Higher genus capillary surfaces in the unit ball of ${\mathbb{R}}^3$

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 Scherk-type 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 n-dimensional 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 disk-type 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 k-dimensional non-degenerate 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 non-zero 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 Costa-Hoffman-Meeks 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 non-trivial 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 non-negative 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 Dirichlet-to-Neumann 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 1-parameter 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.

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].

where ${\mathrm{\Delta }}_{\mathrm{\Sigma }}$ denotes the Laplace-Beltrami 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 Costa-Hoffman-Meeks 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.

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 $\tau cosh{s}_{\tau }=1-O({\tau }^2{ln}^{2}1/\tau )$. 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 $A=\{(r,\theta )\mid |r-1|\le \tau \}$ 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:

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 $〈{\tilde{n}}_{\tau },n_{\tau }〉=1+{\mathcal{O}}_{C_b^{2,\alpha }}({\tau }^2)$ 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.

The cut-off 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.