostaHoffmanMeeks surface
The CostaHoffmanMeeks surface of genus $k\in [1,\dots ,+\mathrm{\infty})$ embedded in ${\mathbb{R}}^{3}$ (see [22]) is denoted by ${M}_{k}$.
 1.
It has one planar end ${E}_{m}$ asymptotic to the horizontal plane ${x}_{3}=0$, one top end ${E}_{t}$ and one bottom end ${E}_{b}$ that are, respectively, asymptotic to the upper end and to the lower end of a catenoid having the ${x}_{3}$axis as axis of rotation. The planar end ${E}_{m}$ is located between the two catenoidal ends.
 2.
It is invariant under the action of the rotation of angle $\frac{2\pi}{k+1}$ about the ${x}_{3}$axis, under the action of the reflection in the ${x}_{2}=0$ plane and under the action of the composition of a rotation of angle $\frac{\pi}{k+1}$ about the ${x}_{3}$axis and the reflection in the ${x}_{3}=0$ plane.
 3.
It intersects the ${x}_{3}=0$ plane in $k+1$ straight lines, which intersect themselves at the origin with angles equal to $\frac{\pi}{k+1}$. The intersection of ${M}_{k}$ with the plane ${x}_{3}=\mathrm{const}$ (≠0) is a single Jordan curve. The intersection of ${M}_{k}$ with the upper half space ${x}_{3}>0$ (resp. with the lower half space ${x}_{3}<0$) is topologically an open annulus.
The parameterization of the end ${E}_{i}$ is denoted by ${X}_{i}$, with $i=t,b,m$, and the parameterization of the corresponding end ${E}_{i,\tau}$ of ${M}_{k,\tau}$ is denoted by ${X}_{i,\tau}$. We recall that ${M}_{k,\tau}$ is the image of ${M}_{k}$ by the homothety of ratio τ.
Now we provide a local description of the surface ${M}_{k,\tau}$ near its ends and we introduce the coordinates that we will use.
2.2 The planar end
It can be shown (see [20]) that the function ${u}_{m}$ can be extended at the origin continuously by using Weierstrass representation. In particular we can prove that ${u}_{m}\in {\mathcal{C}}^{2,\alpha}({\overline{B}}_{{\rho}_{0}})$ and ${u}_{m}={\mathcal{O}}_{{C}_{b}^{2,\alpha}}({x}^{k+1})$, where the ${\mathcal{O}}_{{C}_{b}^{n,\alpha}}(g)$ denotes a function that, together with its partial derivatives of order less than or equal to $n+\alpha $ is bounded by a constant times g. Furthermore, by taking into account the symmetries of the surface, it is possible to show the function ${u}_{m}$, in polar coordinates, has to be collinear to $cos(j(k+1)\theta )$, with $j\u2a7e1$ and odd.
2.3 The catenoidal ends
The catenoid C may be divided in two pieces, denoted by ${C}_{\pm}$, which are defined as the image by ${X}_{c}$ of $({\mathbb{R}}^{\pm}\times {S}^{1})$. For any $\tau >0$, we define the catenoid ${C}_{\tau}$ as the image of C by a homothety of ratio τ. Its parametrization is denoted by ${X}_{c,\tau}:=\tau {X}_{c}$. Of course, by this transformation, the two surfaces correspond to ${C}_{\pm}$. They are denoted by ${C}_{\tau ,\pm}$.
for $(s,\theta )\in (\mathrm{\infty},{s}_{0})\times {S}^{1}$, where ${\sigma}_{t,\tau},{\sigma}_{b,\tau}\in \mathbb{R}$, functions ${w}_{t}$, ${w}_{b}$ tend exponentially fast to 0 as s goes to ±∞ reflecting the fact that the ends are asymptotic to a catenoidal end. More precisely it is known that ${w}_{t}={\mathcal{O}}_{{C}_{b}^{2,\alpha}}(\tau {e}^{(k+1)s})$. Furthermore, taking into account the symmetries of the surface, it is easy to show the functions ${w}_{t}$, ${w}_{b}$, in terms of the $(s,\theta )$ coordinates, have to be collinear to $cos(j(k+1)\theta )$, with $j\in \mathbb{N}$ and must satisfy ${w}_{b}(s,\theta )={w}_{t}(s,\theta \frac{\pi}{k+1})$. Furthermore we have ${\sigma}_{t,\tau}={\sigma}_{b,\tau}$. In the sequel we will omit the indices t, b and we will use the notation ${\sigma}_{\tau}$. We assume that ${\sigma}_{\tau}\u2a7d\kappa {\tau}^{2}$, κ being a constant.
The parametrizations of the three ends of ${M}_{k,\tau}$ induce a decomposition of ${M}_{k,\tau}$ into slightly overlapping components: a compact piece ${M}_{k,\tau}({s}_{0}+1,{\rho}_{0}/2)$ and three noncompact pieces ${X}_{t,\tau}(({s}_{0},\mathrm{\infty})\times {S}^{1})$, ${X}_{b,\tau}((\mathrm{\infty},{s}_{0})\times {S}^{1})$ and ${X}_{m,\tau}({\overline{B}}_{{\rho}_{0}}(0))$.
We define a weighted space of functions on ${M}_{k,\tau}$.
and which are invariant with respect to the reflection in the ${x}_{2}=0$ plane, that is, $w(p)=w(\overline{p})$ for all $p\in {M}_{k,\tau}$, where $\overline{p}:=({x}_{1},{x}_{2},{x}_{3})$ if $p=({x}_{1},{x}_{2},{x}_{3})$, invariant with respect to a rotation of angle $\frac{2\pi}{k+1}$ about the ${x}_{3}$ axis and to the composition of a rotation of angle $\frac{\pi}{k+1}$ about the ${x}_{3}$ axis and the reflection in the ${x}_{3}=0$ plane.
We remark that there is no weight on the middle end. In fact we compactify this end and we consider a weighted space of functions defined on a two ended surface.
The proof of Theorem 1.1 consists of two steps. Firstly we will show that for each choice of the genus k there exists, for τ sufficiently small, a family of functions $u\in {\mathcal{C}}_{\delta}^{2,\alpha}({M}_{k,\tau})$ such that their normal graph over ${M}_{k,\tau}$ satisfies the first equation in (3). To do that we need to find the expression of the mean curvature operator for normal graphs of functions defined on ${M}_{k,\tau}$. This is the aim of following section. Secondly we prove that in the family of solutions described above there is a function satisfying also the capillarity condition in (3).
3 The mean curvature of a graph over ${M}_{k,\tau}$
where ${\mathrm{\Delta}}_{\mathrm{\Sigma}}$ denotes the LaplaceBeltrami operator and ${A}_{\mathrm{\Sigma}}$ is the norm of the second fundamental form on the surface.
As for the majority of minimal surfaces, unfortunately the explicit expression of the mean curvature operator of the CostaHoffmanMeeks surfaces is not known. The knowledge of the geometric behavior of such surfaces (we recall that their ends are asymptotic to the two halves of a catenoid and to a plane) allows us to get information about the operator ${\mathbb{L}}_{{M}_{k,\tau}}$ and more generally of the mean curvature operator at the ends of the surfaces.
3.1 Mean curvature operator at the catenoidal ends
for all $s\in \mathbb{R}$ and all ${v}_{1}$, ${v}_{2}$ such that ${\parallel {v}_{i}\parallel}_{{\mathcal{C}}^{2,\alpha}([s,s+1]\times {S}^{1})}\u2a7d1$. The positive constant c does not depend on s.
3.2 Mean curvature operator at the planar end
where ${\overline{L}}_{u}v$ is a second order linear operator with operator with coefficients in ${\mathcal{O}}_{{C}_{b}^{2,\alpha}}({x}^{k+1})$.
We observe that the operator $\frac{1}{{x}^{4}}{\mathbb{L}}_{{\mathbb{R}}^{2}}={\mathrm{\Delta}}_{0}$ clearly maps the space ${\mathcal{C}}^{2,\alpha}({\overline{B}}_{{\rho}_{0}})$ into the space ${\mathcal{C}}^{0,\alpha}({\overline{B}}_{{\rho}_{0}})$.
3.3 Properties of the Jacobi operator of ${M}_{k,\tau}$
The Jacobi operator of ${M}_{k,\tau}$, up to a multiplicative factor, is asymptotic, respectively, to the operators ${x}^{4}{\mathrm{\Delta}}_{0}$ and ${\mathbb{L}}_{C}$ at the planar end and the catenoidal end.
In this subsection we will describe the mapping properties of an elliptic operator related to ${\mathbb{L}}_{{M}_{k,\tau}}$. It will be used to solve the first equation of (3).
is a bounded linear operator.
As in [21] (see also [20] for the same result in a less symmetric setting), using the nondegeneracy of the CostaHoffmanMeeks surfaces shown in [23, 24], it is possible to show the following result.
Proposition 3.1 If $\delta \in (1,2)$, then the operator ${\mathcal{L}}_{\tau ,\delta}$ is surjective and has a kernel of dimension one. Moreover, there exists a right inverse ${G}_{\tau ,\delta}$ for ${\mathcal{L}}_{\tau ,\delta}$ whose norm is bounded.
4 Construction of a family of solutions to ${H}_{{S}_{u}}=0$
In this section we will prove the existence of a family of embedded minimal surfaces and which are close to the piece of surface ${M}_{k,\tau}$ contained in the unit ball ${B}^{3}$.
The value of ${\rho}_{\tau}$ has been chosen so that the image of $x\in {B}_{\overline{\rho}}(0)$, with $x={\rho}_{\tau}$, by the map ${X}_{0,\tau}(x)=(\tau x/{x}^{2},0)\in {\mathbb{R}}^{3}$ (compare (4)) is the circumference ${\mathrm{\Gamma}}_{m}$ of radius 1 in the horizontal plane ${x}_{3}=0$. Moreover, ${s}_{\tau}$ is the value of s for which $\pm ({\sigma}_{\tau}+\tau s)$ is the height of the curves ${\mathrm{\Gamma}}_{t}$, ${\mathrm{\Gamma}}_{b}$ which are the intersection of the unit sphere with the top and bottom halves of the catenoid parametrized by ${C}_{\tau}$ and translated vertically by $\pm {\sigma}_{\tau}$, respectively.
We define ${M}_{k,\tau}^{T}$ to be equal to ${M}_{k,\tau}$ from which we have removed the image of $({s}_{\tau},+\mathrm{\infty})\times {S}^{1}$ by ${X}_{t,\tau}$, the image of $(\mathrm{\infty},{s}_{\tau})\times {S}^{1}$ by ${X}_{b,\tau}$ and the image of ${B}_{{\rho}_{\tau}}(0)$ by ${X}_{m,\tau}$. The boundary curves of ${M}_{k,\tau}^{T}$ do not lie in the unit sphere but they are in a tubular neighborhood of the curves ${\mathrm{\Gamma}}_{t}$, ${\mathrm{\Gamma}}_{b}$, ${\mathrm{\Gamma}}_{m}$. In the sequel we will use also the cylindrical coordinates $(r,\theta ,z)$ (of course $z={x}_{3}$). The circumferences ${\mathrm{\Gamma}}_{t}$, ${\mathrm{\Gamma}}_{b}$ are contained, respectively, in the horizontal planes $z=\pm ({\sigma}_{\tau}+\tau {s}_{\tau})$ and their vertical projection on the $z=0$ plane is the circumference of radius $\tau cosh{s}_{\tau}=1O({\tau}^{2}{ln}^{2}1/\tau )$. The middle boundary curve of ${M}_{k,\tau}^{T}$ is located in a small neighborhood of ${\mathrm{\Gamma}}_{m}$. Points in the middle boundary curve have a height which can be estimated by $O({\tau}^{k+2})$.
By using (4), (7), and (8) we get easily the following lemma. It describes the region of the surface ${M}_{k,\tau}$ which is a graph over the annular domain $A=\{(r,\theta )\mid r1\le \tau \}$ of the ${x}_{3}=0$ plane.
Here $(r,\theta )$ are the polar coordinates in the ${x}_{3}=0$ plane. The functions ${\mathcal{O}}_{{\mathcal{C}}_{b}^{2,\alpha}}(f)$ are defined in the annulus A and are bounded in ${\mathcal{C}}_{b}^{2,\alpha}$ topology by a constant (independent by f) multiplied by f, where the partial derivatives are computed with respect to the vector fields $r{\partial}_{r}$ and ${\partial}_{\theta}$.
We will make a slight modification to the parametrization of the ends ${E}_{t,\tau}$, ${E}_{b,\tau}$ and ${E}_{m,\tau}$, for s and ρ in a small neighborhood of $\pm {s}_{\tau}$ and ${\rho}_{\tau}$, respectively.
The unit normal vector field to ${M}_{k,\tau}$ is denoted by ${n}_{\tau}$. Firstly we modify the vector field ${n}_{\tau}$ into a transverse unit vector field ${\tilde{n}}_{\tau}$. ${\tilde{n}}_{\tau}$ is a smooth interpolation of the following vector fields defined on different pieces of the surface:

at the top (resp. bottom) catenoidal end, the unit normal vector ${n}_{c}({s}_{\tau},\cdot )$ (resp. ${n}_{c}({s}_{\tau},\cdot )$) for s in a small neighborhood of $s={s}_{\tau}$ (resp. $s={s}_{\tau}$); we recall that ${n}_{c}(\pm {s}_{\tau},\cdot )$ are the unit normal vectors to the translated copy of the halves catenoid parametrized by ${X}_{c,\tau}\pm {\sigma}_{\tau}{e}_{3}$ along the curves ${\mathrm{\Gamma}}_{t}$, ${\mathrm{\Gamma}}_{b}$;

at the middle planar end, the vertical vector field ${e}_{3}$ for ρ in a small neighborhood of $\rho ={\rho}_{\tau}$;

the normal vector field ${n}_{\tau}$ on the remaining part of the surface.
This follows easily from (13) together with the fact that ${u}_{m}$ decays at least like ${x}^{k+1}$ on ${E}_{m,\tau}$.
where ${\mathbb{L}}_{{M}_{k,\tau}^{T}}$ is the Jacobi operator of ${M}_{k,\tau}^{T}$, ${Q}_{\tau}$ is a nonlinear second order differential operator and ${\tilde{L}}_{\tau}$ is a linear operator which takes into account the change of the normal vector field ${n}_{\tau}$ into ${\tilde{n}}_{\tau}$.
The operator ${\tilde{L}}_{\tau}$ is supported in a neighborhood of $\{\pm {s}_{\tau}\}\times {S}^{1}$ and of $\{{\rho}_{\tau}\}\times {S}^{1}$. It is possible to show that the coefficients of ${\tilde{L}}_{\tau}$ are uniformly bounded by a constant times ${\tau}^{2}$. First we observe that $\u3008{\tilde{n}}_{\tau},{n}_{\tau}\u3009=1+{\mathcal{O}}_{{C}_{b}^{2,\alpha}}({\tau}^{2})$ in a neighborhood of $\{\pm {s}_{\tau}\}\times {S}^{1}$ and of $\{{\rho}_{\tau}\}\times {S}^{1}$ and the result of [20] Appendix B show that the change of vector field induces a linear operator whose coefficients are bounded by a constant times ${\tau}^{2}$.
As we will see in the sequel, the function $u\in {\mathcal{C}}_{\delta}^{2,\alpha}({M}_{k,\tau})$ which solves $H({\mathrm{\Sigma}}_{u})=0$, depends nonlinearly by a triple of functions defined on the boundary curves of ${M}_{k,\tau}^{T}$. Here is the definition of the functional space we will consider.
 1.
${\chi}_{+}{H}_{{\phi}_{t}}({s}_{\tau}s,\cdot )$ on the image of ${X}_{t,\tau}$, where ${\chi}_{+}$ is a cutoff function equal to 0 for $s\u2a7d{s}_{0}+1$ and identically equal to 1 for $s\in [{s}_{0}+2,{s}_{\tau}]$;
 2.
${\chi}_{}{H}_{{\phi}_{b}}(s+{s}_{\tau},\cdot )$ on the image of ${X}_{b,\tau}$, where ${\chi}_{}$ is a cutoff function equal to 0 for $s\u2a7e{s}_{0}1$ and identically equal to 1 for $s\in [{s}_{\tau},{s}_{0}2]$;
 3.
${\chi}_{m}{\tilde{H}}_{{\rho}_{\tau},{\phi}_{m}}(\cdot ,\cdot )$ on the image of ${X}_{m,\tau}$, where ${\chi}_{m}$ is a cutoff function equal to 0 for $\rho \u2a7e{\rho}_{0}$ and identically equal to 1 for $\rho \in [{\rho}_{\tau},{\rho}_{0}/2]$;
 4.
zero on the remaining part of the surface ${M}_{k,\tau}^{T}$.
The cutoff functions just introduced must enjoy the same symmetry properties as the functions in ${\mathcal{C}}_{\delta}^{2,\alpha}({M}_{k,\tau})$. $\tilde{H}$ and H are harmonic extension operators introduced, respectively, in Propositions A.1 and A.2.
We will prove that, under appropriates hypotheses, the graph ${\mathrm{\Sigma}}_{u}$ over ${M}_{k,\tau}^{T}$ of the function $u={w}_{\mathrm{\Phi}}+v$, is a surface whose mean curvature vanishes.
A similar equation holds for the bottom end. In the following we will assume $\alpha >0$ and close to zero.
The existence of a solution $v\in {\mathcal{C}}_{\delta}^{2,\alpha}({M}_{k,\tau})$ for (27) is a consequence of the following result, which proves that $T(\mathrm{\Phi},\cdot )$ is a contraction mapping.
where c is a positive constant, for all $v,{v}_{1},{v}_{2}\in {\mathcal{C}}_{\delta}^{2,\alpha}({M}_{k,\tau})$ and satisfying ${\parallel v\parallel}_{{\mathcal{C}}_{\delta}^{2,\alpha}}\u2a7d2{c}_{\kappa}{\tau}^{5/2}$ and for all boundary data ${\mathrm{\Phi}}_{i}=({\phi}_{t,i},{\phi}_{m,i},{\phi}_{b,i})\in {[{\mathcal{C}}^{2,\alpha}({S}^{1})]}_{\mathrm{sym}}^{3}$, $i=1,2$, enjoying the same properties as Φ.
together with the fact that ${s}_{\tau}=ln\tau +ln2+O(\tau )$ and ${\rho}_{\tau}=\tau $, from which ${e}^{2{s}_{\tau}}\le c{\tau}^{2}$.
□
Theorem 4.5 Let $\delta \in (1,2)$, $\alpha \in (0,1/4)$ and $B:=\{w\in {\mathcal{C}}_{\delta}^{2,\alpha}({M}_{k,\tau}){\parallel w\parallel}_{{\mathcal{C}}_{\delta}^{2,\alpha}}\u2a7d2{c}_{\kappa}{\tau}^{5/2}\}$. Then the nonlinear mapping $T(\mathrm{\Phi},\cdot )$ defined above has a unique fixed point v in B.
Proof The previous lemma shows that, if τ is chosen small enough, the nonlinear mapping $T(\mathrm{\Phi},\cdot )$ is a contraction mapping from the ball B of radius $2{c}_{\kappa}{\tau}^{5/2}$ in ${\mathcal{C}}_{\delta}^{2,\alpha}({M}_{k,\tau})$ into itself. This value follows from the estimate of the norm of $T(\mathrm{\Phi},0)$. Consequently thanks to Schäuder fixed point theorem, $T(\mathrm{\Phi},\cdot )$ has a unique fixed point w in this ball. □
This argument provides a new surface ${M}_{k,\tau}^{T}(\mathrm{\Phi})$ whose mean curvature equals zero, which is close to ${M}_{k,\tau}^{T}$ and has three boundary curves.
with ${\rho}_{\tau}=\tau $. All the functions ${V}_{i}$, $i=t,b,m$, depend nonlinearly on $\tau ,\mathrm{\Phi}$.
□
Remark 4.7 In next section we will use previous result to prove Theorem 1.1 under the additional assumption $\delta \in (1,5/4)$. Consequently in (29) it appears a positive power of τ. The previous result can be reformulated as follows: all of the mappings ${V}_{i}(\tau ,\cdot )$ are contracting. Furthermore the norm $\parallel {V}_{i}\parallel $ is $O({\tau}^{\frac{5}{2}})$.
5 Proof of Theorem 1.1
The surface ${M}_{k,\tau}^{T}(\mathrm{\Phi})$ we constructed in previous section, has three boundary curves. Such curves do not lie in the sphere $\partial {B}^{3}$. So we introduce a new surface ${\tilde{M}}_{k,\tau}^{T}(\mathrm{\Phi}):={M}_{k,\tau}^{T}(\mathrm{\Phi})\cap {B}^{3}$.
To prove the main theorem we need to show that there exists Φ such that also the second equation of (3) is satisfied.
Near the boundary curves, the surface ${\tilde{M}}_{k,\tau}^{T}(\mathrm{\Phi})$ is the graph in the direction of the vectors ${n}_{i}$, $i=t,m,b$, over the ends of ${M}_{k,\tau}^{T}$ of the functions ${U}_{i}(r,\theta )$ for $i=t,m,b$.
In other terms ${\tilde{r}}_{i}(\theta )$ is the value of the rvariable for which ${\tilde{U}}_{i}({\tilde{r}}_{i}(\theta ),\theta )$ is the parametrization of a curve on the sphere $\partial {B}^{3}$. More precisely it is one of the boundary curves of ${\tilde{M}}_{k,\tau}^{T}(\mathrm{\Phi})$.
where ${r}_{i}$ denotes ${r}_{\tau}$ for $i=t,b$ and ${r}_{i}=1$ when $i=m$. They are the values taken by r for $s={s}_{\tau}$ and $\rho ={\rho}_{\tau}$.
and ${\tilde{\nu}}_{m}=({\tilde{r}}_{m},0,{U}_{m}({\tilde{r}}_{m},\theta )+O({\tau}^{3}))$.
$v$ denotes the length of the vector v. If ${\mathrm{\Gamma}}_{t}$, ${\mathrm{\Gamma}}_{b}$ are the intersection curves of the asymptotic halves catenoid and $\partial {B}^{3}$, then ${N}_{t}$, ${N}_{b}$ and ${\nu}_{t}$, ${\nu}_{b}$ are, respectively, the normal vectors to the asymptotic halves catenoid and to $\partial {B}^{3}$ along ${\mathrm{\Gamma}}_{t}$, ${\mathrm{\Gamma}}_{b}$. We observe that ${\nu}_{t}$, ${\nu}_{b}$ have unit length. Such normal vectors can be computed as done for ${\tilde{N}}_{t}$, ${\tilde{N}}_{b}$, and ${\tilde{\nu}}_{t}$, ${\tilde{\nu}}_{b}$.
By construction of ${U}_{t}$, ${U}_{b}$ and the fact that ${n}_{t,r},{n}_{b,r}=O({\tau}^{2})$, it follows that ${{\tilde{N}}_{t}}^{2}$, ${{\tilde{N}}_{b}}^{2}$ can be estimated as $1+\frac{{\tau}^{2}}{{\tilde{r}}_{t}^{2}}+O({\tau}^{3})$ and $1+\frac{{\tau}^{2}}{{\tilde{r}}_{b}^{2}}+O({\tau}^{3})$, respectively. If we replace ${\tilde{r}}_{t}(\theta )$, ${\tilde{r}}_{b}(\theta )$ by ${r}_{t}={r}_{b}$, and we set ${U}_{t}={U}_{b}=0$, we get the values of ${{N}_{t}}^{2}$, ${{N}_{b}}^{2}$. In conclusion ${{\tilde{N}}_{t}}^{2}$, ${{\tilde{N}}_{b}}^{2}$ are small perturbations of ${{N}_{t}}^{2}$, ${{N}_{b}}^{2}$.
The equation ${A}_{t}=\frac{{\tilde{N}}_{t}\cdot {\tilde{\nu}}_{t}}{{\tilde{N}}_{t}}\frac{{N}_{t}\cdot {\nu}_{t}}{{N}_{t}}=0$ is equivalent to $({\tilde{N}}_{t}\cdot {\tilde{\nu}}_{t}){N}_{t}=({N}_{t}\cdot {\nu}_{t}){\tilde{N}}_{t}$. In view of previous observations this last equation can be seen as a small perturbation of the simpler equation ${N}_{t}({\tilde{N}}_{t}\cdot {\tilde{\nu}}_{t}{N}_{t}\cdot {\nu}_{t})=0$.
see (33).
The definition of the space ${[{\mathcal{C}}^{1,\alpha}({S}^{1})]}_{\mathrm{sym}}^{3}$ is similar to definition 4.2, with the unique difference of the lower regularity.
We want to show the existence of a solution of ${A}_{\tau}(\mathrm{\Phi})=0$.
Proposition 5.1 There exists ${\kappa}_{0}>0$ such that if $\kappa >{\kappa}_{0}$ then there exists ${\tau}_{0}>0$ for which, for each $\tau \in (0,{\tau}_{0})$, then ${A}_{\tau}(\mathrm{\Phi})=0$ has a solution in ${\mathcal{B}}_{\kappa}$, the ball centered at $(0,0,0)$ and of radius $\kappa {\tau}^{2}$ in ${[{\mathcal{C}}^{2,\alpha}({S}^{1})]}_{\mathrm{sym}}^{3}$.
The operator ${I}_{\tau}$ can be seen as an approximation of ${T}_{\tau}$: indeed we get ${I}_{\tau}$ from ${T}_{\tau}$ omitting some nonlinear terms and evaluating the remaining ones at ${r}_{i}$ instead of ${\tilde{r}}_{i}(\theta )$.
From (32) and Lemma 4.6 we obtain ${\parallel ({A}_{\tau}{I}_{\tau})(\mathrm{\Psi})\parallel}_{{[{\mathcal{C}}^{1,\alpha}({S}^{1})]}_{\mathrm{sym}}^{3}}\u2a7dc{\tau}^{5/2}$, for any $\mathrm{\Psi}\in {\mathcal{B}}_{\kappa}\subset {[{\mathcal{C}}^{1,\alpha}({S}^{1})]}_{\mathrm{sym}}^{3}$. We would like to show existence of a solution to ${A}_{\tau}(\mathrm{\Phi})=0$ by the LeraySchauder degree theory but the nonlinear operator ${I}_{\tau}{A}_{\tau}$ is not compact. We apply the same technique as in Proposition 15 of [25].
where c does not depend on q.
Now we can apply LeraySchauder degree theory to prove the existence of a solution ${\mathrm{\Phi}}_{q}$ to ${A}_{\tau}^{q}(\mathrm{\Phi})=0$ in ${\mathcal{B}}_{\kappa}$ for $\tau \in (0,{\tau}_{0})$, with ${\tau}_{0}$ small enough and $\kappa >{\kappa}_{0}$ with ${\kappa}_{0}$ chosen large enough.
Since the norm of ${\mathrm{\Phi}}_{q}$ is bounded uniformly in q, we can extract a sequence $\{{q}_{j}\}$ converging to 0 such that $\{{\mathrm{\Phi}}_{{q}_{j}}\}$ converges in ${[{\mathcal{C}}^{2,{\alpha}^{\prime}}({S}^{1})]}_{\mathrm{sym}}^{3}$ for any fixed ${\alpha}^{\prime}<\alpha $. Thanks to the continuity of ${A}_{\tau}^{q}$ and to (34), the limit of this sequence converges to a solution of ${A}_{\tau}(\mathrm{\Phi})=0$ for all $\tau \in (0,{\tau}_{0})$. □
The zero of ${A}_{\tau}$ provides the boundary data Φ for which the surface ${\tilde{M}}_{k,\tau}^{T}(\mathrm{\Phi})$ meets $\partial {B}^{3}$ in order to make (33) satisfied. That finishes the proof of Theorem 1.1.
Appendix
Results in this section are about the existence of some harmonic extension operators.
The following result gives a harmonic extension of a function on ${\mathbb{R}}^{2}\setminus {D}_{\overline{\rho}}$.
for some constant $c>0$.
Since $\frac{\overline{\rho}}{\rho}\u2a7d1$, then ${(\frac{\overline{\rho}}{\rho})}^{i}\u2a7d(\frac{\overline{\rho}}{\rho})$, we can conclude that $w(\theta ,\rho )\u2a7dc\phi (\theta )$ and then ${\parallel w\parallel}_{{C}^{2,\alpha}}\u2a7dc{\parallel \phi \parallel}_{{C}^{2,\alpha}}$. □
for some constant $c>0$.
The proof is immediate once we observe that, if $\phi ={\sum}_{j\u2a7e2}{\phi}_{j}cos(j\theta )$, then the solution is ${H}_{\phi}={\sum}_{j\u2a7e2}{e}^{js}{\phi}_{j}cos(j\theta )$.
Declarations
Acknowledgements
This research was supported by Basic Science Research Program through the National Research Foundation of South Korea (NRF) funded by the Ministry of Education, Grant NRF2013R1A1A1013521.
Authors’ Affiliations
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