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 .
We set
and we define to be the value of s such that
(19)
We get
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 4.1 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
(20)
(21)
(22)
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 , 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:
(23)
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:
(24)
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 [20] 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.
(25)
Now we consider the triple of functions ,
(26)
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:
with
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 4.4 Let , and satisfying (26) and enjoying the properties given above. There exist constants and , such that
(28)
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 , 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
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 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 4.6 The function , for , satisfies and
(29)
The function
satisfies
and
(30)
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 .