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 ${B}^{3}$.
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 $O({\tau}^{\frac{5}{2}})$.