### Abstract

We construct the first examples of capillary surfaces of positive genus, embedded
in the unit ball of

**MSC: **
53A10, 35R35, 53C21.

##### Keywords:

minimal surface; perturbation method; nonlinear pde’s### 1 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.

PS de Laplace proved that the height *u* of a capillary surface over a domain

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.

T Young, who considered the case
*i.e.* cylindrical containers) we see that the following additional boundary condition (Young
condition) is satisfied:

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
*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
*n*-dimensional Riemannian submanifold of

Fall and Mercuri in [13] constructed by a perturbation method disk-type minimal surfaces embedded in an infinite
cylinder in

In [15] Fall showed that, given a bounded domain of
*∂*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

In [16] Fall and Mahmoudi showed that if Ω is a domain of
*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.

In this work we show the existence of higher genus minimal capillary surfaces by a
perturbation method. Let
*k*, embedded in

where
*p*;

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

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

**Theorem 1.1***For each*
*there exists*
*positive and small enough*, *such that for each*
*there exists a surface*
*embedded in*
*of genus**k*, *whose boundary*
*is composed by three simple Jordan curves*
*and satisfying*

*Such surfaces are invariant under the action of the rotation of angle*
*about the*
*axis*, *under the action of the reflection in the*
*plane and under the action of the composition of a rotation of angle*
*about the*
*axis and the reflection in the*
*plane*.

We observe that for values of *τ* in the range of validity of our theorem
*τ* is the homothety ratio, this says that, as *τ* tends to 0 the limit of

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

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
*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
*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
*k* connected components, for any finite

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.

### 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

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
*i*th component of

We will adapt to our setting some arguments used in [20,21].

#### 2.1 The scaled Costa-Hoffman-Meeks surface

The Costa-Hoffman-Meeks surface of genus

After suitable rotation and translation,

1. It has one planar end

2. It is invariant under the action of the rotation of angle

3. It intersects the

The parameterization of the end
*τ*.

Now we provide a local description of the surface

#### 2.2 The planar end

The planar end

where

It can be shown (see [20]) that the function
*g*. Furthermore, by taking into account the symmetries of the surface, it is possible
to show the function

#### 2.3 The catenoidal ends

The parametrization of the standard catenoid *C*, whose axis of revolution is the

where
*C* is given by

The catenoid *C* may be divided in two pieces, denoted by
*C* by a homothety of ratio *τ*. Its parametrization is denoted by

Up to some dilation, we can assume that the two ends

for

for
*s* goes to ±∞ reflecting the fact that the ends are asymptotic to a catenoidal end.
More precisely it is known that
*t*, *b* and we will use the notation
*κ* being a constant.

For all

The parametrizations of the three ends of

We define a weighted space of functions on

**Definition 2.1** Given

where

and which are invariant with respect to the reflection in the

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

### 3 The mean curvature of a graph over
M
k
,
τ

It is well known that the mean curvature
*u* over a minimal surface Σ can be decomposed as
*Q* is a nonlinear differential operator of higher order. The operator

where

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

#### 3.1 Mean curvature operator at the catenoidal ends

The surface parametrized by
*w* satisfies the minimal surface equation

*i.e.*

and

Here
*l*, and satisfy

for all
*c* does not depend on *s*.

Finally we observe that the operator

#### 3.2 Mean curvature operator at the planar end

If we linearize the nonlinear equation (5) we obtain

If we consider
*τ*, the Jacobi operator of the plane, that is,
*u* is denoted by

where

Since we assume that

where

We recall that if the function *v* satisfies the equation
*w* must satisfy in such a way the surface parametrized by
*w* over the middle end
*v* by

If we set

We observe that the operator

#### 3.3 Properties of the Jacobi operator of
M
k
,
τ

The Jacobi operator of

In this subsection we will describe the mapping properties of an elliptic operator
related to

The volume form on

that is identically equal to

Finally on

It is possible to check that

is a bounded linear operator.

As in [21] (see also [20] for the same result in a less symmetric setting), using the non-degeneracy of the Costa-Hoffman-Meeks surfaces shown in [23,24], it is possible to show the following result.

**Proposition 3.1***If*
*then the operator*
*is surjective and has a kernel of dimension one*. *Moreover*, *there exists a right inverse*
*for*
*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

We set

and we define
*s* such that

We get

We define

The value of
*s* for which

We define

By using (4), (7), and (8) we get easily the following lemma. It describes the region
of the surface

**Lemma 4.1***There exists*
*such that*, *for all*
*an annular part of the ends*
*and*
*of*
*can be written as vertical graphs over the annulus**A**of the functions*

*Here*
*are the polar coordinates in the*
*plane*. *The functions*
*are defined in the annulus**A**and are bounded in*
*topology by a constant* (*independent by**f*) *multiplied by**f*, *where the partial derivatives are computed with respect to the vector fields*
*and*

We will make a slight modification to the parametrization of the ends
*s* and *ρ* in a small neighborhood of

The unit normal vector field to

• at the top (resp. bottom) catenoidal end, the unit normal vector
*s* in a small neighborhood of

• at the middle planar end, the vertical vector field
*ρ* in a small neighborhood of

• the normal vector field

We observe that at the top end

This follows easily from (10) together with the fact that

This follows easily from (13) together with the fact that

The mean curvature of the graph
*u* in the direction of the vector field
*u* by a second order nonlinear elliptic operator:

where

The operator

As we will see in the sequel, the function

**Definition 4.2** Given

Now we consider the triple of functions

We define

1.

2.

3.

4. zero on the remaining part of the surface

The cut-off functions just introduced must enjoy the same symmetry properties as
the functions in
*H* are harmonic extension operators introduced, respectively, in Propositions A.1 and
A.2.

We will prove that, under appropriates hypotheses, the graph

The equation to solve is

Since we are looking for solutions having the form

The resolution of the previous equation is obtained by the one of the following fixed point problem:

with

where

where

**Remark 4.3** From the definition of

This phenomenon of explosion of the norm does not occur near the catenoidal type ends:

A similar equation holds for the bottom end. In the following we will assume

The existence of a solution

**Proposition 4.4***Let*
*and*
*satisfying* (26) *and enjoying the properties given above*. *There exist constants*
*and*
*such that*

*and*, *for all*

*where**c**is a positive constant*, *for all*
*and satisfying*
*and for all boundary data*
*enjoying the same properties as* Φ.

*Proof* We recall that the Jacobi operator associated to

To obtain this estimate we used the following ones:

(a similar estimate holds for the bottom end) and

together with the fact that

Using the estimates of the coefficients of
*γ* (see (18)), we obtain

As for the last term, we recall that the expression of the operator

In fact

As for the second estimate, we recall that

Then

We observe that from the considerations above it follows that

and

Then

To get the last estimate it suffices to observe that

□

**Theorem 4.5***Let*
*and*
*Then the nonlinear mapping*
*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
*B* of radius
*w* in this ball. □

This argument provides a new surface

The surface

where

with

**Lemma 4.6***The function*
*for*
*satisfies*
*and*

*The function*
*satisfies*
*and*

*Proof* We recall that the functions
*v* for the operator
*v* which holds at the catenoidal type ends. Precisely stated:

for

□

**Remark 4.7** In next section we will use previous result to prove Theorem 1.1 under the additional
assumption
*τ*. The previous result can be reformulated as follows: all of the mappings

### 5 Proof of Theorem 1.1

The surface

To prove the main theorem we need to show that there exists Φ such that also the second equation of (3) is satisfied.

We recall that we modified the immersion of

Near the boundary curves, the surface

As a consequence the top and bottom ends of

where

where

Let
*θ* defined as

In other terms
*r*-variable for which

Using the expression of