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}.
We set
and we define {s}_{\tau} to be the value of s such that
{(\tau coshs)}^{2}+{({\sigma}_{\tau}+\tau s)}^{2}=1.
(19)
We get
{s}_{\tau}=ln\tau +ln2+O(\tau ).
We define {r}_{\tau} so that
{s}_{\tau}=ln\left(\frac{2{r}_{\tau}}{\tau}\right).
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.
Lemma 4.1 There exists {\tau}_{0}>0 such that, for all \tau \in (0,{\tau}_{0}) an annular part of the ends {E}_{t,\tau}, {E}_{b,\tau} and {E}_{m,\tau} of {M}_{k,\tau} can be written as vertical graphs over the annulus A of the functions
{Z}_{t}(r,\theta )={\sigma}_{\tau}+\tau ln\left(\frac{2r}{\tau}\right)+{\mathcal{O}}_{{\mathcal{C}}_{b}^{2,\alpha}}\left({\tau}^{3}\right),
(20)
{Z}_{b}(r,\theta )={Z}_{t}(r,\theta \frac{\pi}{k+1}),
(21)
{Z}_{m}(r,\theta )={\mathcal{O}}_{{\mathcal{C}}_{b}^{2,\alpha}}\left(\tau {\left(\frac{r}{\tau}\right)}^{(k+1)}\right).
(22)
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.
We observe that at the top end {E}_{t,\tau}, we can give the following estimate:
\left{\tau}^{2}{cosh}^{2}s({\mathbb{L}}_{{M}_{k,\tau}}v{\left({\tau}^{2}{cosh}^{2}s\right)}^{1}({\partial}_{ss}v+{\partial}_{\theta \theta}v))\right\u2a7dc\left{\left({cosh}^{2}s\right)}^{1}v\right.
(23)
This follows easily from (10) together with the fact that {w}_{t} decays at least like {({cosh}^{2}s)}^{1} on {E}_{t,\tau}. Similar considerations hold at the bottom end {E}_{b,\tau}. Near the middle planar end {E}_{m,\tau}, we observe that the following estimate holds:
\left{\tau}^{2}{x}^{4}({\mathbb{L}}_{{M}_{k,\tau}}v{x}^{4}{\tau}^{2}{\mathrm{\Delta}}_{0}v)\right\u2a7dc\left{x}^{2k+3}\mathrm{\nabla}v\right.
(24)
This follows easily from (13) together with the fact that {u}_{m} decays at least like {x}^{k+1} on {E}_{m,\tau}.
The mean curvature of the graph {\mathrm{\Sigma}}_{u} of a function u in the direction of the vector field {\tilde{n}}_{\tau} is the image of u by a second order nonlinear elliptic operator:
2H({\mathrm{\Sigma}}_{u})={\mathbb{L}}_{{M}_{k,\tau}^{T}}u+{\tilde{L}}_{\tau}u+{Q}_{\tau}(u),
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.
Definition 4.2 Given k\u2a7e1, \alpha \in (0,1), the space {[{\mathcal{C}}^{n,\alpha}({S}^{1})]}_{\mathrm{sym}}^{3} is defined to be the space of triples of functions \mathrm{\Phi}=({\phi}_{t},{\phi}_{m},{\phi}_{b}) such that {\phi}_{j}\in {\mathcal{C}}^{n,\alpha}({S}^{1}) and even, {\phi}_{t} is collinear to cos(j(k+1)\theta ), with j\u2a7e1; {\phi}_{m} is collinear to cos(l(k+1)\theta ), with l\u2a7e1 and odd, {\phi}_{b}={\phi}_{t}(\theta \frac{\pi}{k+1}), and whose norm, defined below, is finite.
{\parallel \mathrm{\Phi}\parallel}_{{[{\mathcal{C}}^{n,\alpha}({S}^{1})]}_{\mathrm{sym}}^{3}}:={\parallel {\phi}_{t}\parallel}_{{\mathcal{C}}^{n,\alpha}({S}^{1})}+{\parallel {\phi}_{m}\parallel}_{{\mathcal{C}}^{n,\alpha}({S}^{1})}+{\parallel {\phi}_{b}\parallel}_{{\mathcal{C}}^{n,\alpha}({S}^{1})}.
(25)
Now we consider the triple of functions \mathrm{\Phi}=({\phi}_{t},{\phi}_{m},{\phi}_{b})\in {[{\mathcal{C}}^{2,\alpha}({S}^{1})]}_{\mathrm{sym}}^{3},
{\parallel \mathrm{\Phi}\parallel}_{{[{\mathcal{C}}^{2,\alpha}({S}^{1})]}_{\mathrm{sym}}^{3}}\u2a7d\kappa {\tau}^{2}.
(26)
We define {w}_{\mathrm{\Phi}} to be the function equal to

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.
The equation to solve is
H({\mathrm{\Sigma}}_{u})=0.
Since we are looking for solutions having the form u={w}_{\mathrm{\Phi}}+v, we can write it as
{\mathbb{L}}_{{M}_{k,\tau}^{T}}({w}_{\mathrm{\Phi}}+v)+{\tilde{L}}_{\tau}({w}_{\mathrm{\Phi}}+v)+{Q}_{\tau}({w}_{\mathrm{\Phi}}+v)=0.
The resolution of the previous equation is obtained by the one of the following fixed point problem:
v=T(\mathrm{\Phi},v)
(27)
with
T(\mathrm{\Phi},v)={G}_{\tau ,\delta}\circ {\mathcal{E}}_{\tau}\left(\gamma ({\tilde{L}}_{\tau}({w}_{\mathrm{\Phi}}+v){\mathbb{L}}_{{M}_{k,\tau}^{T}}{w}_{\mathrm{\Phi}}{Q}_{\tau}({w}_{\mathrm{\Phi}}+v))\right),
where \delta \in (1,2), the operator {G}_{\tau ,\delta} is defined in Proposition 3.1 and {\mathcal{E}}_{\tau} is a linear extension operator such that
{\mathcal{E}}_{\tau}:{\mathcal{C}}_{\delta}^{0,\alpha}\left({M}_{k,\tau}^{T}\right)\u27f6{\mathcal{C}}_{\delta}^{0,\alpha}({M}_{k,\tau}),
where {\mathcal{C}}_{\delta}^{0,\alpha}({M}_{k,\tau}^{T}) denotes the space of functions of {\mathcal{C}}_{\delta}^{0,\alpha}({M}_{k,\tau}) restricted to {M}_{k,\tau}^{T}. It is defined by {\mathcal{E}}_{\tau}v=v in {M}_{k,\tau}^{T}, {\mathcal{E}}_{\tau}v=0 in the image of [{s}_{\tau}+1,+\mathrm{\infty})\times {S}^{1} by {X}_{t,\tau}, in the image of (\mathrm{\infty},{s}_{\tau}1]\times {S}^{1} by {X}_{b,\tau} and in the image of {B}_{{\rho}_{\tau}/2} by {X}_{m,\tau}. Finally {\mathcal{E}}_{\tau}v is an interpolation of these values in the remaining part of {M}_{k,\tau} such that
\begin{array}{c}({\mathcal{E}}_{\tau}v)\circ {X}_{t,\tau}(s,\theta )=(1+{s}_{\tau}s)(v\circ {X}_{t,\tau}({s}_{\tau},\theta )),\phantom{\rule{1em}{0ex}}\text{for}(s,\theta )\in [{s}_{\tau},{s}_{\tau}+1]\times {S}^{1},\hfill \\ ({\mathcal{E}}_{\tau}v)\circ {X}_{b,\tau}(s,\theta )=(1+{s}_{\tau}+s)(v\circ {X}_{b,\tau}({s}_{\tau},\theta )),\phantom{\rule{1em}{0ex}}\text{for}(s,\theta )\in [{s}_{\tau}1,{s}_{\tau}]\times {S}^{1},\hfill \\ ({\mathcal{E}}_{\tau}v)\circ {X}_{m,\tau}(\rho ,\theta )=(\frac{2}{{\rho}_{\tau}}\rho 1)(v\circ {X}_{m,\tau}({\rho}_{\tau},\theta ))\phantom{\rule{1em}{0ex}}\text{for}(\rho ,\theta )\in [{\rho}_{\tau}/2,{\rho}_{\tau}]\times {S}^{1}.\hfill \end{array}
Remark 4.3 From the definition of {\mathcal{E}}_{\tau}, if suppv\cap ({B}_{{\rho}_{\tau}}{B}_{{\rho}_{\tau}/2})\ne \mathrm{\varnothing}, then
{\parallel ({\mathcal{E}}_{\tau}v)\circ {X}_{m,\tau}\parallel}_{{\mathcal{C}}^{0,\alpha}({\overline{B}}_{{\rho}_{0}})}\u2a7dc{\rho}_{\tau}^{\alpha}{\parallel v\circ {X}_{m,\tau}\parallel}_{{\mathcal{C}}^{0,\alpha}({B}_{{\rho}_{0}}{B}_{{\rho}_{\tau}})}.
This phenomenon of explosion of the norm does not occur near the catenoidal type ends:
{\parallel ({\mathcal{E}}_{\tau}v)\circ {X}_{t,\tau}\parallel}_{{\mathcal{C}}^{0,\alpha}([{s}_{0},+\mathrm{\infty})\times {S}^{1})}\u2a7dc{\parallel v\circ {X}_{t,\tau}\parallel}_{{\mathcal{C}}^{0,\alpha}([{s}_{0},{s}_{\tau}]\times {S}^{1})}.
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.
Proposition 4.4 Let \delta \in (1,2), \alpha \in (0,1/4) and \mathrm{\Phi}=({\phi}_{t},{\phi}_{m},{\phi}_{b})\in {[{\mathcal{C}}^{2,\alpha}({S}^{1})]}_{\mathrm{sym}}^{3} satisfying (26) and enjoying the properties given above. There exist constants {c}_{\kappa}>0 and {\tau}_{\kappa}>0, such that
{\parallel T(\mathrm{\Phi},0)\parallel}_{{\mathcal{C}}_{\delta}^{2,\alpha}({M}_{k,\tau})}\u2a7d{c}_{\kappa}{\tau}^{5/2}
(28)
and, for all \tau \in (0,{\tau}_{\kappa}),
\begin{array}{c}{\parallel T(\mathrm{\Phi},{v}_{2})T(\mathrm{\Phi},{v}_{1})\parallel}_{{\mathcal{C}}_{\delta}^{2,\alpha}({M}_{k,\tau})}\u2a7dc{\tau}^{3/2}{\parallel {v}_{2}{v}_{1}\parallel}_{{\mathcal{C}}_{\delta}^{2,\alpha}({M}_{k,\tau})},\hfill \\ {\parallel T({\mathrm{\Phi}}_{2},v)T({\mathrm{\Phi}}_{1},v)\parallel}_{{\mathcal{C}}_{\delta}^{2,\alpha}({M}_{k,\tau})}\u2a7dc{\tau}^{3/2}{\parallel {\mathrm{\Phi}}_{2}{\mathrm{\Phi}}_{1}\parallel}_{{[{\mathcal{C}}^{2,\alpha}({S}^{1})]}_{\mathrm{sym}}^{3}},\hfill \end{array}
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.
Proof We recall that the Jacobi operator associated to {M}_{k,\tau}, is asymptotic (up to a multiplication by 1/{\tau}^{2}) to the Jacobi operator of the catenoid (respectively, of the plane) plane at the catenoidal ends (respectively, at the planar end). The function {w}_{\mathrm{\Phi}} is identically zero far from the ends where the explicit expression of {\mathbb{L}}_{{M}_{k,\tau}} is not known: this is the reason for our particular choice in the definition of {w}_{\mathrm{\Phi}}. Then from the definition of {w}_{\mathrm{\Phi}} and thanks to Proposition 3.1 we obtain the estimate
\begin{array}{c}{\parallel {\mathcal{E}}_{\tau}(\gamma {\mathbb{L}}_{{M}_{k,\tau}}{w}_{\mathrm{\Phi}})\parallel}_{{\mathcal{C}}_{\delta}^{0,\alpha}({M}_{k,\tau})}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel (\gamma {\mathbb{L}}_{{M}_{k,\tau}^{T}}({\partial}_{s}^{2}+{\partial}_{\theta}^{2}))({w}_{\mathrm{\Phi}}\circ {X}_{t,\tau})\parallel}_{{\mathcal{C}}_{\delta}^{0,\alpha}([{s}_{0}+1,{s}_{\tau}]\times {S}^{1})}\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+{\parallel (\gamma {\mathbb{L}}_{{M}_{k,\tau}^{T}}({\partial}_{s}^{2}+{\partial}_{\theta}^{2}))({w}_{\mathrm{\Phi}}\circ {X}_{b,\tau})\parallel}_{{\mathcal{C}}_{\delta}^{0,\alpha}([{s}_{\tau},{s}_{0}1]\times {S}^{1})}\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+{\rho}_{\tau}^{\alpha}{\parallel (\gamma {\mathbb{L}}_{{M}_{k,\tau}^{T}}{\mathrm{\Delta}}_{0})({w}_{\mathrm{\Phi}}\circ {X}_{m,\tau})\parallel}_{{\mathcal{C}}^{0,\alpha}([{\rho}_{\tau},{\rho}_{0}]\times {S}^{1})}\hfill \\ \phantom{\rule{1em}{0ex}}\u2a7dc{\parallel {cosh}^{2}s({w}_{\mathrm{\Phi}}\circ {X}_{t,\tau})\parallel}_{{\mathcal{C}}_{\delta}^{0,\alpha}([{s}_{0}+1,{s}_{\tau}]\times {S}^{1})}+c{\parallel {cosh}^{2}s({w}_{\mathrm{\Phi}}\circ {X}_{b,\tau})\parallel}_{{\mathcal{C}}_{\delta}^{0,\alpha}([{s}_{\tau},{s}_{0}1]\times {S}^{1})}\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+c{\tau}^{\alpha}{\parallel {\rho}^{2k+3}\mathrm{\nabla}({w}_{\mathrm{\Phi}}\circ {X}_{m,\tau})\parallel}_{{\mathcal{C}}^{0,\alpha}([{\rho}_{\tau},{\rho}_{0}]\times {S}^{1})}\hfill \\ \phantom{\rule{1em}{0ex}}\u2a7d{c}_{\kappa}{\tau}^{4}+{c}_{\kappa}{\tau}^{5/2}\u2a7d{c}_{\kappa}{\tau}^{5/2}.\hfill \end{array}
To obtain this estimate we used the following ones:
\begin{array}{c}\underset{[{s}_{0}+1,{s}_{\tau}]\times {S}^{1}}{sup}{e}^{\delta s}{\parallel {cosh}^{2}s({w}_{\mathrm{\Phi}}\circ {X}_{t,\tau})\parallel}_{{\mathcal{C}}^{0,\alpha}([s,s+1]\times {S}^{1})}\hfill \\ \phantom{\rule{1em}{0ex}}\u2a7dc\underset{[{s}_{0}+1,{s}_{\tau}]\times {S}^{1}}{sup}{e}^{\delta s}{e}^{2({s}_{\tau}s)}{e}^{2s}{\parallel {\varphi}_{t}\parallel}_{{\mathcal{C}}^{2,\alpha}({S}^{1})}\hfill \\ \phantom{\rule{1em}{0ex}}\u2a7dc{e}^{2{s}_{\tau}}{\parallel {\varphi}_{t}\parallel}_{{\mathcal{C}}^{2,\alpha}({S}^{1})}\u2a7d{c}_{\kappa}{\tau}^{4}\hfill \end{array}
(a similar estimate holds for the bottom end) and
\begin{array}{c}{\rho}_{\tau}^{\alpha}{\parallel {\rho}^{2k+3}\mathrm{\nabla}({w}_{\mathrm{\Phi}}\circ {X}_{m,\tau})\parallel}_{{C}^{0,\alpha}([{\rho}_{\tau},{\rho}_{0}]\times {S}^{1})}\hfill \\ \phantom{\rule{1em}{0ex}}\u2a7dc{\tau}^{\alpha}{\rho}_{\tau}{\parallel {\phi}_{m}\parallel}_{{\mathcal{C}}^{2,\alpha}({S}^{1})}\u2a7d{c}_{\kappa}{\tau}^{5/2}\hfill \end{array}
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}.
Using the estimates of the coefficients of {\tilde{L}}_{\tau} and the definition of γ (see (18)), we obtain
\begin{array}{rcl}{\parallel {\mathcal{E}}_{\tau}(\gamma {\tilde{L}}_{\tau}{w}_{\mathrm{\Phi}})\parallel}_{{\mathcal{C}}_{\delta}^{0,\alpha}({M}_{k,\tau})}& \u2a7d& c{\tau}^{2}{\parallel {w}_{\mathrm{\Phi}}\circ {X}_{t,\tau}\parallel}_{{\mathcal{C}}_{\delta}^{0,\alpha}([{s}_{0}+1,{s}_{\tau}]\times {S}^{1})}\\ +c{\tau}^{2}{\parallel {w}_{\mathrm{\Phi}}\circ {X}_{b,\tau}\parallel}_{{\mathcal{C}}_{\delta}^{0,\alpha}([{s}_{\tau},{s}_{0}1]\times {S}^{1})}\\ +c{\tau}^{2\alpha}{\parallel {w}_{\mathrm{\Phi}}\circ {X}_{m,\tau}\parallel}_{{\mathcal{C}}^{0,\alpha}([{\rho}_{\tau},{\rho}_{0}]\times {S}^{1})}\u2a7d{c}_{\kappa}{\tau}^{4\alpha}.\end{array}
As for the last term, we recall that the expression of the operator {Q}_{\tau} depends on the type of end we are considering (see (17) and (11)). We have
{\parallel {\mathcal{E}}_{\tau}(\gamma {Q}_{\tau}({w}_{\mathrm{\Phi}}))\parallel}_{{\mathcal{C}}_{\delta}^{0,\alpha}({M}_{k,\tau})}\u2a7d{c}_{\kappa}{\tau}^{5/2}.
In fact
\begin{array}{c}{\parallel {\mathcal{E}}_{\tau}(\gamma {Q}_{\tau}({w}_{\mathrm{\Phi}}))\parallel}_{{\mathcal{C}}_{\delta}^{0,\alpha}({M}_{k,\tau})}\hfill \\ \phantom{\rule{1em}{0ex}}\u2a7dc\tau {\parallel \frac{{w}_{\mathrm{\Phi}}}{\tau coshs}\circ {X}_{t,\tau}\parallel}_{{\mathcal{C}}_{\delta /2}^{2,\alpha}([{s}_{0}+1,{s}_{\tau}]\times {S}^{1})}^{2}+c\tau {\parallel \frac{{w}_{\mathrm{\Phi}}}{\tau coshs}\circ {X}_{b,\tau}\parallel}_{{\mathcal{C}}_{\delta /2}^{2,\alpha}([{s}_{\tau},{s}_{0}1]\times {S}^{1})}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+c{\tau}^{(12\alpha )}{\parallel \frac{{x}^{2}}{\tau}{w}_{\mathrm{\Phi}}\circ {X}_{m,\tau}\parallel}_{{\mathcal{C}}^{2,\alpha}([{\rho}_{\tau},{\rho}_{0}]\times {S}^{1})}^{2}\u2a7d{c}_{\kappa}{\tau}^{5/2}.\hfill \end{array}
As for the second estimate, we recall that
T(\mathrm{\Phi},v):={G}_{\tau ,\delta}\circ {\mathcal{E}}_{\tau}\left(\gamma ({\tilde{L}}_{\tau}({w}_{\mathrm{\Phi}}+v){\mathbb{L}}_{{M}_{k,\tau}}{w}_{\mathrm{\Phi}}{Q}_{\tau}({w}_{\mathrm{\Phi}}+v))\right).
Then
\begin{array}{c}{\parallel T(\mathrm{\Phi},{v}_{2})T(\mathrm{\Phi},{v}_{1})\parallel}_{{\mathcal{C}}_{\delta}^{2,\alpha}({M}_{k,\tau})}\hfill \\ \phantom{\rule{1em}{0ex}}\u2a7d{\parallel {\mathcal{E}}_{\tau}(\gamma {\tilde{L}}_{\tau}({v}_{2}{v}_{1}))\parallel}_{{\mathcal{C}}_{\delta}^{0,\alpha}({M}_{k,\tau})}+{\parallel {\mathcal{E}}_{\tau}\left(\gamma ({Q}_{\tau}({w}_{\mathrm{\Phi}}+{v}_{1}){Q}_{\tau}({w}_{\mathrm{\Phi}}+{v}_{2}))\right)\parallel}_{{\mathcal{C}}_{\delta}^{0,\alpha}({M}_{k,\tau})}.\hfill \end{array}
We observe that from the considerations above it follows that
{\parallel {\mathcal{E}}_{\epsilon}(\gamma {\tilde{L}}_{\tau}({v}_{2}{v}_{1}))\parallel}_{{\mathcal{C}}_{\delta}^{0,\alpha}({M}_{k,\tau})}\u2a7dc{\tau}^{2}{\parallel {v}_{2}{v}_{1}\parallel}_{{\mathcal{C}}_{\delta}^{2,\alpha}({M}_{k,\tau})}
and
\begin{array}{c}{\parallel {\mathcal{E}}_{\tau}\left(\gamma ({Q}_{\tau}({w}_{\mathrm{\Phi}}+{v}_{1}){Q}_{\tau}({w}_{\mathrm{\Phi}}+{v}_{2}))\right)\parallel}_{{\mathcal{C}}_{\delta}^{0,\alpha}({M}_{k,\tau})}\hfill \\ \phantom{\rule{1em}{0ex}}\u2a7dc{\parallel {v}_{2}{v}_{1}\parallel}_{{\mathcal{C}}_{\delta}^{2,\alpha}({M}_{k,\tau})}(\tau {\parallel \frac{{w}_{\mathrm{\Phi}}}{\tau coshs}\circ {X}_{t,\tau}\parallel}_{{\mathcal{C}}^{0,\alpha}([{s}_{0}+1,{s}_{\tau}]\times {S}^{1})}\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\tau {\parallel \frac{{w}_{\mathrm{\Phi}}}{\tau coshs}\circ {X}_{b,\tau}\parallel}_{{\mathcal{C}}^{0,\alpha}([{s}_{\tau},{s}_{0}1]\times {S}^{1})}+{\tau}^{12\alpha}{\parallel \frac{{x}^{2}}{\tau}{w}_{\mathrm{\Phi}}\circ {X}_{m,\tau}\parallel}_{{\mathcal{C}}^{0,\alpha}([{\rho}_{\tau},{\rho}_{0}]\times {S}^{1})})\hfill \\ \phantom{\rule{1em}{0ex}}\u2a7d{c}_{\kappa}{\tau}^{3/2}{\parallel {v}_{2}{v}_{1}\parallel}_{{\mathcal{C}}_{\delta}^{2,\alpha}({M}_{k,\tau})}.\hfill \end{array}
Then
{\parallel T(\mathrm{\Phi},{v}_{2})T(\mathrm{\Phi},{v}_{1})\parallel}_{{\mathcal{C}}_{\delta}^{2,\alpha}({M}_{k,\tau})}\u2a7dc{\tau}^{3/2}{\parallel {v}_{2}{v}_{1}\parallel}_{{\mathcal{C}}_{\delta}^{2,\alpha}({M}_{k,\tau})}.
To get the last estimate it suffices to observe that
\begin{array}{c}{\parallel T({\mathrm{\Phi}}_{2},v)T({\mathrm{\Phi}}_{1},v)\parallel}_{{\mathcal{C}}_{\delta}^{2,\alpha}({M}_{k,\tau})}\hfill \\ \phantom{\rule{1em}{0ex}}\u2a7d{\parallel {\mathcal{E}}_{\tau}(\gamma {\tilde{L}}_{\tau}({w}_{{\mathrm{\Phi}}_{2}}{w}_{{\mathrm{\Phi}}_{1}}))\parallel}_{{\mathcal{C}}_{\delta}^{0,\alpha}({M}_{k,\tau})}+{\parallel {\mathcal{E}}_{\epsilon}\left(\gamma ({Q}_{\epsilon}({w}_{{\mathrm{\Phi}}_{2}}+v){Q}_{\tau}({w}_{{\mathrm{\Phi}}_{1}}+v))\right)\parallel}_{{\mathcal{C}}_{\delta}^{0,\alpha}({M}_{k,\tau})}\hfill \\ \phantom{\rule{1em}{0ex}}\u2a7dc{\tau}^{3/2}{\parallel {\mathrm{\Phi}}_{2}{\mathrm{\Phi}}_{1}\parallel}_{{[{\mathcal{C}}^{2,\alpha}({S}^{1})]}_{\mathrm{sym}}^{3}}+c{\parallel v\parallel}_{{\mathcal{C}}_{\delta}^{0,\alpha}({M}_{k,\tau})}{\parallel {\mathrm{\Phi}}_{2}{\mathrm{\Phi}}_{1}\parallel}_{{[{\mathcal{C}}^{2,\alpha}({S}^{1})]}_{\mathrm{sym}}^{3}}\hfill \\ \phantom{\rule{1em}{0ex}}\u2a7dc{\tau}^{3/2}{\parallel {\mathrm{\Phi}}_{2}{\mathrm{\Phi}}_{1}\parallel}_{{[{\mathcal{C}}^{2,\alpha}({S}^{1})]}_{\mathrm{sym}}^{3}}.\hfill \end{array}
□
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.
The surface {M}_{k,\tau}^{T}(\mathrm{\Phi}) is, close to its upper and lower boundary curve, the graph over the catenoidal ends in the direction given by the vector {\tilde{n}}_{\tau} of the functions
\begin{array}{c}{U}_{t}(r,\theta )={H}_{{\phi}_{t}}({s}_{\tau}ln\frac{2r}{\tau},\theta )+{V}_{t}(r,\theta ),\hfill \\ {U}_{b}(r,\theta )={U}_{t}(r,\theta \frac{\pi}{k+1}),\hfill \end{array}
where {s}_{\tau}=ln\tau +ln2+O(\tau ). Nearby the middle boundary the surface is the vertical graph of
{U}_{m}(r,\theta )={\tilde{H}}_{{\rho}_{\tau},{\phi}_{m}}(\frac{\tau}{r},\theta )+{V}_{m}(r,\theta ),
with {\rho}_{\tau}=\tau. All the functions {V}_{i}, i=t,b,m, depend nonlinearly on \tau ,\mathrm{\Phi}.
Lemma 4.6 The function {V}_{i}(\tau ,{\phi}_{i}), for i=t,b, satisfies {\parallel {V}_{i}(\tau ,{\phi}_{i})(\cdot ,\cdot )\parallel}_{{\mathcal{C}}^{2,\alpha}({\overline{B}}_{1}{B}_{3/4})}\u2a7dc{\tau}^{4\delta} and
{\parallel {V}_{i}(\tau ,{\phi}_{i,2})(\cdot ,\cdot ){V}_{i}(\tau ,{\phi}_{i,1})(\cdot ,\cdot )\parallel}_{{\mathcal{C}}^{2,\alpha}({\overline{B}}_{1}{B}_{3/4})}\u2a7dc{\tau}^{3/2\delta}{\parallel {\phi}_{i,2}{\phi}_{i,1}\parallel}_{{\mathcal{C}}^{2,\alpha}({S}^{1})}.
(29)
The function
{V}_{m}(\tau ,{\phi}_{m})
satisfies
{\parallel {V}_{m}(\tau ,{\phi}_{m})(\cdot ,\cdot )\parallel}_{{\mathcal{C}}^{2,\alpha}({\overline{B}}_{1}{B}_{3/4})}\u2a7dc{\tau}^{5/2}
and
{\parallel {V}_{m}(\tau ,{\phi}_{m,2})(\cdot ,\cdot ){V}_{m}(\tau ,{\phi}_{m,1})(\cdot ,\cdot )\parallel}_{{\mathcal{C}}^{2,\alpha}({\overline{B}}_{1}{B}_{3/4})}\u2a7dc{\tau}^{3/2}{\parallel {\phi}_{m,2}{\phi}_{m,1}\parallel}_{{\mathcal{C}}^{2,\alpha}({S}^{1})}.
(30)
Proof We recall that the functions {V}_{t}, {V}_{b}, {V}_{m} are the restrictions to {E}_{t,\tau}, {E}_{b,\tau}, {E}_{m,\tau} of a fixed point v for the operator T(\mathrm{\Phi},\cdot ). The estimates of their norm are a consequence of Proposition 4.4. Observe that to derive the estimate of the norm of {V}_{t} and {V}_{b} we use the better estimate for the norm of the fixed point v which holds at the catenoidal type ends. Precisely stated: {\parallel v\circ {X}_{i}\parallel}_{{\mathcal{C}}_{\delta}^{2,\alpha}}\u2a7d{c}_{\kappa}{\tau}^{4} with i=t,b. Then (29) follows from
\begin{array}{c}{\parallel {V}_{i}(\tau ,{\phi}_{i,2})(\cdot ,\cdot ){V}_{i}(\tau ,{\phi}_{i,1})(\cdot ,\cdot )\parallel}_{{\mathcal{C}}^{2,\alpha}({\overline{B}}_{1}{B}_{3/4})}\hfill \\ \phantom{\rule{1em}{0ex}}\u2a7dc{e}^{\delta {s}_{\tau}}{\parallel (T({\mathrm{\Phi}}_{2},{V}_{i})T({\mathrm{\Phi}}_{1},{V}_{i}))\circ {X}_{i,\tau}\parallel}_{{\mathcal{C}}_{\delta}^{2,\alpha}({\mathrm{\Omega}}_{i}\times {S}^{1})},\hfill \end{array}
for i=t,b, with {\mathrm{\Omega}}_{t}=[{s}_{0},{s}_{\tau}] and {\mathrm{\Omega}}_{b}=[{s}_{\tau},{s}_{0}]. To get the estimate (30) we observe that
\begin{array}{c}{\parallel {V}_{m}(\tau ,{\phi}_{m,2})(\cdot ,\cdot ){V}_{m}(\tau ,{\phi}_{m,1})(\cdot ,\cdot )\parallel}_{{\mathcal{C}}^{2,\alpha}({\overline{B}}_{1}{B}_{3/4})}\hfill \\ \phantom{\rule{1em}{0ex}}\u2a7dc{\parallel (T({\mathrm{\Phi}}_{2},{V}_{m})T({\mathrm{\Phi}}_{1},{V}_{m}))\circ {X}_{m,\tau}\parallel}_{{\mathcal{C}}^{2,\alpha}([{\rho}_{\tau},{\rho}_{0}]\times {S}^{1})}.\hfill \end{array}
□
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}}).