We construct the first examples of capillary surfaces of positive genus, embedded in the unit ball of 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 Costa-Hoffman-Meeks minimal surfaces.
MSC: 53A10, 35R35, 53C21.
Keywords:minimal surface; perturbation method; nonlinear pde’s
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 satisfies the differential equation
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 , understood that the capillary surface meets the tube (or more generally the container) making an angle, called contact angle, which depends on the liquid and on the material which composes the container and not on the gravity. For liquids in tubes (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 , for a survey on more recent discoveries about capillarity.
Existence and uniqueness for the solution of capillarity problem for graphs over domains of (also in the more general form where , for an assigned function f), has been extensively studied in the past, see e.g. Gerhardt , Lieberman , Simon and Spruck , Spruck , Uraltseva .
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  showed the existence of Scherk-type capillary minimal graphs.
Very recently, Calle and Shahriyari in  have solved the prescribed mean curvature equation with a boundary contact angle condition. They show the existence of graphs over domains in , where is a n-dimensional Riemannian submanifold of . In  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  constructed by a perturbation method disk-type minimal surfaces embedded in an infinite cylinder in and which intersect its boundary orthogonally. In  they extended this result to Riemannian manifolds.
In  Fall showed that, given a bounded domain of 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 , which, after scaling and translation, converges to a hemisphere of radius 1 as .
In  Fall and Mahmoudi showed that if Ω is a domain of 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.
In this work we show the existence of higher genus minimal capillary surfaces by a perturbation method. Let be the unit ball centered at the origin of . For each and , with small enough, there exists a surface of genus k, embedded in with non-empty boundary which consists in three simple closed curves , , which lie in and such that
where denotes the mean curvature at the point p; and denote, respectively, the unit normal vector to the surface and to at . The functions are decreasing smooth and non-zero for . 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 . As we will explain in Section 2.1, is asymptotic to a top half catenoid, to a bottom half catenoid and to a horizontal plane. The functions are defined to be the values of the scalar product we obtain if we replace by the two halves catenoid and the plane. In particular .
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 are denoted by .
Theorem 1.1For each , there exists positive and small enough, such that for each there exists a surface embedded in , of genusk, 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 . In other terms the surface cannot make a constant angle equal to with along , . We point out that . As τ is the homothety ratio, this says that, as τ tends to 0 the limit of 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  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 is a totally geodesic disk or a surfaces of genus 1 with boundary having at most 3 connected components. Consequently, at least for , 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 , 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 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 of genus 0 with boundary having k connected components, for any finite . 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 . 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 over Σ has vanishing mean curvature and the scalar product of the unit normal vectors, , equals at each point of the ith component of , with .
2.1 The scaled Costa-Hoffman-Meeks surface
The Costa-Hoffman-Meeks surface of genus embedded in (see ) is denoted by .
After suitable rotation and translation, enjoys the following properties.
1. It has one planar end asymptotic to the horizontal plane , one top end and one bottom end that are, respectively, asymptotic to the upper end and to the lower end of a catenoid having the -axis as axis of rotation. The planar end is located between the two catenoidal ends.
2. It is 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.
3. It intersects the plane in straight lines, which intersect themselves at the origin with angles equal to . The intersection of with the plane (≠0) is a single Jordan curve. The intersection of with the upper half space (resp. with the lower half space ) is topologically an open annulus.
The parameterization of the end is denoted by , with , and the parameterization of the corresponding end of is denoted by . We recall that is the image of by the homothety of ratio τ.
Now we provide a local description of the surface near its ends and we introduce the coordinates that we will use.
2.2 The planar end
The planar end of the surface can be parametrized by
where . Here is fixed small enough. In the sequel, where necessary, we will consider on also the polar coordinates . The function satisfies the minimal surface equation, which has the following form:
It can be shown (see ) that the function can be extended at the origin continuously by using Weierstrass representation. In particular we can prove that and , where the denotes a function that, together with its partial derivatives of order less than or equal to is bounded by a constant times g. Furthermore, by taking into account the symmetries of the surface, it is possible to show the function , in polar coordinates, has to be collinear to , with and odd.
2.3 The catenoidal ends
The parametrization of the standard catenoid C, whose axis of revolution is the -axis, is denoted by . We have
where . The unit normal vector field to C is given by
The catenoid C may be divided in two pieces, denoted by , which are defined as the image by of . For any , we define the catenoid as the image of C by a homothety of ratio τ. Its parametrization is denoted by . Of course, by this transformation, the two surfaces correspond to . They are denoted by .
Up to some dilation, we can assume that the two ends and of are asymptotic to some translated copy of the two halves of the catenoid parametrized by in the vertical direction. Therefore, and can be parametrized, respectively, by
for , where , functions , 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 . Furthermore, taking into account the symmetries of the surface, it is easy to show the functions , , in terms of the coordinates, have to be collinear to , with and must satisfy . Furthermore we have . In the sequel we will omit the indices t, b and we will use the notation . We assume that , κ being a constant.
For all and , we define
The parametrizations of the three ends of induce a decomposition of into slightly overlapping components: a compact piece and three noncompact pieces , and .
We define a weighted space of functions on .
Definition 2.1 Given , and , the space is defined to be the space of functions in for which the following norm is finite:
and which are invariant with respect to the reflection in the plane, that is, for all , where if , invariant with respect to a rotation of angle about the axis and to the composition of a rotation of angle about the axis and the reflection in the 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 such that their normal graph over 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 . 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
It is well known that the mean curvature of the normal graph of a function u over a minimal surface Σ can be decomposed as , where denotes a linear second order elliptic operator and Q is a nonlinear differential operator of higher order. The operator is known under the name of Jacobi operator and it is defined as the linearized of the mean curvature operator. For a minimal surface Σ in its expression is
where denotes the Laplace-Beltrami operator and 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 and more generally of the mean curvature operator at the ends of the surfaces.
3.1 Mean curvature operator at the catenoidal ends
The surface parametrized by is minimal if and only if the function w satisfies the minimal surface equation
being the Jacobi operator of the catenoid, i.e.
Here , are nonlinear second order differential operators which are bounded in , for every l, and satisfy , , together with
for all and all , such that . The positive constant c does not depend on s.
Finally we observe that the operator maps the functional space
3.2 Mean curvature operator at the planar end
If we linearize the nonlinear equation (5) we obtain
If we consider we get an operator which equals, up to a multiplication by τ, the Jacobi operator of the plane, that is, . The graph surface of the function u is denoted by and its mean curvature by . Then , the mean curvature of the graph of the function , in terms of , is
Since we assume that is a minimal surface, we have . So we get the following equation:
where is a second order linear operator with operator with coefficients in .
We recall that if the function v satisfies the equation with then the graph of the function is minimal. Now we are interested in finding the equation which a function w must satisfy in such a way the surface parametrized by , that is the graph of w over the middle end , is minimal. That is equivalent to require that the graph of is minimal. Then we can obtain the wanted equation by replacing v by in (15). So we get
If we set to simplify the notation, we can write this equation in the following way:
We observe that the operator clearly maps the space into the space .
3.3 Properties of the Jacobi operator of
The Jacobi operator of , up to a multiplicative factor, is asymptotic, respectively, to the operators and at the planar end and the catenoidal end.
In this subsection we will describe the mapping properties of an elliptic operator related to . It will be used to solve the first equation of (3).
The volume form on is denoted by . In the parameterization of the ends introduced above, such form can be written as and near the catenoidal type ends and as near the middle end. Now we can define globally on a smooth function
that is identically equal to on and equal to (resp. , ) on the end (resp. , ). They are defined in such a way that for , we have, respectively,
Finally on we have
It is possible to check that
is a bounded linear operator.
Proposition 3.1If , 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
In this section we will prove the existence of a family of embedded minimal surfaces and which are close to the piece of surface contained in the unit ball .
and we define to be the value of s such that
We define so that
The value of has been chosen so that the image of , with , by the map (compare (4)) is the circumference of radius 1 in the horizontal plane . Moreover, is the value of s for which is the height of the curves , which are the intersection of the unit sphere with the top and bottom halves of the catenoid parametrized by and translated vertically by , respectively.
We define to be equal to from which we have removed the image of by , the image of by and the image of by . The boundary curves of do not lie in the unit sphere but they are in a tubular neighborhood of the curves , , . In the sequel we will use also the cylindrical coordinates (of course ). The circumferences , are contained, respectively, in the horizontal planes and their vertical projection on the plane is the circumference of radius . The middle boundary curve of is located in a small neighborhood of . Points in the middle boundary curve have a height which can be estimated by .
By using (4), (7), and (8) we get easily the following lemma. It describes the region of the surface which is a graph over the annular domain of the plane.
Lemma 4.1There exists such that, for all an annular part of the ends , and of can be written as vertical graphs over the annulusAof the functions
Here are the polar coordinates in the plane. The functions are defined in the annulusAand are bounded in topology by a constant (independent byf) multiplied byf, 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 , and , for s and ρ in a small neighborhood of and , respectively.
The unit normal vector field to is denoted by . Firstly we modify the vector field into a transverse unit vector field . 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 (resp. ) for s in a small neighborhood of (resp. ); we recall that are the unit normal vectors to the translated copy of the halves catenoid parametrized by along the curves , ;
• at the middle planar end, the vertical vector field for ρ in a small neighborhood of ;
• the normal vector field on the remaining part of the surface.
We observe that at the top end , we can give the following estimate:
This follows easily from (10) together with the fact that decays at least like on . Similar considerations hold at the bottom end . Near the middle planar end , we observe that the following estimate holds:
This follows easily from (13) together with the fact that decays at least like on .
The mean curvature of the graph of a function u in the direction of the vector field is the image of u by a second order nonlinear elliptic operator:
where is the Jacobi operator of , is a nonlinear second order differential operator and is a linear operator which takes into account the change of the normal vector field into .
The operator is supported in a neighborhood of and of . It is possible to show that the coefficients of are uniformly bounded by a constant times . First we observe that in a neighborhood of and of and the result of  Appendix B show that the change of vector field induces a linear operator whose coefficients are bounded by a constant times .
As we will see in the sequel, the function which solves , depends nonlinearly by a triple of functions defined on the boundary curves of . Here is the definition of the functional space we will consider.
Definition 4.2 Given , , the space is defined to be the space of triples of functions such that and even, is collinear to , with ; is collinear to , with and odd, , and whose norm, defined below, is finite.
Now we consider the triple of functions ,
We define to be the function equal to
1. on the image of , where is a cut-off function equal to 0 for and identically equal to 1 for ;
2. on the image of , where is a cut-off function equal to 0 for and identically equal to 1 for ;
3. on the image of , where is a cut-off function equal to 0 for and identically equal to 1 for ;
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 . and H are harmonic extension operators introduced, respectively, in Propositions A.1 and A.2.
We will prove that, under appropriates hypotheses, the graph over of the function , is a surface whose mean curvature vanishes.
The equation to solve is
Since we are looking for solutions having the form , we can write it as
The resolution of the previous equation is obtained by the one of the following fixed point problem:
where , the operator is defined in Proposition 3.1 and is a linear extension operator such that
where denotes the space of functions of restricted to . It is defined by in , in the image of by , in the image of by and in the image of by . Finally is an interpolation of these values in the remaining part of such that
Remark 4.3 From the definition of , if , then
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 and close to zero.
The existence of a solution for (27) is a consequence of the following result, which proves that is a contraction mapping.
Proposition 4.4Let , and satisfying (26) and enjoying the properties given above. There exist constants and , such that
and, for all ,
wherecis 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 , is asymptotic (up to a multiplication by ) to the Jacobi operator of the catenoid (respectively, of the plane) plane at the catenoidal ends (respectively, at the planar end). The function is identically zero far from the ends where the explicit expression of is not known: this is the reason for our particular choice in the definition of . Then from the definition of and thanks to Proposition 3.1 we obtain the estimate
To obtain this estimate we used the following ones:
(a similar estimate holds for the bottom end) and
together with the fact that and , from which .
Using the estimates of the coefficients of and the definition of γ (see (18)), we obtain
As for the last term, we recall that the expression of the operator depends on the type of end we are considering (see (17) and (11)). We have
As for the second estimate, we recall that
We observe that from the considerations above it follows that
To get the last estimate it suffices to observe that
Theorem 4.5Let , and . Then the nonlinear mapping defined above has a unique fixed pointvinB.
Proof The previous lemma shows that, if τ is chosen small enough, the nonlinear mapping is a contraction mapping from the ball B of radius in into itself. This value follows from the estimate of the norm of . Consequently thanks to Schäuder fixed point theorem, has a unique fixed point w in this ball. □
This argument provides a new surface whose mean curvature equals zero, which is close to and has three boundary curves.
The surface is, close to its upper and lower boundary curve, the graph over the catenoidal ends in the direction given by the vector of the functions
where . Nearby the middle boundary the surface is the vertical graph of
with . All the functions , , depend nonlinearly on .
Lemma 4.6The function , for , satisfies and
The function satisfies and
Proof We recall that the functions , , are the restrictions to , , of a fixed point v for the operator . The estimates of their norm are a consequence of Proposition 4.4. Observe that to derive the estimate of the norm of and we use the better estimate for the norm of the fixed point v which holds at the catenoidal type ends. Precisely stated: with . Then (29) follows from
for , with and . To get the estimate (30) we observe that
Remark 4.7 In next section we will use previous result to prove Theorem 1.1 under the additional assumption . Consequently in (29) it appears a positive power of τ. The previous result can be reformulated as follows: all of the mappings are contracting. Furthermore the norm is .
5 Proof of Theorem 1.1
The surface we constructed in previous section, has three boundary curves. Such curves do not lie in the sphere . So we introduce a new 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 in in order to have the normal vector to equal to the normal vectors in a neighborhood of its top and bottom boundary curves and equal to in a neighborhood of the middle boundary curve. Precisely, at the catenoidal type ends, from (6), in a neighborhood of the boundary curves equals the vector fields (here we use the basis )
Near the boundary curves, the surface is the graph in the direction of the vectors , , over the ends of of the functions for .
As a consequence the top and bottom ends of , near the boundary curves, can be parametrized as follows:
Let be the function of θ defined as
In other terms is the value of the r-variable for which is the parametrization of a curve on the sphere . More precisely it is one of the boundary curves of .
Using the expression of we get the following estimate: