# Hierarchy structures in finite index CMC surfaces

William H. Meeks III This research was partially supported by CNPq - Brazil,
grant no. 400966/2014-0.  Joaquín Pérez Research of both authors was partially
supported by MINECO/MICINN/FEDER grant no. PID2020-117868GB-I00, regional
grant P18-FR-4049, and by the “Maria de Maeztu” Excellence Unit IMAG,
reference CEX2020-001105-M, funded by MCINN/AEI/10.13039/501100011033/
CEX2020-001105-M.

###### Abstract

Given ${\varepsilon}_{0}>0$, $I\in\mathbb{N}\cup\\{0\\}$ and $K_{0},H_{0}\geq
0$, let $X$ be a complete Riemannian $3$-manifold with injectivity radius
$\mbox{\rm Inj}(X)\geq{\varepsilon}_{0}$ and with the supremum of absolute
sectional curvature at most $K_{0}$, and let $M\looparrowright X$ be a
complete immersed surface of constant mean curvature $H\in[0,H_{0}]$ with
index at most $I$. For such $M\looparrowright X$, we prove Structure Theorem
1.2 which describes how the interesting ambient geometry of the immersion is
organized locally around at most $I$ points of $M$ where the norm of the
second fundamental form takes on large local maximum values.

Mathematics Subject Classification: Primary 53A10, Secondary 49Q05, 53C42

Key words and phrases: Constant mean curvature, finite index $H$-surfaces,
area estimates for constant mean curvature surfaces, curvature estimates for
one-sided stable minimal surfaces.

###### Contents

  1. 1 Introduction
  2. 2 Index of $1$-sided $H$-immersions, harmonic coordinates and multi-valued graphs
  3. 3 Finitely branched minimal surfaces in $\mathbb{R}^{3}$ of finite index
  4. 4 Almost flat annular $H$-multi-graphs of bounded multiplicity
  5. 5 The proof of Structure Theorem 1.2
    1. 5.1 The case of uniformly bounded second fundamental form and the proof of item (E) of Theorem 1.2 in the general case
    2. 5.2 Stable pieces of $H$-surfaces in $\Lambda$ and their curvature estimate
    3. 5.3 Strategy of the proof of Theorem 1.2
    4. 5.4 Proof of items 1, 2, 3 of Theorem 1.2 for $I=1$
      1. 5.4.1 Local pictures around points where $|A_{M}|>t$, for $t$ sufficiently large
      2. 5.4.2 Local pictures have a uniform size
    5. 5.5 Proofs of items 1, 2, 3 of Theorem 1.2 for $I=I_{0}+1$
      1. 5.5.1 Local pictures around points where $|A_{M}|>t$, for $t$ sufficiently large
      2. 5.5.2 How to proceed if the (first) local pictures fail to have a uniform size
      3. 5.5.3 Finding an $s_{0}$-th local picture with a uniform size
    6. 5.6 Counting index, genus and total spinning for local hierarchies
      1. 5.6.1 The hierarchy associated to a $\Delta$-type piece
    7. 5.7 Proofs of items (A)-…-(D) of Theorem 1.2
  6. 6 Sequential compactness results in $\Lambda$ for $X$ fixed
  7. 7 Appendix A: Curvature estimates for stable $H$-surfaces
  8. 8 Appendix B: Some results in reference [17] used in this paper

##  1 Introduction

Let $X$ denote a complete Riemannian $3$-manifold with positive injectivity
radius $\mbox{\rm Inj}(X)$ and bounded absolute sectional curvature. Let $M$
be a complete immersed surface in $X$ of constant mean curvature $H\geq 0$,
which we call an $H$-surface in $X$. The Jacobi operator of $M$ is the
Schrödinger operator

$L=\Delta+|A_{M}|^{2}+\mbox{Ric}(N),$

where $\Delta$ is the Laplace-Beltrami operator on $M$, $|A_{M}|^{2}$ is the
square of the norm of its second fundamental form and $\mbox{Ric}(N)$ denotes
the Ricci curvature of $X$ in the direction of the unit normal vector $N$ to
$M$; the index of $M$ is the index of $L$,

$\mbox{Index}(M)=\lim_{R\to\infty}\mbox{Index}(B_{M}(p,R)),$

where $B_{M}(p,R)$ is the intrinsic metric ball in $M$ of radius $R>0$
centered at a point $p\in M$, and $\mbox{Index}(B_{M}(p,R))$ is the number of
negative eigenvalues of $L$ on $B_{M}(p,R)$ with Dirichlet boundary
conditions. Here, we have assumed that the immersion is two-sided (this holds
in particular if $H>0$). In the case, $H=0$ and the immersion is one-sided,
then the index is defined in a similar manner using compactly supported
variations in the normal bundle; see Definition 2.1 for details.

The primary goal of this paper is to describe the structure of complete
immersed $H$-surfaces $F\colon M\looparrowright X$ (also called
$H$-immersions) which have a fixed bound $I\in\mathbb{N}\cup\\{0\\}$ on their
index and a fixed upper bound $H_{0}$ for their constant mean curvatures $H$,
in certain small intrinsic neighborhoods of points with sufficiently large
norm $|A_{M}|$ of their second fundamental forms, see Theorem 1.2. When $M$
has non-empty boundary, we will assume, after a choice of some
${\varepsilon}_{0}\in(0,\mbox{\rm Inj}(X))$, that there is an upper bound
$A_{0}$ of $|A_{M}|$ in the intrinsic ${\varepsilon}_{0}$-neighborhood of the
boundary of $M$. Theorem 1.2 plays an important theoretical role in
understanding global properties of such surfaces in much the same way that the
local structure theorems of Colding-Minicozzi [7, 8] (for embedded minimal
surfaces) and of Meeks-Tinaglia [21] (for embedded $H$-surfaces with $H>0$)
play a fundamental role in understanding global properties of complete
embedded $H$-surfaces of finite genus, especially in the case where
$X=\mathbb{R}^{3}$. However, we point out that the results in this paper do
not depend on the results for embedded $H$-surfaces of Colding-Minicozzi and
Meeks-Tinaglia; for applications of Theorem 1.2 to the global theory of finite
index $H$-surfaces in Riemannian 3-manifolds, see [18].

In the sequel, we will denote by $B_{X}(x,r)$ (resp. $\overline{B}_{X}(x,r)$)
the open (resp. closed) metric ball centered at a point $x\in X$ of radius
$r>0$. For a Riemannian surface $M$ with smooth compact boundary $\partial M$,
$\kappa(M)=\int_{\partial M}\kappa_{g}$ will stand for the total geodesic
curvature of $\partial M$, where $\kappa_{g}$ denotes the pointwise geodesic
curvature of $\partial M$ with respect to the inward pointing unit conormal
vector of $M$ along $\partial M$.

######  Definition 1.1.

For every $I\in\mathbb{N}\cup\\{0\\}$, ${\varepsilon}_{0}>0$, and
$H_{0},A_{0},K_{0}\geq 0$, we denote by

$\Lambda=\Lambda(I,H_{0},{\varepsilon}_{0},A_{0},K_{0})$

the space of all $H$-immersions $F\colon M\looparrowright X$ satisfying the
following conditions:

  1. (A1)

$X$ is a complete Riemannian 3-manifold with injectivity radius $\mbox{\rm
Inj}(X)\geq{\varepsilon}_{0}$ and absolute sectional curvature bounded from
above by $K_{0}$.

  2. (A2)

$M$ is a complete surface with smooth boundary (possibly empty) and when
$\partial M\neq\varnothing$, there are points in $M$ of distance greater than
${\varepsilon}_{0}$ from $\partial M$.

  3. (A3)

$H\in[0,H_{0}]$ and $F$ has index at most $I$.

  4. (A4)

If $\partial M\neq\varnothing$, then for any ${\varepsilon}\in(0,\infty]$ we
let $U(\partial M,{\varepsilon})=\\{x\in M\mid d_{M}(x,\partial
M)<{\varepsilon}\\}$ be the open intrinsic ${\varepsilon}$-neighborhood of
$\partial M$. Then, $|A_{M}|$ is bounded from above by $A_{0}$ in $U(\partial
M,{\varepsilon}_{0})$.

Suppose that $(F\colon M\looparrowright X)\in\Lambda$ and $\partial
M\neq\varnothing$. For any positive
${\varepsilon}_{1}\leq{\varepsilon}_{2}\in[0,\infty]$, let

$U(\partial M,{\varepsilon}_{1},{\varepsilon}_{2})=U(\partial
M,{\varepsilon}_{2})\setminus\overline{U(\partial
M,{\varepsilon}_{1})},\quad\overline{U}(\partial
M,{\varepsilon}_{1},{\varepsilon}_{2})=\overline{U(\partial
M,{\varepsilon}_{2})}\setminus U(\partial M,{\varepsilon}_{1}).$

When $\partial M=\varnothing$, we define $U(\partial
M,{\varepsilon}_{1},\infty)=\overline{U}(\partial M,{\varepsilon}_{1},\infty)$
as $M$.

In the next result we will make use of harmonic coordinates $\varphi_{x}\colon
U\to B_{X}(x,r)$ defined on an open subset $U$ of $\mathbb{R}^{3}$ containing
the origin, taking values in a geodesic ball $B_{X}(x,r)$ centered at a point
$x\in X$ of positive radius $r$ less than the injectivity radius of $X$ at $x$
and with a $C^{1,{\alpha}}$ control of the ambient metric on $X$, see
Definition 2.2 for details.

######  Theorem 1.2 (Structure Theorem for finite index $H$-surfaces).

Given ${\varepsilon}_{0}>0$, $K_{0},H_{0},A_{0}\geq 0$,
$I\in\mathbb{N}\cup\\{0\\}$ and $\tau\in(0,\pi/10]$, there exist
$A_{1}\in[A_{0},\infty)$, ${\delta}_{1},{\delta}\in(0,{\varepsilon}_{0}/2]$
with ${\delta}_{1}\leq{\delta}/2$, such that the following hold:

For any $(F\colon M\looparrowright
X)\in\Lambda=\Lambda(I,H_{0},{\varepsilon}_{0},A_{0},K_{0})$, there exists a
(possibly empty) finite collection
$\mathcal{P}_{F}=\\{p_{1},\ldots,p_{k}\\}\subset U(\partial
M,{\varepsilon}_{0},\infty)$ of points, $k\leq I$, and numbers
$r_{F}(1),\ldots,{r_{F}}(k)\in[{\delta}_{1},\frac{{\delta}}{2}]$ with
$r_{F}(1)>4r_{F}(2)>\ldots>4^{k-1}r_{F}(k)$, satisfying the following:

  1. 1.

Portions with concentrated curvature: Given $i=1,\ldots,k$, let $\Delta_{i}$
be the component of
$F^{-1}(\overline{B}_{X}(F(p_{i}),r_{F}(i)))$ containing $p_{i}$. Then:

    1. a.

$\Delta_{i}\subset\overline{B}_{M}(p_{i},\frac{5}{4}r_{F}(i))$ (in particular,
$\Delta_{i}$ is compact).

    2. b.

$\Delta_{i}$ has smooth boundary and
$F(\partial\Delta_{i})\subset\partial\overline{B}_{X}(F(p_{i}),r_{F}(i))$.

    3. c.

$B_{M}(p_{i},\frac{7}{5}r_{F}(i))\cap
B_{M}(p_{j},\frac{7}{5}r_{F}(j))=\varnothing$ for $i\neq j$. In particular,
the intrinsic distance between $\Delta_{i},\Delta_{j}$ is greater than
$\frac{3}{10}{\delta}_{1}$ for every $i\neq j$.

    4. d.

$|A_{M}|(p_{i})=\max_{\Delta_{i}}|A_{M}|=\max\\{|A_{M}|(p)\ :\ p\in
M\setminus\cup_{j=1}^{i-1}B_{M}(p_{j},\frac{5}{4}r_{F}(j))\\}\geq A_{1}$, see
Figure 1.

Figure 1: The second fundamental form concentrates inside the intrinsic
compact regions $\Delta_{i}$ (in red), each of which is mapped through the
immersion $F$ to a surface inside the extrinsic ball in $X$ centered at
$F(p_{i})$ of radius $r_{F}(i)>0$, with
$F(\partial\Delta_{i})\subset\partial\overline{B}_{X}(F(p_{i}),r_{F}(i))$.
Although the boundary $\partial\Delta_{i}$ might not be at constant intrinsic
distance from the ‘center’ $p_{i}$, $\Delta_{i}$ lies entirely inside the
intrinsic ball centered at $p_{i}$ of radius $\frac{5}{4}r_{F}(i)$. The
intrinsic open balls $B_{M}(p_{i},\frac{7}{5}r_{F}(i))$ are pairwise disjoint.

    5. e.

The index $\mbox{\rm Index}(\Delta_{i})$ of $\Delta_{i}$ is positive.

  2. 2.

Transition annuli: For $i=1,\ldots,k$ fixed, let $e(i)\in\mathbb{N}$ be the
number of boundary components of $\Delta_{i}$. Then, there exist planar disks
$\mathbb{D}_{1},\ldots,\mathbb{D}_{e(i)}\subset T_{F(p_{i})}X$ of radius
$2r_{F}(i)$ centered at the origin in $T_{F(p_{i})}X$, such that if we denote
by

$P_{i,h}=\varphi_{F(p_{i})}(\mathbb{D}_{h}),\quad h\in\\{1,\ldots,e(i)\\},$

(here $\varphi_{F(p_{i})}$ denotes a harmonic chart centered at $F(p_{i})$,
see Definition 2.2), then

$F(\Delta_{i})\cap[\overline{B}_{X}(F(p_{i}),r_{F}(i))\setminus
B_{X}(F(p_{i}),r_{F}(i)/2)]$

consists of $e(i)$ annular multi-graphs111See Definition 2.3 for this notion
of multi-graphs. $G_{i,1},\ldots,G_{i,e(i)}$ over their projections to
$P_{i,1},\ldots,P_{i,e(i)}$, with multiplicities $m_{i,1},\ldots
m_{i,e(i)}\in\mathbb{N}$ respectively, and whose related graphing functions
$u$ satisfy

$\frac{|u(x)|}{|x|}+|\nabla u|(x)\leq\tau,$ (1.1)

where we have taken coordinates $x$ in each of the $P_{i,h}$ and denoted by
$|x|$ the extrinsic distance to $F(p_{i})$ in the ambient metric of $X$, see
Figure 2.

Figure 2: The transition annuli: On the right, one has the extrinsic
representation in $X$ of one of the annular multi-graphs $G$ in
$F(\Delta_{1})\cap[\overline{B}_{X}(F(p_{1}),r_{F}(1))\setminus
B_{X}(F(p_{1}),r_{F}(1)/2)]$; in this case, the multiplicity of the multi-
graph is 3. On the left, one has the intrinsic representation of the same
annulus (shadowed); there is one such annular multi-graph for each boundary
component of $\Delta_{i}$.

  3. 3.

Region with uniformly bounded curvature: $|A_{M}|<A_{1}$ on
$\widetilde{M}:=M\setminus\bigcup_{i=i}^{k}\mbox{\rm Int}(\Delta_{i})$.

Moreover, the following additional properties hold:

  1. A.

$\sum_{i=1}^{k}I(\Delta_{i})\leq I$, where $I(\Delta_{i})=\mbox{\rm
Index}(\Delta_{i})$.

  2. B.

Geometric and topological estimates: Given $i=1,\ldots,k$, let
$m(i):=\sum_{h=1}^{e(i)}m_{i,h}$ be the total spinning of the boundary of
$\Delta_{i}$, let $g(\Delta_{i})$ denote the genus of $\Delta_{i}$ (in the
case $\Delta_{i}$ is non-orientable, $g(\Delta_{i})$ denotes the genus of its
oriented cover222If $\Sigma$ is a compact non-orientable surface and
$\widehat{\Sigma}\stackrel{{\scriptstyle 2:1}}{{\to}}\Sigma$ denotes the
oriented cover of $\Sigma$, then the genus of $\widehat{\Sigma}$ plus 1 equals
the number of cross-caps in $\Sigma$.). Then, $m(i)\geq 2$ and the following
upper estimates hold:

    1. (a)

If $I(\Delta_{i})=1$, then $\Delta_{i}$ is orientable, $g(\Delta_{i})=0$, and
$(e(i),m(i))\in\\{(2,2),(1,3)\\}$.

    2. (b)

If $\Delta_{i}$ is orientable and $I(\Delta_{i})\geq 2$, then $m(i)\leq
3I(\Delta_{i})-1$, $e(i)\leq 3I(\Delta_{i})-2$, and $g(\Delta_{i})\leq
3I(\Delta_{i})-4$.

    3. (c)

If $\Delta_{i}$ is non-orientable, then $I(\Delta_{i})\geq 2$, $m(i)\leq
3I(\Delta_{i})-1$, $e(i)\leq 3I(\Delta_{i})-2$ and $g(\Delta_{i})\leq
6I(\Delta_{i})-8$.

    4. (d)

$\chi(\Delta_{i})\geq-6I(\Delta_{i})+2m(i)+e(i)$, and thus,
$\chi(\cup_{i=1}^{k}\Delta_{i})\geq-6I+2S+e$, where

$e=\sum_{i=1}^{k}e(i),\qquad S=\sum_{i=1}^{k}m(i).$

    5. (e)

$|\kappa(\Delta_{i})-2\pi m(i)|\leq\frac{\tau}{m(i)}$, and so, the total
geodesic curvature $\kappa(\widetilde{M})$ of $\widetilde{M}$ along
$\partial\widetilde{M}\setminus\partial M$ satisfies
$\left|\kappa(\widetilde{M})+2\pi S\right|\leq\frac{\tau}{2}k$, and so,

$2\pi S-\frac{\tau}{2}k\leq\sum_{i=1}^{k}\kappa(\Delta_{i})\leq 2\pi
S+\frac{\tau}{2}k.$ (1.2)

    6. (f)

$-\int_{\Delta_{i}}K>3\pi,$ and so,

$-\int_{\cup_{i=1}^{k}\Delta_{i}}K=-2\pi\chi(\cup_{i=1}^{k}\Delta_{i})+\int_{\cup_{i=1}^{k}\partial\Delta_{i}}\kappa_{g}>3k\pi.$
(1.3)

  3. C.

Genus estimate outside the concentration of curvature: If $M$ is orientable,
$k\geq 1$ and the genus $g(M)$ of $M$ is finite, then the genus
$g(\widetilde{M})$ of $\widetilde{M}$ satisfies

$0\leq g(M)-g(\widetilde{M})\leq 3I-2.$

  4. D.

Area estimate outside the concentration of curvature: If $k\geq 1$, then

$\mbox{\rm Area}(\widetilde{M})\geq 14\pi\sum_{i=1}^{k}m(i)r_{F}(i)^{2}\geq
2\pi\sum_{i=1}^{k}m(i)r_{F}(i)^{2}\geq\mbox{\rm
Area}\left(\bigcup_{i=1}^{k}\Delta_{i}\right)\geq k\pi{\delta}_{1}^{2}.$

  5. E.

There exists a $C>0$, depending on ${\varepsilon}_{0},K_{0},H_{0}$ and
independent of $I$, such that

$\mbox{\rm Area}(M)\geq\left\\{\begin{array}[]{ll}{\displaystyle
C\max\\{1,\mbox{\rm Radius}(M)\\}}&\mbox{if $\partial M\neq\varnothing$,}\\\
{\displaystyle C\max\\{1,\mbox{\rm Diameter}(M)\\}}&\mbox{if $\partial
M=\varnothing$,}\end{array}\right.$ (1.4)

where

$\displaystyle\mbox{\rm Radius}(M)$ $\displaystyle=$ $\displaystyle\sup_{x\in
M}d_{M}(x,\partial M)\in(0,\infty]\quad\mbox{if }\partial M\neq\varnothing,$
$\displaystyle\mbox{\rm Diameter}(M)$ $\displaystyle=$
$\displaystyle\sup_{x,y\in M}d_{M}(x,y)\qquad\qquad\quad\;\mbox{if }\partial
M=\varnothing.$

In particular, if $M$ has infinite radius or if $M$ has empty boundary and it
is non-compact, then its area is infinite.

The proof of the Structure Theorem 1.2 is carried out in Section 5, and it
will be done by induction on the index bound $I$. In the case $I=0$ , Theorem
1.2 is obtained by using curvature estimates for stable $H$-surfaces, and the
arguments in this special case generalize to the case where for a given
$I\in\mathbb{N}$, there exists a uniform curvature estimate for the immersions
in $\Lambda=\Lambda(I,H_{0},{\varepsilon}_{0},A_{0},K_{0})$, see Section 5.1.
A non-trivial step in the proof of Theorem 1.2 involves an analysis of the
local pictures on different scales for a sequence of complete
$H_{n}$-immersions $F_{n}\colon M_{n}\looparrowright X_{n}$ with
$H_{n}\in[0,H_{0}]$ and $\mbox{Index}({F_{n}})\leq I$, such that
$\\{\sup|A_{M_{n}}|\\}_{n}$ is unbounded (these local pictures are limits of
the $F_{n}$ after blowing up on certain scales). Although non-trivial, this
analysis is simpler in the case $I=1$ because in this case there is only one
scale for the local pictures of $F_{n}$; this case is studied in Section 5.4.
The analysis of these local pictures in the general case is carried out in
Section 5.5, and it is based on the lower bounds obtained by Chodosh and
Maximo [6] and Karpukhin [13] for the index of a possibly branched, complete
immersed minimal surface $\Sigma$ in $\mathbb{R}^{3}$ with finite total
curvature, in terms of its genus, total branching order, and the number of its
ends counted with multiplicity. After coming back to the original scale, these
complexity estimates will give upper bounds for the total geodesic curvature
of the boundary of the portion $\widetilde{M}$ of $M$ defined in item 3 of
Theorem 1.2, as well as to give lower bounds in item C on the genus of
$\widetilde{M}$ in terms of $I$ and the genus of $M$ when $M$ is orientable.
These geometric and topological bounds are obtained in Sections 5.6 and 5.7.
What this analysis demonstrates is that there is an organized hierarchy-type
structure in the geometry of a complete, immersed $H$-surface $F\colon
M\looparrowright X$ near points of large, almost-maximal norm of the second
fundamental form of the immersion, from which the title of the paper is
derived; this hierarchy structure of $F$ around such special points is
described explicitly in Section 5.6 and plays an essential role in the proofs
of our main results.

A key step in the proof of Theorem 1.2 is to obtain curvature estimates for a
large portion of the $H$-surface $(F\colon M\looparrowright X)\in\Lambda$ in
that theorem. These curvature estimates are obtained in Section 5.2 and they
are based on related curvature estimates for stable $H$-surfaces developed in
Appendix A (Section 7).

Observe that (1.4) is a lower bound for the area of an $H$-surface in a
Riemannian 3-manifold $X$, described in terms of an upper bound for its
absolute mean curvature function $|H|$, a lower bound of the injectivity
radius of $X$ and an upper bound of the sectional curvature of $X$. This area
estimate is proven in Section 5.7 and follows from a more general area
estimate and an intrinsic monotonicity of volume formula for $n$-dimensional
submanifolds with bounded length of their mean curvature vectors in
$m$-dimensional Riemannian manifolds $X$ that have a lower bound for their
injectivity radius and an upper bound for the sectional curvature of $X$. Both
of these auxiliary results are proven in our paper [17], and we include their
statements (without proofs) in this paper for the sake of completeness, see
Propositions 8.1 and 8.3 in Appendix (B) (Section 8). In Proposition 8.4 of
the same appendix we state explicit scale invariant weak chord-arc estimates
for finitely branched minimal surfaces of finite total curvature in
$\mathbb{R}^{3}$ in terms of the index and total branching (also proven in
[17]); these chord-arc estimates are applied in the proof of item 1 in Theorem
1.2.

An important theoretical consequence of Structure Theorem 1.2 is the existence
of compactness results for $H$-surfaces of bounded index in $X$. More
specifically, in Section 6 we state and prove some compactness results for
sequences of complete immersions with constant mean curvature in
$\Lambda=\Lambda(I,H_{0},{\varepsilon}_{0},A_{0},K_{0})$, as described in
Theorem 1.2, in the particular case that the immersions are defined on
connected surfaces without boundary, the ambient space $X$ is independent of
the element in the sequence and the image of each immersion in the sequence
intersects a fixed compact subdomain of $X$. In this case, the limit object
that we encounter (after passing to a subsequence) is a complete, possibly
finitely branched immersion of constant mean curvature at most $H_{0}$ and
index at most $I$.

In regards to the just mentioned compactness results in Section 6, it is worth
mentioning the related paper [3] by Bourni, Sharp and Tinaglia, where they
give weak compactness results for a sequence of embedded CMC hypersurfaces in
a compact Riemannian manifold of dimension $m$ with $3\leq m\leq 7$, provided
that their areas and Morse indices are bounded. As they remark in [3], their
results were motivated by the derivation of the genus-dependent area bounds
for triply periodic CMC surfaces properly embedded in $\mathbb{R}^{3}$ by
Meeks and Tinaglia in [22]; also the results in [3] and in our present paper
are motivated other recent works [1, 2, 3, 4, 5, 15, 26], which together help
to describe the geometry of finite index CMC surfaces $M$ embedded in closed
Riemannian 3-manifolds and relationships between index, area and genus of such
an $M$.

In [18] we give applications of Theorem 1.2 to understand global properties of
immersed $H$-surfaces $M\looparrowright X$ of fixed finite index $I$,
especially results related to the area and diameter of such an $M$ when it is
compact without boundary; in particular, we deduce that the area of such an
$M$ (resp. the diameter) grows at least linearly (resp. logarithmically) with
the genus.

Acknowledgments: We thank Otis Chodosh for explaining to us his work [6] with
Davi Maximo on lower bounds for the index of a complete branched minimal
surface in $\mathbb{R}^{3}$ of finite total curvature, in terms of its genus
and number of ends counted with multiplicity, and how the analysis by
Karpukhin [13] can be used to allow finitely many branch points.

##  2 Index of $1$-sided $H$-immersions, harmonic coordinates and multi-valued
graphs

In Theorem 1.2 we referred to the index of 1-sided minimal immersions,
harmonic coordinates and finitely valued multi-graphs. We will devote this
section to give some details about these notions.

######  Definition 2.1.

Given a $1$-sided minimal codimension-one immersion $F\colon M\looparrowright
X$ in a Riemannian manifold $X$, let $\widetilde{M}\to M$ be the two-sided
cover of $M$ and let $\tau\colon\widetilde{M}\to\widetilde{M}$ be the
associated deck transformation of order 2. Denote by $\widetilde{\Delta}$,
$|\widetilde{A}|^{2}$ the Laplacian and squared norm of the second fundamental
form of $\widetilde{M}$, and let $N\colon\widetilde{M}\to TX$ be a unitary
normal vector field. The index of $F$ is defined as the number of negative
eigenvalues of the elliptic, self-adjoint operator
$\widetilde{\Delta}+|\widetilde{A}|^{2}+\mbox{Ric}(N,N)$ defined over the
space of compactly supported smooth functions
$\phi\colon\widetilde{M}\to\mathbb{R}$ such that $\phi\circ\tau=-\phi$.

######  Definition 2.2.

Given a (smooth) Riemannian manifold $X$, a local chart $(x_{1},\ldots x_{n})$
defined on an open set $U$ of $X$ is called harmonic if $\Delta x_{i}=0$ for
all $i=1,\ldots n$.

Following Definition 5 in [11], we make the next definition. Given $Q>1$ and
$\alpha\in(0,1)$, we define the $C^{1,\alpha}$-harmonic radius at a point
$x_{0}\in X$ as the largest number $r=r(Q,{\alpha})(x_{0})$ so that in the
geodesic ball $B_{X}(x_{0},r)$ of center $x_{0}$ and radius $r$, there is a
harmonic coordinate chart such that the metric tensor $g$ of $X$ is
$C^{1,{\alpha}}$-controlled in these coordinates. Namely, if $g_{ij}$,
$i,j=1,\ldots,n$, are the components of $g$ in these coordinates, then

  1. 1.

$Q^{-1}{\delta}_{ij}\leq g_{ij}\leq Q\,{\delta}_{ij}$ as bilinear forms, and

  2. 2.

${\displaystyle\sum_{{\beta}=1}^{3}r\sup_{y}|\frac{\partial g_{ij}}{\partial
x_{{\beta}}}(y)|+\sum_{{\beta}=1}^{3}r^{1+{\alpha}}\sup_{y\neq
z}\frac{\left|\frac{\partial g_{ij}}{\partial x_{{\beta}}}(y)-\frac{\partial
g_{ij}}{\partial x_{{\beta}}}(z)\right|}{d_{X}(y,z)^{{\alpha}}}}\leq Q-1$.

The $C^{1,{\alpha}}$-harmonic radius $r(Q,{\alpha})(X)$ of $X$ is now defined
by

$r(Q,{\alpha})(X)=\inf_{x_{0}\in X}r(Q,{\alpha})(x_{0}).$

If the absolute sectional curvature of $X$ is bounded by some constant
$K_{0}>0$ and Inj$(X)\geq{\varepsilon}_{0}>0$, then Theorem 6 in [11] implies
that given $Q>1$ and ${\alpha}\in(0,1)$, there exists
$C=C(Q,{\alpha},{\varepsilon}_{0},K_{0})$ (observe that $C$ does not depend on
$X$) such that $r(Q,{\alpha})(X)\geq C$.

######  Definition 2.3.

Let $f\colon\Sigma\looparrowright\mathbb{R}^{3}$ be an immersed annulus, $P$ a
plane passing through the origin and $\Pi\colon\mathbb{R}^{3}\to P$ the
orthogonal projection. Given $m\in\mathbb{N}$, let $\sigma_{m}\colon P_{m}\to
P^{*}=P\setminus\\{\vec{0}\\}$ be the $m$-sheeted covering space of $P^{*}$.
We say that $\Sigma$ is an $m$-valued graph over $P$ if
$\vec{0}\not\in(\Pi\circ f)(\Sigma)$, the induced map $(\Pi\circ f)_{*}\colon
H_{1}(\Sigma)=\mathbb{Z}\to H_{1}(P^{*})=\mathbb{Z}$ satisfies $|(\Pi\circ
f)_{*}(1)|=m$ and $\Pi\circ f\colon\Sigma\to P^{*}$ has a smooth injective
lift $\widetilde{f}\colon\Sigma\to P_{m}$ through $\sigma_{m}$; in this case,
we say that $\Sigma$ has multiplicity $m$ as a multi-graph.

Given $Q>1$ and ${\alpha}\in(0,1)$, let $X$ be a Riemannian 3-manifold and
$(x_{1},x_{2},x_{3})$ be a harmonic chart for $X$ defined on $B_{X}(x_{0},r)$,
$x_{0}\in X$, $r>0$, where the metric tensor $g$ of $X$ is
$C^{1,{\alpha}}$-controlled in the sense of Definition 2.2. Let $P\subset
B_{X}(x_{0},r)$ be the image by this harmonic chart of the intersection of a
plane in $\mathbb{R}^{3}$ passing through the origin with the domain of the
chart. In this setting, the notion of $m$-valued graph over $P$ generalizes
naturally to an immersed annulus

$f\colon\Sigma\looparrowright B_{X}(x_{0},r),$

where the projection $\Pi$ refers to the harmonic coordinates. If
$f\colon\Sigma\looparrowright B_{X}(x_{0},r)$ is an $m$-valued graph over $P$
and $u$ is the corresponding graphing function that expresses $f(\Sigma)$, we
can consider the gradient $\nabla u$ with respect to the metric on $P$ induced
by the ambient metric of $X$. Both $u$ and $|\nabla u|$ depend on the choice
of harmonic coordinates around $x_{0}$ (and they also depend on $Q$), but if
$\frac{|u(x)|}{|x|}+|\nabla u|(x)\leq\tau$ for some $\tau\in(0,\pi/10]$ and
$Q>1$ sufficiently close to $1$, then $\frac{|u(x)|}{|x|}+|\nabla u|(x)<2\tau$
for any other choice of harmonic chart around $x_{0}$ with this restriction of
$Q$.

##  3 Finitely branched minimal surfaces in $\mathbb{R}^{3}$ of finite index

In the process of finding local pictures of $H$-immersions as in the proof of
Theorem 1.2, we will find complete, non-flat, finitely branched minimal
surfaces in $\mathbb{R}^{3}$. We will devote this section to obtain some
properties of these surfaces which will be useful in the sequel.

######  Definition 3.1.

Let $\Sigma$ be a smooth surface endowed with a conformal class of metrics. We
say that a harmonic map $f\colon\Sigma\to\mathbb{R}^{3}$ is a (possibly non-
orientable) branched minimal surface if it is a conformal immersion outside of
a locally finite set of points $\mathcal{B}_{\Sigma}\subset\Sigma$, where $f$
fails to be an immersion. Points in $\mathcal{B}_{\Sigma}$ are called branch
points of $f$. It is well-known (see e.g. Micallef and White [23, Theorem
1.4]) that given $p\in\mathcal{B}_{\Sigma}$, there exist a conformal
coordinate $(\overline{\mathbb{D}},z)$ for $\Sigma$ centered at $p$ (here
$\overline{\mathbb{D}}$ is the closed unit disk in the plane), a
diffeomorphism $u$ of $\overline{\mathbb{D}}$ and a rotation $\phi$ of
$\mathbb{R}^{3}$ such that $\phi\circ f\circ u$ has the form

$z\mapsto(z^{q},x(z))\in\mathbb{C}\times\mathbb{R}\sim\mathbb{R}^{3}$

for $z$ near $0$, where $q\in\mathbb{N}$, $q\geq 2$, $x$ is of class $C^{2}$,
and $x(z)=o(|z|^{q})$. The branching order $B(p)\in\mathbb{N}$ is defined to
be $q-1$. The total branching order of $f$ is

$B(\Sigma):=\sum_{p\in\mathcal{B}_{\Sigma}}B(p).$

The next result is a generalization of the well-known Jorge-Meeks formula [12]
to the case of a possibly branched and non-orientable complete minimal surface
$\Sigma\looparrowright\mathbb{R}^{3}$ of finite total curvature and finite
branching order. It is well-known that each of the (finitely many) ends of
$\Sigma$ is a multi-graph of finite multiplicity over the exterior of a disk
in the plane passing through the origin and orthogonal to the extended value
of the unoriented Gauss map of $\Sigma$. We will use the language the total
spinning of $\Sigma$ to describe the sum of these multiplicities; for
instance, the classical Henneberg and Enneper surfaces each have one end and
total spinning equal to three.

######  Proposition 3.2.

Let $\Sigma\looparrowright\mathbb{R}^{3}$ be a complete, finitely connected
and finitely branched minimal surface with finite total curvature, $e$ ends
with total spinning $S$, and total branching order $B(\Sigma)$. Then:

$\frac{1}{2\pi}\int_{\Sigma}K+S-B(\Sigma)=\chi(\overline{\Sigma})-e=\chi(\Sigma),$
(3.1)

where $K\colon\Sigma\setminus\mathcal{B}_{\Sigma}\to(-\infty,0]$ is the
Gaussian curvature function and $\overline{\Sigma}$ denotes the
conformal333Observe that $\Sigma$ admits an atlas whose changes of coordinates
are conformal or anti-conformal. compactification of $\Sigma$. Furthermore, if
$G\colon\overline{\Sigma}\to\mathbb{P}^{2}$ denotes the extended unoriented
Gauss map of $\Sigma$, then the degree of $G$ satisfies

$\deg(G)=\frac{1}{2\pi}\int_{\Sigma}K\equiv\chi(\overline{\Sigma})\mod 2.$
(3.2)

In particular,

$S-B(\Sigma)\equiv e\mod 2.$ (3.3)

###### Proof.

To prove each of the statements in the above proposition it suffices to
consider the special case that $\Sigma$ is connected, which we will assume
holds for the remainder of the proof.

We first prove (3.2). Note that the total curvature $\int_{\Sigma}K$ equals
$2\pi\deg(G)$. First consider the case that $\deg(G)\neq 0$. By Theorem 1 in
[16], $\deg(G)\equiv\chi(\overline{\Sigma})\mod 2$, which proves that (3.2)
holds in this case. If the degree of the Gauss map is zero, then the image of
the branched immersion is a flat plane, and we can view $\overline{\Sigma}$ as
a connected, finitely branched covering of the sphere. Hence,
$\overline{\Sigma}$ is orientable with even Euler characteristic. Thus, (3.2)
holds in all cases.

Using Gauss-Bonnet in the compact portion of $\Sigma$ obtained by removing
pairwise disjoint disks around its ends (viewed as points in
$\overline{\Sigma}$) and the branch points of $\Sigma$, and taking the radii
of the removed disks going to zero, we obtain equation (3.1). Taking classes
mod 2 in (3.1) and using (3.2), we obtain (3.3). ∎

We next recall a fundamental lower bound for the index $I(f)$ of a connected,
complete, possibly finitely branched minimal surface
$f\colon\Sigma\looparrowright\mathbb{R}^{3}$ with finite total curvature,
which is due to Chodosh and Maximo [6] and to Karpukhin [13]:

$3I(f)\geq\left\\{\begin{array}[]{ll}{\displaystyle
2g(\Sigma)+2\sum_{j=1}^{e}(d_{j}+1)-2B-5}&\mbox{if $\Sigma$ is orientable,}\\\
{\displaystyle g(\widetilde{\Sigma})+2\sum_{j=1}^{e}(d_{j}+1)-2B-4}&\mbox{if
$\Sigma$ is non-orientable,}\end{array}\right.$ (3.4)

where $g(\Sigma)$ is the genus of $\Sigma$ if $\Sigma$ is orientable (resp.
$g(\widetilde{\Sigma})$ is the genus of the orientable cover
$\widetilde{\Sigma}$ of $\Sigma$ if $\Sigma$ is not orientable), $e$ and $B$
are respectively the number of ends and the total branching order of $\Sigma$,
and for each end $E_{j}$ of $\Sigma$, $d_{j}$ is the multiplicity of $E_{j}$
as a multi-graph over the limiting tangent plane of $E_{j}$.

Inequality (3.4) has not been explicitly stated in the literature, so an
explanation is in order. Ros [24] proved that $3I(f)\geq 2g(\Sigma)$ using
harmonic square integrable $1$-forms on $\Sigma$ for a minimal immersion
$f\colon\Sigma\looparrowright\mathbb{R}^{3}$ with finite total curvature, in
order to produce test functions for the index operator of $f$. Chodosh and
Maximo [6, Theorem 1] improved Ros’ technique with an enlarged space of
harmonic $1$-forms which admit certain singularities at the ends of $\Sigma$
that take care of the spinning (multiplicity) of each end of such an immersion
$f$, obtaining a simplified version of (3.4) without the term $-2B$. Finally,
Karpukhin [13, Proposition 2.3 and Remark 2.4] included the study of branch
points although he made use of the original space of $L^{2}(\Sigma)$ harmonic
$1$-forms considered by Ros. Formula (3.4) is the combined inequality that one
can deduce from [6] and [13].

######  Remark 3.3.

  1. 1.

If $\Sigma$ is orientable and the index of $f$ is even, then all summands in
(3.4) except for the $-5$ in the RHS are even. Therefore, the inequality still
holds after adding $1$ to the RHS of (3.4).

  2. 2.

Inequality (3.4) can be expressed in a unified way regardless of the
orientability character of $\Sigma$, if we use the Euler characteristic.
Recall that if $\Sigma$ is orientable, then its Euler characteristic is
$\chi=\chi(\Sigma)=2-2g(\Sigma)-e$, while if $\Sigma$ is non-orientable, the
Euler characteristic of its orientable cover is
$\chi(\widetilde{\Sigma})=2-2g(\widetilde{\Sigma})-2e$ where $e$ is the number
of ends of $\Sigma$, and so, $\chi=\chi(\Sigma)=1-g(\Sigma)-e$. Thus, (3.4)
reduces to

$3I(f)\geq-\chi+2S+e-2B-3,$ (3.5)

where $S=\sum_{j=1}^{r}d_{j}$ is the total spinning of the ends of $f$
(sometimes we will refer to $S$ as the total spinning of $f$).

######  Lemma 3.4.

Let $f\colon\Sigma\looparrowright\mathbb{R}^{3}$ be a complete, connected,
non-flat, finitely branched minimal surface with branch point set
$\mathcal{B}_{\Sigma}\subset\Sigma$. Then:

  1. 1.

If $f$ is stable, then $\Sigma$ is non-orientable and
$f(\mathcal{B}_{\Sigma})$ contains more than 1 point.

  2. 2.

If $\Sigma$ non-orientable with $f(\mathcal{B}_{\Sigma})$ consisting of at
most one point in $\mathbb{R}^{3}$, then $I(f)\geq 2$; in particular, if
$\Sigma$ has exactly one branch point, then $I(f)\geq 2$.

###### Proof.

Assume that $f\colon\Sigma\looparrowright\mathbb{R}^{3}$ is stable. Also,
suppose for the moment that $\Sigma$ is orientable. Let
$g\colon\overline{\Sigma}\to\mathbb{S}^{2}$ be the Gauss map extended to the
conformal compactification $\overline{\Sigma}=\Sigma\cup\mathcal{E}$ of
$\Sigma$ obtained after adding the set $\mathcal{E}$ of its ends. Let
$\mathcal{C}\subset\overline{\Sigma}$ be the set of branch points of $g$. Let
$\Omega(\varepsilon)\subset\mathbb{S}^{2}$ be the complement of the union of a
pairwise disjoint collection of open $\varepsilon$-disks around the points in
the finite set $g(\mathcal{E}\cup\mathcal{B}_{\Sigma}\cup\mathcal{C})$. For
$\varepsilon>0$ sufficiently small, the Schrödinger operator $\Delta+2$ has
negative first Dirichlet eigenvalue on $\bf\Omega(\varepsilon)$, where
$\Delta$ is the spherical Laplacian. Since
$g|_{g^{-1}(\Omega({\varepsilon}))}\colon
g^{-1}(\Omega({\varepsilon}))\to\Omega({\varepsilon})$ is a finite covering,
then each component of $g^{-1}(\Omega({\varepsilon}))$ is a smooth, compact
unstable domain. This contradicts that $f$ is stable, which proves that
$\Sigma$ is non-orientable.

Since $\Sigma$ is non-orientable and $f$ is stable, then the main result in
Ros [24] implies that $\mathcal{B}_{\Sigma}\neq\varnothing$. To finish the
proof of item 1, suppose that $f(\mathcal{B}_{\Sigma})$ is a single point in
$\mathbb{R}^{3}$ (say the origin) and we will find a contradiction. The area
density of $\Sigma$ at the origin is at least $B(\Sigma)+l$, where $B(\Sigma)$
is the total branching order of $f$ and $l$ is the cardinality of
$\mathcal{B}_{\Sigma}$. Using the monotonicity formula for minimal surfaces,
the total spinning $S$ of the ends of $f$ is at least $B(\Sigma)+l+1$. Using
(3.4), since $g(\widetilde{\Sigma})\geq 0$ and $e\geq 1$, we have

$\displaystyle 3I(f)$ $\displaystyle\geq
g(\widetilde{\Sigma})+2\sum_{j=1}^{e}(d_{j}+1)-2B(\Sigma)-4$
$\displaystyle\geq 2S+2e-2B(\Sigma)-4\geq 2S-2B(\Sigma)-2\geq 2l>0,$ (3.6)

which contradicts that $\Sigma$ is stable and proves item 1 of the lemma.

To prove item 2, assume that $\Sigma$ is non-orientable and
$f(\mathcal{B}_{\Sigma})$ contains at most one point. If $f$ is unbranched,
then, by Theorem 1.8 in [6], the index of $f$ is at least 2. So assume that
$\mathcal{B}_{\Sigma}\neq\varnothing$. If $I(f)=1$, then the calculation in
(3.6) implies $l=e=1$ and the total spinning $S$ of the ends of $f$ is
$B(\Sigma)+l+1=B(\Sigma)+2$. But this implies that $S-B(\Sigma)$ is even and
$e$ is odd, which contradicts the last statement of Proposition 3.2. Hence, by
item 1 of the lemma, $I(f)\geq 2$. ∎

##  4 Almost flat annular $H$-multi-graphs of bounded multiplicity

For the next lemma we will need the following notation. For $0<R_{1}<R_{2}$,
we let

$\mathbb{A}(R_{1},R_{2})=\\{x\in\mathbb{R}^{3}\ |\ R_{1}\leq|x|\leq R_{2}\\}.$

Observe that the statement of the next Lemma 4.1 is invariant under
homotheties centered at the origin.

######  Lemma 4.1.

Given $\tau\in(0,\pi/10]$ and $L_{0}>0$, there exists
${\alpha}_{1}\in(0,\tau]$ such that the following property holds. Take
${\alpha}\in(0,{\alpha}_{1}]$, $0<R_{1}\leq R_{2}/2$, and a compact immersed
annulus $\Sigma\subset\mathbb{A}(R_{1},R_{2})$ with
$\partial\Sigma\subset\partial\mathbb{A}(R_{1},R_{2})$, satisfying the
following conditions:

  1. (B1)

$\Sigma$ makes an angle greater than or equal to $\frac{\pi}{2}-{\alpha}$ with
every sphere $\mathbb{S}^{2}(r)$ of radius $r\in[R_{1},R_{2}]$ centered at the
origin.

  2. (B2)

Given $R\in[R_{1},R_{2}/2]$, the image of $\Sigma\cap\mathbb{A}(R,2R)$ through
the Gauss map of $\Sigma$ is contained in the closed spherical neighborhood of
radius ${\alpha}$ centered at some point $v(R)\in\mathbb{S}^{2}(1)$.

  3. (B3)

$\mbox{\rm Length}(\Sigma\cap\mathbb{S}^{2}(R_{1}))<L_{0}R_{1}$.

Then there exist $m\in\mathbb{N}$, $m\leq\frac{L_{0}+1}{2\pi}$, such that for
any $R\in[R_{1},R_{2}/2]$, $\Sigma\cap\mathbb{A}(R,2R)$ consists of an
$m$-valued graph with respect to its projection to the plane $v(R)^{\perp}$
orthogonal to $v(R)$, of a function $u$ that satisfies

$\frac{|u(x)|}{|x|}+|\nabla u|(x)<\frac{\tau}{2}$

at every point $x$ in its domain of definition. Furthermore, for each
$R\in[R_{1},R_{2}]$ the following properties hold:

  1. (C1)

$|\mbox{\rm Length}(\Sigma\cap\mathbb{S}^{2}(R))-2\pi mR|<f_{1}({\alpha})R$,
where $f_{1}=f_{1}({\alpha})\in(0,\tau]$ is a function that tends to zero as
${\alpha}\to 0$.

  2. (C2)

The intrinsic distance between the two boundary components of
$\Sigma\cap\mathbb{A}(R_{1},R)$ is at most $\sqrt{1+\tau^{2}/4}\ (R-R_{1})$.

  3. (C3)

$|\mbox{\rm Area}(\Sigma\cap\mathbb{A}(R_{1},R))-\pi
m(R^{2}-R_{1}^{2})|<f_{2}({\alpha})(R-R_{1})$, where
$f_{2}=f_{2}({\alpha})\in(0,\tau]$ is a function that tends to zero as
${\alpha}\to 0$.

###### Proof.

The first step in the proof consists of showing that for $\tau\in(0,\pi/10]$
and $L_{0}>0$ given, items (C1) and (C2) hold in the range
$R\in[R_{1},2R_{1}]$ for some choice of $m\in\mathbb{N}$,
$m\leq\frac{L_{0}+1}{2\pi}$ depending on a compact immersed annulus $\Sigma$
satisfying (B1), (B2), (B3) provided that $0<{\alpha}\leq{\alpha}_{1}$ and
${\alpha}_{1}$ is sufficiently small.

Observe that if $0<{\alpha}\leq{\alpha}_{1}<\pi/4$, condition (B2) above for
$R=R_{1}$ implies that $\Sigma\cap\mathbb{A}(R_{1},2R_{1})$ is a multi-graph
with respect to its projection to the plane $v(R_{1})^{\perp}$. We call $u$ to
the related graphing function, and $m\in\mathbb{N}$ to its multiplicity.
Taking ${\alpha}_{1}$ sufficiently small, (B2) guarantees that $|\nabla u|$
can be made arbitrarily small. By condition (B1), the almost orthogonality of
$\Sigma$ with spheres $\mathbb{S}^{2}(R)$ with $R\in[R_{1},2R_{1}]$ implies
that if ${\alpha}_{1}$ is sufficiently small, we have that
$\frac{|u(x)|}{|x|}$ can also be made arbitrarily small. In particular, we
have that $\frac{|u(x)|}{|x|}+|\nabla u|(x)<\frac{\tau}{2}$ in
$\Sigma\cap\mathbb{A}(R_{1},2R_{1})$ if ${\alpha}_{1}$ is sufficiently small
in terms of $\tau$. Similar arguments show that the length of
$\Sigma\cap\mathbb{S}^{2}(R_{1})$ differs from $2\pi mR_{1}$ by a function of
${\alpha}$ that tends to zero as ${\alpha}\to 0$ (in particular, $2\pi m\leq
L_{0}+1$ provided that ${\alpha}_{1}$ is sufficiently small). Thus (C1) holds
in $[R_{1},2R_{1}]$ for some function $f_{1}({\alpha})$ that tends to zero as
${\alpha}\to 0$.

Regarding the validity of (C2) in the range $[R_{1},2R_{1}]$, given
$R\in[R_{1},2R_{1}]$ and given a point $x\in\Sigma\cap\mathbb{S}^{2}(R_{1})$,
let $\Pi_{x}\subset\mathbb{R}^{3}$ be the plane passing through the origin
that contains both $v(R_{1})$ and $x$; without loss of generality, assume
$\Pi_{x}$ is the $(x_{1},x_{3})$-plane and $v(R_{1})=(0,0,1)$. Let ${\Gamma}$
be the component of $\Sigma\cap\mathbb{A}(R_{1},R)\cap\Pi_{x}$ that passes
through $x$ and note that ${\Gamma}$ is a smooth embedded arc that can be
parameterized using polar coordinates in $\Pi_{x}$ by
${\Gamma}(r)=(r,\theta(r))$, $r\in[R_{1},R]$. Next assume that ${\alpha}_{1}$
is chosen less than or equal to $\arcsin(\tau/2)$ and we will prove that (C2)
holds. Property (B1) implies that the angle between ${\Gamma}^{\prime}(r)$ and
the radial outward pointing unit vector field $\partial_{r}$ is at most
${\alpha}_{1}$, which implies
$|{\Gamma}^{\prime}(r)|\leq\sqrt{1+\sin^{2}({\alpha}_{1})}\leq\sqrt{1+\tau^{2}/4}$.
Therefore, (C2) holds in $[R_{1},2R_{1}]$.

The second step in the proof consists of demonstrating that (C1) and (C2) hold
for every $R\in[R_{1},R_{2}]$. To see this, it suffices to iterate a finite
number of times the above arguments replacing $R_{1}$ by $2R_{1}$,
$4R_{1},\ldots,2^{k}R_{1}$, where $k\in\mathbb{N}$ is the first positive
integer such that $2^{k}R_{1}>R_{2}/2$. Then we conclude that (C1) and (C2)
hold in $[R_{1},R_{2}/2]$, and by iterating once again replacing $R_{1}$ by
$R_{2}/2$ we get that (C1) and (C2) hold in $[R_{1},R_{2}]$.

Finally, (C3) holds for every $R\in[R_{1},R_{2}]$ by (C1) and the co-area
formula. ∎

######  Remark 4.2.

Since the statement of Lemma 4.1 is invariant under re-scalings of the ambient
metric, we conclude that last lemma holds if we replace the ambient space
$\mathbb{R}^{3}$ by a sufficiently small closed geodesic ball
$\overline{B}_{X}(x,R_{2})$ centered at any point $x\in X$ of radius
$R_{2}\in(0,{\varepsilon}_{0}/2)$ (using harmonic coordinates, see Definition
2.2 and recall that ${\varepsilon}_{0}>0$ is a lower bound for Inj$(X)$) in
the Riemannian 3-manifold $X$, with the following changes:

  1. (D1)

We replace the notion of Gauss map in hypothesis (B2) of Lemma 4.1 by parallel
translation of the unit normal vector to $\Sigma$ at a point
$q\in\Sigma\cap\overline{B}_{X}(x,R_{2})$ along the corresponding radial
geodesic arc joining the point $x$ to $q$.

  2. (D2)

We replace the upper bound in conclusion (C2) of Lemma 4.1 by
$\sqrt{1+\tau^{2}/3}$ times the extrinsic distance in $X$ between the two
boundary components of $[\overline{B}_{X}(x,R_{2})\setminus B_{X}(x,R_{1})]$
(here $0<R_{1}\leq R_{2}/2$).

######  Definition 4.3.

Fix $\tau\in(0,\pi/10]$. Let ${\delta}_{2}\in(0,{\varepsilon}_{0}]$ be such
that Remark 4.2 holds for any choice of extrinsic radii $R_{1},R_{2}$ with
$0<R_{1}\leq R_{2}/2<R_{2}\leq{\delta}_{2}$ in $X$. Fix such $R_{1},R_{2}$,
choose $\tau_{1}\in(0,\tau]$, and let $m\in\mathbb{N}$ be an integer to be
fixed later. Given $x\in X$, we consider the collection

${\mathcal{G}}(x;R_{1},R_{2},\tau_{1},m)$

of multi-graphical (immersed) $H$-annuli
$G\subset\overline{B}_{X}(x,R_{2})\setminus B_{X}(x,R_{1})$ (here
$H\in[0,H_{0}]$) with multiplicity $m(G)\leq m$, such that $G$ is “almost
flat” in terms of $\tau_{1}$, in the sense that $G$ satisfies the following
properties:

  1. (E1)

$G$ is an immersed $H$-annulus in $X$, whose boundary $\partial
G\subset\partial B_{X}(x,R_{1})\cup\partial B_{X}(x,R_{2})$ consists of two
closed curves, one on each ambient geodesic sphere, and $G$ is the graph over
its projection to a “planar” disk $P=\varphi_{x}(\mathbb{D}_{2R_{2}})$ (this
map $\varphi_{x}$ gives harmonic coordinates around $x$), where
$\mathbb{D}_{2R_{2}}\subset T_{x}X$ is a planar disk of radius $2R_{2}$
centered at the origin in $T_{x}X$, of a function $u$ defined on a domain
$\Omega$ of the $m(G)$-sheeted cover of the annulus
$\mathbb{D}_{2R_{2}}\setminus\\{0\\}$.

  2. (E2)

Given $y$ in $P$, denote by $|y|$ the distance to the point $x$ in the ambient
metric of $X$. Then, the graphing function that defines $G$ satisfies

$\frac{|u(y)|}{|y|}+|\nabla u|(y)\leq\tau_{1}\quad\mbox{in $\Omega$}.$

######  Lemma 4.4.

In the situation of Definition 4.3, there exist
${\delta}_{3}\in(0,{\delta}_{2}]$ and $\tau_{1}\in(0,\tau]$ such that for
every $r\in(0,{\delta}_{3}]$ and $G\in\mathcal{G}(x;r/2,r,\tau_{1},m)$, the
geodesic curvature function of $G\cap B_{X}(x,\frac{3}{4}r)$ along its
intersection with $\partial B_{X}(x,\frac{3}{4}r)$ is everywhere positive and
its integral, the total geodesic curvature $\kappa(G)$, satisfies

$\left|\kappa(G)-2\pi m(G)\right|\leq\frac{\tau}{m}.$ (4.1)

Furthermore, every such graph $G$ is stable.

###### Proof.

Suppose that $G_{n}\in\mathcal{G}(x;r_{n}/2,r_{n},\tau_{n},m)$ has
$(r_{n},\tau_{n})\to(0,0)$. Since $\tau_{n}\to 0$, the image of the “Gauss
map” of $G_{n}$ (in the sense of item (D1) in Remark 4.2) is arbitrarily
small. After re-scaling the ambient metric on $X$ by $1/r_{n}$, we find
related multi-graphs $G_{n}^{*}=\frac{1}{r_{n}}G_{n}$ with constant mean
curvature (which is arbitrarily small if $n$ is taken sufficiently large). For
$n$ sufficiently large, $G_{n}^{*}$ is stable (and $G_{n}$ as well). This
implies that there exist ${\delta}^{\prime}_{3}\in(0,{\delta}_{2}]$ and
$\tau^{\prime}_{1}\in(0,\tau]$ such that for every
$r\in(0,{\delta}^{\prime}_{3}]$ and
$G\in\mathcal{G}(x;r/2,r,\tau^{\prime}_{1},m)$, $G$ is stable.

From this point on, we will additionally assume that
$r_{n}\in(0,{\delta}^{\prime}_{3}]$, $\tau_{n}\in(0,\tau^{\prime}_{1}]$ while
(4.1) fails to hold for each $n$. Curvature estimates for stable constant mean
curvature surfaces then imply that there exists $C>0$ (independent of $n$)
such that the norm of the second fundamental form of the intersection of
$G_{n}^{*}$ with $A_{n}^{*}(\frac{5}{8},\frac{7}{8})$ is less than $C$ for all
$n$, where

${\textstyle
A_{n}^{*}(\frac{5}{8},\frac{7}{8}):=\frac{1}{r_{n}}[\overline{B}_{X}(x,\frac{7}{8}r_{n}))\setminus
B_{X}(x,\frac{5}{8}r_{n})].}$

Since $\tau_{n}\to 0$, we conclude that the $G_{n}^{*}\cap
A_{n}^{*}(\frac{5}{8},\frac{7}{8})$ converge as $n\to\infty$ to a flat multi-
graph $G^{*}$ in $\mathbb{R}^{3}$ over the annulus of inner radius $5/8$ and
outer radius $7/8$ (and the convergence $G_{n}^{*}\to G^{*}$ is smooth in the
interior of $G^{*}$), with some multiplicity $m^{*}$ at most $m$ (thus the
multiplicity $m(G_{n})$ of $G_{n}$ equals $m^{*}$ for $n$ large enough).
Clearly, the total geodesic curvature of $G^{*}$ along its intersection with
the sphere $\partial\mathbb{B}(3/4)$ is $2\pi m^{*}$. Since the convergence of
the $G_{n}^{*}$ to $G^{*}$ is smooth in Int$(G^{*})$, we have that
$\kappa(G_{n})=\kappa(G_{n}^{*})$ converges as $n\to\infty$ to $2\pi m^{*}$,
which equals $2\pi m(G_{n})$ for $n$ large enough. Since $\frac{\tau}{m}>0$,
then (4.1) holds for $n$ large enough, which is contrary to our hypothesis,
and so, the lemma is proved. ∎

######  Definition 4.5.

Fix $L_{0}>0$ and $m\in\mathbb{N}$, $m\leq\frac{L_{0}+1}{2\pi}$. Let
${\alpha}_{1}={\alpha}_{1}(L_{0})\in(0,\tau]$ be the value given by Lemma 4.1
(recall that $\tau\in(0,\pi/10]$ is fixed). Choose
${\delta}_{3}\in(0,{\delta}_{2}]$, $\tau_{1}\in(0,{\alpha}_{1}]$ given by
Lemma 4.4 such that (4.1) holds for every
$G\in\mathcal{G}(x;{\delta}_{3}/2,{\delta}_{3},\tau_{1},m)$.

Observe that both ${\delta}_{3}$ and $\tau_{1}$ depend on the values of
$L_{0}$ and $m$. We will describe later how to choose $L_{0},m$ that give rise
to ${\delta}_{3},\tau_{1}$ by Lemma 4.4, in order to define the values of
${\delta}_{1},{\delta}$ that appear in Theorem 1.2.

##  5 The proof of Structure Theorem 1.2

Consider numbers ${\varepsilon}_{0}>0$, $K_{0},H_{0},A_{0}\in[0,\infty)$,
$I\in\mathbb{N}\cup\\{0\\}$ and $\tau\in(0,\pi/10]$ and let
$\Lambda=\Lambda(I,H_{0},{\varepsilon}_{0},A_{0},K_{0})$ be the space of CMC
immersions given in Definition 1.1.

###  5.1 The case of uniformly bounded second fundamental form and the proof
of item (E) of Theorem 1.2 in the general case

Suppose that the norms of the second fundamental forms of all immersions
$F\in\Lambda$ are bounded by a constant $A_{1}$ independent of $F$ (clearly
one can assume $A_{1}\geq A_{0}$). In this case, Theorem 1.2 holds with the
choices $k=0$ (there are no radii $r_{F}(i)$ or components $\Delta_{i}$),
$2{\delta}_{1}={\delta}={\delta}_{3}$ (this ${\delta}_{3}$ is given by
Definition 4.5 for $(L_{0},m)=(2\pi+1,1)$), and $M=\widetilde{M}$, because:

  1. (F1)

Items 1, 2, (A), (B) and (D) of Theorem 1.2 are vacuous.

  2. (F2)

Item 3 of Theorem 1.2 holds by assumption and item (C) reduces to
$g(M)=g(\widetilde{M})$.

  3. (F3)

We next prove that item (E) of Theorem 1.2 holds without the assumption that
the norms of the second fundamental forms of all immersions $F\in\Lambda$ are
bounded by a constant $A_{1}$ independent of $F$; this will complete the proof
of item (E) in the general case. In order to find the constant
$C=C({\varepsilon}_{0},K_{0},H_{0})>0$ that satisfies item (E), we distinguish
two cases.

    1. (F3.A)

First suppose that $\partial M\neq\varnothing$. By item (A2) in the definition
of $\Lambda$, there exists a point $p_{0}\in\mbox{\rm Int}(M)$ such that
$B_{M}(p_{0},{\varepsilon}_{0})$ is contained in the interior of $M$. By
inequality (8.5) in Proposition 8.3,

$\mathrm{Area}(M)\geq\mathrm{Area}(B_{M}(p_{0},{\varepsilon}_{0}))\geq
C_{A}{\varepsilon}_{0},$ (5.1)

where the constants $r_{2}=r_{2}({\varepsilon}_{0},K_{0},H_{0})>0$,
$C_{A}=\min\\{{\varepsilon}_{0},\frac{r_{2}^{2}}{{\varepsilon}_{0}}\\}>0$ are
given by Proposition 8.3. Given any $y\in M$ such that $d_{M}(y,\partial
M)\geq{\varepsilon}_{0}$, then (8.4) in Proposition 8.3 gives

$\mathrm{Area}(M)\geq\mathrm{Area}(B_{M}(y,d_{M}(y,\partial M)))\geq
C_{A}\,d_{M}(y,\partial M).$ (5.2)

Define $C_{0}=\min\\{C_{A}{\varepsilon}_{0},C_{A}\\}>0$, which only depends
${\varepsilon}_{0},K_{0},H_{0}$ but not on $I$. We claim that

$\mbox{\rm Area}(M)\geq C_{0}\max\\{1,\mbox{Radius}(M)\\}$ (5.3)

(which proves item (E) of Theorem 1.2 in this case (F3.A)): if
$\mbox{Radius}(M)\leq 1$, then our claim follows from (5.1). If
$\mbox{Radius}(M)>1$, then our claim follows from (5.2) since
$\mbox{Radius}(M)=\sup\\{d_{M}(y,\partial M)\mid d_{M}(y,\partial
M)\geq{\varepsilon}_{0}\\}$.

    2. (F3.B)

Next assume that $\partial M=\varnothing$. Since the sectional curvature of
$X$ is bounded from above by $K_{0}$, then the Ricci curvature of $X$ is
bounded from above by $2K_{0}$. It follows that there exists
${\varepsilon}_{1}={\varepsilon}_{1}(K_{0},H_{0})>0$ such that for any point
$x\in X$, the geodesic spheres of radius at most ${\varepsilon}_{1}$ are
embedded with mean curvature greater than $H_{0}$. By the mean curvature
comparison principle, for any point $p\in M$, there is a least one other point
$q\in M$ such that the extrinsic distance
$d_{X}(F(p),F(q))>{\varepsilon}_{1}$, and hence, the intrinsic distance
$d_{M}(p,q)>{\varepsilon}_{1}$. Define

$C_{A}^{1}=\min\left\\{{\varepsilon}_{1},\frac{r_{2}^{2}}{{\varepsilon}_{1}}\right\\}>0,$

where $r_{2}=r_{2}({\varepsilon}_{0},K_{0},H_{0})>0$ is given by Proposition
8.3, and let

$C_{1}=\min\\{C_{A}^{1}{\varepsilon}_{1},C_{A}^{1}\\}.$

Observe that $C_{A}^{1},C_{1}$ depend only on ${\varepsilon}_{0},K_{0},H_{0}$
but not on $I$. We claim that

$\mbox{\rm Area}(M)\geq C_{1}\max\\{1,\mbox{\rm Diameter}(M)\\}$ (5.4)

(which proves item E of Theorem 1.2 in this case (F3.B)).

To prove that (5.4) holds, first note that if $\mbox{\rm Diameter}(M)=\infty$,
then $M$ is non-compact and it has infinite area by Corollary 8.2.

Assume now that $\mbox{\rm Diameter}(M)<\infty$. Since $M$ is compact, the
Hopf-Rinow theorem ensures that there exist points $p,q\in M$ such that
$\mbox{Diameter}(M)=d_{M}(p,q)$. Notice that for $n\in\mathbb{N}$ such that
$\mbox{\rm Diameter}(M)>\frac{1}{n}$, the triangle inequality implies

${\textstyle\mbox{\rm Diameter}(M)-\frac{1}{n}=\mbox{\rm Radius}(M\setminus
B_{M}(q,\frac{1}{n})),}$

and so,

$\mbox{\rm Diameter}(M)=\lim_{n\to\infty}\mbox{\rm Radius}(M\setminus
B_{M}(q,\textstyle{\frac{1}{n}})).$ (5.5)

By our choice of ${\varepsilon}_{1}$ and for $n$ sufficiently large, the point
$p\in M\setminus B_{M}(q,\frac{1}{n})$ is at distance at least
${\varepsilon}_{1}$ from $\partial(M\setminus B_{M}(q,\frac{1}{n}))$, and so,
in this case the restriction of the immersion $F\colon M\looparrowright X$ to
$M\setminus B_{M}(q,\frac{1}{n})$ satisfies the hypotheses of Proposition 8.3.
Therefore, by Proposition 8.3 and (5.3) with $C_{0}$ replaced by $C_{1}$,

$\mbox{\rm Area}(M\setminus B_{M}(q,\textstyle{\frac{1}{n}}))\geq
C_{1}\max\\{1,\mbox{\rm Radius}(M\setminus
B_{M}(q,\textstyle{\frac{1}{n}}))\\},$ (5.6)

hence

$\displaystyle\mbox{\rm Area}(M)$ $\displaystyle=$
$\displaystyle\lim_{n\to\infty}\mbox{\rm Area}(M\setminus
B_{M}(q,\textstyle{\frac{1}{n}}))$
$\displaystyle\stackrel{{\scriptstyle\eqref{7.6}}}{{\geq}}$
$\displaystyle\lim_{n\to\infty}C_{1}\max\\{1,\mbox{\rm Radius}(M\setminus
B_{M}(q,\textstyle{\frac{1}{n}}))\\}$
$\displaystyle\stackrel{{\scriptstyle\eqref{7.5}}}{{=}}$ $\displaystyle
C_{1}\max\\{1,\mbox{\rm Diameter}(M)\\},$

which proves (5.4) holds.

From (F3.A) and (F3.B) we deduce that item (E) of Theorem 1.2 holds for the
value $C=\min\\{C_{0},C_{1}\\}$, regardless of whether or not the norms of the
second fundamental forms of all immersions $F\in\Lambda$ are bounded.

In the sequel we will assume that there is no uniform bound for the norms of
the second fundamental forms of surfaces in $\Lambda$.

###  5.2 Stable pieces of $H$-surfaces in $\Lambda$ and their curvature
estimate

By Theorem 7.1 and with the notation there, there exists a universal constant
$C_{s}>0$ such that given a stable $H$-immersion $F\colon M\looparrowright X$,

$|A_{M}|(p)\leq\frac{C_{s}}{\min\\{{\varepsilon}_{0},d_{M}(p,\partial
M),\frac{\pi}{2\sqrt{K_{0}}}\\}},\quad\forall p\in M.$ (5.7)

Define $\widehat{C}_{s}\colon(0,{\varepsilon}_{0}]\to(0,\infty)$ by

$\widehat{C}_{s}({\varepsilon})=1+\max\\{A_{0},\frac{2C_{s}}{\min\\{{\varepsilon},\frac{\pi}{\sqrt{K_{0}}}\\}}\\},\quad{\varepsilon}\in(0,{\varepsilon}_{0}].$
(5.8)

It follows that if $F\colon M\looparrowright X$ lies in
$\Lambda=\Lambda(I,H_{0},{\varepsilon}_{0},A_{0},K_{0})$ and $p\in M$
satisfies $|A_{M}|(p)>\widehat{C}_{s}({\varepsilon})$, then $p\in U(\partial
M,{\varepsilon}_{0},\infty)$ and the intrinsic ball centered at $p$ of radius
${\varepsilon}/2$ is unstable.

######  Lemma 5.1.

Let $F\colon M\looparrowright X$ be an element in $\Lambda$ and
${\varepsilon}\in(0,{\varepsilon}_{0}]$ such that
$\sup|A_{M}|>\widehat{C}_{s}({\varepsilon})$. Then, there exists a finite
subset $\\{q_{1},\ldots,q_{k}\\}\subset U(\partial
M,{\varepsilon}_{0},\infty)$ with $1\leq k=k(F)\leq I$, such that

  1. 1.

$|A_{M}|$ achieves its maximum in $M$ at $q_{1}$, and for $i=2,\ldots,k$,
$|A_{M}|$ achieves its maximum in
$M\setminus[B_{M}(q_{1},{\varepsilon})\cup\ldots\cup
B_{M}(q_{i-1},{\varepsilon})]$ at $q_{i}$.

  2. 2.

For each $i=1,\ldots,k$, $|A_{M}|(q_{i})>\widehat{C}_{s}({\varepsilon})$, and
so, the pairwise disjoint intrinsic balls $B_{M}(q_{i},{\varepsilon}/2)$ are
unstable.

  3. 3.

$|A_{M}|\leq\widehat{C}_{s}({\varepsilon})$ in
$M\setminus[B_{M}(q_{1},{\varepsilon})\cup\ldots\cup
B_{M}(q_{k},{\varepsilon})]$, and so, $|A_{M}|$ is bounded on $M$.

###### Proof.

Since $\sup|A_{M}|>\widehat{C}_{s}({\varepsilon})$, we can find
$q^{\prime}_{1}\in M$ such that
$|A_{M}|(q_{1}^{\prime})>\widehat{C}_{s}({\varepsilon})$. In particular, the
intrinsic disk $B_{M}(q_{1}^{\prime},{\varepsilon}/2)$ is unstable. We now
distinguish two possibilities: if $|A_{M}|\leq\widehat{C}_{s}({\varepsilon})$
on $M\setminus B_{M}(q_{1}^{\prime},{\varepsilon})$, then $|A_{M}|$ is
globally bounded on $M$. Otherwise, there exists $q^{\prime}_{2}\in M\setminus
B_{M}(q_{1}^{\prime},{\varepsilon})$ such that
$|A_{M}|(q_{2}^{\prime})>\widehat{C}_{s}({\varepsilon})$. In particular,
$B_{M}(q_{2}^{\prime},{\varepsilon}/2)$ is unstable. Observe that
$B_{M}(q_{1}^{\prime},{\varepsilon}/2)$,
$B_{M}(q^{\prime}_{2},{\varepsilon}/2)$ are disjoint. Again we discuss two
possibilities depending on whether or not
$|A_{M}|\leq\widehat{C}_{s}({\varepsilon})$ on
$M\setminus[B_{M}(q_{1}^{\prime},{\varepsilon})\cup
B_{M}(q_{2}^{\prime},{\varepsilon})]$. In the first case, $|A_{M}|$ is bounded
on $M$; in the second case, we repeat the argument of finding a point
$q_{3}^{\prime}\in M\setminus[B_{M}(q_{1}^{\prime},{\varepsilon})\cup
B_{M}(q_{2}^{\prime},{\varepsilon})]$ such that
$|A_{M}|(q_{3}^{\prime})>\widehat{C}_{s}({\varepsilon})$,
$B_{M}(q_{3}^{\prime},{\varepsilon}/2)$ is unstable and the collection
$\\{B_{M}(q_{i}^{\prime},{\varepsilon}/2)\ |\ i=1,2,3\\}$ is pairwise
disjoint. Since the index of $F$ is at most $I$, we cannot repeat this process
of finding pairwise disjoint unstable domains more than $I$ times, say that we
can do it $k^{\prime}\leq I$ times. Therefore, we conclude that
$|A_{M}|\leq\widehat{C}_{s}({\varepsilon})$ in
$M\setminus[B_{M}(q^{\prime}_{1},{\varepsilon})\cup\ldots\cup
B_{M}(q^{\prime}_{k^{\prime}},{\varepsilon})]$; in particular $|A_{M}|$ is
bounded on $M$. We next replace $q^{\prime}_{1}$ by a maximum $q_{1}$ of
$|A_{M}|$ in $M$ (which occurs in the compact set
$\overline{B}_{M}(q^{\prime}_{1},{\varepsilon})\cup\ldots\cup\overline{B}_{M}(q^{\prime}_{k^{\prime}},{\varepsilon})$),
$q^{\prime}_{2}$ by a maximum $q_{2}$ of $|A_{M}|$ in
$W_{1}=[\overline{B}_{M}(q^{\prime}_{1},{\varepsilon})\cup\ldots\cup\overline{B}_{M}(q^{\prime}_{k^{\prime}},{\varepsilon})]\setminus
B_{M}(q_{1},{\varepsilon})$ (if $|A_{M}|$ restricted to $W_{1}$ is greater
than $\widehat{C}_{s}({\varepsilon})$), and repeat the process to obtain a
finite set of points $\\{q_{1},\ldots,q_{k}\\}$. Observe that the number $k$
of these points cannot be greater than $I$. Now the lemma holds. ∎

###  5.3 Strategy of the proof of Theorem 1.2

Given $t\geq\widehat{C}_{s}({\varepsilon}_{0})$, let $\Lambda_{t}$ be the
subset of $\Lambda$ consisting of those immersions $F\colon M\looparrowright
X$ such that $\sup\\{|A_{M}|(p)\ |\ p\in M\\}>t$. Similar arguments as those
in Section 5.1 show that Theorem 1.2 holds for immersions in
$\Lambda\setminus\Lambda_{t}$, with the choices $k=0$, $A_{1}=t$,
$2{\delta}_{1}={\delta}={\delta}_{3}$ given by Definition 4.5 for
$(L_{0},m)=(2\pi+1,1)$ and $M=\widetilde{M}$. So the theorem will be proven if
we show that it holds for immersions in $\Lambda_{t}$ for some large
$t\geq\widehat{C}_{s}({\varepsilon}_{0})$.

Observe that if $I=0$, then $\widehat{C}_{s}({\varepsilon}_{0})$ is a uniform
bound for the norm of the second fundamental forms of surfaces in $\Lambda$,
and the theorem holds in this case.

The strategy to prove the theorem consists of proving the following two steps:

  1. Step 1.

Items 1, 2, 3 of the theorem hold. This will be proven by induction on $I$, by
analyzing local pictures of a sequence of immersions $\\{F_{n}\colon
M_{n}\looparrowright X_{n}\\}_{n}\subset\Lambda$ whose second fundamental
forms blow up as $n\to\infty$. We will do this in Sections 5.4 and 5.5.

  2. Step 2.

If items 1, 2, 3 of the theorem hold, then items (A)-…-(D) also hold for a
possibly larger choice of $A_{1}$ (recall that we proved item (E) of Theorem
1.2 in item (F3) of Section 5.1). For this part, we will verify that the
induction argument in Step 1 can be carried out so that items (A)-…-(D) hold
for $F_{n}$ with $n$ large enough. This step will be done in Section 5.7,
which in turn needs some results in Section 5.6.

Our next goal is to complete Step 1. Although not strictly needed in the
induction process, we first explain the arguments needed to prove the case
$I=1$ since they will help clarify why items 1, 2, 3 of the theorem hold for
$I+1$ provided that they hold for $I$.

###  5.4 Proof of items 1, 2, 3 of Theorem 1.2 for $I=1$

Assume $I=1$. By previous arguments, we can assume that for each
$n>\widehat{C}_{s}({\varepsilon}_{0})$ there exists an $H_{n}$-immersion
$F_{n}\colon M_{n}\looparrowright X_{n}$ in $\Lambda$ such that
$\sup|A_{M_{n}}|>n$ with $H_{n}\in[0,H_{0}]$. We will next describe the local
picture of any such sequence $\\{F_{n}\\}_{n}$ around points of concentrated
norm of their second fundamental forms. As $I=1$, Lemma 5.1 gives that for
each $n>\widehat{C}_{s}({\varepsilon}_{0})$, there is a point $p_{1}(n)\in
U(\partial M_{n},{\varepsilon}_{0},\infty)$ where $|A_{M_{n}}|$ achieves its
maximum and $|A_{M_{n}}|\leq\widehat{C}_{s}({\varepsilon}_{0})$ in
$M_{n}\setminus B_{M_{n}}(p_{1}(n),{\varepsilon}_{0})$.

####  5.4.1 Local pictures around points where $|A_{M}|>t$, for $t$
sufficiently large

Next we will adapt some arguments in [19] to this immersed setting. Given
$n>\widehat{C}_{s}({\varepsilon}_{0})$, observe that the (unique) maximum of
the function
$h_{n}\colon\overline{B}_{M_{n}}(p_{1}(n),{\varepsilon}_{0})\to[0,\infty)$
given by

$h_{n}=|A_{M_{n}}|\,d_{M_{n}}(\cdot,\partial
B_{M_{n}}(p_{1}(n),{\varepsilon}_{0}))$ (5.9)

is attained at $p_{1}(n)$. Define ${\lambda}_{n}=|A_{M_{n}}|(p_{1}(n))$.
Following the arguments at the beginning of the proof of Theorem 1 in [19], we
have that:

  1. (G1)

${\lambda}_{n}$ tends to infinity as $n\to\infty$.

  2. (G2)

For $r>0$ fixed, the sequence of extrinsic balls
$\\{{\lambda}_{n}B_{X_{n}}(F_{n}(p_{1}(n)),r/{\lambda}_{n})\\}_{n}$ converges
$C^{1,{\alpha}}$, ${\alpha}\in(0,1)$, as $n\to\infty$ to the open ball
$\mathbb{B}(r)$ of radius $r$ centered at the origin $\vec{0}$ in
$\mathbb{R}^{3}$ with its usual metric, where we have used harmonic
coordinates in $X_{n}$ centered at $p_{1}(n)$ and identified $p_{1}(n)$ with
$\vec{0}$.

  3. (G3)

The intrinsic balls ${\lambda}_{n}B_{M_{n}}(p_{1}(n),r/{\lambda}_{n})$ can be
considered to be a sequence of pointed immersions with constant mean curvature
$H_{n}/{\lambda}_{n}$ (observe that $H_{n}/{\lambda}_{n}$ is arbitrarily small
for $n$ sufficiently large) and non-empty topological boundary.

  4. (G4)

For $n$ large, the immersed surface
${\lambda}_{n}B_{M_{n}}(p_{1}(n),r/{\lambda}_{n})$ passes through $\vec{0}$
with norm of its second fundamental form equal to $1$ at this point.
Furthermore, the norms of the second fundamental forms of
${\lambda}_{n}B_{M_{n}}(p_{1}(n),r/{\lambda}_{n})$ are everywhere less than or
equal to 1.

  5. (G5)

After extracting a subsequence, the
${\lambda}_{n}B_{M_{n}}(p_{1}(n),r/{\lambda}_{n})$ converge $C^{1,{\alpha}}$
as mappings to a relatively compact pointed minimal immersion
$f_{r}\colon\Sigma(r)\looparrowright\mathbb{B}(r)$ that passes through
$\vec{0}$, with bounded Gaussian curvature and index at most $1$,
$|A_{\Sigma(r)}|(\vec{0})=1$ and $|A_{\Sigma(r)}|\leq 1$ on $\Sigma(r)$.

  6. (G6)

Defining $\Sigma=\bigcup_{r\geq 1}\Sigma(r)$ and
$f\colon\Sigma\looparrowright\mathbb{R}^{3}$ by $f|_{\Sigma(r)}=f_{r}$, we
produce a complete pointed minimal immersion with index at most $1$,
$\vec{0}\in\Sigma$, $|A_{\Sigma}|(\vec{0})=1$ and $|A_{\Sigma}|\leq 1$ on
$\Sigma$.

Since $f$ is not flat at the origin, then the index of $f$ is 1. In this
setting, López and Ros [14] proved that if $\Sigma$ is orientable, then $f$ is
either a catenoid or an Enneper minimal surface. Theorem 1.8 in Chodosh and
Maximo [6] gives that $\Sigma$ must be orientable.

We next show that items 1, 2, 3 of Theorem 1.2 hold in this case $I=1$ with
the choice $k=1$. Observe that the multiplicity of the end of the Enneper
surface is $m=3$, and the total multiplicity of the ends of a catenoid is $2$.
This motivates the choice of $L_{0}$ in the next paragraph. We next explain
how to choose the constants $A_{1}$ and ${\delta}_{1},{\delta}$ that appear in
the main statement of Theorem 1.2.

Let ${\alpha}_{1}={\alpha}_{1}(\tau)\in(0,\tau]$ be the constant given by
Lemma 4.1 for $L_{0}=6\pi+1$; observe that the length of the intersection of a
catenoid or an Enneper minimal surface with a sphere $\mathbb{S}^{2}(R)$ of
sufficiently large radius $R$ is less than $L_{0}R$.

We can also pick a smallest $R>0$ (only depending on $\tau$) so that the
following properties hold:

  1. (H0)

The index of $f(\Sigma)\cap\mathbb{B}(R/3)$ is $1$.

  1. (H1)

$f(\Sigma)\setminus\mathbb{B}(R/3)$ consists of one or two multi-graphs over
its projection to a plane $\Pi\subset\mathbb{R}^{3}$ that passes though
$\vec{0}$; here $\Pi$ is the limit tangent plane at infinity for $f$.

  2. (H2)

The image through the Gauss map of $f$ of each component $C_{j}$ of
$f(\Sigma)\setminus\mathbb{B}(R/3)$ is contained in the spherical neighborhood
of radius ${\alpha}_{1}/2$ centered at a point $v\in\mathbb{S}^{2}(1)$
perpendicular to $\Pi$ (thus, $C_{j}$ satisfies condition (B2) of Lemma 4.1
with $R_{1}=R/3$ and ${\alpha}={\alpha}_{1}/2$).

  3. (H3)

$f(\Sigma)$ makes an angle greater than $\frac{\pi}{2}-\frac{{\alpha}_{1}}{2}$
with every sphere $\mathbb{S}^{2}(r)$ of radius $r\geq R/3$ centered at the
origin (so, $C_{j}$ satisfies condition (B1) of Lemma 4.1 with $R_{1}=R/3$ and
${\alpha}={\alpha}_{1}/2$).

  4. (H4)

The length of each component of the intersection of $f(\Sigma)$ with any
sphere $\mathbb{S}^{2}(r)$ centered at the origin and radius $r\geq R/3$ is
less than $(L_{0}-\frac{1}{2})r$ (hence each component of
$f(\Sigma)\setminus\mathbb{B}(R/3)$ satisfies condition (B3) of Lemma 4.1 with
$R_{1}=R/3$).

Applying the estimate (8.7) in Proposition 8.4 with $I=1$ and $B=0$, we
deduce:

  1. (H5)

By item 2 of Proposition 8.4, the intrinsic distance in the pullback metric by
$f$ from $\vec{0}\in\Sigma$ to any point in the boundary of
$f^{-1}(\overline{\mathbb{B}}(R/2))$ is at most $\widehat{C}\frac{R}{2}$,
where $\widehat{C}$ is defined there. Observe that (8.6) is not enough to
estimate this intrinsic distance, since it only gives that the intrinsic
distance in the pullback metric by $f$ from $\vec{0}\in\Sigma$ to the boundary
of $f^{-1}(\overline{\mathbb{B}}(R/2))$ is at most
$\widehat{L}\frac{R}{2}=\frac{\sqrt{3}}{2}R$.

######  Definition 5.2.

Given $r\in[R/2,4R]$, we denote by ${\Delta}_{n}(r)\subset M_{n}$ the
connected component of
$({\lambda}_{n}F_{n})^{-1}({\lambda}_{n}\overline{B}_{X_{n}}(F_{n}(p_{1}(n)),\frac{r}{{\lambda}_{n}}))$
that contains $p_{1}(n)$.

Properties (H0)-…-(H5) and the convergence in (G5)-(G6) imply that for
${\lambda}_{n}$ large (in particular, for $n$ sufficiently large), the
immersion ${\lambda}_{n}F_{n}$ satisfies:

  1. (I0)

The index of $({\lambda}_{n}F_{n})|_{{\Delta}_{n}(R/2)}$ equals $1$.

  1. (I1)

$({\lambda}_{n}F_{n})({\Delta}_{n}(4R)\setminus{\Delta}_{n}(R/2))$ consists of
one or two multi-graphs over their projections to $\Pi$. We let
$\widetilde{G}_{n}$ denote any of these multi-graphs inside
$({\lambda}_{n}F_{n})({\Delta}_{n}(4R)\setminus{\Delta}_{n}(R/2))$.

  2. (I2)

The image of $\widetilde{G}_{n}$ through the “Gauss map” of
${\lambda}_{n}F_{n}$ (defined through ambient parallel translation, see Remark
4.2) is contained in the spherical neighborhood of radius ${\alpha}_{1}$
centered at $v$ (here we have identified $\mathbb{R}^{3}$ with the tangent
space to ${\lambda}_{n}X_{n}$ at $F_{n}(p_{1}(n))$).

  3. (I3)

$\widetilde{G}_{n}$ makes an angle greater than $\frac{\pi}{2}-{\alpha}_{1}$
with every geodesic sphere $\widetilde{S}(r)$ in ${\lambda}_{n}X_{n}$ centered
at $F_{n}(p_{1}(n))$ of radius $r\in[R/2,4R]$.

  4. (I4)

$\mbox{\rm Length}[\widetilde{G}_{n}\cap\widetilde{S}(R/2)]<L_{0}R/2$.

  5. (I5)

The intrinsic distance in the pullback metric by ${\lambda}_{n}F_{n}$ on
$M_{n}$, from $p_{1}(n)$ to any point in the boundary of $\Delta_{n}(R/2)$ is
at most $(\widehat{C}/2+1)R$.

Back in the original scale, observe that $\Delta_{n}(r)\subset
F_{n}^{-1}(\overline{B}_{X}(F_{n}(p_{1}(n)),\frac{r}{{\lambda}_{n}}))$ for any
$r\in[R/2,4R]$, and the following properties hold for $n$ sufficiently large:

  1. (J0)

The index of $F_{n}|_{\Delta_{n}(R/2)}$ equals $1$.

  1. (J1)

$F_{n}(\Delta_{n}(4R)\setminus\Delta_{n}(R/2))$ is a union of one or two
multi-graphs over their projections to $\Pi$. We let $G_{n}$ denote any of
these multi-graphs.

  2. (J2)

The image of $G_{n}$ through the “Gauss map” of $F_{n}$ is contained in the
spherical neighborhood of radius ${\alpha}_{1}$ centered at $v$.

  3. (J3)

$G_{n}$ makes an angle greater than $\frac{\pi}{2}-{\alpha}_{1}$ with every
geodesic sphere $S(r)$ in $X_{n}$ centered at $F_{n}(p_{1}(n))$ of radius
$r\in[\frac{R}{2{\lambda}_{n}},\frac{4R}{{\lambda}_{n}}]$.

  4. (J4)

$\mbox{\rm Length}[G_{n}\cap
S(\frac{R}{2{\lambda}_{n}})]<L_{0}\frac{R}{2{\lambda}_{n}}$.

  5. (J5)

The intrinsic distance in the pullback metric by $F_{n}$ on $M_{n}$, from
$p_{1}(n)$ to any point in the boundary of $\Delta_{n}(R/2)$ is at most
$\frac{1}{{\lambda}_{n}}(\widehat{C}/2+1)R$.

Therefore, given $r\in[\frac{R}{2{\lambda}_{n}},\frac{2R}{{\lambda}_{n}}]$,
$G_{n}\cap[\overline{B}_{X}(F_{n}(p_{1}(n)),2r)\setminus
B_{X}(F_{n}(p_{1}(n)),r)]$ satisfies the hypotheses (B1), (B2), (B3) of Lemma
4.1 with the choices $L_{0}=6\pi+1$, inner radius $r$, outer radius $2r$ and
${\alpha}={\alpha}_{1}$. Our next step will be demonstrating that the outer
radius for which the hypotheses of Lemma 4.1 hold for $F_{n}$, is bounded from
below by some positive constant, independent of the sequence.

####  5.4.2 Local pictures have a uniform size

######  Proposition 5.3.

There exists ${\delta}_{4}\in(0,{\delta}_{3}]$ (this
${\delta}_{3}\in(0,{\delta}_{2}]$ was given in Definition 4.5 for the choices
$L_{0}=6\pi+1$ and $m=3$) such that the hypotheses of Lemma 4.1 hold for
annular enlargements of the multi-graphs $G_{n}$ between the geodesic spheres
in $X$ centered at $F_{n}(p_{1}(n))$ of radii $R_{1}=\frac{R}{2{\lambda}_{n}}$
and $R_{2}={\delta}_{4}$, and with the choice ${\alpha}=\tau_{1}$ for
hypotheses (B1), (B2) (this $\tau_{1}\in(0,{\alpha}_{1}]$ was also introduced
in Definition 4.5).

###### Proof.

Define $r_{n}$ as the supremum of the extrinsic radii $r\geq 4R/{\lambda}_{n}$
such that annular enlargements of the $G_{n}$ satisfy conditions (B1), (B2),
(B3) of Lemma 4.1 for the choices $L_{0}=6\pi+1$, inner radius
$R_{1}=\frac{R}{2{\lambda}_{n}}$, outer radius $R_{2}=r$ and
${\alpha}={\alpha}_{1}$. We will prove the proposition by contradiction, so
suppose $r_{n}\to 0$ as $n\to\infty$.

Rescale $F_{n}$ by expanding the ambient metric of $X_{n}$ by the factor
$1/r_{n}$ centered at $F_{n}(p_{1}(n))$ and denote the resulting sequence of
rescaled immersions by $\frac{1}{r_{n}}F_{n}\colon
M_{n}\looparrowright\frac{1}{r_{n}}X_{n}$. Our goal is to understand the limit
of (a subsequence of) $\\{\frac{1}{r_{n}}F_{n}\\}_{n}$.

Notice that $4R\leq{\lambda}_{n}r_{n}$ must go to infinity as $n\to\infty$:
otherwise, $\frac{1}{r_{n}}F_{n}$ is rescaled from $F_{n}$ on the scale of the
second fundamental form, and in that case we have proved that the
subsequential limit of the $\frac{1}{r_{n}}F_{n}$ is a catenoid or an Enneper
minimal surface, each of whose ends satisfies Lemma 4.1 for every outer
radius— see properties (H1),(H2),(H3) above —, contradicting the definition as
a supremum of $r_{n}$.

As ${\lambda}_{n}r_{n}\to\infty$, then property (J0) implies that
$\frac{1}{r_{n}}F_{n}$ has index zero away from the origin for $n$ large; more
precisely, the following property holds:

  1. $(\diamondsuit)$

For any $s>0$ and for every $n\in\mathbb{N}$ sufficiently large (depending
only on $s$), the portion of $\frac{1}{r_{n}}F_{n}(M_{n})$ outside of the
intrinsic ball of radius $s$ centered at $F_{n}(p_{1}(n))$ is stable.

By curvature estimates for stable $H$-surfaces, we deduce that the sequence
$\\{\frac{1}{r_{n}}F_{n}\\}_{n}$ has locally bounded second fundamental form
in $\mathbb{R}^{3}\setminus\\{\vec{0}\\}$.

Applying Lemma 4.1 (see also Remark 4.2) to $\frac{1}{r_{n}}F_{n}$ with
${\alpha}=\tau_{1}$, we conclude that for $n$ large, the image of
$\frac{1}{r_{n}}F_{n}$ contains an immersed annulus
$\Omega_{n}(\frac{1}{2},1)$ in the annular region
$\mathbb{A}(\frac{1}{2},1)=\\{x\in\mathbb{R}^{3}\ |\ \frac{1}{2}\leq|x|\leq
1\\}$, and $\Omega_{n}(\frac{1}{2},1)$ is an $m^{\prime}$-valued graph with
respect to its projection to a plane $v_{1}^{\perp}$ passing through the
origin (the multiplicity $m^{\prime}$ of this graph does not depend on $n$
after passing to a subsequence; in fact $m^{\prime}=1$ or $3$; similarly, the
plane $v_{1}^{\perp}$ is independent of $n$). Observe that by definition of
$r_{n}$, either $\Omega_{n}(\frac{1}{2},1)$ makes an angle of
$\frac{\pi}{2}-\tau_{1}$ with $\mathbb{S}^{2}(1)$ at some point of
$\Omega_{n}(\frac{1}{2},1)\cap\mathbb{S}^{2}(1)$, or the Gauss map image of
$\Omega_{n}(\frac{1}{2},1)$ contains two points at spherical distance
$\tau_{1}$ apart.

After passing to a subsequence, $\Omega_{n}(\frac{1}{2},1)$ converges smoothly
as $n\to\infty$ to an immersed minimal annulus $A$ in
$\mathbb{A}(\frac{1}{2},1)$ which is a multi-graph of multiplicity
$m^{\prime}$ with respect to $v_{1}^{\perp}$ and either $A$ makes an angle of
$\frac{\pi}{2}-\tau_{1}$ with $\mathbb{S}^{2}(1)$ or the Gauss map image of
$A$ contains two points at spherical distance $\tau_{1}$ apart. In particular,
$A$ cannot be contained in a plane passing through the origin.

Repeating the same reasoning in $\mathbb{A}(2^{-k},2^{-k+1})$ for every
$k\in\mathbb{N}$ and using a diagonal argument, we conclude that (a
subsequence of) the $\frac{1}{r_{n}}F_{n}$ converges smoothly in
$\mathbb{A}(0,1)=\mathbb{B}(1)\setminus\\{\vec{0}\\}$ to an immersed minimal
punctured disk $D^{*}$ that has $\vec{0}$ in its closure, such that $A\subset
D^{*}$. As $\\{\frac{1}{r_{n}}F_{n}\\}_{n}$ has locally bounded second
fundamental form in $\mathbb{R}^{3}\setminus\\{\vec{0}\\}$, then (a
subsequence of) the $\frac{1}{r_{n}}F_{n}$ converge smoothly to a minimal
immersion $\widetilde{D}$ in $\mathbb{R}^{3}\setminus\\{\vec{0}\\}$ such that
$D^{*}\subset\widetilde{D}$, and $\widetilde{D}$ is complete away from
$\vec{0}$, in the sense that divergent arcs in $\widetilde{D}$ either have
infinite length or diverge to $\vec{0}$. Clearly $\widetilde{D}$ has $\vec{0}$
in its closure. Since $\frac{1}{r_{n}}F_{n}$ is stable away from the origin,
then $\widetilde{D}$ is stable. In this setting and when $\widetilde{D}$ is
two-sided, $\widetilde{D}$ extends smoothly to a plane passing through
$\vec{0}$ (by Lemma 3.3 in Meeks-Pérez-Ros [20], see also Colding-Minicozzi
[8]). This contradicts the fact that $A$ cannot be contained in a plane
passing through the origin. In the case that $\widetilde{D}$ is one-sided,
then we can view $\widetilde{D}$ as a branched stable minimal immersion with
branch locus at the origin (with finite branching order); in this setting,
item 1 of Lemma 3.4 gives a contradiction. These contradictions finish the
proof of Proposition 5.3. ∎

######  Definition 5.4.

Consider the ${\delta}_{4}\in(0,{\delta}_{3}]$ given by Proposition 5.3. Then
we define

${\delta}:={\delta}_{4}/2,\quad{\delta}_{1}={\delta}/2.$

We will show that this is a valid choice for the ${\delta}_{1},{\delta}$
appearing in Theorem 1.2 in the case $I=1$.

We finish this section by showing how to deduce items 1, 2, 3 of Theorem 1.2
in this case of $I=1$ (this is part of Step 1 in our strategy of proof of
Theorem 1.2 explained in Section 5.3). We first explain how to choose the
value of $A_{1}\in[A_{0},\infty)$ that appears in the main statement of the
theorem. In Section 5.3 we saw that it suffices to prove items 1, 2, 3 of
Theorem 1.2 for immersions in $\Lambda_{t}$ for some large
$t>\widehat{C}_{s}({\delta}_{1}/2)$. Choose
$t>\widehat{C}_{s}({\delta}_{1}/2)$ sufficiently large so that:

  1. (K1)

$\frac{R}{t}(\widehat{C}/2+1)\leq\frac{{\delta}_{1}}{10}$) (recall that $R$
was defined just before items (H0)-…-(H5) only depending on $\tau$ and
$\widehat{C}$ was given in item 2 of Proposition 8.4 as a function of $I,B$,
which in this case where $I=1,B=0$ give
$\widehat{C}=4\sqrt{3}+\frac{11}{2}\pi$, see also item (H5)).

  2. (K2)

For every $(F\colon M\looparrowright X)\in\Lambda_{t}$, Lemma 5.1 applied to
$F$ for ${\varepsilon}={\varepsilon}_{0}$ implies that there exists a point
$p_{1}\in U(\partial M,{\varepsilon}_{0},\infty)$ such that

$|A_{M}|(p_{1})=\max\\{|A_{M}|(p)\ |\ p\in M\\},$

and if $t$ is sufficiently large, then the description in items (J0)-…-(J5)
holds for $F$ with $p_{1}(n),{\lambda}_{n}$ replaced by $p_{1}$,
$|A_{M}|(p_{1})$ respectively.

Define $A_{1}=t$. Next we will prove items 1, 2, 3 of Theorem 1.2 for
immersions in $\Lambda_{t}$. Given $(F\colon M\looparrowright
X)\in\Lambda_{t}$, define $r_{F}(1)$ to be ${\delta}_{1}$, and $\Delta_{1}$ to
be the component of $F^{-1}(\overline{B}_{X}(F(p_{1}),r_{F}(1))$ that contains
$p_{1}$. Let $S_{F}(\frac{R}{2t})$ denote the extrinsic geodesic sphere in $X$
centered at $F(p_{1})$ with radius $\frac{R}{2t}$. Let $q$ be a point in
$\partial\Delta_{1}$. Then,

$\begin{array}[]{rclr}d_{M}(p_{1},q)&\leq&{\displaystyle\max_{x\in\partial\Delta_{1}\cap
F^{-1}(S_{F}(\frac{R}{2t}))}}d_{M}(p_{1},x)+d_{M}(\Delta_{1}\cap
F^{-1}(S_{F}(\frac{R}{2t})),q)&\\\
\rule[-3.60004pt]{0.0pt}{14.22636pt}&\leq&\frac{R}{t}(\widehat{C}/2+1)+d_{M}(\Delta_{1}\cap
F^{-1}(S_{F}(\frac{R}{2t})),q).&\mbox{(By (J5), as
$|A_{M}|(p_{1})\geq{t}$)}\end{array}$

By properties (J2), (J3) and (J4) and by Proposition 5.3, we can apply Lemma
4.1 to each of the annular portions of $\Delta_{1}$ with the choices
$R_{1}=\frac{R}{2t}$ and $R_{2}=r_{F}(1)$ (observe that
$\frac{R}{2t}\leq\frac{R}{t}(\widehat{C}/2+1)\leq\frac{{\delta}_{1}}{10}<\frac{r_{F}(1)}{2}$).
Using item (C2) of Lemma 4.1 (see also item (D2) of Remark 4.2) in the second
term of the last RHS, we get

$\begin{array}[]{rclr}d_{M}(p_{1},q)&\leq&\frac{R}{t}(\widehat{C}/2+1)+\sqrt{1+\frac{\tau^{2}}{3}}\,(r_{F}(1)-\frac{R}{2t})&\\\
\rule[-3.60004pt]{0.0pt}{14.22636pt}&<&\frac{{\delta}_{1}}{10}+\sqrt{1+\frac{\tau^{2}}{3}}\,r_{F}(1)&\qquad\mbox{(By
(K1))}\\\
\rule[-3.60004pt]{0.0pt}{14.22636pt}&=&\left(\frac{1}{10}+\sqrt{1+\frac{\tau^{2}}{3}}\right)r_{F}(1).&\end{array}$

Since $\tau\leq\pi/10$, then $d_{M}(p_{1},q)<\frac{5}{4}r_{F}(1)$. This proves
item 1(a) of Theorem 1.2.

Item 1(b) follows from the definition of $\Delta_{1}$. Observe that item 1(c)
is vacuous because $k=1$. Item 1(d) holds because $F\in\Lambda_{t}$ and
$A_{1}=t$. Item 1(e) follows from (J0) (see also (K2)), which finishes the
proof of item 1 of Theorem 1.2. Item 2 follows from Lemma 4.1.

Next we show item 3. Given $q\in\widetilde{M}=M-\mbox{\rm Int}(\Delta_{1})$,
let ${\gamma}\subset M$ be an arc joining $p_{1}$ with $q$. Let
${\gamma}_{1}\subset{\gamma}$ be the smallest subarc of ${\gamma}$ that joins
$p_{1}$ with some point $q_{1}\in\partial\Delta_{1}$. By definition of
$\Delta_{1}$, $F(q_{1})$ is at extrinsic distance $r_{F}(1)$ from $F(p_{1})$,
and thus, length$({\gamma})\geq\mbox{length}({\gamma}_{1})\geq
r_{F}(1)={\delta}_{1}$ for every arc ${\gamma}$ joining $p_{1}$ with $q$.
Therefore, $d_{M}(p_{1},q)\geq{\delta}_{1}$. As $q$ is any point in
$\widetilde{M}$, we conclude that $\widetilde{M}\subset
M-B_{M}(p_{1},{\delta}_{1})$. Hence, item 3 of Theorem 1.2 will be proved if
we check that $|A_{M}|\leq A_{1}$ in $M-B_{M}(p_{1},{\delta}_{1})$. Applying
item 3 of Lemma 5.1 to ${\varepsilon}={\delta}_{1}$ (this is possible since
${\delta}_{1}\leq{\varepsilon}_{0}$ and $\sup|A_{M}|\geq
t>\widehat{C}_{s}({\delta}_{1}/2)\geq\widehat{C}_{s}({\delta}_{1})$), we
conclude that $|A_{M}|\leq\widehat{C}_{s}({\delta}_{1})$ in
$M-B_{M}(p_{1},{\delta}_{1})$. Since $\widehat{C}_{s}$ is non-increasing, we
have
$\widehat{C}_{s}({\delta}_{1})\leq\widehat{C}_{s}({\delta}_{1}/2)<t=A_{1}$, so
item 3 of Theorem 1.2 holds.

Thus, items 1, 2, 3 of Theorem 1.2 hold in this case $I=1$.

###  5.5 Proofs of items 1, 2, 3 of Theorem 1.2 for $I=I_{0}+1$

Assume that items 1, 2, 3 of Theorem 1.2 hold for $I=I_{0}$ and we will prove
the same items hold for $I=I_{0}+1$.

By the arguments in the first paragraph of Section 5.3, we can assume that for
each $n>\widehat{C}_{s}({\varepsilon}_{0})$ there exists an $H_{n}$-immersion
$F_{n}\colon M_{n}\looparrowright X_{n}$ in
$\Lambda(I_{0}+1,H_{0},{\varepsilon}_{0},A_{0},K_{0})$ such that
$\sup|A_{M_{n}}|>n$. By Lemma 5.1, for each
$n>\widehat{C}_{s}({\varepsilon}_{0})$ there exists a finite set

$\\{p_{1}(n),\ldots,p_{m(n)}(n)\\}\subset U(\partial
M_{n},{\varepsilon}_{0},\infty),$

$m(n)\leq I_{0}+1$, such that

  1. (L1)

$|A_{M_{n}}|$ achieves its maximum in $M_{n}$ at $p_{1}(n)$ and for
$i=2,\ldots,m(n)$, $|A_{M_{n}}|$ achieves its maximum in
$M_{n}\setminus[B_{M_{n}}(p_{1}(n),{\varepsilon}_{0})\cup\ldots\cup
B_{M_{n}}(p_{i-1}(n),{\varepsilon}_{0})]$ at $p_{i}(n)$.

  2. (L2)

For each $i=1,\ldots,m(n)$,
$|A_{M_{n}}|(p_{i}(n))>\widehat{C}_{s}({\varepsilon}_{0})$, and so, the
pairwise disjoint intrinsic balls $B_{M_{n}}(p_{i}(n),{\varepsilon}_{0}/2)$
are unstable.

  3. (L3)

$|A_{M_{n}}|\leq\widehat{C}_{s}({\varepsilon}_{0})$ in
$M_{n}\setminus[B_{M_{n}}(p_{1}(n),{\varepsilon}_{0})\cup\ldots\cup
B_{M_{n}}(p_{m(n)}(n),{\varepsilon}_{0})]$.

####  5.5.1 Local pictures around points where $|A_{M}|>t$, for $t$
sufficiently large

Given $n>\widehat{C}_{s}({\varepsilon}_{0})$, consider the function
$h_{n}\colon\overline{B}_{M_{n}}(p_{1}(n),{\varepsilon}_{0})\to[0,\infty)$
given by (5.9). As in the case $I=1$, the maximum of $h_{n}$ occurs at
$p_{1}(n)$. Let ${\lambda}_{n}=|A_{M_{n}}|(p_{1}(n))$. Then, properties
(G1)-…-(G6) hold with the only change in (G5) (resp. in (G6)) that $f_{r}$
(resp. $f$) has index at most $I_{0}+1$. In the sequel, will use the same
notation as in (G1)-…-(G6).

Unlike what we had in the case $I=1$, we do not dispose of a classification
result for the possible limit minimal immersion $f$ in this current setting.
Still, we can estimate some aspects of its geometry; observe that $f$ has
finite total curvature, since it has finite index (see Fischer-Colbrie [9] for
the orientable case, and see the last paragraph of the proof of Theorem 17 in
Ros [24] for the non-orientable case). Therefore, $f$ is proper, the domain
$\Sigma$ of $f$ has finite genus and finitely many ends, each of which is
mapped by $f$ to a multi-graph over the exterior of a disk in a plane of
$\mathbb{R}^{3}$ passing through the origin, with finite multiplicity. We will
denote by $e\geq 1$ the number of ends of $f$, and $d_{1},\ldots,d_{e}\geq 1$
the multiplicities of these ends. Hence, $\sum_{j=1}^{e}d_{j}$ is the total
spinning of the ends. Also, $g(\Sigma),I(f)$ will stand for the genus of
$\Sigma$ and the index of $f$, respectively.

######  Claim 5.5 (Lower bound for the total spinning plus the number of the
ends of $f$).

$\sum_{j=1}^{e}(d_{j}+1)\geq 4.$ (5.10)

###### Proof.

If all the ends of $f$ are embedded, then $e\geq 2$ (as $f$ is not flat) and
$d_{j}=1$ for each $j=1,\ldots,e$. Thus, $\sum_{j=1}^{e}(d_{j}+1)=2e\geq 4$.

If $f$ has at least one non-embedded end, then the monotonicity formula for
minimal surfaces implies that the area growth of $f$ at infinity is at least
that of three planes (again because $f$ is not flat). Therefore, in this case,
$\sum_{j=1}^{r}d_{j}\geq 3$ and the claim follows. ∎

######  Claim 5.6 (Upper bound for the genus of $\Sigma$).

If $\Sigma$ is orientable, then $2g(\Sigma)\leq 3I(f)-3$. If $\Sigma$ is non-
orientable, then $g(\widetilde{\Sigma})\leq 3I(f)-4$, where
$g(\widetilde{\Sigma})$ is the genus of the orientable cover
$\widetilde{\Sigma}$ of $\Sigma$.

###### Proof.

This follows directly from equations (3.4) and (5.10), after observing that
the total branching order $B(\Sigma)$ of $f$ is zero. ∎

######  Claim 5.7 (Upper bound for the total spinning of $f$).

$2\sum_{j=1}^{e}d_{j}\leq\left\\{\begin{array}[]{ll}3I(f)+3&\mbox{if $\Sigma$
is orientable,}\\\ 3I(f)+2&\mbox{if $\Sigma$ is non-
orientable.}\end{array}\right.$

###### Proof.

This follows directly from (3.4) since $e\geq 1$ and $g(\Sigma)\geq 0$ if
$\Sigma$ is orientable (resp. $g(\widetilde{\Sigma})\geq 0$ if $\Sigma$ is
non-orientable). ∎

Recall that we have fixed $\tau\in(0,\pi/10]$. Let
${\alpha}_{1}={\alpha}_{1}(\tau)\in(0,\tau]$ be the constant given by Lemma
4.1 for $L_{0}=3\pi(I_{0}+2)+1$; observe that the total length $L^{f}({r})$ of
the intersection of $f(\Sigma)$ with a sphere $\mathbb{S}^{2}({r})$ of
sufficiently large radius ${r}$ is less than $L_{0}{r}$; this follows since
for ${r}$ large,

$\frac{L^{f}({r})}{r}\sim
2\pi\sum_{j=1}^{e}d_{j}\stackrel{{\scriptstyle\mbox{\footnotesize{(Claim
\ref{eqts})}}}}{{\leq}}\pi[3I(f)+3]\leq\pi[3(I_{0}+1)+3].$ (5.11)

We can also pick a smallest $R>0$ (only depending on $\tau$) so that the
following properties hold (compare with properties (H0)-…-(H5) above):

  1. (H0)’

The index of $f(\Sigma)\cap\mathbb{B}(R/3)$ is $I(f)$.

  1. (H1)’

$f(\Sigma)\setminus\mathbb{B}(R/3)$ consists of $e$ multi-graphs over their
projections to planes $\Pi_{j}\subset\mathbb{R}^{3}$ passing though $\vec{0}$,
$j=1,\ldots,e$.

  2. (H2)’

The image through the Gauss map of $f$ of each component $C_{j}$ of
$f(\Sigma)\setminus\mathbb{B}(R/3)$ is contained in the spherical neighborhood
of radius ${\alpha}_{1}/2$ centered at a point $v_{j}\in\mathbb{S}^{2}(1)$
perpendicular to $\Pi_{j}$ (thus, $C_{j}$ satisfies item (B2) of Lemma 4.1
with $R_{1}=R/3$ and ${\alpha}={\alpha}_{1}/2$).

  3. (H3)’

$f(\Sigma)$ makes an angle greater than $\frac{\pi}{2}-\frac{{\alpha}_{1}}{2}$
with every sphere $\mathbb{S}^{2}({r})$ of radius ${r}\geq R/3$ centered at
the origin (so, $C_{j}$ satisfies item (B1) of Lemma 4.1 with $R_{1}=R/3$ and
${\alpha}={\alpha}_{1}/2$).

  4. (H4)’

The total length of the intersection of $f(\Sigma)$ with any sphere
$\mathbb{S}^{2}({r})$ centered at the origin and radius ${r}\geq R/3$ is less
than $(L_{0}-\frac{1}{2}){r}$ (hence $C_{j}$ satisfies item (B3) of Lemma 4.1
with $R_{1}=R/3$).

Applying the last sentence in item 2 of Proposition 8.4 with $I=I_{0}+1$ and
$B=0$, we deduce that:

  1. (H5)’

The intrinsic distance in the pullback metric by $f$ from $\vec{0}\in\Sigma$
to any point in the boundary of $f^{-1}(\overline{\mathbb{B}}(R/2))$ is at
most $a(I_{0})R$, where

$a(I_{0})=\frac{\widehat{C}(I_{0}+1,0)}{2}=\sqrt{6}(3I_{0}+1)\sqrt{I_{0}+2}+\frac{\pi}{4}(6I_{0}+11).$
(5.12)

Given $r\in[\frac{R}{2},4R]$, let ${\Delta}_{n}(r)$ be the domain inside
$M_{n}$ given by Definition 5.2, related to the $f,R$ above. Properties
(H0)’-…-(H5)’ imply that for ${\lambda}_{n}$ large, the immersion
${\lambda}_{n}F_{n}$ satisfies (compare with properties (I0)-…-(I5) above):

  1. (I0)’

The index of $({\lambda}_{n}F_{n})|_{\Delta_{n}(R/2)}$ equals $I(f)$.

  1. (I1)’

$({\lambda}_{n}F_{n})(\Delta_{n}(4R)\setminus\Delta_{n}(R/2))$ can be
considered to be a union of $e$ multi-graphs over their projections to the
$\Pi_{j}$, $j=1,\ldots,e$. We denote these multi-graphs by
$\widetilde{G}_{n}(1),\ldots,\widetilde{G}_{n}(e)$.

  2. (I2)’

For $j=1,\ldots,e$, the image of $\widetilde{G}_{n}(j)$ through the “Gauss
map” of ${\lambda}_{n}F_{n}$ (defined through ambient parallel translation,
see Remark 4.2) is contained in the spherical neighborhood of radius
${\alpha}_{1}$ centered at $v_{j}$ (here we have identified $\mathbb{R}^{3}$
with the tangent space to ${\lambda}_{n}X$ at $F_{n}(p_{1}(n))$).

  3. (I3)’

$\widetilde{G}_{n}(j)$ makes an angle greater than
$\frac{\pi}{2}-{\alpha}_{1}$ with every geodesic sphere $\widetilde{S}({r})$
in ${\lambda}_{n}X_{n}$ centered at $F_{n}(p_{1}(n))$ of radius
${r}\in[R/2,4R]$.

  4. (I4)’

$\mbox{\rm Length}[\widetilde{G}_{n}(j)\cap\widetilde{S}(R/2)]<L_{0}R/2$.

  5. (I5)’

The intrinsic distance in the pullback metric by ${\lambda}_{n}F_{n}$ on
$M_{n}$, from $p_{1}(n)$ to any point of the boundary of $\Delta_{n}(R/2)$ is
at most $[a(I_{0})+1]R$.

Back in the original scale, we have that $\Delta_{n}({r})\subset
F_{n}^{-1}(\overline{B}_{X_{n}}(F_{n}(p_{1}(n)),{r}/{\lambda}_{n}))$ for all
${r}\in[\frac{R}{2},4R]$ and the following properties hold for $n$
sufficiently large:

  1. (J0)’

The index of $F_{n}|_{\Delta_{n}(R/2)}$ equals $I(f)$.

  1. (J1)’

$F_{n}(\Delta_{n}(4R)\setminus\Delta_{n}(R/2))$ is a union of $e$ multi-graphs
over their projections to the $\Pi_{j}$, $j=1,\ldots,e$. We call
$G_{n}(1),\ldots,G_{n}(e)$ to these multi-graphs.

  2. (J2)’

For $j=1,\ldots,e$, the image of $G_{n}(j)$ through the “Gauss map” of $F_{n}$
is contained in the spherical neighborhood of radius ${\alpha}_{1}$ centered
at $v_{j}$.

  3. (J3)’

$G_{n}(j)$ makes an angle greater than $\frac{\pi}{2}-{\alpha}_{1}$ with every
geodesic sphere $S({r})$ in $X_{n}$ centered at $F_{n}(p_{1}(n))$ of radius
${r}\in[\frac{R}{2{\lambda}_{n}},\frac{4R}{{\lambda}_{n}}]$.

  4. (J4)’

$\mbox{\rm Length}[G_{n}(j)\cap
S(\frac{R}{2{\lambda}_{n}})]<L_{0}\frac{R}{2{\lambda}_{n}}$.

  5. (J5)’

The intrinsic distance in the pullback metric by $F_{n}$ on $M_{n}$, from
$p_{1}(n)$ to any point of the boundary of $\Delta_{n}(R/2)$ is at most
$\frac{R}{{\lambda}_{n}}[a(I_{0})+1]$.

Therefore, given ${r}\in[\frac{R}{2{\lambda}_{n}},\frac{2R}{{\lambda}_{n}}]$,
$G_{n}(j)\cap[\overline{B}_{X_{n}}(F_{n}(p_{1}(n)),2{r})\setminus
B_{X_{n}}(F_{n}(p_{1}(n)),{r})]$ satisfies the hypotheses (B1), (B2), (B3) of
Lemma 4.1 with the choices $L_{0}=3\pi(I_{0}+2)+1$, inner extrinsic radius
${r}$, outer extrinsic radius $2{r}$ and ${\alpha}={\alpha}_{1}$.

####  5.5.2 How to proceed if the (first) local pictures fail to have a
uniform size

######  Definition 5.8.

Define ${r}_{n}$ as the supremum of the extrinsic radii ${r}\geq
4R/{\lambda}_{n}$ such that for all $j=1,\ldots,e$, annular enlargements
$\widehat{G}_{n}(j)$ of the $G_{n}(j)$ satisfy conditions (B1), (B2), (B3) of
Lemma 4.1 for the choices $L_{0}=3\pi(I_{0}+2)+1$, inner extrinsic radius
$R_{1}=\frac{R}{2{\lambda}_{n}}$, outer extrinsic radius $R_{2}={r}_{n}$ and
${\alpha}={\alpha}_{1}$.

Figure 3: Schematic representation of the extrinsic geometry of the immersion
$(F_{n}\colon M_{n}\looparrowright
X_{n})\in\Lambda=\Lambda(I_{0}+1,H_{0},{\varepsilon}_{0},A_{0},K_{0})$ around
a point $p_{1}(n)$ where the maximum of $|A_{M_{n}}|$ in $M_{n}$ is achieved.
Here, ${\lambda}_{n}=|A_{M_{n}}|(p_{1}(n))$ tends to infinity and
${\lambda}_{n}F_{n}$ converges as $n\to\infty$ to the complete minimal
immersion $f\colon\Sigma\looparrowright\mathbb{R}^{3}$ with finite total
curvature. Horizontal distances in the figure represent extrinsic distances in
$X_{n}$ measured from $F_{n}(p_{1}(n))$. For $n$ large enough and in the range
of extrinsic radii between $\frac{R}{2{\lambda}_{n}}$ and $r_{n}\geq
4R/{\lambda}_{n}$, $F_{n}$ consists of $e$ multi-graphical pieces
$\widehat{G}_{n}(1),\ldots,\widehat{G}_{n}(e)$, where $e$ is the number of
ends of $f$.

######  Remark 5.9.

  1. 1.

Unlike what happened in the case $I=1$ (Section 5.4), we can no longer insure
that the outer extrinsic radius $r_{n}$ is bounded from below by some positive
constant independent of $n$ (i.e. Proposition 5.3 does not necessarily hold in
our setting). The reason for this difference is that in our current situation,
the estimate $I(f)\leq I_{0}+1$ is not necessarily an equality (as it was when
$I=1$), and thus, with the notation in the proof of Proposition 5.3, we cannot
insure that if $r_{n}\to 0$ as $n\to\infty$, then $\frac{1}{r_{n}}F_{n}$ has
index zero away from the origin for $n$ large.

  2. 2.

If $r_{n}\to 0$ as $n\to\infty$ and $\frac{1}{r_{n}}F_{n}$ has index zero away
from the origin for $n$ large in the sense that property $(\diamondsuit)$
above holds, then the arguments in the proof of Proposition 5.3 lead to a
contradiction. Hence we conclude that one of the two following excluding
possibilities holds:

    1. (2.A)

$\\{r_{n}\\}_{n}$ is bounded away from zero, with this lower bound being
independent of the sequence $\\{F_{n}\\}_{n}\subset\Lambda$. In this case,
Proposition 5.3 holds (now ${\delta}_{3}\in(0,{\delta}_{2}]$ is given by
Definition 4.5 for the choices $L_{0}=3\pi(I_{0}+2)+1$ and $m$ being 1 plus
the integer part of $\frac{1}{2}[3(I_{0}+1)+3]$, see equation (5.11) which
estimates the total spinning of $f$ by above; also see Proposition 5.16
below). In this case, we can apply Proposition 5.17 below to conclude the
proof of items 1, 2, 3 of Theorem 1.2.

    2. (2.B)

There exists some sequence $\\{F_{n}\\}_{n}\subset\Lambda$ (with associated
base points $p_{1}(n)$) such that $r_{n}\to 0$ and $\frac{1}{r_{n}}F_{n}$
fails to have index zero away from the origin for $n$ large, in the sense that
the property $(\diamondsuit)$ above fails.

Assume that we are in case (2.B) above. Roughly speaking, we will show that
the immersions $\frac{1}{r_{n}}F_{n}$ converge as $n\to\infty$ to a possibly
finitely branched, complete minimal immersion
$f_{2}\colon\Sigma_{2}\looparrowright\mathbb{R}^{3}$ away from finitely many
points where curvature blows-up. Furthermore, $\Sigma_{2}$ is finitely
connected and its Morse index is at most $(I_{0}+1)-I(f_{1})\leq I_{0}$. This
compactness result is delicate and we will divide its proof into the following
two steps.

  1. (M1)

Describe the behavior of the immersions $\frac{1}{r_{n}}F_{n}$ near the origin
as $n\to\infty$. We will do this in Lemmas 5.10 and 5.11.

  2. (M2)

Analyze the global convergence of the $\frac{1}{r_{n}}F_{n}$ (after passing to
a subsequence) to a complete, finitely branched minimal immersion
$f_{2}\colon\Sigma_{2}\looparrowright\mathbb{R}^{3}$ with finite total
curvature. We will do this in Proposition 5.13.

The proof of the next lemma follows easily from the behavior of the blow-down
limit of any of the $e$ ends of the complete minimal immersion
$f=f_{1}\colon\Sigma\looparrowright\mathbb{R}^{3}$ defined just after items
(L1)-(L3).

######  Lemma 5.10.

Relabel as $e_{1}=e$ the number of ends of $f_{1}$. Suppose $r_{n}\to 0$ as
$n\to\infty$. Then, after choosing a subsequence, each of the finite number of
extended and scaled multi-graphs
$(\frac{1}{r_{n}}F_{n})|_{\widehat{G}_{n}(j)}$, considered to be a mapping on
an open annulus, converges as $n\to\infty$ to a conformal minimal immersion of
a punctured disk

$f_{2,j}\colon\mathbb{D}^{*}=\\{z\in\mathbb{C}\mid
0<|z|<1\\}\looparrowright\mathbb{R}^{3}$

(here $j\in\\{1,\ldots,e_{1}\\}$ refers to the j-th end of $f_{1}$) with
$f_{2,j}(\mathbb{D}^{*})\subset\mathbb{B}(1)\setminus\\{\vec{0}\\}$.
Furthermore, for each such $j$,

  1. 1.

$f_{2,j}$ extends analytically to a possibly branched minimal disk
$\overline{f}_{2,j}\colon\mathbb{D}=\mathbb{D}^{*}\cup\\{0\\}\looparrowright\mathbb{R}^{3}$
with $\overline{f}_{2,j}(0)=\vec{0}$;

  2. 2.

The branching order of $\overline{f}_{2,j}$ at $0$ is one less than the
multiplicity of the associated sequence of multi-graphs
$(\frac{1}{r_{n}}F_{n})|_{\widehat{G}_{n}(j)}$. Such multiplicity (which is
independent of $n$ large) coincides with the spinning of associated j-th end
of $f_{1}\colon\Sigma\looparrowright\mathbb{R}^{3}$.

Let $\mathcal{D}=\\{D_{1},\ldots,D_{e_{1}}\\}$ be the set of parameter domains
of the associated branched minimal disks
$\\{\overline{f}_{2,1},\ldots,\overline{f}_{2,e_{1}}\\}$ given by item 1 of
Lemma 5.10, and consider the map
$F_{\infty}\colon\cup\mathcal{D}\looparrowright\mathbb{B}(1)$ be defined by

$F_{\infty}|_{D_{i}}=\overline{f}_{2,i},\quad i=1,\ldots,e_{1}.$

Observe that $\cup\mathcal{D}$ (disjoint union) can be considered to be a
smooth surface. Let $\mathcal{S}(0)\subset\cup\mathcal{D}$ be the finite set
of centers of the disks $D_{i}$, $i=1,\ldots,e_{1}$. Consider the quotient
space $\widehat{\mathcal{D}}$ of $\cup\mathcal{D}$ where each of the elements
in $\mathcal{S}(0)$ identifies to one point that we denote by
$\widehat{0}\in\widehat{\mathcal{D}}$, and every other point of
$\cup\mathcal{D}$ only identifies with itself. Let

$\pi\colon\cup\mathcal{D}\to\widehat{\mathcal{D}}$

be the related quotient map, that is, $\pi|_{\mathcal{S}(0)}$ is the constant
map equal to $\widehat{0}$, and the restriction of $\pi$ to
$(\cup\mathcal{D})\setminus\mathcal{S}(0)$ is injective. After endowing
$\widehat{\mathcal{D}}$ with the quotient topology, $\widehat{\mathcal{D}}$ is
a path-connected topological space and

$\widehat{\mathcal{S}}(0):=\pi(\mathcal{S}(0))=\\{\widehat{0}\\}.$ (5.13)

Furthermore, $\widehat{\mathcal{D}}\setminus\widehat{\mathcal{S}}(0)$ is a
smooth immersed surface. In what follows, we will at times consider the
induced well-defined continuous map
$F_{\infty}\colon\widehat{\mathcal{D}}\looparrowright\mathbb{B}(1)$, which we
denote the same way.

The next statement can be viewed as an direct consequence of Lemma 5.10.

######  Lemma 5.11.

In the above situation, the following properties hold:

  1. 1.

$F_{\infty}$ restricted to
$F_{\infty}^{-1}(\mathbb{B}(1)\setminus\mathbb{B}(\frac{1}{2}))$ consists of
$e_{1}$ multi-graphs.

  2. 2.

The sequence of immersions $\frac{1}{r_{n}}F_{n}$ restricted to the component
$\Delta_{2,n}\subset M_{n}$ of

${\textstyle(\frac{1}{r_{n}}F_{n})^{-1}(B_{\frac{1}{r_{n}}X_{n}}(\vec{0},1/2))}$

that contains $p_{1}(n)$, converges as $n\to\infty$ to $F_{\infty}$, where we
consider $F_{\infty}\colon\widehat{\mathcal{D}}\looparrowright\mathbb{B}(1)$
to be defined on the quotient space $\widehat{\mathcal{D}}$.

  3. 3.

The convergence in item 2 is smooth away from $\mathcal{S}(0)$, or from
$\widehat{\mathcal{S}}(0)$ when we consider $F_{\infty}$ to be defined on
$\widehat{\mathcal{D}}$.

Lemma 5.11 describes the convergence of (a subsequence of) the
$\frac{1}{r_{n}}F_{n}$ in a neighborhood of $\widehat{\mathcal{S}}(0)$, to a
family $F_{\infty}\colon\widehat{\mathcal{D}}\looparrowright\mathbb{B}(1)$ of
minimal disks branched at the origin, and finishes step (M1) above.

Step (M2) needs two ingredients, which are Lemma 5.12 and Proposition 5.13
below. The first one relies on the validity of Theorem 1.2 for $I=I_{0}$ (by
the induction hypothesis), while in Proposition 5.13 we will construct the
complete, finitely branched minimal immersion
$f_{2}\colon\Sigma\looparrowright\mathbb{R}^{3}$ of finite total Gaussian
curvature, which is the limit of a subsequence of the $\frac{1}{r_{n}}F_{n}$
as a consequence of Lemma 5.12.

We remark that the surfaces $M_{n}$ and the associated points $p_{1}(n)$ in
the next theorem are not the same surfaces and points that we have been using
previously in this section with this notation; so the reader should keep in
mind this abuse of notation when reading the next result.

######  Lemma 5.12.

Consider a sequence $(\widetilde{F}_{n}\colon
M_{n}\looparrowright\widetilde{X}_{n})\in\Lambda(I_{0},H_{0},{\varepsilon}_{0},A_{0},K_{0})$
such that the following properties hold.

  1. (N1)

$\\{\max_{M_{n}}|A_{\widetilde{F}_{n}}|\\}_{n}$ is not bounded from above. In
particular, after passing to a subsequence we can assume there exists
$p_{1}(n)\in M_{n}$ such that

$\max_{M_{n}}|A_{\widetilde{F}_{n}}|=|A_{\widetilde{F}_{n}}|(p_{1}(n))>\max\\{n,A_{1}\\}\quad\mbox{for
every $n\in\mathbb{N}$,}$

where $A_{1}\in[A_{0},\infty)$ is given in the statement of Theorem 1.2 for
$I=I_{0}$ (which can be applied by the induction hypothesis); observe that the
existence of $p_{1}(n)$ is guaranteed by Lemma 5.1.

  2. (N2)

In harmonic coordinates centered at $\widetilde{F}_{n}(p_{1}(n))$, hence
$\widetilde{F}_{n}(p_{1}(n))=\vec{0}$ for all $n\in\mathbb{N}$, the metrics on
$\widetilde{X}_{n}$ converge uniformly in the $C^{0}$ norm to the flat metric
on $\mathbb{R}^{3}$, and the (constant) mean curvatures of the
$\widetilde{F}_{n}$ converge to zero as $n\to\infty$.

Let $\Delta_{1}(n)$ be the component of
$\widetilde{F}_{n}^{-1}(\overline{B}_{\widetilde{X}_{n}}(\widetilde{F}_{n}(p_{1}(n),r_{\widetilde{F}_{n}}(1)))$
described in item 1 of Theorem 1.2 and let
$\Delta_{1}(n,\frac{2}{3})=\Delta_{1}(n)\cap\widetilde{F}_{n}^{-1}(\overline{B}_{\widetilde{X}_{n}}(\widetilde{F}_{n}(p_{1}(n)),\frac{2}{3}r_{\widetilde{F}_{n}}(1)))$.
Then, after replacing by a subsequence, the following hold:

  1. 1.

$\\{r_{\widetilde{F}_{n}}(1)\\}_{n\in\mathbb{N}}$ converges to a positive
number $r\in[{\delta}_{1},\frac{{\delta}}{2}]$, where
${\delta}_{1},{\delta}\in(0,{\varepsilon}_{0}/2]$ are given by Theorem 1.2.

  2. 2.

Let $b$ be the number of boundary components of $\Delta_{1}(n)$, which is
independent of $n$. Then, the $b$ multi-graphs

$\widetilde{F}_{n}(\Delta_{1}(n))\cap[\overline{B}_{\widetilde{X}_{n}}(\widehat{F}_{n}(p_{1}(n)),r_{\widetilde{F}_{n}}(1))\setminus
B_{\widetilde{X}_{n}}(\widehat{F}_{n}(p_{1}(n)),\textstyle{\frac{1}{2}}r_{\widetilde{F}_{n}}(1))]$

described in item 2 of Theorem 1.2 converge as $n\to\infty$ to $b$ minimal
multi-graphs in
$\overline{\mathbb{B}}(\vec{0},r)\setminus\mathbb{B}(\vec{0},r/2)$, each of
which satisfies the same estimate (1.1) as the multi-graphs in the sequence
that converge to it.

  3. 3.

There exists $J\in\mathbb{N}$, $J\leq I_{0}$, ${\varepsilon}_{1}\in(0,r)$ and
a finite set for each $n\in\mathbb{N}$

$\textstyle{Q(n)=\\{q_{1}(n)=p_{1}(n),q_{2}(n),\ldots,q_{J}(n)\\}\subset
B_{M_{n}}\left(p_{1}(n),\frac{2}{3}r\right),}$

such that

    1. (a)

$|A_{\widetilde{F}_{n}}|(q_{i}(n))>\max\\{n,A_{1}\\}$, for all $i=1,\ldots,J$
and for each $n\in\mathbb{N}$ (compare to item 1.d of Theorem 1.2).

    2. (b)

Given $i,j\in 1,\ldots,J$ with $i\neq j$, the intrinsic distance in $M_{n}$
between $q_{i}(n)$ and $q_{j}(n)$ is at least ${\varepsilon}_{1}$ (compare to
item 1.c of Theorem 1.2).

    3. (c)

Given $s\in\mathbb{N}$, $\\{|A_{\widetilde{F}_{n}}|\\}_{n}$ is uniformly
bounded in
$B_{M_{n}}(p_{1}(n),\frac{2}{3}r)\setminus\cup_{i=1}^{J}B_{M_{n}}(q_{i}(n),\frac{{\varepsilon}_{1}}{3s})$
(compare to item 3 of Theorem 1.2).

    4. (d)

There exist (not necessarily distinct) points
$x_{1}=\vec{0},x_{2},\ldots,x_{J}\in\mathbb{B}(\vec{0},\frac{2}{3}r)$ (this is
the ball in $\mathbb{R}^{3}$ with its flat metric) such that when viewed in
harmonic coordinates in $\widetilde{X}_{n}$ centered at $p_{1}(n)$, the points
$\widetilde{F}_{n}(q_{i}(n))$ converge as $n\to\infty$ to $x_{i}$, for each
$i=1,\ldots,J$.

  4. 4.

For $s\in\mathbb{N}$ large and fixed, and for each $i\in\\{1,\ldots,J\\}$,
there exist ${\delta}_{i}(s),{\delta}_{i}(1,s),r_{i}(n,s)$ with
$0<{\delta}_{i}(1,s)\leq
r_{i}(n,s)\leq{\delta}_{i}(s)/2<{\delta}_{i}(s)<\frac{2{\varepsilon}_{1}}{3s}$,
such that the following hold. Let $A_{i}(n,s)$ be the component of
$\widetilde{F}_{n}^{-1}(B_{\widetilde{X}_{n}}(\widetilde{F}_{n}(q_{i}(n)),r_{i}(n,s)))$
that contains $q_{i}(n)$. Then, there exists $s_{0}\in\mathbb{N}$ such that
for each integer $s\geq s_{0}$, there exists $N(s)\in\mathbb{N}$ so that for
$n\geq N(s)$:

    1. (a)

The positive numbers $r_{i}(n,s)$ converge as $n\to\infty$ to some
$r_{i}(s)\in[{\delta}_{i}(1,s),{\delta}_{i}(s)/2]$.

    2. (b)

$A_{i}(n,s)$ is compact with smooth non-empty boundary and

$\widetilde{F}_{n}(\partial A_{i}(n,s))\subset\partial
B_{\widetilde{X}_{n}}(\widetilde{F}_{n}(q_{i}(n)),r_{i}(n,s))$

(compare to item 1.b of Theorem 1.2).

    3. (c)

The number $\widetilde{e}_{i}\in\mathbb{N}$ of boundary components of
$A_{i}(n,s)$ is independent of $n,s$ and the restriction of
$\widetilde{F}_{n}$ to an annular neighborhood of each boundary component of
$A_{i}(n,s)$ is a multi-graph of positive integer multiplicity $m_{h,i}$
independent of $n,s$ (here $h\in\\{1,\ldots,\widetilde{e}_{i}\\}$), whose
related graphing function $u=u_{n,s}$ satisfies inequality (1.1) for $n,s$
sufficiently large, where $x$ expresses harmonic coordinates in
$B_{\widetilde{X}_{n}}(\widetilde{F}_{n}(q_{i}(n)),\frac{{\varepsilon}_{1}}{2})$
(compare to item 2 of Theorem 1.2); the union of these annular neighborhoods
of $\partial A_{i}(n,s)$ can be taken to be
$A_{i}(n,s)\setminus\widetilde{F}_{n}^{-1}(B_{\widetilde{X}_{n}}(\widetilde{F}_{n}(q_{i}(n)),\textstyle{\frac{r_{i}(n,s)}{2}}))$.

    4. (d)

The $\widetilde{F}_{n}$ restricted to
$\Delta_{1}(n,\frac{2}{3})\setminus\cup_{i=1}^{J}A_{i}(n,s)$ converge smoothly
as $n\to\infty$ to a minimal immersion

$F_{\infty,s}\colon
M_{s}\looparrowright\textstyle{\mathbb{B}(\vec{0},\frac{2}{3}r)}$

of a compact surface $M_{s}$ with boundary, and
$F_{\infty,s}(M_{s})\cap[\overline{\mathbb{B}}(\vec{0},\frac{2}{3}r)\setminus\mathbb{B}(\vec{0},\frac{1}{2}r)]$
consists of the intersection of the limiting multi-graphs appearing in item 2
with
$\overline{\mathbb{B}}(\vec{0},\frac{2}{3}r)\setminus\mathbb{B}(\vec{0},\frac{1}{2}r)$.

    5. (e)

The boundary $\partial M_{s}$ decomposes into $J+1$ collections of curves

$\\{{\alpha}_{1},\ldots,{\alpha}_{b}\\},\quad\\{{\beta}_{1,i}(s),\ldots,{\beta}_{\widetilde{e}_{i},i}(s)\\}_{i=1,\ldots,J}$

(recall that $b$ is the number of boundary components of $\Delta_{1}(n)$)
where
$F_{\infty,s}({\alpha}_{h})\subset\partial\mathbb{B}(\vec{0},\frac{2}{3}r)$
for each $h=1,\ldots,b$, and
$F_{\infty,s}({\beta}_{l,i}(s))\subset\partial\mathbb{B}(x_{i},r_{i}(s))$ for
some $i=1,\ldots,J$ and for every $l=1,\ldots,\widetilde{e}_{i}$.

  5. 5.

There exists an infinite strictly increasing sequence

$\mathfrak{S}=\\{s_{1},s_{2},\ldots,s_{j},\ldots\\}\subset\mathbb{N}$

such that for each $j\in\mathbb{N}$, for $n$ sufficiently large depending on
$j$,

$A_{i}(n,s_{j+1})\subset\mbox{\rm Int}(A_{i}(n,s_{j}))\;\mbox{\rm and
}\;\mbox{\rm Index}(A_{i}(n,s_{j}))=\mbox{\rm Index}(A_{i}(n,s_{1})).$

In particular, for each $j\in\mathbb{N}$ and $n$ sufficiently large depending
on $j$, $A_{i}(n,s_{1})\setminus A_{i}(n,s_{j})$ is stable.

  6. 6.

For each $s_{j}\in\mathfrak{S}$ defined in item 5, $M_{s_{j+1}}\subset
M_{s_{j}}$ and $F_{\infty,s_{j+1}}|_{M_{s}}=F_{\infty,s_{j}}$. Then
$M_{\infty}=\cup_{s_{j}\in\mathfrak{S}}M_{s_{j}}$ is a compact Riemann surface
with boundary, punctured in $e:=\sum_{i=1}^{J}\widetilde{e}_{i}$ points
$\\{P_{1,i},\ldots,P_{\widetilde{e}_{i},i}\\}_{i=1,\ldots,J}$, and the
immersion $F_{\infty}\colon M_{\infty}\looparrowright\mathbb{R}^{3}$ given by
$F_{\infty}|_{M_{s}}=F_{\infty,s}$ extends to a finitely branched minimal
immersion

$\overline{F}_{\infty}\colon
M_{\infty}\cup\\{P_{1,i},\ldots,P_{\widetilde{e}_{i},i}\\}_{i=1,\ldots,J}\looparrowright\textstyle{\mathbb{B}(\vec{0},\frac{2}{3}r)},$

such that
$\overline{F}_{\infty}(\\{P_{1,i},\ldots,P_{\widetilde{e}_{i},i}\\})=\\{x_{i}\\}$,
$i=1,\ldots,J$, and the branch points of $\overline{F}_{\infty}$ are contained
in the set $\\{P_{1,i},\ldots,P_{\widetilde{e}_{i},i}\ |\ i=1,\ldots,J\\}$.

  7. 7.

For $i\in\\{1,\ldots,J\\}$ fixed and ${\varepsilon}>0$ sufficiently small and
fixed, the branching contribution $B_{i}\in\mathbb{N}\cup\\{0\\}$ to
$\overline{F}_{\infty}$ from $\\{P_{1,i},\ldots,P_{\widetilde{e}_{i},i}\\}$ is
$B_{i}=S_{i}-\widetilde{e}_{i}$, where

$S_{i}=\sum_{h=1}^{\widetilde{e}_{i}}m_{h,i}$ (5.14)

is the total spinning of the boundary curves of $\widetilde{F}_{n}$ restricted
to the component $\Delta(i,n,{\varepsilon})$ of
$\widetilde{F}_{n}^{-1}(B_{\widetilde{X}_{n}}(\widetilde{F}_{n}(q_{i}(n)),{\varepsilon}))$
containing $q_{i}(n)$ (for $n$ sufficiently large, $S_{i}$ is independent of
$n$). Furthermore,

$S_{i}\leq 3I(\Delta(i,n,{\varepsilon}));$ (5.15)

(here $I(\Delta(i,n,{\varepsilon}))$ is the index of
$\Delta(i,n,{\varepsilon})$) and so, the total branching of
$\overline{F}_{\infty}$ is at most
$\sum_{i=1}^{J}(S_{i}-\widetilde{e}_{i})\leq 3I_{0}-J$.

###### Proof.

Since $[{\delta}_{1},\frac{{\delta}}{2}]$ is compact, after replacing by a
subsequence, the sequence
$\\{r_{\widetilde{F}_{n}}(1)\\}_{n}\subset[{\delta}_{1},\frac{{\delta}}{2}]$
given by Theorem 1.2 converges to a positive number
$r\in[{\delta}_{1},\frac{{\delta}}{2}]$. The convergence stated in item 2 of
the multi-graphs

$\widetilde{F}_{n}(\Delta_{1}(n))\cap[\overline{B}_{\widetilde{X}_{n}}(\widehat{F}_{n}(p_{1}(n)),r_{\widetilde{F}_{n}}(1))\setminus
B_{\widetilde{X}_{n}}(\widehat{F}_{n}(p_{1}(n)),\textstyle{\frac{1}{2}}r_{\widetilde{F}_{n}}(1))]$

to minimal multi-graphs in
$\overline{\mathbb{B}}(\vec{0},r)\setminus\mathbb{B}(\vec{0},\frac{1}{2}r)$ is
standard by curvature estimates for CMC graphs. This gives items 1 and 2 of
the lemma.

We next prove that item 3 holds. To find the finite set $Q(n)$, we will
proceed as follows. Suppose for the moment that, after replacing by a
subsequence, for each $s\in\mathbb{N}$, $|A_{\widetilde{F}_{n}}|$ is uniformly
bounded in $B_{M_{n}}(p_{1}(n),\frac{2}{3}r)\setminus
B_{M_{n}}(p_{1}(n),\frac{2}{3s}r)$. In this case, the set
$Q(n):=\\{q_{1}(n)=p_{1}(n)\\}$ is easily seen to satisfy item 3 of the lemma
with the choice ${\varepsilon}_{1}=\frac{2}{3}r$. Otherwise, after replacing
by a subsequence, there exists an $s_{1}\in\mathbb{N}$ and a point
$q_{2}(n)\in B_{M_{n}}(p_{1}(n),\frac{2}{3}r)\setminus
B_{M_{n}}(p_{1}(n),\frac{2}{3s_{1}}r)$ such that
$|A_{\widetilde{F}_{n}}|(q_{2}(n))>\max\\{n,10A_{1}\\}$. If after replacing by
a subsequence, $\\{A_{\widetilde{F}_{n}}\\}_{n}$ is uniformly bounded in
$B_{M_{n}}(p_{1}(n),\frac{2}{3}r)\setminus\cup_{i=1}^{2}B_{M_{n}}(q_{i}(n),\frac{2}{3s}r)]$
for each $s\in\mathbb{N}$, then the set $Q(n):=\\{q_{1}(n),q_{2}(n)\\}$
satisfies item 3 of the lemma with
${\varepsilon}_{1}=\frac{1}{2}d_{M_{n}}(q_{1}(n),q_{2}(n))$, since after
replacing by another subsequence, $\widetilde{F}_{n}(q_{2}(n))$ converges as
$n\to\infty$ to some $x_{2}\in\mathbb{B}(\vec{0},\frac{2}{3}r)$ (note that
$x_{2}$ might be $\vec{0}$). Continuing inductively, we arrive at two sets of
points

$Q(n)=\\{q_{1}(n)=p_{1}(n),q_{2}(n),\ldots,q_{J}(n)\\},\quad\\{x_{1}=\vec{0},x_{2},\ldots,x_{J}\\}\subset\mathbb{B}(\vec{0},{\textstyle\frac{2}{3}r}),$

satisfying item 3 of the lemma with respect to

${\varepsilon}_{1}=\frac{1}{2}\min\\{d_{M_{n}}(q_{i}(n),q_{j}(n))\ |\
i,j=1,\ldots,J,\ i\neq j\\}.$

Here, $J\leq I_{0}$ because the index of $B_{M_{n}}(p_{1}(n),\frac{2}{3}r)$ is
at most $I_{0}$. This finishes the proof of item 3 of the lemma.

Regarding item 4, we make the following two observations:

  1. (O1)

For each $s\in\mathbb{N}$, there is a uniform upper bound $A_{2}(s)\geq A_{0}$
on the norm of the second fundamental forms of the immersions
$\widetilde{F}_{n}$ restricted to

$\bigcup_{i=1}^{J}\left[\overline{B}_{M_{n}}(q_{i}(n),\textstyle{\frac{{\varepsilon}_{1}}{3s}})\setminus
B_{M_{n}}(q_{i}(n),\textstyle{\frac{{\varepsilon}_{1}}{4s}})\right].$

(this follows from the already proven item 3(c) of this lemma).

  2. (O2)

For each $n\in\mathbb{N}$, let $\widehat{F}_{n,s}$ be the restriction of
$\widetilde{F}_{n}$ to
$\bigcup_{i=1}^{J}\overline{B}_{M_{n}}(q_{i}(n),\textstyle{\frac{{\varepsilon}_{1}}{3s}})$.
Then, observation (O1) implies that for each $n\in\mathbb{N}$,
$\widehat{F}_{n,s}$ lies in the space
$\Lambda(I_{0},H_{0},{\varepsilon}_{1}/12s,A_{2}(s),K_{0})$.

We next apply Theorem 1.2 to
$\widehat{F}_{n,s}\in\Lambda(I_{0},H_{0},{\varepsilon}_{1}/12s,A_{2}(s),K_{0})$
(which can be applied by the induction hypothesis), from where one has a
corresponding constant $\widehat{A}_{1}(s)\geq A_{1}$ that replaces the
previous constant $A_{1}$ and where the choice of $\tau$ is the same as
previously considered. Assume that $n$ is chosen sufficiently large, so that
given $i=1,\ldots,J$, the point $q_{i}(n)$ satisfies that the maximum of the
norm of the second fundamental form of $\widehat{F}_{n,s}$ in
$\overline{B}_{M_{n}}(q_{i}(n),\frac{{\varepsilon}_{1}}{3s})$ is achieved at
$q_{i}(n)$, with value greater than $10\widehat{A}_{1}(s)$ (by item 1(d) of
Theorem 1.2). Another consequence of Theorem 1.2 applied to
$\widehat{F}_{n,s}$ is that for $n$ large and for each $i=1,\ldots,J$, we have
associated positive numbers

${\delta}_{i}(s),{\delta}_{i}(1,s),r_{\widehat{F}_{n,s}}(i,s)$

with ${\delta}_{i}(1,s),{\delta}_{i}(s),r_{\widehat{F}_{n,s}}(i,s)$) playing
the respective roles of the related numbers ${\delta}_{1},{\delta},r_{F}(i)$
in Theorem 1.2, where

$0<{\delta}_{i}(1,s)\leq
r_{\widehat{F}_{n,s}}(i,s)\leq{\delta}_{i}(s)/2<{\delta}_{i}(s)<\frac{2{\varepsilon}_{1}}{3s}\quad\mbox{for
all $n$.}$ (5.16)

We also have a $\Delta$-type domain
$\Delta_{i}(q_{i}(n),r_{\widehat{F}_{n,s}}(i))$ defined by item 1 of Theorem
1.2, that is, $\Delta_{i}(q_{i}(n),r_{\widehat{F}_{n,s}}(i))$ is the component
of
$\widehat{F}_{n,s}^{-1}(\overline{B}_{\widetilde{X}_{n}}(\widehat{F}_{n,s}(q_{i}(n)),r_{\widehat{F}_{n,s}}(i)))$
containing $q_{i}(n)$, so that the conclusions of Theorem 1.2 hold for these
${\delta}_{i}(s),{\delta}_{i}(1,s),r_{\widehat{F}_{n,s}}(i,s),\Delta_{i}(q_{i}(n),r_{\widehat{F}_{n,s}}(i))$.
In particular,

$\displaystyle\Delta_{i}(q_{i}(n),r_{\widehat{F}_{n,s}}(i))$
$\displaystyle\subset$ $\displaystyle
B_{M_{n}}(q_{i}(n),\textstyle{\frac{{\varepsilon}_{1}}{3s}}),$
$\displaystyle\widehat{F}_{n,s}\left[\partial\Delta_{i}(q_{i}(n),r_{\widehat{F}_{n,s}}(i))\right]$
$\displaystyle\subset$ $\displaystyle\partial
B_{\widetilde{X}_{n}}(\widehat{F}_{n,s}(q_{i}(n)),r_{\widehat{F}_{n,s}}(i)).$
(5.17)

We next check that the domains and numbers

$A_{i}(n,s):=\Delta_{i}(q_{i}(n),r_{\widehat{F}_{n,s}}(i)),\qquad
r_{i}(n,s):=r_{\widehat{F}_{n,s}}(i)$

satisfy items 4(a)-…-4(e) stated in the lemma. Item 4(a) follows directly from
(5.16), and item 4(b) follows from (5.17). Regarding item 4(c), since
$A_{i}(n,s)$ is compact, its boundary has a finite number
$\widetilde{e}_{i}(n,s)\in\mathbb{N}$ of components. By item 2 of Theorem 1.2,
each boundary component of $A_{i}(n,s)$ admits an annular neighborhood which
is a multi-graph of positive integer multiplicity $m_{h,i}(n,s)\in\mathbb{N}$
(the index $h$ parameterizes the set of boundary components of $A_{i}(n,s)$).
The fact that both the number of such boundary components and the
multiplicities $m_{h,i}(n,s)$ can be considered to be independent of $n,s$
(after passing to a subsequence in $n$) follow from the fact the
$m_{h,i}(n,s)$ are bounded independently of $n$, which in turn can be deduced
from the following inequality (see item B(a) of Theorem 1.2):

$m_{h,i}(n,s)\leq 3\,\mbox{Index}(A_{i}(n,s))\leq 3\,I_{0},\quad\mbox{for all
$n$.}$

Now, the rest of properties stated in item 4(c) of the lemma are direct
consequences of Theorem 1.2 applied to $\widehat{F}_{n,s}$.

The convergence statement in item 4(d) follows from standard curvature
estimates for CMC immersions. The last sentence in item 4(d) follows from
uniqueness of the limit as $n\to\infty$ of the $\widetilde{F}_{n}$ restricted
to
$\Delta_{1}(n,\frac{2}{3})\setminus\cup_{i=1}^{J}\Delta_{i}(q_{i}(n),r_{\widehat{F}_{n}}(i))$
and from the already proven item 2 of this lemma. Item 4(e) holds by
construction, which finishes the proof of item 4 of the lemma.

Regarding item 5, choose $s_{1}^{\prime}=1$ and $s_{2}^{\prime}\in\mathbb{N}$,
$s_{2}^{\prime}>1$, such that
$\frac{{\varepsilon}_{1}}{3s_{2}^{\prime}}<{\delta}_{i}(1,1)$. Assuming
$s_{1}^{\prime},\ldots,s_{j}^{\prime}$ defined, choose
$s_{j+1}^{\prime}\in\mathbb{N}$ such that $s_{j+1}^{\prime}>s_{j}^{\prime}$
and
$\frac{{\varepsilon}_{1}}{3s_{j+1}^{\prime}}<{\delta}_{i}(s_{j}^{\prime},1)$.
This inequality implies that $A_{i}(n,s^{\prime}_{j+1})\subset\mbox{\rm
Int}(A_{i}(n,s^{\prime}_{j}))$ and so, $\mbox{\rm
Index}(A_{i}(n,s^{\prime}_{j+1}))\leq\mbox{\rm
Index}(A_{i}(n,s^{\prime}_{j}))$. Since $\mbox{\rm
Index}(A_{i}(n,s^{\prime}_{1}))$ is finite, there exists $j_{0}\in\mathbb{N}$
such that $\mbox{\rm Index}(A_{i}(n,s^{\prime}_{j}))=\mbox{\rm
Index}(A_{i}(n,s^{\prime}_{j_{0}}))$ for all $j\geq j_{0}$. Now label
$s_{j}=s^{\prime}_{j+j_{0}}$ for each $j\in\mathbb{N}$, and item 5 of the
lemma is proved.

By item 4(d), for each $j\in\mathbb{N}$ the restrictions of
$\widetilde{F}_{n}$ to
$\Delta_{1}(n,\frac{2}{3})\setminus\cup_{i=1}^{J}A_{i}(n,s_{j})$ converge
smoothly as $n\to\infty$ to a minimal immersion $F_{\infty,s_{j}}\colon
M_{s_{j}}\looparrowright\textstyle{\mathbb{B}(\vec{0},\frac{2}{3}r)}$ of a
compact surface $M_{s_{j}}$ with boundary. Since
$A_{i}(n,s_{j+1})\subset\mbox{\rm Int}(A_{i}(n,s_{j}))$ then
$M_{s_{j+1}}\subset M_{s_{j}}$ for each $j$, and by uniqueness of the limit we
have $F_{\infty,s_{j+1}}|_{M_{s}}=F_{\infty,s_{j}}$ for each $j$.

By a standard diagonal argument in $n$ and $s_{j}$, the map

$F_{\infty}\colon
M_{\infty}=\bigcup_{s_{j}\in\mathfrak{S}}M_{s_{j}}\looparrowright\textstyle{\mathbb{B}(\vec{0},\textstyle{\frac{2}{3}r})},$

given by $F_{\infty}|_{M_{s_{j}}}=F_{\infty,s_{j}}$ for each $j\in\mathbb{N}$,
is a minimal immersion with finite area, defined on a surface $M_{\infty}$ of
finite genus: the bound on the genus of $M_{\infty}$ is the same bound as on
the genus of the surface $\Delta_{1}(n)$, which, by item B of Theorem 1.2, is
at most $6I(\Delta_{1}(n))-8\leq 6I_{0}-8$ if the index $I(\Delta_{1}(n))\geq
2$; if $I(\Delta_{1}(n))=1$, then the genus of $\Delta_{1}(n)$ is zero.
Observe that $M_{\infty}$ has at least $J$ annular ends, and the number $e$ of
these ends of $M_{\infty}$ is finite (at most $3I_{0}-1$ by item B of Theorem
1.2). Furthermore, the image by $\widehat{F}_{\infty}$ of these ends of
$M_{\infty}$ is
$\\{x_{1},\ldots,x_{J}\\}\subset\mathbb{B}(\vec{0},\frac{2}{3}r)$. By
regularity results in [10], $M_{\infty}$ is a compact Riemann surface with $b$
boundary components, punctured in $e:=\sum_{i=1}^{J}\widetilde{e}_{i}$ points,
and we can denote the set of ends of $M_{\infty}$ by

$\\{P_{1,i},\ldots,P_{\widetilde{e}_{i},i}\\}_{i=1,\ldots,J},$

in such a way that the immersion $F_{\infty}$ extends to a finitely branched
minimal immersion

$\overline{F}_{\infty}\colon
M\cup\\{P_{1,i},\ldots,P_{\widetilde{e}_{i},i}\\}_{i=1,\ldots,J}\looparrowright\textstyle{\mathbb{B}(\vec{0},\frac{2}{3}r)},$
(5.18)

such that
$\overline{F}_{\infty}(\\{P_{1,i},\ldots,P_{\widetilde{e}_{i},i}\\})=\\{x_{i}\\}$,
$i=1,\ldots,J$, and by construction, the set of branch points of
$\overline{F}_{\infty}$ is contained in the set
$\\{P_{1,i},\ldots,P_{\widetilde{e}_{i},i}\ |\ i=1,\ldots,J\\}$. This proves
item 6 of the lemma.

Finally, we prove item 7. Observe that the branching order
$B(P_{h,i})\in\mathbb{N}\cup\\{0\\}$ of $\overline{F}_{\infty}$ at $P_{h,i}$
equals

$B(P_{h,i})=m_{h,i}-1,$ (5.19)

where $m_{h,i}\in\mathbb{N}$ is the multiplicity defined in item 4(c) above.
By adding this in the set $\\{P_{1,i},\ldots,P_{\widetilde{e}_{i},i}\\}$, we
deduce that the branching contribution $B_{i}\in\mathbb{N}\cup\\{0\\}$ to
$\overline{F}_{\infty}$ from this set is $B_{i}=S_{i}-\widetilde{e}_{i}$,
where $S_{i}=\sum_{h=1}^{\widetilde{e}_{i}}m_{h,i}$, and thus, (5.14) is
proved. Finally, the estimate (5.15) for the total spinning of
$\Delta(i,n,{\varepsilon})$ (for a sufficiently small ${\varepsilon}>0$)
follows from item B of Theorem 1.2. This finishes the proof of the lemma. ∎

We now come back to item (M2) above. Using the notation in Lemma 5.11,
suppose, after choosing a subsequence, that the $\frac{1}{r_{n}}F_{n}$
restricted to $\Delta_{2,n}$ converge to a family

$F_{\infty}\colon\widehat{\mathcal{D}}\looparrowright\mathbb{B}(1)$ (5.20)

of minimal disks branched at the origin as described in Lemmas 5.10 and 5.11.
Thus, the desired (global) limit
$f_{2}\colon\Sigma_{2}\looparrowright\mathbb{R}^{3}$ of the
$\frac{1}{r_{n}}F_{n}$ is already constructed in a neighborhood of
$\widehat{\mathcal{S}}(0)$ in $\widehat{\mathcal{D}}$ (see equation (5.13)),
where a non-trivial part of the index of $\frac{1}{r_{n}}F_{n}$ is collapsing
(namely, this collapsing index is $I(f_{1})>0$); since the remaining index of
$\frac{1}{r_{n}}F_{n}$ is at most $(I_{0}+1)-I(f_{1})\leq I_{0}$, we are
allowed to apply Lemma 5.12 to $\frac{1}{r_{n}}F_{n}$. We next make this
paragraph and the previously alluded to global convergence in item (M2)
rigorous.

######  Proposition 5.13.

In the situation above, let $\widetilde{F}_{n}\colon
M_{n}\looparrowright\widetilde{X}_{n}$ be $\frac{1}{r_{n}}F_{n}\colon
M_{n}\looparrowright\frac{1}{r_{n}}X_{n}$. Then, after replacing by a
subsequence, there exist $R_{0}\geq 10,{\varepsilon}_{2}\in(0,{\delta}_{1}]$
and a collection of points
$Q_{2}(n)=\\{q_{1}(n)=p_{1}(n),q_{2}(n),\ldots,q_{J}(n)\\}\subset
B_{M_{n}}(p_{1}(n),R_{0})$, $J\leq I_{0}$, such that:

  1. 1.

For any $R>R_{0}$, $\\{|A_{\widetilde{F}_{n}}|\\}_{n}$ is uniformly bounded in
$B_{M_{n}}(p_{1}(n),R)\setminus B_{M_{n}}(p_{1}(n),R_{0})$.

  2. 2.

$d_{M_{n}}(q_{i}(n),q_{j}(n))\geq{\varepsilon}_{2}$ for each $n\in\mathbb{N}$
and $i\neq j\in\\{1,2,\ldots,J\\}$.

  3. 3.

For each $i\in\\{1,2,\ldots,J\\}$ and $m\in\mathbb{N}$ with
$\frac{1}{m}<{\varepsilon}_{2}$,

$|A_{\widetilde{F}_{n}}|(q_{i}(n))>n=\max\\{|A_{\widetilde{F}_{n}}|(x):x\in
B_{M_{n}}(q_{i}(n),1/m)\\}$

and there exists $A_{2}(m)>1$ such that $|A_{\widetilde{F}_{n}}|<A_{2}(m)$ in
$B_{M_{n}}(p_{1}(n),R_{0})\setminus\cup_{i=1}^{J}B_{M_{n}}(q_{i}(n),1/m)$.

  4. 4.

There exist (not necessarily distinct) points
$x_{1}=\vec{0},x_{2},\ldots,x_{J}\in\mathbb{B}(\vec{0},R_{0})$ (here
$\mathbb{B}(\vec{0},R)$ denotes the ball centered the origin with radius $R>0$
in $\mathbb{R}^{3}$ with its flat metric) such that when viewed in harmonic
coordinates in $\widetilde{X}_{n}$ centered at $\widetilde{F}_{n}(p_{1}(n))$,
the points $\widetilde{F}_{n}(q_{i}(n))$ converge as $n\to\infty$ to $x_{i}$,
for each $i=1,2,\ldots,J$.

  5. 5.

For almost all $R>R_{0}$ and for $m$ sufficiently large, the
$\widetilde{F}_{n}$ restricted to

${\textstyle\overline{B}_{M_{n}}(p_{1}(n),R)\setminus\cup_{i=1}^{J}B_{M_{n}}(q_{i}(n),\frac{1}{m})}$

converge smoothly as $n\to\infty$ to a minimal immersion $F_{\infty,m,R}\colon
M_{m,R}\looparrowright\overline{\mathbb{B}}(\vec{0},R)$ of a compact surface
with boundary $M_{m,R}$. Furthermore, $M_{m,R}\subset M_{m+1,R^{\prime}}$ and
$F_{\infty,m+1,R^{\prime}}|_{M_{m,R}}=F_{\infty,m,R}$ whenever
$R^{\prime}>R>R_{0}$.

  6. 6.

Define

$\Sigma_{2}^{*}:=\bigcup_{\stackrel{{\scriptstyle
m\in\mathbb{N}}}{{R>R_{0}}}}M_{m,R},\qquad
f_{2}^{*}\colon\Sigma_{2}^{*}\looparrowright\mathbb{R}^{3},\
f_{2}^{*}|_{M_{m,R}}=F_{\infty,m,R}.$

Then, $\Sigma_{2}^{*}$ is a (possibly disconnected) open Riemann surface and
$f_{2}^{*}$ is a minimal immersion. Furthermore, the conformal completion
$\overline{\Sigma}_{2}$ of $\Sigma_{2}^{*}$ has the structure of a compact
Riemann surface,
$\overline{\Sigma}_{2}\setminus\Sigma_{2}^{*}=\mathcal{S}(f_{2})\cup\mathcal{E}_{2}$
is a finite set, and
$f_{2}^{*}\colon\Sigma_{2}^{*}\looparrowright\mathbb{R}^{3}$ extends through
$\mathcal{S}(f_{2})$ to a finitely branched, complete minimal immersion
$f_{2}\colon\Sigma_{2}=\Sigma_{2}^{*}\cup\mathcal{S}(f_{2})\looparrowright\mathbb{R}^{3}$
with finite total curvature, where

    1. (a)

$\mathcal{S}(f_{2})$ is the disjoint union of the finite set
$\mathcal{S}(\vec{0})=\\{P_{1,1},\ldots P_{e_{1},1}\\}\subset
f_{2}^{-1}(\\{x_{1}=\vec{0}\\})$ that appears in Lemma 5.11, together with the
closely related finite sets

$\mathcal{S}(x_{i})=\\{P_{1,i},\ldots,P_{b_{i},i}\\}\subset
f_{2}^{-1}(\\{x_{i}\\}),\quad i=2,\ldots J.$

Furthermore, the set of branch points of $f_{2}$ is contained in
$\mathcal{S}(f_{2})$ and its branch locus (image) is contained in
$\\{x_{1}=\vec{0},x_{2},\ldots,x_{J}\\}\subset\mathbb{B}(R_{0})$.

    2. (b)

The set of ends of $f_{2}$ is $\mathcal{E}_{2}=\\{E_{1},\ldots,E_{e_{2}}\\}$.

    3. (c)

The map $F_{\infty}$ given in (5.20) coincides with $f_{2}$ in a neighborhood
of $\mathcal{S}(x_{1}=\vec{0})$ in $\Sigma_{2}$.

  7. 7.

The total branching order $B(f_{2})$ of $f_{2}$ can be estimated from above as
follows:

$B(f_{2})\leq 3[I_{0}+1-\mbox{\rm Index}(f_{1})]-J\leq 3I_{0}-1.$ (5.21)

  8. 8.

The following properties hold for some $R>3R_{0}$:

    1. (a)

The index of $f_{2}^{-1}(\mathbb{B}(R/3))$ is $I(f_{2})$ (compare to property
(H0)’ above).

    2. (b)

$f_{2}(\Sigma_{2})\setminus\mathbb{B}(R/3)$ consists of $e_{2}$ multi-graphs
over their projections to planes $\Pi_{j}\subset\mathbb{R}^{3}$ passing though
$\vec{0}$, $j=1,\ldots,e_{2}$ (compare to property (H1)’). Furthermore, each
of these end representatives contains no non-trivial geodesic arcs with
boundary points in the boundary of $\Sigma_{2}\setminus
f_{2}^{-1}(\mathbb{B}(R/3))$.

    3. (c)

The image through the Gauss map of $f_{2}$ of each component $C_{j}$ of
$f_{2}(\Sigma_{2})\setminus\mathbb{B}(R/3)$ is contained in the spherical
neighborhood of radius ${\alpha}_{1}/2$ centered at a point
$v_{j}\in\mathbb{S}^{2}(1)$ perpendicular to $\Pi_{j}$, where
${\alpha}_{1}={\alpha}_{1}(\tau)\in(0,\tau]$ is the constant given by Lemma
4.1 for $L_{0}=3\pi(I_{0}+2)+1$ (thus, $C_{j}$ satisfies item (B2) of Lemma
4.1 with $R_{1}=R/3$ and ${\alpha}={\alpha}_{1}/2$, compare to item (H2)’).

    4. (d)

$f_{2}(\Sigma_{2})$ makes an angle greater than
$\frac{\pi}{2}-\frac{{\alpha}_{1}}{2}$ with every sphere $\mathbb{S}^{2}({r})$
of radius $r\geq R/3$ centered at the origin (so, $C_{j}$ satisfies item (B1)
of Lemma 4.1 with $R_{1}=R/3$ and ${\alpha}={\alpha}_{1}/2$, compare to
(H3)’).

    5. (e)

The total length of the intersection of $f_{2}(\Sigma_{2})$ with any sphere
$\mathbb{S}^{2}({r})$ centered at the origin and radius $r\geq R/3$ is less
than $(L_{0}-\frac{1}{2})r$ (hence $C_{j}$ satisfies item (B3) of Lemma 4.1
with $R_{1}=R/3$, compare to (H4)’).

    6. (f)

For all $n\in\mathbb{N}$, the component $\Delta_{2,n}(R/3)$ of
$\widetilde{F}_{n}^{-1}(B_{\widetilde{X}_{n}}(\widetilde{F}_{n}(p_{1}(n)),R/3))$
that contains $p_{1}(n)$ has index at least $I(f_{1})+I(f_{2})+(J-1)$ and if
$J=1$, then $I(f_{2})>0$. In particular, $I(\Delta_{2,n}(R/3))>I(f_{1})$.

###### Proof.

Recall the notation and statement of Lemma 5.11. By assumption, the
$\widetilde{F}_{n}$ restricted to $\Delta_{2,n}$ converge to $F_{\infty}$
given by equation (5.20). Since the restriction of $F_{\infty}$ to
$F_{\infty}^{-1}(\mathbb{B}(1)\setminus\mathbb{B}(\frac{1}{2}))$ consists of
$e_{1}$ multi-graphs (here $e_{1}$ is the number of ends of $f_{1}$), then
$\widetilde{F}_{n}(\Delta_{2,n})$ is graphical in the region
$B_{\widetilde{X}_{n}}(\widetilde{F}(p_{1}(n)),1)\setminus
B_{\widetilde{X}_{n}}(\widetilde{F}(p_{1}(n)),\frac{1}{2})$, and thus, the
surfaces

${\textstyle
M^{\prime}_{n}=M_{n}\setminus[\Delta_{2,n}\cap\widetilde{F}_{n}^{-1}(B_{\widetilde{X}_{n}}(\widetilde{F}(p_{1}(n)),\frac{1}{2})]}$

have uniform curvature estimates in a fixed sized
${\varepsilon}^{\prime}_{0}$-neighborhood of its boundary (for some
${\varepsilon}_{0}^{\prime}\in(0,{\varepsilon}_{0}]$). Let
$F^{\prime}_{n}\colon M_{n}^{\prime}\looparrowright\widetilde{X}_{n}$ be the
restriction of $\widetilde{F}_{n}$ to $M_{n}^{\prime}$ and for all
$n\in\mathbb{N}$, we can consider $F_{n}^{\prime}$ to be an element in a fixed
related space $\Lambda^{\prime}$ except that the index of the immersions in

$\Lambda^{\prime}=\Lambda(I_{0},H_{0},{\varepsilon}_{0}^{\prime},A_{0},K_{0})$

is at most $I_{0}$. By induction, we can suppose that Theorem 1.2 holds for
the subspace $\Lambda^{\prime}$.

The construction of the finite set $\\{q_{2}(n),\ldots,q_{J}(n)\\}\subset
B_{M_{n}}(p_{1}(n),R_{0})$, $J\leq I_{0}$, appearing in the statement of the
proposition, follows exactly the same arguments used to prove the existence of
the related set $Q(n)$ given in item 3 of Lemma 5.12. Similarly, items 2, 3
and 4 of the proposition can be deduced from the same reasoning as items 3(b),
3(c) and 3(d) of Lemma 5.12 respectively; in particular, we use the number
${\delta}_{1}\in(0,{\varepsilon}_{0}/2]$ defined in item 1 of Lemma 5.12 in
order to find ${\varepsilon}_{2}\in(0,{\delta}_{1}]$ satisfying item 2 of the
proposition. We leave the details to the reader.

The existence of the number $R_{0}\geq 10$ and item 1 of the proposition
follows from the fact that the number $J$ of sequences