# Geometry of CMC surfaces of finite index

William H. Meeks III∗ Joaquín Pérez Research of both authors was partially
supported by MINECO/MICINN/FEDER grant no. PID2020-117868GB-I00, regional
grants P18-FR-4049 and A-FQM-139-UGR18, 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 $r_{0}>0$, $I\in\mathbb{N}\cup\\{0\\}$ and $K_{0},H_{0}\geq 0$, let $X$
be a complete Riemannian $3$-manifold with injectivity radius $\mbox{\rm
Inj}(X)\geq r_{0}$ and with the supremum of absolute sectional curvature at
most $K_{0}$, and let $M\looparrowright X$ be a complete immersed surface with
index at most $I$ of constant mean curvature $H\in[0,H_{0}]$. We will obtain
geometric estimates for such an $M\looparrowright X$ as a consequence of the
Hierarchy Structure Theorem in [3]. The Hierarchy Structure Theorem (see
Theorem 2.2 below) will be applied to understand global properties of
$M\looparrowright X$, especially results related to the area and diameter of
$M$. By item E of Theorem 2.2, the area of such a non-compact
$M\looparrowright X$ is infinite. We will improve this area result by proving
the following when $M$ is connected; here $g(M)$ denotes the genus of the
orientable cover of $M$:

  1. 1.

There exists $C=C(I,r_{0},K_{0},H_{0})>0$ such that $\mbox{\rm Area}(M)\geq
C(g(M)+1)$.

  2. 2.

There exists $C^{\prime}>0$ independent of $I,r_{0},K_{0},H_{0}$ and
$G=G(I,r_{0},K_{0},H_{0})\in\mathbb{N}$ such that if $g(M)\geq G$, then
$\mbox{\rm Area}(M)\geq C^{\prime}(g(M)+1)$.

  3. 3.

If the scalar curvature $\rho$ of $X$ satisfies $3H^{2}+\frac{1}{2}\rho\geq c$
in $X$ for some $c>0$, then there exist $A,D>0$ depending on
$c,I,r_{0},K_{0},H_{0}$ such that $\mbox{\rm Area}(M)\leq A$ and $\mbox{\rm
Diameter}(M)\leq D$. Hence, $M$ is compact and, by item 1, $g(M)\leq A/C-1.$

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, Hierarchy Structure
Theorem, Bishop-Cheeger-Gromov relative volume comparison theorem, area of
hyperbolic annuli.

##  1 Introduction

Throughout the paper, $X$ denotes 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 $M$, $|A_{M}|$ is 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 2.3 for details.

The primary goal of this paper is to apply the Hierarchy Structure Theorem 2.2
(proven in [3]) to understand certain global properties of closed constant
mean curvature surfaces in Riemannian 3-manifolds. Theorem 2.2 describes the
geometric 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 curvature $H\geq 0$, in certain small intrinsic
neighborhoods of points with sufficiently large norm $|A_{M}|$ of their second
fundamental forms.

Our application of Theorem 2.2 is stated below; it provides lower bounds for
the areas and intrinsic diameters of immersed closed $H$-surfaces $M$ in $X$
of finite index in terms of their genera, when the indices and the constant
mean curvatures of the surfaces are bounded from above by fixed constants. We
also provide upper bound area estimates for $B_{M}(x,r)$ in $M$ for every
$x\in M$ and $r>0$, independently on whether or not $M$ is compact but
depending on upper bounds for $H$ and its index.

In the case that $M$ is non-orientable, the genus $g(M)$ of $M$ is the genus
of its oriented cover.

######  Theorem 1.1 (Area and diameter estimates).

For $r_{0}>0$, $K_{0},H_{0}\geq 0$, consider all complete Riemannian
3-manifolds $X$ with injectivity radius $\mbox{\rm Inj}(X)\geq r_{0}$ and
absolute sectional curvature bounded from above by $K_{0}$, and let
$\lambda=\max\\{1,\frac{1}{r_{0}},\sqrt{K_{0}},H_{0}\\}$. Let $M$ be a
complete immersed $H$-surface in $X$ with empty boundary, $H\in[0,H_{0}]$,
index at most $I\in\mathbb{N}\cup\\{0\\}$ and genus $g(M)$, which in the
language of Theorem 2.2 implies $M\in\Lambda=\Lambda(I,H_{0},r_{0},1,K_{0})$
with additional chosen constant $\tau=\pi/10$. Then:

  1. 0.

The area of $M$ is greater than $C_{A}/{\lambda}^{2}$, where

$C_{A}:=\pi\,\left(\frac{\pi}{4}\right)^{2}e^{-\frac{\pi}{2}-1+\frac{\pi}{4}}\approx
0.325043,$

and if $M$ is compact, the extrinsic diameter of each component of $M$ is
greater than $\frac{\pi}{4{\lambda}}$.

  2. 1.

(Item 1 in the abstract). There exists $C_{1}(I)>0$, independent of
$M,r_{0},K_{0},H_{0}$, such that:

$\mathrm{Area}(M)\geq\frac{C_{1}(I)}{\lambda^{2}}(g(M)+1).$ (1.1)

  3. 2.

    1. (a)

Suppose that one of the following two conditions holds:

      1. (i)

$I=0$, i.e., $M$ is stable.

      2. (ii)

$I\geq 1$ and $k=0$ with the notation of Theorem 2.2 (in particular,
$M=\widetilde{M}$).

Depending on whether condition (i) or (ii) holds, we introduce the following
constant $K_{1}=K_{1}(I)$. If condition (i) holds, let $C_{s}\geq 2\pi$ be the
universal curvature estimate for stable $H$-surfaces described in Theorem 3.5
below, and let $K_{1}:=-1-\frac{1}{2}C_{s}^{2}$. If condition (ii) holds, let
$K_{1}=K_{1}(I)=-1-\frac{1}{2}A_{1}^{2}$ where $A_{1}=A_{1}(I)\geq 1$ is the
constant given by Theorem 2.2 for the space $\Lambda(I,1,1,1,1)$.

For all $x\in M$ and $r>0$,

$\mbox{\rm
Area}(B_{M}(x,r))\leq\frac{2\pi}{-K_{1}{\lambda}^{2}}\left[\mathop{\rm
cosh}\nolimits\left({\lambda}\sqrt{-K_{1}}r\right)-1\right],$ (1.2)

and if $M$ is connected, then

$\mbox{\rm Diameter}(M)\geq\frac{1}{\sqrt{K_{1}}{\lambda}}\mathop{\rm
arccosh}\nolimits\left[\frac{-K_{1}C_{1}(I)}{2\pi}(g(M)+1)+1\right].$ (1.3)

    2. (b)

Suppose $k\geq 1$ (in particular, $I\geq 1$). Let
${\delta}={\delta}(I)\in(0,\frac{1}{2}]$ be the constant described in Theorem
2.2 for the space $\Lambda(I,1,1,1,1)$, and let
$\Delta_{1},\ldots,\Delta_{k}$, $k\leq I$, be the smooth compact domains
associated to $M$ introduced in item 1 of Theorem 2.2. There exists
$A_{2}(I)\geq 6$, independent of $M,r_{0},K_{0},H_{0}$, such that for all
$x\in M$ and all $r>0$, the following estimates hold:

$\mbox{\rm
Area}(B_{M}(x,r)\setminus\cup_{i=1}^{k}\Delta_{i})\leq\frac{2(6\pi+1)}{{\lambda}^{2}A_{2}(I)}I\left[2[\mathop{\rm
cosh}\nolimits({\lambda}\sqrt{A_{2}(I)}r)-1]+\frac{\sinh({\lambda}\sqrt{A_{2}(I)}r)}{A_{2}(I)^{1/2}}\right],$
(1.4) $\mbox{\rm
Area}(B_{M}(x,r))\leq\frac{I}{{\lambda}^{2}}\left[\frac{2(6\pi+1)}{A_{2}(I)}\left(2[\mathop{\rm
cosh}\nolimits({\lambda}\sqrt{A_{2}(I)}r)-1]+\frac{\sinh({\lambda}\sqrt{A_{2}(I)}r)}{A_{2}(I)^{1/2}}\right)+\frac{3\pi}{8}\right],$
(1.5)

and if $M$ is connected, then

$\mbox{\rm Diameter}(M)\geq\frac{1}{{\lambda}\sqrt{A_{2}(I)}}\mathop{\rm
arccosh}\nolimits\left[\frac{C_{1}(I)}{20I}(g(M)+1)\right].$ (1.6)

  4. 3.

(Item 2 in the abstract). Let $C=\pi/(3+4C_{s}+4C_{s}^{2})$. There exists a
$G(I)\in\mathbb{N}$, so that whenever $g(M)\geq G(I)$, then:

$\mathrm{Area}(M)\geq\frac{C}{{\lambda}^{2}}(g(M)+1).$ (1.7)

  5. 4.

(Item 3 in the abstract). If the scalar curvature $\rho$ of $X$ satisfies
$3H^{2}+\frac{1}{2}\rho\geq c$ in $X$ for some $c>0$, then $M$ is compact, and
there exists $A_{3}(I,c)>0$ such that

$\mbox{\rm Area}(M)\leq\frac{A_{3}(I,c)}{{\lambda}^{2}}\qquad\mbox{\rm
Diameter}(M)\leq\frac{4\pi(I+1)}{{\lambda}\sqrt{3c}},\qquad
g(M)\leq\frac{A_{3}(I,c)}{C_{1}(I)}-1.$ (1.8)

In the proof of Theorem 1.1 the estimates for the constants $C_{1}(I)$,
$C_{2}(I)$, $G(I)$, $A_{2}(I)$, $A_{3}(I,c)$ will be given in terms of the
related constants $A_{1}(I)$, ${\delta}(I)$ given in the Hierarchy Structure
Theorem for the space $\Lambda(I,1,1,1,1)$ described in Definition 2.1.

In [8], Saturnino obtained, independently, related bounds for the genus of a
closed, connected, embedded surface of constant mean curvature in a closed
three-dimensional Riemannian manifold. Such bounds depend on the ambient
geometry, and on the index and area of the surface.

Acknowledgments: The authors would like to thank Harold Rosenberg for his
thoughts and discussions of our initial attempts at understanding the
existence of the linear area estimates given in item 1 of Theorem 1.1.

##  2 The Hierarchy Structure Theorem

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 2.1.

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 is at least one point in $M$ of distance
${\varepsilon}_{0}$ 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 radius $r\in(0,\mbox{\rm Inj}_{X}(x))$ (here, $\mbox{\rm
Inj}_{X}(x)$ stands for the injectivity radius of $X$ at $x$) and with a
$C^{1,{\alpha}}$ control of the ambient metric on $X$, see Definition 2.4 for
details.

######  Theorem 2.2 (Structure Theorem for finite index $H$-surfaces [3]).

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.

$|A_{M}|(p_{i})=\max_{\Delta_{i}}|A_{M}|=\max\\{|A_{M}|(p)\ :\ p\in
M\setminus\cup_{j=1}^{i-1}B_{M}(p_{j},\frac{5}{4}r_{F}(j))\\}\geq 10A_{1}$,
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 2.4), 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 2.5 for this notion
of multi-graph. $G_{i,1},\ldots,G_{i,e(i)}$ 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

$\frac{|u(x)|}{|x|}+|\nabla u|(x)\leq\tau,$ (2.1)

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: $|A_{M}|\leq A_{1}$ 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 $I(\Delta_{i})\geq 2$ and $\Delta_{i}$ is orientable, 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,

$2\pi S-\frac{\tau}{2}k\leq\sum_{i=1}^{k}\kappa(\Delta_{i})\leq 2\pi
S+\frac{\tau}{2}k.$ (2.2)

    6. (f)

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

$-\int_{\cup_{i=1}^{k}\Delta_{i}}K=-2\pi\chi(\cup_{i=1}^{k}\Delta_{i})+\int_{\cup_{i=1}^{k}\partial\Delta_{i}}\kappa_{g}>3k\pi.$
(2.3)

  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

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

  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,\mathrm{Radius}(M)\\}}&\mbox{if $\partial M\neq\varnothing$,}\\\
{\displaystyle C\max\\{1,\mathrm{Diameter}(M)\\}}&\mbox{if $\partial
M=\varnothing$,}\end{array}\right.$ (2.4)

where

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

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

######  Definition 2.3.

Given a $1$-sided minimal immersion $F\colon M\looparrowright X$, 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 2.4.

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 [1], 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 on 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,

  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$(X)\geq r_{0}>0$, then Theorem 6 in [1] implies that given
$Q>1$ and ${\alpha}\in(0,1)$ there exists $C=C(Q,{\alpha},r_{0},K_{0})$
(observe that $C$ does not depend on $X$) such that $r(Q,{\alpha})(X)\geq C$.

######  Definition 2.5.

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
$\vec{0}\not\in(\Pi\circ f)(\Sigma)$ 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 degree $m$ as a multi-
graph.

Given $Q>1$ and ${\alpha}\in(0,1)$, let $X$ be a Riemannian 3-manifold and
$(x_{1},x_{2},x_{3})$ 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 2.4. 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 $\frac{|u(x)|}{|x|}+|\nabla u|<\tau$ for some
$\tau\in(0,\pi/10]$ and $Q>1$ sufficiently close to $1$, then
$\frac{|u(x)|}{|x|}+|\nabla u|<2\tau$ for any other choice of harmonic chart
around $x_{0}$ with this restriction of $Q$.

##  3 The proof of Theorem 1.1

This section is dedicated to the proof of Theorem 1.1. Note the complete
surfaces considered in this theorem have empty boundary. Let $F\colon
M\looparrowright X$ be an immersion as in the statement of Theorem 1.1.

We will use the notation in Theorem 2.2 and fix $\tau=\pi/10$. Notice that as
the boundary of $M$ is empty, then we may consider the $H$-immersion $F\colon
M\looparrowright X$ of index $I$ described in Theorem 1.1 to be an element of
$\Lambda(I,H_{0},r_{0},1,K_{0})$, where $H_{0},r_{0},K_{0}$ are given in the
hypotheses of Theorem 1.1.

###  3.1 Normalizing the space $\Lambda$

After scaling the Riemannian metric of $X$ by the square root of

$\lambda=\max\\{1,\frac{1}{r_{0}},\sqrt{K_{0}},H_{0}\\},$ (3.1)

one obtains a new Riemannian manifold $X^{\prime}$; note that this scaling of
the metric scales arc length in $X$ by the factor $\lambda\geq 1$, and that
the metric of $M$ induced by the isometric immersion $F$ creates an associated
isometric immersion $F^{\prime}\colon M\looparrowright X^{\prime}$ such that
$F^{\prime}(p)=F(p)$ for each $p\in M$. After this homothetic change of the
metric, we can consider $F^{\prime}\colon M\looparrowright X^{\prime}$ to be
an immersion satisfying the following properties:

  1. 1.

$\mbox{\rm Inj}(X^{\prime})\geq 1$.

  2. 2.

The absolute sectional curvature of $X^{\prime}$ is less than or equal to 1.

  3. 3.

$F^{\prime}$ is an isometric immersion of constant mean curvature
$H^{\prime}\in[0,1]$.

  4. 4.

$(F^{\prime}\colon M\looparrowright X^{\prime})\in\Lambda(I,1,1,1,1)$.

  5. 5.

$\mathrm{Area}(F)=\lambda^{2}\,\mathrm{Area}(F^{\prime}).$

  6. 6.

$\mathrm{Diameter}(F)=\lambda\,\mathrm{Diameter}(F^{\prime}).$

Items 5 and 6 above allow us to easily convert estimates on the area of
subdomains and lengths of curves in the domain of $F$ to areas and lengths of
the corresponding domains and curves in the domain of $F^{\prime}$, and
thereby, these conversion formulae reduce the proofs of statements given in
Theorem 1.1 for $F\in\Lambda(I,H_{0},r_{0},1,K_{0})$ to the corresponding
estimates for $F^{\prime}$ in $\Lambda(I,1,1,1,1)$. Thus, for the remainder of
the proof of Theorem 1.1, we will assume $F\colon M\looparrowright X$ lies in
$\Lambda(I,1,1,1,1)$, and refer to Area$(M)$, Diameter$(M)$ for those with
respect to the induced metric by $F$.

###  3.2 Proof of item  ‣ 1.1 of the Theorem

Consider an element $(F\colon M\looparrowright X)\in\Lambda(I,1,1,1,1)$. If
$M$ is non-compact, then the last sentence in item E of Theorem 2.2 states
that $M$ has infinite area, which proves that the inequality Area$(M)\geq
C_{A}$ in item  ‣ 1.1 holds vacuously (for any choice of $C_{A}>0$). If
moreover $M$ is connected, then there exists is a geodesic ray in $M$, i.e.,
an embedded, length-minimizing unit-speed geodesic arc
${\gamma}\colon[0,\infty)\to M$; in particular, the diameter of $M$ is
infinite, and thus the second statement in item  ‣ 1.1 also holds vacuously.

For the remainder of this section we will assume that $M$ is compact.

######  Lemma 3.1.

Given $x_{0}\in M$, let $M(x_{0})$ be the component of $M$ containing $x_{0}$.
Then, $M(x_{0})$ is not contained in the closed extrinsic ball
$\overline{B}_{X}(x_{0},\pi/4)$ (in particular, $\partial B_{M}(x_{0},\pi/4)$
is not empty).

######  Remark 3.2.

Observe that if the lemma holds, then the extrinsic diameter of $M$ is greater
than $\pi/4$ (in particular, the intrinsic diameter has the same lower bound),
which proves the second statement in item  ‣ 1.1 of Theorem 1.1.

###### Proof of Lemma 3.1.

Fix a point $x_{0}\in M$ and let $r\in(0,\pi/4)$. Since the injectivity radius
of $X$ is at least 1, all the distance spheres $\partial B_{X}(x_{0},r)$ with
$r\in(0,1)$ are geodesic spheres. By comparison results and since the absolute
sectional curvature of $X$ is bounded by 1, the second fundamental form of
$\partial B_{X}(x_{0},r)$ has normal curvatures greater than $1$. Assume that
$M(x_{0})$ is contained in $\overline{B}_{X}(x_{0},r)$. As $M(x_{0})$ is
compact, then there exists a largest $r_{1}\in(0,r]$ such that
$M(x_{0})\subset\overline{B}_{X}(x_{0},r_{1})$, and there exists $x\in
M(x_{0})\cap\partial B_{X}(x_{0},r_{1})$. This implies that all the normal
curvatures of $M$ at $x$ are greater than $1$, which implies that the mean
curvature of $M$ is greater than 1, which contradicts that $F\colon
M\looparrowright X$ lies in $\Lambda(I,1,1,1,1)$. This contradiction proves
that $M(x_{0})$ cannot be contained in $\overline{B}_{X}(x_{0},r)$. Since this
holds for every $r\in(0,\pi/4)$ and $M$ is compact, we conclude that
$M(x_{0})$ cannot be contained in $B_{X}(x_{0},\pi/4)$. In fact, $M(x_{0})$
cannot be contained in $\overline{B}_{X}(x_{0},\pi/4)$ (otherwise the maximum
principle for the mean curvature operator would imply that $M(x_{0})=\partial
B_{X}(x_{0},r)$, which contradicts that $x_{0}\in M(x_{0})$). Now the lemma is
proved. ∎

Using [2, Proposition 2.5 and item 3 of Remark 2.5] with $R_{1}=a=H_{0}=1$,
for each $p\in\mbox{\rm Int}(M)$ we have

$\mathrm{Area}[B_{M}(p,r)]\geq
E(r):=\pi\,r^{2}e^{-2r-1+r\cot(r)}\quad\mbox{for every }r\in(0,\pi/4].$ (3.2)

Therefore, Area$(M)\geq\mathrm{Area}[B_{M}(p,\pi/4)]=E(\pi/4)\approx
0.325043$. This completes the proof of item  ‣ 1.1 of Theorem 1.1.

###  3.3 Proof of item 1 of the theorem

Consider an element $(F\colon M\looparrowright X)\in\Lambda(I,1,1,1,1)$. If
$M$ is non-compact, then the last sentence in item E of Theorem 2.2 states
that $M$ has infinite area, which vacuously implies item 1 of the theorem
holds (for any choice of $C_{1}(I)$). Henceforth, assume $M$ is compact.

Let $M=M_{1}\cup\ldots\cup M_{b}$, $b\in\mathbb{N}$, be the decomposition of
$M$ in connected components. Assume inequality (1.1) holds for each $M_{i}$.
Then, we can find $C_{1,i}(I)>0$ such that

$\mbox{Area}(M_{i})\geq C_{1,i}(I)(g(M_{i})+1)$ (3.3)

(because the index of each $M_{i}$ is at most $I$), where $g(M_{i})$ is the
genus of $M_{i}$. Define $C_{1}(I)=\min\\{C_{1,1}(I),\ldots,C_{1,b}(I)\\}>0$.
Thus, adding up (3.3) in $i=1,\ldots,b$, we deduce that (1.1) holds for $M$.
Henceforth, it suffices to prove that (1.1) holds under the additional
assumption that $M$ is connected.

The region $\widetilde{M}\subset M$ defined in item 3 of Theorem 2.2 for the
space $\Lambda(I,1,1,1,1)$ produces a uniform bound $A_{1}=A_{1}(I)\geq 1$
from above on the norm the second fundamental form of $\widetilde{M}$. Let us
define

$K_{1}=K_{1}(I):=-1-\frac{1}{2}A_{1}^{2}.$ (3.4)

Since $A_{1}\geq 1$, then $K_{1}\leq-\frac{3}{2}$. The Gauss equation gives

$K=K_{X}(TM)+\mathop{\rm det}\nolimits(A_{M}),$ (3.5)

where $K$ denotes the Gaussian curvature of $M$ and $K_{X}(TM)$ is the
sectional curvature of $X$ for the tangent plane to $M$. Since the absolute
sectional curvature of $X$ is bounded by 1, $H^{2}\geq\mathop{\rm
det}\nolimits(A)$ and $H\in[0,1]$, we have the following upper and lower
estimates for $K$ in $\widetilde{M}$:

$K_{1}\leq-1-\frac{1}{2}|A_{M}|^{2}\leq-1+\mathop{\rm det}\nolimits(A_{M})\leq
K\leq 1+\mathop{\rm det}\nolimits(A_{M})\leq 1+H^{2}\leq 2.$ (3.6)

####  3.3.1 Item 1 holds when $k=0$.

We first show that item 1 of Theorem 1.1 holds in the special case that the
integer $k$ defined in Theorem 2.2 is zero. To see this, observe that
$\widetilde{M}=M$, and thus, (3.6) ensures that $M$ has Gaussian curvature
bounded from below by $K_{1}$ and from above by 2. Let $\widehat{M}$ be the
orientable cover of $M$.

Suppose $g(M)=0$ (recall that $M$ was assumed to be compact and connected).
Applying to $\widehat{M}$ the Gauss-Bonnet theorem, we have

$2\cdot\mbox{Area}(\widehat{M})\geq\int_{\widehat{M}}K=4\pi.$

If $M$ is non-orientable, then
$\mbox{Area}(M)=\frac{1}{2}\mbox{Area}(\widehat{M})\geq\pi$. This inequality
also holds in the case $M$ is orientable (in fact, $M=\widehat{M}$ and so,
$\mbox{Area}(M)=\mbox{Area}(\widehat{M})\geq 2\pi$). Therefore, inequality
(1.1) holds with $C_{1}(I)=\pi$ if $g(M)=0$ and $k=0$.

Suppose now that $g(M)\geq 2$. Hence, Gauss-Bonnet applied to $\widehat{M}$
gives

$-K_{1}\cdot\mbox{Area}(\widehat{M})\geq-\int_{\widehat{M}}K=-2\pi\chi(\widehat{M})=4\pi(g(\widehat{M})-1)=4\pi(g(M)-1)\geq\frac{4\pi}{3}(g(M)+1).$

If $M$ is non-orientable, then
$\mbox{Area}(M)=\frac{1}{2}\mbox{Area}(\widehat{M})\geq\frac{2\pi}{3|K_{1}|}(g(M)+1)$.
This inequality also holds in the case $M$ is orientable (in fact,
$M=\widehat{M}$ and thus,
$\mbox{Area}(M)=\mbox{Area}(\widehat{M})\geq\frac{4\pi}{3|K_{1}|}(g(M)+1)$).
Therefore, inequality (1.1) holds with $C_{1}(I)=\frac{2\pi}{3|K_{1}|}$ if
$g(M)\geq 2$ and $k=0$.

By the already proven item  ‣ 1.1 of Theorem 1.1, the area of $M$ is at least
$C_{A}$. In particular if $g(M)=1$ (i.e., $M$ is a torus or a Klein bottle),
then one can still obtain a lower bound estimate for the area of $M$ by

$\mbox{Area}(M)\geq C_{A}=\frac{C_{A}}{2}(g(M)+1).$

Therefore, inequality (1.1) holds with $\displaystyle
C_{1}(I)=\frac{C_{A}}{2}$ if $g(M)=1$ and $k=0$.

Finally we consider the minimum of the constants
$\pi,\frac{2\pi}{3|K_{1}|},\frac{C_{A}}{2}$ obtained in the three cases above.
As observed previously, $|K_{1}|\geq\frac{3}{2}$, and so,

$C_{3}=C_{3}(I)=\min\\{\pi,\frac{2\pi}{3|K_{1}|},\frac{C_{A}}{2}\\}=\min\\{\frac{2\pi}{3|K_{1}|},\frac{C_{A}}{2}\\},$

we deduce that (1.1) holds with $C_{3}$, instead of $C_{1}$, for connected
compact $M$ when $k=0$.

####  3.3.2 Item 1 holds when $k\geq 1$.

Assume that $k\geq 1$ (in particular, $I\geq 1$) and we will obtain a constant
$C_{4}=C_{4}(I)\in(0,\pi{\delta}_{1}^{2})$ that satisfies

$\mbox{Area}(M)\geq C_{4}(g(M)+1),$ (3.7)

which will complete the proof of item 1 of Theorem 1.1 after setting
$C_{1}(I)=\min\\{C_{3}(I),C_{4}(I)\\}$. We will need the following two claims.

######  Claim 3.3.

If $g(M)\geq 12I-3$, then inequality (3.7) holds with constant
$C^{\prime}_{4}(I)=\frac{\pi}{|K_{1}(I)|}$.

######  Claim 3.4.

If $g(M)<12I-3$. then inequality (3.7) holds with
$C^{\prime\prime}_{4}(I)=\frac{C_{A}}{12I-3}$.

###### Proof of Claim 3.3.

We start applying the Gauss-Bonnet formula and (3.6):

$|K_{1}|\cdot\mbox{\rm
Area}(\widetilde{M})\geq\left|\int_{\widetilde{M}}K\right|=\left|\int_{{M}}K-\int_{\cup_{i=1}^{k}\Delta_{i}}K\right|.$
(3.8)

On the other hand, calling $g=g(M)$,

$\displaystyle\int_{{M}}K-\int_{\cup_{i=1}^{k}\Delta_{i}}K$ $\displaystyle=$
$\displaystyle
2\pi(\chi({M})-2\pi\chi(\cup_{i=1}^{k}\Delta_{i})+\int_{\partial(\cup_{i=1}^{k}\Delta_{i})}\kappa_{g}$
(3.9) $\displaystyle\leq$ $\displaystyle 2\pi(1-g)+2\pi(6I-2S-e)+2\pi S+\tau
k,$

where in the last inequality we have used item B(d) of Theorem 2.2 and (2.2).
Since $\tau\leq\pi/10$ in Theorem 2.2 and $S+e\geq 4k$ (this last inequality
follows since $e(i)\geq 1$ and $m(i)\geq 2$, and if $e(i)=1$ then $m(i)\geq
3$), we can bound (3.9) from above by $2\pi(1-g+6I-4k+\frac{k}{20})$, which in
turn is at most $2\pi(-g+6I-2)$ because $k\geq 1$. Therefore,

$\int_{{M}}K-\int_{\cup_{i=1}^{k}\Delta_{i}}K\leq-\pi(2g-12I+4).$ (3.10)

Since $g\geq 12I-3$ by hypothesis, then the RHS of (3.10) is at most
$-\pi(g+1)$, and thus, we conclude that

$\int_{{M}}K-\int_{\cup_{i=1}^{k}\Delta_{i}}K\leq-\pi(g+1).$ (3.11)

Now, (3.11) and (3.8) give

$|K_{1}|\cdot\mbox{\rm Area}(\widetilde{M})\geq\pi(g+1),$ (3.12)

from where Claim 3.3 follows. ∎

###### Proof of Claim 3.4.

By the already proven item  ‣ 1.1 of Theorem 1.1, we have
$\mathrm{Area}(M)\geq C_{A}$, which is $\geq\frac{C_{A}}{12I-3}(g(M)+1)$
$g(M)<12I-3$. This finishes the proof of Claim 3.4. ∎

Once Claims 3.3 and 3.4 are proved, we will conclude that inequality (3.7)
holds in all cases with $k\geq 1$ with
$C_{4}(I)=\min\\{C^{\prime}_{4}(I),C^{\prime\prime}_{4}(I)\\}$. This completes
the proof of item 1 of Theorem 1.1.

###  3.4 Proof of item 2a of the theorem

It is worth recalling some aspects related to curvature estimates for complete
stable $H$-surfaces $\Sigma$ (possibly with boundary) in complete Riemannian
three-manifolds of absolute sectional curvature at most 1 and injectivity
radius at most 1; such curvature estimates are independent on the value of the
(constant) mean curvature. Rosenberg, Toubiana and Souam [7, Main Theorem]
proved that there exists a universal constant $C^{\prime}_{s}>0$ such that if
$\Sigma$ is two-sided, then for any point $p\in\Sigma$ of distance at least 1
from $\partial\Sigma$, then $|A_{M}|(p)\leq C^{\prime}_{s}$. In [3], we
generalized this curvature estimate to include the case where $M$ is not
necessarily two-sided. Namely, we proved the following statement.

######  Theorem 3.5 (Curvature estimate for stable $H$-surfaces [3]).

There exists $C^{\prime\prime}_{s}\geq 2\pi$ such that given $K_{0}>0$ and a
complete Riemannian 3-manifold $(Y,g)$ of bounded sectional curvature $|K|\leq
K_{0}$, then for any immersed one-sided stable minimal surface
$M\looparrowright Y$ and for any $p\in M$,

$|A_{M}|(p)\leq\frac{C^{\prime\prime}_{s}}{\min\\{\mbox{\rm
Inj}_{Y}(p),d_{M}(p,\partial M),\frac{\pi}{2\sqrt{K_{0}}}\\}}.$ (3.13)

Let $C_{s}:=\max\\{C^{\prime}_{s},C^{\prime\prime}_{s}\\}$. Given
${\varepsilon}_{0}>0$, $K_{0}\geq 0$, if $X$ is a complete Riemannian
3-manifold with injectivity radius at least ${\varepsilon}_{0}$ and bounded
sectional curvature $|K|\leq K_{0}$, and $F\colon M\looparrowright X$ is a
stable $H$-immersion, then

$|A_{M}|(p)\leq\frac{C_{s}}{\min\\{{\varepsilon}_{0},d_{M}(p,\partial
M),\frac{\pi}{2\sqrt{K_{0}}}\\}}.$ (3.14)

Consider an element $(F\colon M\looparrowright X)\in\Lambda(I=0,1,1,1,1)$, in
particular, $M$ is stable. Particularizing (3.14) to the case
${\varepsilon}_{0}=1$, $\partial M=\varnothing$, $K_{0}=1$, we get that
$|A_{M}|\leq C_{s}$ in $M$. By the same argument using the Gauss equation as
in (3.4) and (3.6), we deduce that the Gaussian curvature $K$ of $M$ satisfies
$K\geq K_{1}:=-1-\frac{1}{2}C_{s}^{2}$ in $M$. In this setting, the Bishop-
Cheeger-Gromov relative volume comparison theorem (see e.g. [5, Lemma 36])
implies that for every $x\in M$ and $r>0$,

$\mbox{Area}(B_{M}(x,r))\leq\mbox{Area}(\mathbb{B}_{K_{1}}(r))=\frac{2\pi}{-K_{1}}\left[\mathop{\rm
cosh}\nolimits\left(\sqrt{-K_{1}}r\right)-1\right],$ (3.15)

where $\mathbb{B}_{K_{1}}(r)$ denotes the metric ball of radius $r$ in the
hyperbolic plane of curvature $K_{1}<0$. This finishes the proof of inequality
(1.2) provided that $I=0$ (that is assuming condition (i) in item 2a holds).

Now suppose $(F\colon M\looparrowright X)\in\Lambda(I,1,1,1,1)$, $I\geq 1$. To
prove (1.2) provided that condition (ii) in item 2 holds, observe that in this
case $M=\widetilde{M}$. By (3.6), $K\geq K_{1}$ where
$K_{1}=-1-\frac{1}{2}A_{1}^{2}$ and $A_{1}=A_{1}(I)\geq 1$ is given by Theorem
2.2 for the space $\Lambda(I,1,1,1,1)$. Applying the above arguments to this
new choice of the constant $K_{1}$, we get that (3.15) holds, which proves
(1.2) provided that (ii) in item 2 holds.

In order to show (1.3) (regardless of whether condition (i) or (ii) in item 2
holds), assume $M$ is connected. Observe that we can assume that $M$ is
compact (otherwise its diameter is infinite by the argument in the first
paragraph of Section 3.2 and (1.3) holds vacuously). Choose a point $x\in M$.
Taking $r=\mbox{Diameter}(M):=D$ in (3.15) and using the already proven
inequality (1.1), we get

$C_{1}(I)(g(M)+1)\leq\mbox{Area}(M)=\mbox{Area}(B_{M}(x,D))\stackrel{{\scriptstyle\eqref{3.4a}}}{{\leq}}\frac{2\pi}{-K_{1}}\left[\mathop{\rm
cosh}\nolimits\left(\sqrt{-K_{1}}D\right)-1\right],$

or equivalently,

$D\geq\frac{1}{\sqrt{K_{1}}}\mathop{\rm
arccosh}\nolimits\left[\frac{-K_{1}C_{1}(I)}{2\pi}(g(M)+1)+1\right],$

which proves inequality (1.3), and so, finishes the proof of item 2a.

###  3.5 Area growth of collar neighborhoods of $\widetilde{M}$ if $k\geq 1$.

######  Definition 3.6.

For a complete surface $\Sigma$ with boundary $\partial\Sigma$ and for any
$r>0$, let

$\Sigma(r)=\\{x\in\Sigma\mid d_{\Sigma}(x,\partial\Sigma)\leq r\\}$

be the collar neighborhood of $\partial\Sigma$ in $\Sigma$ of radius $r$.

Consider an element $(F\colon M\looparrowright X)\in\Lambda(I,1,1,1,1)$.
Assume $k\geq 1$ with the notation of Theorem 2.2. Hence, $\widetilde{M}$ is a
surface with smooth boundary. For later uses, next we will give an upper
estimate for the area growth of the collar neighborhood $\widetilde{M}(r)$ of
$\widetilde{M}$, $r>0$.

######  Proposition 3.7.

Let $c_{1},\ldots,c_{e}$ be the set of components of $\partial\widetilde{M}$.
Choose for each $i\in\\{1,\ldots,e\\}$ a parametrization by arc length
${\gamma}_{i}\colon[0,L_{i}]\to c_{i}$ with associated geodesic curvature
function $\kappa_{i}(t)$ with respect to the inward pointing unit conormal
vector $\eta=\eta(t)$ of $\widetilde{M}$ along $\partial\widetilde{M}$. Then,
for each $i\in\\{1,\ldots,e\\}$:

  1. 1.

$\kappa_{i}(t)$ is negative in $[0,L_{i}]$.

  2. 2.

There exists a complete annulus $\Sigma_{i}$ with boundary, with constant
Gaussian curvature $K_{1}$ (this constant is defined in (3.4)), whose boundary
is parameterized by arc length by
$\widehat{{\gamma}}_{i}=\widehat{{\gamma}}_{i}(t)\colon[0,L_{i}]\to\partial\Sigma_{i}$,
and such that the geodesic curvature function $\widehat{\kappa}_{i}$ of
$\partial\Sigma_{i}$ with respect to the inward pointing unit conormal vector
of $\Sigma_{i}$ along $\partial\Sigma_{i}$ satisfies
$\widehat{\kappa}_{i}(t)=\kappa_{i}(t)$ for all $t\in[0,L_{i}]$. Furthermore,
$\Sigma_{i}$ is unique up to isometry.

  3. 3.

For each $r>0$ we have

$\mbox{\rm Area}(\overline{}\widetilde{M}(r))\leq\sum_{i=1}^{e}\mbox{\rm
Area}(\Sigma_{i}(r))=\kappa(\widetilde{M})\frac{1-\mathop{\rm
cosh}\nolimits(\sqrt{-K_{1}}r)}{-K_{1}}+L\frac{\sinh(\sqrt{-K_{1}}r)}{(-K_{1})^{3/2}}.$
(3.16)

where
$\kappa(\widetilde{M})=\int_{\partial\widetilde{M}}\kappa_{g}=\sum_{i=1}^{e}\kappa_{o}(t)\,dt$
is the total geodesic curvature of $\partial M$ with respect to the inward
pointing unit conormal vector of $\widetilde{M}$ along
$\partial\widetilde{M}$, and $L=\sum_{i=1}^{e}L_{i}$ is the length of
$\partial\widetilde{M}$.

###### Proof.

Item 1 follows from the proof of the Hierarchy Structure Theorem 2.2 in [3];
specifically see Lemma 6.4 in [3].

Item 2 follows directly from the following two facts. First, given $\ell>0$
and a smooth function $\kappa\colon[0,\ell]\to(-\infty,0)$, standard geometry
of curves in the hyperbolic plane $\mathbb{H}^{2}(K_{1})$ with constant
Gaussian curvature $K_{1}$ ensures that there exists a smooth unit speed curve
${\alpha}\colon[0,\ell]\to\mathbb{H}^{2}(-K_{1})$ such that $\kappa(t)$ is the
geodesic curvature of ${\alpha}$ at ${\alpha}(t)$, for all $t\in[0,\ell]$;
furthermore, ${\alpha}$ is unique up to isometries of $\mathbb{H}^{2}(K_{1})$.
Second, if $n\colon[0,\ell]\to U\mathbb{H}^{2}(K_{1})$ is the unit normal
vector to ${\alpha}$ pointing to its non-convex side (here
$U\mathbb{H}^{2}(K_{1})$ denotes the unit tangent bundle to
$\mathbb{H}^{2}(K_{1})$), then the map
$\phi\colon[0,\ell]\times[0,\infty)\to\mathbb{H}^{2}(K_{1})$

$\phi(t,r)=\exp_{{\alpha}(t)}(r\,n(t)),\quad(t,r)\in[0,\ell]\times[0,\infty)$
(3.17)

is a submersion, where $\exp\colon
U\mathbb{H}^{2}(K_{1})\to\mathbb{H}^{2}(K_{1})$ is the exponential map. $\phi$
induces a hyperbolic metric $g_{h}$ on $[0,L]\times[0,\infty)$ so that for
each $t_{0}\in[0,\ell]$, the curve of the form
$r\in[0,\infty)\mapsto\phi(t_{0},r)$ is a unitary geodesic orthogonal to the
arc ${\alpha}([0,L])$ at ${\alpha}(t_{0})$; in particular, after identifying
the two geodesic arcs $\phi(\\{0\\}\times[0,\infty))$ and
$\phi(\\{\ell\\}\times[0,\infty))$ by a hyperbolic isometry, we obtain a
quotient hyperbolic annulus $(\Sigma_{{\alpha}},g_{h})$ with the properties
desired in item 2, in the special case that $\ell=L_{i}$ and
$\kappa=\kappa_{i}$.

To prove of item 3 of the proposition, first observe that since $K\geq-K_{1}$
on $\widetilde{M}(r)$ by (3.6), we can use relative volume comparison
arguments see e.g. [5, Lemma 36]) to deduce that

$\mbox{Area}(\widetilde{M}(r))\leq\sum_{i=1}^{e}\mbox{Area}(\Sigma_{i}(r)).$

It remains to prove that for all $i=1,\ldots,e$ and $r>0$, the following holds

$\mbox{Area}(\Sigma_{i}(r))=\frac{1}{-K_{1}}\left[(1-\mathop{\rm
cosh}\nolimits(\sqrt{-K_{1}}r))\int_{0}^{L_{i}}\kappa(s)\,ds+\frac{L_{i}}{\sqrt{-K_{1}}}\sinh(\sqrt{-K_{1}}r)\right].$
(3.18)

######  Claim 3.8.

Let ${\alpha}\colon[0,\ell]\to\mathbb{H}^{2}(-1)$ a smooth arc parameterized
by arc length, with negative geodesic curvature function $\kappa=\kappa(s)$.
Consider the complete hyperbolic annulus with boundary
$(\Sigma_{{\alpha}},g_{h})$ constructed in (3.17) in terms of ${\alpha}$ with
$K_{1}=-1$. Given $r>0$, let ${\alpha}_{r}\colon[0,\ell]\to\Sigma_{{\alpha}}$
be the equidistant arc to ${\alpha}$ at distance $r$ on the non-convex side of
${\alpha}$. Then:

  1. 1.

The geodesic curvature function $\kappa_{r}$ of ${\alpha}_{r}$ is given by

$\kappa_{r}(s)=\frac{\kappa(s)-\mathop{\rm tanh}\nolimits(r)}{1-\mathop{\rm
tanh}\nolimits(r)\kappa(s)},\quad\forall s\in[0,\ell].$ (3.19)

  2. 2.

Let $\Sigma_{{\alpha}}(r)\subset\Sigma_{{\alpha}}$ be the domain enclosed by
${\alpha},{\alpha}_{r}$ and the two geodesics of $\Sigma_{{\alpha}}$ that join
the extrema of ${\alpha},{\alpha}_{r}$ (so that these geodesics are orthogonal
to both ${\alpha},{\alpha}_{r}$ at their extrema). Then,

$\mbox{\rm Area}(\Sigma_{{\alpha}}(r))=(1-\mathop{\rm
cosh}\nolimits(r))\int_{0}^{\ell}\kappa(s)\,ds+\ell\sinh(r),$ (3.20)

where $\ell=L({\alpha})$ is the length of ${\alpha}$.

  3. 3.

If we replace $\mathbb{H}^{2}(-1)$ by $\mathbb{H}^{2}(K_{1})$, then (3.20)
becomes

$\mbox{\rm Area}(\Sigma_{{\alpha}}(r))=\frac{1}{-K_{1}}\left[(1-\mathop{\rm
cosh}\nolimits(\sqrt{-K_{1}}r))\int_{0}^{\ell}\kappa(s)\,ds+\frac{\ell}{\sqrt{-K_{1}}}\sinh(\sqrt{-K_{1}}r)\right].$
(3.21)

###### Proof of the claim.

Recall that $\Sigma_{{\alpha}}$ submerses into $\mathbb{H}^{2}(-1)$ through
the map $\phi$ given in (3.17). In particular,
$\Sigma_{{\alpha}}\setminus\partial\Sigma_{{\alpha}}$ is locally isometric to
$\mathbb{H}^{1}(-1)$. This property clearly allows us to prove the claim
assuming that $\Sigma_{{\alpha}}(r)$ embeds into $\mathbb{H}^{2}(-1)$: for
item 1 of the claim this is obvious, while for items 2 and 3 we can divide
$[0,\ell]$ into a partition $0=s_{0}<s_{1}<\ldots<s_{n}=\ell$ such that if we
denote by ${\alpha}_{i}={\alpha}|_{[s_{i-1},s_{i}]}$, $i=1,\ldots,n$ and we
apply the same procedure as with $\Sigma_{{\alpha}}$ to construct $n$
“rectangles” $\Sigma_{{\alpha}_{i}}(r)$, then each $\Sigma_{{\alpha}_{i}}(r)$
embeds into $\mathbb{H}^{2}(-1)$. In this way, both equations (3.20) and
(3.21) will follow by adding up the corresponding equalities over the
rectangles $\Sigma_{{\alpha}_{1}}(r),\ldots,\Sigma_{{\alpha}_{n}}(r)$, which
only intersect along geodesics in their boundaries. Therefore, for the
remainder of this proof we will assume that $\Sigma_{{\alpha}}(r)$ is embedded
in $\mathbb{H}^{2}(-1)$.

We will use the model of $\mathbb{H}^{2}(-1)$ as the upper sheet of a
hyperboloid in the Lorentz-Minkowski space
$\mathbb{L}^{3}=(\mathbb{R}^{3},\langle,\rangle_{L}=dx_{1}^{2}+dx_{2}^{2}-dx_{3}^{2})$.
In this model, $\mathbb{H}^{2}(-1)=\\{x\in\mathbb{L}^{3}\ |\ \langle
x,x\rangle_{L}=-1,\ x_{3}>0\\}$, and the induced metric by
$\langle,\rangle_{L}$ on $\mathbb{H}^{2}(-1)$ is positive definite and has
constant Gaussian curvature $-1$. Given $x\in\mathbb{H}^{2}(-1)$, the tangent
plane $T_{x}\mathbb{H}^{3}$ identifies to $\langle
x\rangle^{\perp}\subset\mathbb{L}^{3}$. Given $x\in\mathbb{H}^{2}(-1)$ and
$v\in\langle x\rangle^{\perp}$, the unique geodesic in $\mathbb{H}^{2}(-1)$
with initial conditions ${\gamma}(0)=x$, ${\gamma}^{\prime}(0)=v$ is

${\gamma}(t)={\gamma}(t,x,v)=\mathop{\rm
cosh}\nolimits(|v|t))x+\frac{\sinh(|v|t)}{|v|}v,$ (3.22)

and the parallel transport along ${\gamma}(\cdot,x,v)$ from $0$ to $t$ is
given by

$\tau_{0}^{t}\colon T_{x}\mathbb{H}^{2}(-1)=\langle x\rangle^{\perp}\to
T_{{\gamma}(t)}\mathbb{H}^{2}=\langle{\gamma}(t)\rangle^{\perp},\quad\tau_{0}^{t}(w)=w+\frac{\langle
v,w\rangle_{L}}{|v|^{2}}(\dot{{\gamma}}(t)-v),$ (3.23)

where $\dot{{\gamma}}(t)=\frac{d{\gamma}}{dt}$.

Given $s\in[0,\ell]$, the equidistant curve ${\alpha}_{r}$ is

${\alpha}_{r}(s)=\exp_{{\alpha}(s)}(r\mathcal{R}{\alpha}^{\prime}(s))={\gamma}(r,{\alpha}(s),\mathcal{R}{\alpha}^{\prime}(s))=\mathop{\rm
cosh}\nolimits(r){\alpha}(s)+\sinh(r)\mathcal{R}{\alpha}^{\prime}(s),$

where $\mathcal{R}$ is the rotation of angle $\pi/2$ in each tangent plane to
$\mathbb{H}^{2}(-1)$ so that $\mathcal{R}{\alpha}^{\prime}$ points to the non-
convex side of ${\alpha}$. ${\alpha}_{r}^{\prime}(s)=J_{s}(r)$ is the value at
$t=r$ of the unique Jacobi field $J_{s}=J_{s}(t)$ along the geodesic
$t\mapsto{\gamma}(t,{\alpha}(s),J{\alpha}^{\prime}(s))$ with initial
conditions

$J_{s}(0)={\alpha}^{\prime}(s),\quad\frac{DJ_{s}}{dt}(0)=\frac{D(\mathcal{R}{\alpha}^{\prime})}{ds}(s)=-\kappa(s){\alpha}^{\prime}(s),$
(3.24)

where $\\{{\alpha}^{\prime},n_{{\alpha}}=\mathcal{R}{\alpha}^{\prime}\\}$ is
the Frenet dihedron for ${\alpha}$. Since both $J_{s}(0),\frac{DJ_{s}}{dt}(0)$
are orthogonal to $\dot{{\gamma}}(0)$, we deduce that $J_{s}(t)$ is everywhere
orthogonal to $\dot{{\gamma}}(t)$. In particular,
$J_{s}(t)=f_{s}(t)\tau_{0}^{t}({\alpha}^{\prime}(s))$, where $f_{s}(t)$ is a
solution of the ODE $\ddot{f}-f=0$. Imposing (3.24), we have $f_{s}(0)=1$,
$\dot{f_{s}}(0)=-\kappa(s)$, and thus

$f_{s}(t)=\mathop{\rm cosh}\nolimits(t)-\kappa(s)\sinh(t).$ (3.25)

Therefore, an orthogonal basis of the tangent and normal line to
${\alpha}_{r}$ is

$\displaystyle{\alpha}_{r}^{\prime}(s)$ $\displaystyle=$ $\displaystyle
J_{s}(r)=f_{s}(r)\tau_{0}^{r}({\alpha}^{\prime}(s))\stackrel{{\scriptstyle\eqref{3.18a}}}{{=}}f_{s}(r){\alpha}^{\prime}(s),\hskip
56.9055pt\mbox{(not necessarily unitary)}$ $\displaystyle n_{{\alpha}_{r}}(s)$
$\displaystyle=$
$\displaystyle\tau_{0}^{r}(n_{{\alpha}}(s))\stackrel{{\scriptstyle\eqref{3.18a}}}{{=}}n_{{\alpha}}(s)=\mathcal{R}\dot{{\gamma}}(r)=\sinh(r){\alpha}(s)+\mathop{\rm
cosh}\nolimits(r)\mathcal{R}{\alpha}^{\prime}(s),\quad\mbox{(unitary)}$

Hence by the Frenet equations for ${\alpha}_{r}$, we will obtain the negative
of the geodesic curvature $\kappa_{r}(s)$ of ${\alpha}_{r}$ by taking the
derivative w.r.t. $s$ to $n_{{\alpha}_{r}}(s)$ and dividing by
$|{\alpha}_{r}^{\prime}(s)|=f_{s}(r)$:

$-\kappa_{r}(s)=\frac{\frac{d}{ds}(n_{{\alpha}_{r}}(s))}{f_{s}(r)}=\frac{\sinh(r)-\mathop{\rm
cosh}\nolimits(r)\kappa(s)}{\mathop{\rm cosh}\nolimits(r)-\kappa(s)\sinh(r)},$

which proves the first item of the claim.

As for item 2, observe that the interior of $\Sigma_{{\alpha}}(r)$ is
topologically a disk, and that $\partial\Sigma_{{\alpha}}(r)$ contains four
cusps, in each of which the exterior angle to $\Sigma_{{\alpha}}(r)$ along its
boundary is $\pi/2$. Applying the Gauss-Bonnet Theorem to
$\Sigma_{{\alpha}}(r)$, we get

$0=-\mbox{Area}(\Sigma_{{\alpha}}(r))+\int_{{\alpha}}\kappa+\int_{{\alpha}_{r}}\kappa_{r}=-\mbox{Area}(\Sigma_{{\alpha}}(r))+\int_{0}^{\ell}\kappa(s)\,ds-\int_{0}^{\ell}\kappa_{r}(s)\,ds.$
(3.26)

Using (3.19),

$\displaystyle\int_{0}^{\ell}\kappa_{r}(s)\,ds$ $\displaystyle=$
$\displaystyle\int_{0}^{\ell}\frac{\kappa(s)-\mathop{\rm
tanh}\nolimits(r)}{1-\mathop{\rm
tanh}\nolimits(r)\kappa(s)}|{\alpha}_{r}^{\prime}(s)|\,ds\stackrel{{\scriptstyle\eqref{3.21}}}{{=}}\int_{0}^{\ell}(\mathop{\rm
cosh}\nolimits(r)\kappa(s)-\sinh(r))\,ds$ (3.27) $\displaystyle=$
$\displaystyle\mathop{\rm
cosh}\nolimits(r)\int_{0}^{\ell}\kappa(s)\,ds-\sinh(r)\ell.$

(3.26) and (3.27) give (3.20), which finishes the proof of item 2 of the
claim. Item 3 follows from (3.20) after an elementary rescaling argument. ∎

Equation (3.18) follows directly from (3.21) with the obvious change of
notation $\Sigma_{i}(r)=\Sigma_{{\alpha}}(r)$, $L_{i}=\ell$. This finishes the
proof of Proposition 3.7. ∎

###  3.6 Proof of item 2b of the theorem

Consider an element $(F\colon M\looparrowright X)\in\Lambda(I,1,1,1,1)$.
Assume $k\geq 1$ (hence $I\geq 1$) with the notation of Theorem 2.2. By
(3.16), we have for each $r>0$

$\mbox{\rm Area}(\overline{}\widetilde{M}(r))\leq f(r),$ (3.28)

where $f$ is the increasing function

$f(r)=\frac{\kappa(\widetilde{M})}{K_{1}}[\mathop{\rm
cosh}\nolimits(\sqrt{-K_{1}}r)-1]+L\frac{\sinh(\sqrt{-K_{1}}r)}{(-K_{1})^{3/2}}.$

######  Lemma 3.9.

Given $x\in M$ and $r>0$, we have

$\mbox{\rm Area}\left[B_{M}(x,r)\setminus(\cup_{i=1}^{k}\Delta_{i})\right]\leq
f(2r).$

###### Proof.

Suppose first that $x\in\cup_{i=1}^{k}\Delta_{i}$. Then,
$B_{M}(x,r)\subset\widetilde{M}(r)\cup\left(\cup_{i=1}^{k}\Delta_{i}\right)$,
and thus,
$B_{M}(x,r)\setminus\left(\cup_{i=1}^{k}\Delta_{i}\right)\subset\widetilde{M}(r)$.
Hence,

$\mbox{Area}\left[B_{M}(x,r)\setminus\left(\cup_{i=1}^{k}\Delta_{i}\right)\right]\leq\mbox{Area}(\widetilde{M}(r))\stackrel{{\scriptstyle\eqref{3.24a}}}{{\leq}}f(r)<f(2r).$

Now suppose $x\in\mbox{Int}(\widetilde{M})$ and let $d>0$ be the distance from
$x$ to $\cup_{i=1}^{k}\Delta_{i}$. We distinguish two cases, depending on
whether $r\leq d$ or not.

If $r\leq d$, then since $K\geq K_{1}$ in $\widetilde{M}$ and
$B_{M}(x,r)\subset\widehat{M}$, the Bishop-Cheeger-Gromov relative volume
comparison theorem implies

$\mbox{Area}(B_{M}(x,r))\leq\mbox{Area}(\mathbb{B}_{K_{1}}(r))=\frac{2\pi}{-K_{1}}\left[\mathop{\rm
cosh}\nolimits\left(\sqrt{-K_{1}}r\right)-1\right]\stackrel{{\scriptstyle(\star)}}{{<}}f(r)<f(2r),$

where $\mathbb{B}_{K_{1}}(r)$ denotes the metric ball of radius $r$ in the
hyperbolic plane of curvature $K_{1}$, and in $(\star)$ we have used that
$|\kappa(\widetilde{M})|>2\pi$.

If $r>d$, then the triangle inequality ensures that
$B_{M}(x,r)\setminus\left(\cup_{i=1}^{k}\Delta_{i}\right)\subset\widetilde{M}(2r)$,
and thus,

$\mbox{Area}\left[B_{M}(x,r)\setminus\left(\cup_{i=1}^{k}\Delta_{i}\right)\right]\leq\mbox{Area}(\widetilde{M}(2r))\stackrel{{\scriptstyle\eqref{3.24a}}}{{\leq}}f(2r),$

which finishes the proof of the lemma. ∎

Now we are ready to prove inequality (1.4). First, observe that (2.2) implies
that

$-2\pi S-\tau I\leq\kappa(\widetilde{M})\leq-2\pi S+\tau I,$

and so,

$0<\frac{\kappa(\widetilde{M})}{K_{1}}\leq\frac{2\pi S+\tau
I}{-K_{1}}\stackrel{{\scriptstyle(\star)}}{{\leq}}\frac{6\pi+\tau}{-K_{1}}I\leq\frac{6\pi+1}{-K_{1}}I,$
(3.29)

where in $(\star)$ we have used that $S\leq 3I$ (this follows from item B of
Theorem 2.2).

Second, we can estimate from above the length $L$ of $\partial\widetilde{M}$
as follows: each component $c_{j}$ of $\partial\widetilde{M}$ is contained in
the boundary of a certain compact set $\Delta_{i}$, and the length
$L(\partial\Delta_{i})$ of $\partial\Delta_{i}$ can be estimated from above
using [3, item (C1) of Lemma 6.1] (also see [3, Remark 6.2]) as

$L(\partial\Delta_{i})\leq(2\pi m(i)+1)r_{F}(i)\leq\frac{2\pi
m(i)+1}{2}{\delta},$ (3.30)

where $m(i)$ is the total spinning of the boundary of $\Delta_{i}$ ($m(i)$ was
introduced in item B of Theorem 2.2) and
${\delta}\leq{\varepsilon}_{0}/2=1/2$, ${\delta}_{1}\leq\frac{{\delta}}{2}$,
$r_{F}(i)\in[{\delta}_{1},{\delta}/2]$ were introduced in the main statement
of Theorem 2.2. Adding up (3.30) in the set
$\\{\Delta_{1},\ldots,\Delta_{k}\\}$, we get

$L\leq\frac{2\pi S+k}{2}{\delta}\leq\frac{6\pi
I+I}{2}{\delta}=\frac{(6\pi+1){\delta}}{2}I\leq\frac{6\pi+1}{4}I$ (3.31)

Finally, Lemma 3.9, (3.29) and (3.31) give

$\displaystyle\mbox{\rm
Area}\left[B_{M}(x,r)\setminus(\cup_{i=1}^{k}\Delta_{i})\right]$
$\displaystyle\stackrel{{\scriptstyle\mbox{\footnotesize(Lemma~{}\ref{lema3.8})}}}{{\leq}}$
$\displaystyle f(2r)=\frac{\kappa(\widetilde{M})}{K_{1}}[\mathop{\rm
cosh}\nolimits(2\sqrt{-K_{1}}r)-1]+L\frac{\sinh(2\sqrt{-K_{1}}r)}{(-K_{1})^{3/2}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{3.25},\eqref{3.27}}}{{\leq}}$
$\displaystyle\frac{6\pi+1}{-K_{1}}I[\mathop{\rm
cosh}\nolimits(2\sqrt{-K_{1}}r)-1]+\frac{6\pi+1}{4}I\frac{\sinh(2\sqrt{-K_{1}}r)}{(-K_{1})^{3/2}}$
$\displaystyle=$ $\displaystyle\frac{2(6\pi+1)}{A_{2}(I)}I\left[2[\mathop{\rm
cosh}\nolimits(\sqrt{A_{2}(I)}r)-1]+\frac{\sinh(\sqrt{A_{2}(I)}r)}{A_{2}(I)^{1/2}}\right],$

where we have defined $A_{2}(I)=-4K_{1}(I)\geq 6$. This proves (1.4).

In order to see that inequality (1.5) holds, just observe that by item D of
Theorem 2.2,

$\mbox{\rm Area}(\cup_{i=1}^{k}\Delta_{i})\leq
2\pi\sum_{i=1}^{k}m(i)r^{2}_{F}(i)\leq 2\pi
S\frac{{\delta}^{2}}{4}\leq\frac{\pi S}{8}\leq\frac{3\pi}{8}I,$ (3.32)

hence

$\displaystyle\mbox{\rm Area}\left[B_{M}(x,r)\right]$ $\displaystyle\leq$
$\displaystyle\mbox{\rm
Area}\left[B_{M}(x,r)\setminus(\cup_{i=1}^{k}\Delta_{i})\right]+\mbox{\rm
Area}(\cup_{i=1}^{k}\Delta_{i})$ (3.33) $\displaystyle\leq$
$\displaystyle\frac{2(6\pi+1)}{A_{2}(I)}I\left[2[\mathop{\rm
cosh}\nolimits(\sqrt{A_{2}(I)}r)-1]+\frac{\sinh(\sqrt{A_{2}(I)}r)}{A_{2}(I)^{1/2}}\right]+\frac{3\pi}{8}I$
$\displaystyle=$ $\displaystyle
I\left[\frac{2(6\pi+1)}{A_{2}(I)}\left(2[\mathop{\rm
cosh}\nolimits(\sqrt{A_{2}(I)}r)-1]+\frac{\sinh(\sqrt{A_{2}(I)}r)}{A_{2}(I)^{1/2}}\right)+\frac{3\pi}{8}\right],$

from where one deduces (1.5).

To finish this section, we prove (1.6). Let us denote by $h(r)$ the RHS of
(3.33). Then, taking $r=D:=\mbox{Diameter}(M)$ we have $B_{M}(x,D)=M$ for any
$x\in M$, and so,

$C_{1}(I)(g+1)\stackrel{{\scriptstyle\eqref{1.1}}}{{\leq}}\mbox{Area}(M)=\mbox{Area}\left[B_{M}(x,D)\right]\leq
h(D).$ (3.34)

Since $\sinh(t)\leq\mathop{\rm cosh}\nolimits(t)$,

$\displaystyle h(D)$ $\displaystyle\leq$ $\displaystyle
I\left[\frac{2(6\pi+1)}{A_{2}(I)}\left(2[\mathop{\rm
cosh}\nolimits(\sqrt{A_{2}(I)}D)-1]+\frac{\mathop{\rm
cosh}\nolimits(\sqrt{A_{2}(I)}D)}{A_{2}(I)^{1/2}}\right)+\frac{3\pi}{8}\right]$
(3.35) $\displaystyle\stackrel{{\scriptstyle(A_{2}(I)\geq 6)}}{{\leq}}$
$\displaystyle I\left[\frac{6\pi+1}{3}\left(2[\mathop{\rm
cosh}\nolimits(\sqrt{A_{2}(I)}D)-1]+\frac{\mathop{\rm
cosh}\nolimits(\sqrt{A_{2}(I)}D)}{\sqrt{6}}\right)+\frac{3\pi}{8}\right]$
$\displaystyle=$ $\displaystyle
I\left[\frac{6\pi+1}{3}\left(\left(2+\frac{1}{\sqrt{6}}\right)\mathop{\rm
cosh}\nolimits(\sqrt{A_{2}(I)}D)-2\right)+\frac{3\pi}{8}\right]$
$\displaystyle<$ $\displaystyle
I\frac{6\pi+1}{3}\left(2+\frac{1}{\sqrt{6}}\right)\mathop{\rm
cosh}\nolimits(\sqrt{A_{2}(I)}D).$

(3.34) and (3.35) give

$C_{1}(I)(g+1)\leq
I\frac{6\pi+1}{3}\left(2+\frac{1}{\sqrt{6}}\right)\mathop{\rm
cosh}\nolimits(\sqrt{A_{2}(I)}D),$

from where inequality (1.6) follows directly. This completes the proof of item
2 of the theorem.

###  3.7 Proof of item 3 of the theorem

Consider an element $(F\colon M\looparrowright X)\in\Lambda(I,1,1,1,1)$. As in
previous sections, we may assume that $M$ is compact and connected, and let
$g=g(M)$ be the genus of $M$.

######  Claim 3.10.

If $I=0$, then (1.7) holds with $G(0)=0$.

###### Proof.

Since $M$ is stable, we have $|A_{M}|\leq C_{s}$ in $M$ by (3.14). Thus, the
Gauss equation implies that the Gaussian curvature $K$ of $M$ satisfies
$K\geq-1-\frac{1}{2}C_{s}^{2}$. Using the Gauss-Bonnet theorem,

$\textstyle{(1+\frac{1}{2}C_{s}^{2})}\mbox{Area}(M)\geq-\int_{M}K=-2\pi\chi(M)\geq
2\pi(g-1).$

Hence,

$\mbox{Area}(M)\geq\frac{2\pi}{1+\frac{1}{2}C_{s}^{2}}(g-1),$

which is strictly bigger than $\frac{\pi}{3+4C_{s}+4C_{2}^{2}}(g+1)$ when
$g\geq 2$. Consequently, (1.7) holds whenever $g\geq 2$. To finish the proof
of the claim it remains to check that (1.7) holds for $g=0,1$, which we do
next.

$\mbox{Area}(M)\stackrel{{\scriptstyle\mbox{\footnotesize(item~{}\ref{It0})}}}{{\geq}}C_{A}\stackrel{{\scriptstyle\mbox{\footnotesize(a)}}}{{>}}\frac{2\pi}{3+4C_{s}+4C_{2}^{2}}\stackrel{{\scriptstyle\mbox{\footnotesize(b)}}}{{\geq}}\frac{2\pi}{3+4C_{s}+4C_{2}^{2}}(g+1)$

where in (a) we have used that $C_{s}\geq 2\pi$, and in (b) that $g\leq 1$.
Now the claim is proved. ∎

By Claim 3.10, it remains to prove item 3 of the theorem assuming $I\geq 1$.
The additional assumption $g\geq 12I-3$ guarantees, by (3.11), that

$\int_{\widetilde{M}}K=\int_{{M}}K-\int_{\cup_{i=1}^{k}\Delta_{i}}K\leq-\pi(g+1).$
(3.36)

By Lemma 7.1 in [3], there exists a positive constant $\widehat{C}_{s}(1)$,
which in our setting is $1+2C_{s}$, such that if
$\sup|A_{M}|>\widehat{C}_{s}(1)$, then there exists a nonempty finite subset
$\\{q_{1},\ldots,q_{n}\\}\subset M$ with $1\leq n\leq I$, such that

  1. 1.

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

  2. 2.

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

  3. 3.

$|A_{M}|\leq\widehat{C}_{s}(1)$ in $M\setminus[B_{M}(q_{1},1)\cup\ldots\cup
B_{M}(q_{n},1)]$.

We next define a partition of the surface $\widetilde{M}$ that appears in item
(3) of Theorem 2.2.

  * •

If $\sup|A_{M}|\leq\widehat{C}_{s}(1)$, let $\widetilde{M}_{1}=\varnothing$
and $\widetilde{M}_{2}=\widetilde{M}$.

  * •

Otherwise, let
$\widetilde{M}_{1}=\widetilde{M}\cap[\cup_{i=1}^{n}B_{M}(q_{i},1)]$ and
$\widetilde{M}_{2}=\widetilde{M}\setminus[\cup_{i=1}^{n}B_{M}(q_{i},1)]$.

In particular, the second fundamental form of the surface $\widetilde{M}_{2}$
satisfies $|A_{\widetilde{M}_{2}}|\leq\widehat{C}_{s}(1)$.

By the discussion around inequality (3.6), the Gaussian curvature function of
$\widetilde{M}_{1}$ satisfies $K_{\widetilde{M}_{1}}\geq K_{1}$ (where
$K_{1}=K_{1}(I)\leq-3/2$ is defined in (3.4)), and the Gaussian curvature
function of $\widetilde{M}_{2}$ satisfies
$K_{\widetilde{M}_{2}}\geq-1-\frac{1}{2}\widehat{C}_{s}(1)^{2}$. Also, by
inequalities (1.2) and (1.5), there exists an explicit function
$h\colon\mathbb{N}\to(0,\infty)$ such that

$\mbox{Area}[\widetilde{M}\cap[\cup_{i=1}^{n}B_{M}(q_{i},1)]\leq\sum_{i=1}^{n}\mbox{Area}B_{M}(q_{i},1)\leq
h(I).$ (3.37)

Therefore,

${\textstyle
K_{1}(I)h(I)-\left[1+\frac{1}{2}\widehat{C}_{s}(1)^{2}\right]\mbox{\rm
Area}(\widetilde{M}_{2})}\stackrel{{\scriptstyle\eqref{3.37}}}{{\leq}}\int_{\widetilde{M}_{1}}K_{\widetilde{M}_{1}}+\int_{\widetilde{M}_{2}}K_{\widetilde{M}_{2}}=\int_{\widetilde{M}}K\stackrel{{\scriptstyle\eqref{3.7a}}}{{\leq}}-\pi(g+1).$
(3.38)

Solving for the area of $\widetilde{M}_{2}$, we have

$\mbox{\rm
Area}(\widetilde{M}_{2})\geq\frac{\pi(g+1)+K_{1}(I)h(I)}{1+\frac{1}{2}\widehat{C}_{s}(1)^{2}}.$
(3.39)

After setting
$C:=\frac{\pi/2}{1+\frac{1}{2}\widehat{C}_{s}(1)^{2}}=\frac{\pi}{3+4C_{s}+4C_{s}^{2}}$,
we get the estimate

$\mbox{\rm Area}(M)\geq\mbox{\rm Area}(\widetilde{M})\geq\mbox{\rm
Area}(\widetilde{M}_{2})\geq
C(g+1)+\frac{\frac{\pi}{2}(g+1)+K_{1}(I)h(I)}{1+\frac{1}{2}\widehat{C}_{s}(1)^{2}}.$
(3.40)

Define

$G(I):=\max\left\\{12I-3,\left\lceil\frac{-2K_{1}(I)h(I)}{\pi}\right\rceil-1\right\\}\in\mathbb{N},$

where for a real number $x$, we denote by $\lceil x\rceil$ the smallest
integer that is not smaller than $x$ (also known as the ceiling function at
$x$). Then, whenever $g(M)\geq G(I)$ we have that the second term in the RHS
of (3.40) is non-negative, which completes the proof of item 3 of Theorem 1.1.

###  3.8 Proof of item 4 of the theorem

Recall that in Section 3.1 we normalized the space $\Lambda$, passing from an
$H$-immersion $(F\colon M\looparrowright X)\in\Lambda(I,H_{0},r_{0},1,K_{0})$
to the immersion $(F^{\prime}\colon M\looparrowright
X^{\prime})\in\Lambda(I,1,1,1,1)$, where ${\lambda}$ is given by (3.1) and
$X^{\prime}$ is the Riemannian manifold obtained after scaling the original
metric of $X$ by $\sqrt{{\lambda}}$. Observe that $F^{\prime}$ has mean
curvature $H^{\prime}=H/{\lambda}$ and $X^{\prime}$ has scalar curvature
$\rho^{\prime}=\rho/{\lambda}^{2}$.

Suppose that the scalar curvature $\rho$ of $X$ satisfies
$3H^{2}+\frac{1}{2}\rho\geq c$ in $X$ for some $c>0$, where $H$ is the mean
curvature of an immersion $(F\colon M\looparrowright
X)\in\Lambda(I,H_{0},r_{0},1,K_{0})$. In this setting, Rosenberg [6] (see also
[4, Theorem 2.12]) proved that every stable subdomain $\Omega\subset M$
satisfies

$d_{M}(x,\partial\Omega)\leq\frac{2\pi}{\sqrt{3c}}:=R_{c},$ (3.41)

for all $x\in\Omega$. Since $3(H^{\prime})^{2}+\frac{1}{2}\rho^{\prime}\geq
c^{\prime}:=c/{\lambda}^{2}$, the estimate (3.41) applied to the same stable
subdomain $\Omega$ viewed inside the domain of $F^{\prime}$ gives that the
intrinsic distance in the metric induced by $F^{\prime}$ from any $x\in\Omega$
to $\partial\Omega$ is at most
$\frac{2\pi}{\sqrt{3c^{\prime}}}=\frac{2\pi{\lambda}}{\sqrt{3c}}$. This linear
scaling on the upper bound for the intrinsic radius of stable subdomains
allows us to reduce item 4 of Theorem 1.1 to the following statement.

######  Proposition 3.11.

Let $F\colon M\looparrowright X$ be an $H$-immersion in $\Lambda(I,1,1,1,1)$.
If the scalar curvature $\rho$ of $X$ satisfies $3H^{2}+\frac{1}{2}\rho\geq c$
in $X$ for some $c>0$, then $M$ is compact, and there exists $A_{3}(I,c)>0$
such that

$\mbox{\rm Area}(M)\leq A_{3}(I,c)\qquad\mbox{\rm Diameter}(M)\leq
2(I+1)R_{c},\qquad g(M)\leq\frac{A_{3}(I,c)}{C_{1}(I)}-1.$ (3.42)

###### Proof.

We first show that $M$ is compact. Arguing by contradiction, suppose $M$ is
non-compact. Since $M$ is complete, there is a geodesic ray in $M$, i.e., an
embedded, length-minimizing unit-speed geodesic arc
${\gamma}\colon[0,\infty)\to M$. Consider the infinite collection

$\mathcal{C}(n)=\\{B_{M}({\gamma}(2jn),n)\ |\ j\in\mathbb{N}\cup\\{0\\}\\}$

of pairwise disjoint open intrinsic balls in $M$. Since the index of $M$ is at
most $I$, then the subcollection of unstable balls in $\mathcal{C}(n)$ is
finite. This implies that $M$ contains stable balls of arbitrarily large
radius, a property which contradicts that for $r>R_{c}$, $B_{M}(x,r)$ cannot
be stable as follows from (3.41). Therefore, $M$ is compact.

We will divide the proof of (3.42) into two claims.

######  Claim 3.12.

Diameter$(M)\leq 2(I+1)R_{c}$ (i.e., the second inequality in (3.42) holds);
in particular, $M$ is compact.

###### Proof of Claim 3.12.

Arguing by contradiction, suppose that there exist points $p,q\in M$ at
intrinsic distance $L:=d_{M}(p,q)>2(I+1)R_{c}$. Let ${\Gamma}\colon[0,L]\to M$
be a geodesic arc parameterized by arc length, such that ${\Gamma}(0)=p$ and
${\Gamma}(L)=q$. Choose $R>R_{c}$ such that $2(I+1)R\leq L$. Consider the
following collection of $I+1$ pairwise disjoint open intrinsic balls in $M$,
see Figure 3.

$\mathcal{C}^{\prime}(n)=\\{B_{M}\left({\Gamma}((2j-1)R),R\right)\ |\
j=1,\ldots,I+1\\}.$

[h]

Figure 3: The pairwise disjoint collection of metric balls in
$\mathcal{C}^{\prime}(n)$. Observe that the boundary of the last ball in the
chain, $\partial B_{M}({\Gamma}((2I+1)R,R)$, intersects the image of
${\Gamma}$ at the points ${\Gamma}(2IR)$, ${\Gamma}(2(I+1)R)$, and that
$2(I+1)R\leq L$ by construction.

Since the index of $M$ is at most $I$ and the $I+1$ balls in
$\mathcal{C}^{\prime}(n)$ are pairwise disjoint, we deduce that at least one
of these balls is stable. This contradicts that $R>R_{c}$ and (3.41). This
contradiction proves the claim. ∎

######  Claim 3.13.

The first and third inequalities in (3.42) hold for
$A_{3}(I,c)=\frac{\mathop{\rm
cosh}\nolimits[2A_{2}(I)(I+1)R_{c}]-1}{A_{2}(I)^{2}}$.

###### Proof of Claim 3.13.

Take $x\in M$. By the already proven inequalities (1.2) and (1.5), there
exists an explicit function
$\widetilde{h}\colon(\mathbb{N}\cup\\{0\\})\times(0,\infty)\to(0,\infty)$
which is increasing in the variable $r$, such that
$\mbox{Area}(B_{M}(x,r))\leq\widetilde{h}(I,r)$ for all $r>0$. Applying this
estimate to the choice $r=D:=\mbox{Diameter}(M)<\infty$ (observe that
$M=B_{M}(x,D)$) and using that $\widehat{h}(I,r)$ is increasing in $r$, we get

$\mbox{\rm
Area}(M)\leq\widetilde{h}(I,D)\stackrel{{\scriptstyle\mbox{\footnotesize(Claim~{}\ref{claim3.8})}}}{{\leq}}\widetilde{h}(I,2(I+1)R_{c}):=A_{3}(I,c).$
(3.43)

and thus, the first inequality in (3.42) holds. As for the second one, it is
clearly equivalent to proving that $C_{1}(I)(g(M)+1)\leq A_{3}(I,c)$. Applying
(1.1) we have

$C_{1}(I)(g(M)+1)\leq\mbox{Area}(M)\stackrel{{\scriptstyle\eqref{3.20}}}{{\leq}}A_{3}(I,c),$

and the proof of the claim is complete. ∎

Claims 3.12 and 3.13 prove (3.42), which finishes the proof of Proposition
3.11, and consequently item 4 of Theorem 1.1 is also proved. ∎

######  Remark 3.14.

Since the function $\widehat{h}=\widehat{h}(I,r)$ appearing in the proof of
Claim 3.13 is increasing in $r$, and $R_{c}$ is decreasing in $c$, we deduce
that $A_{3}(I,c)=\widetilde{h}(I,2(I+1)R_{c})$ is decreasing in $c$. This
indicates that if we relax the hypothesis $3H^{2}+\frac{1}{2}\rho\geq c$ in
Proposition 3.11 by taking $c\to 0^{+}$, then the estimates for the area,
diameter and genus of $M$ in Proposition 3.11 get worse (in fact, $\lim_{c\to
0^{+}}A_{3}(I,c)=\lim_{c\to 0^{+}}R_{c}=\infty$).

William H. Meeks, III at<EMAIL_ADDRESS>
Mathematics Department, University of Massachusetts, Amherst, MA 01003 Joaquín
Pérez at<EMAIL_ADDRESS>
Department of Geometry and Topology and Institute of Mathematics (IMAG),
University of Granada, 18071, Granada, Spain

## References

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  * [2] W. H. Meeks III and J. Pérez.  Geometry of branched minimal surfaces of finite index.  Preprint at https://arxiv.org/abs/2211.03529.
  * [3] W. H. Meeks III and J. Pérez.  Hierarchy structures in finite index CMC surfaces.  Work in progress.
  * [4] W. H. Meeks III, J. Pérez, and A. Ros.  Stable constant mean curvature surfaces.  In Handbook of Geometrical Analysis, volume 1, pages 301–380. International Press, edited by Lizhen Ji, Peter Li, Richard Schoen and Leon Simon, ISBN: 978-1-57146-130-8, 2008.  MR2483369, Zbl 1154.53009.
  * [5] Peter Petersen.  Riemannian geometry, volume 171 of Graduate Texts in Mathematics.  Springer, Cham, third edition, 2016.  MR3469435, Zbl 1417.53001.
  * [6] H. Rosenberg.  Constant mean curvature surfaces in homogeneously regular $3$-manifolds.  Bull. Austral. Math. Soc., 74:227–238, 2006.  MR2260491 (2007g:53009), Zbl 1104.53057.
  * [7] H. Rosenberg, R. Souam, and E. Toubiana.  General curvature estimates for stable $H$-surfaces in $3$-manifolds and applications.  J. Differential Geom., 84(3):623–648, 2010.  MR2669367, Zbl 1198.53062.
  * [8] A. B. Saturnino.  On the Genus and Area of Constant Mean Curvature Surfaces with Bounded Index.  J. Geom. Anal., 31:11971–211987, 2021.  MR4322559, Zbl 1486.53076.


# Effect of symmetry energy on nuclear stopping and its relation to the
production of light charged fragments

Sanjeev Kumar  Suneel Kumar<EMAIL_ADDRESS>School of Physics and
Material Science, Thapar University, Patiala-147004, Punjab (India)
Rajeev K. Puri  Department of Physics, Panjab University, Chandigarh-160014
(India)

###### Abstract

We present a complete systematics (excitation function, impact parameter,
system size, isospin asymmetry and equations of state dependences) of global
stopping and fragments production for heavy-ion reactions in the energy range
between 50 and 1000 MeV/nucleon in the presence of symmetry energy and
isospin-dependent cross-section. It is observed that the degree of stopping
depends weakly on the symmetry energy and strongly on the isospin-dependent
cross-section. On the other hand, the symmetry energy and isospin-dependent
cross-section has an effect of the order of more than 10$\%$ on the emission
of light charged particles (LCP’s). It means that nuclear stopping and LCP’s
can be used as a tool to get the information of isospin-dependent cross-
section. Interestingly, the LCP’s emission in the presence of symmetry energy
is found to be highly correlated with the global stopping.

###### pacs:

25.70.-z, 25.70.Pq, 21.65.Ef

††preprint: APS/123-QED

##  I Introduction

One of the goals of heavy-ion collisions (HIC) at intermediate energies is to
extend the knowledge of hot and dense nuclear matter to the extreme
conditions. In the past, these studies were focused on multifragmentation,
that constitutes fragments of all sizesSing00 . Additional promising
observable for the understanding of nuclear equation of state is the
anisotropy in the momentum distribution that includes the directed in-plane
flow (bounce off ) as well as out of plane flow (squeeze out) Stoc86 ; Aich91
. The absolute values of the flow results from the interplay between the
attractive mean field and repulsive nucleon-nucleon scatterings. This
interplay is also responsible for the transition from a fused state to total
disassemble one. The another phenomena linked with the above interplay is the
global stopping of nuclear matter. Recently, Puri and co-workersDhaw06 , tried
to correlate the multifragmentation with global nuclear stopping. Their
findings revealed that light charged particles acts in a similar fashion like
anisotropy ratio. They, however, did not take isospin of the system into
account.
Following the recent development of radioactive beam facilities in many parts
of the world, it became possible to study the neutron (or proton) rich nuclear
collisions at intermediate energies. Therefore, for a meaningful
investigation, one should include the isospin dependence of the field. As
pointed out by Bauer Baur88 , nuclear stopping at intermediate energies is
determined by the mean field as well as by the in-medium NN cross-sections.
Unfortunately, his calculations were silent about the symmetry potential. The
recent work of many authors Li95 ; Liu01 ; Feng02  suggest that the degree of
approaching isospin equilibration helps to probe the nuclear stopping in
heavy-ion collisions. In Ref.Liu01 , isospin dependence of cross-section was
investigated in nuclear stopping. In a recent communication Feng02 , authors
studied the behavior of excitation function $Q_{zz}/nucleon$ and concluded
that $Q_{zz}/nucleon$ can provide information about the isospin dependence in
term of cross-sections. Several more studies have also focused in the recent
years on the isospin degree of freedom Liu04 .
We wish to focus on the systematic study of isospin dependence and will focus
on the relation between light charged particles and equilibration of the
reaction using isospin-dependent quantum molecular dynamics.
This study is done within the framework of isospin-dependent quantum molecular
dynamics model that is explained in section-II. The results are presented in
section-III. We present summary in section-IV.

##  II ISOSPIN-DEPENDENT QUANTUM MOLECULAR DYNAMICS (IQMD) MODEL

The isospin-dependent quantum molecular dynamics (IQMD)Hart89  model treats
different charge states of nucleons, deltas and pions explicitlyHart03 , as
inherited from the VUU model Krus85 . The IQMD model has been used
successfully for the analysis of large number of observables from low to
relativistic energies Hart89 ; Hart03 ; Hart02 . The isospin degree of freedom
enters into the calculations via both cross-sections and mean fieldKrus85 ;
Bass95 . The details about the elastic and inelastic cross-sections for
proton-proton and neutron-neutron collisions can be found in Ref.Hart89 ;
Hart02 .
In this model, baryons are represented by Gaussian-shaped density
distributions

$f_{i}(\vec{r},\vec{p},t)=\frac{1}{\pi^{2}\hbar^{2}}~{}e^{-(\vec{r}-\vec{r_{i}}(t))^{2}\frac{1}{2L}}~{}e^{-(\vec{p}-\vec{p_{i}}(t))^{2}\frac{2L}{\hbar^{2}}}.$
(1)

Nucleons are initialized in a sphere with radius $R=1.12A^{1/3}$ fm, in
accordance with the liquid drop model. Each nucleon occupies a volume of
$h^{3}$, so that phase space is uniformly filled. The initial momenta are
randomly chosen between 0 and Fermi momentum($p_{F}$). The nucleons of target
and projectile interact via two and three-body Skyrme forces and Yukawa
potential. The isospin degree of freedom is treated explicitly by employing a
symmetry potential and explicit Coulomb forces between protons of colliding
target and projectile. This helps in achieving correct distribution of protons
and neutrons within nucleus.
The hadrons propagate using Hamilton equations of motion:

$\frac{d{r_{i}}}{dt}~{}=~{}\frac{d\it{\langle~{}H~{}\rangle}}{d{p_{i}}}~{}~{};~{}~{}\frac{d{p_{i}}}{dt}~{}=~{}-\frac{d\it{\langle~{}H~{}\rangle}}{d{r_{i}}},$
(2)

with

$\displaystyle\langle~{}H~{}\rangle$ $\displaystyle=$
$\displaystyle\langle~{}T~{}\rangle+\langle~{}V~{}\rangle$ (3)
$\displaystyle=$
$\displaystyle\sum_{i}\frac{p_{i}^{2}}{2m_{i}}+\sum_{i}\sum_{j>i}\int
f_{i}(\vec{r},\vec{p},t)V^{\it ij}({\vec{r}^{\prime},\vec{r}})$
$\displaystyle\times
f_{j}(\vec{r}^{\prime},\vec{p}^{\prime},t)d\vec{r}d\vec{r}^{\prime}d\vec{p}d\vec{p}^{\prime}.$

The baryon-baryon potential $V^{ij}$, in the above relation, reads as:

$\displaystyle V^{ij}(\vec{r}^{\prime}-\vec{r})$ $\displaystyle=$
$\displaystyle V^{ij}_{Skyrme}+V^{ij}_{Yukawa}+V^{ij}_{Coul}+V^{ij}_{sym}$ (4)
$\displaystyle=$
$\displaystyle\left[t_{1}\delta(\vec{r}^{\prime}-\vec{r})+t_{2}\delta(\vec{r}^{\prime}-\vec{r})\rho^{\gamma-1}\left(\frac{\vec{r}^{\prime}+\vec{r}}{2}\right)\right]$
$\displaystyle+~{}t_{3}\frac{exp(|\vec{r}^{\prime}-\vec{r}|/\mu)}{(|\vec{r}^{\prime}-\vec{r}|/\mu)}~{}+~{}\frac{Z_{i}Z_{j}e^{2}}{|\vec{r}^{\prime}-\vec{r}|}$
$\displaystyle+t_{6}\frac{1}{\varrho_{0}}T_{3}^{i}T_{3}^{j}\delta(\vec{r_{i}}^{\prime}-\vec{r_{j}}).$

Here $Z_{i}$ and $Z_{j}$ denote the charges of $i^{th}$ and $j^{th}$ baryon,
and $T_{3}^{i}$, $T_{3}^{j}$ are their respective $T_{3}$ components (i.e. 1/2
for protons and -1/2 for neutrons). Meson potential consists of Coulomb
interaction only. The parameters $\mu$ and $t_{1},.....,t_{6}$ are adjusted to
the real part of the nucleonic optical potential. For the density dependence
of nucleon optical potential, standard Skyrme-type parameterization is
employed. The choice of equation of state (or compressibility) is still
controversial one. Many studies advocate softer matter, whereas, much more
believe the matter to be harder in nature Krus85 ; Mage00 . We shall use both
hard (H) and soft (S) equations of state that have compressibilities of 380
and 200 MeV, respectively.

The binary nucleon-nucleon collisions are included by employing the collision
term of well known VUU-BUU equation Stoc86 ; Krus85 . The binary collisions
are done stochastically, in a similar way as are done in all transport models.
During the propagation, two nucleons are supposed to suffer a binary collision
if the distance between their centroids

$|r_{i}-r_{j}|\leq\sqrt{\frac{\sigma_{tot}}{\pi}},\sigma_{tot}=\sigma(\sqrt{s},type),$
(5)

”type” denotes the ingoing collision partners (N-N, N-$\Delta$, N-$\pi$,..).
In addition, Pauli blocking (of the final state) of baryons is taken into
account by checking the phase space densities in the final states. The final
phase space fractions $P_{1}$ and $P_{2}$ which are already occupied by other
nucleons are determined for each of the scattering baryons. The collision is
then blocked with probability

$P_{block}~{}=~{}1-(1-P_{1})(1-P_{2}).$ (6)

The delta decays are checked in an analogous fashion with respect to the phase
space of the resulting nucleons.

##  III Results and Discussion

The global stopping in heavy-ion collisions has been studied with the help of
many different variables. In earlier studies, one used to relate the rapidity
distribution with global stopping. The rapidity distribution can be defined as
Dhaw06 ; Wong94 :

$Y(i)=\frac{1}{2}ln\frac{E(i)+p_{z}(i)}{E(i)-p_{z}(i)},$ (7)

where $E(i)$ and $p_{z}(i)$ are, respectively, the total energy and
longitudinal momentum of $i^{th}$ particle. For a complete stopping, one
expects a single Gaussian shape. Obviously, narrow Gaussian indicate better
thermalization compared to broader Gaussian.
The second possibility to probe the degree of stopping is the anisotropy ratio
(R) Liu01 :

$R=\frac{2}{\pi}\frac{\left(\sum_{i}|p_{\perp}(i)|\right)}{\left(\sum_{i}|p_{\parallel}(i)|\right)},$
(8)

where, summation runs over all nucleons. The transverse and longitudinal
momenta are $p_{\perp}(i)$ = $\sqrt{p_{x}^{2}(i)+p_{y}^{2}(i)}$ and
$p_{\parallel}(i)=p_{z}(i)$, respectively. Naturally, for a complete stopping,
$R$ should be close to unity.
Another quantity, which is indicator of nuclear stopping and has been used
recently, is the quadrupole moment $Q_{zz}$, defined asLiu01 :

$Q_{zz}=\sum_{i}\left(2p_{z}^{2}(i)-p_{x}^{2}(i)-p_{y}^{2}(i)\right).$ (9)

Naturally, for a complete stopping, $Q_{zz}$ should be close to 0.
In the present analysis, thousands of event were simulated for the neutron-
rich reaction of ${}_{54}Xe^{131}~{}+~{}_{54}Xe^{131}$ at incident energies
between 50 and 1000 MeV/nucleon using hard equation of state along with energy
dependent Cugnon and constant nucleon-nucleon cross-sectionsKuma98 . Moreover,
to see the effect of compressibilities on nuclear stopping and fragmentation,
soft equation of state is also used in Fig.5. The geometry of the collision
was varied between the most-central to peripheral one. The role of symmetry
energy is studied by simulating the above reaction with and without this term.
As stated above, we plan to study the degree of stopping and emission of
fragments using symmetry energy and isospin-dependent cross-section. We shall
also correlate the degree of stopping with the emission of light charged
particles as is also done in Ref.Dhaw06 . The fragments are constructed within
minimum spanning tree (MST) method Sing00 , which binds nucleons if they are
with in a distance of 4 fm.

Figure 1:  The final phase space of a single event for the reaction of
${}_{54}Xe^{131}~{}+~{}_{54}Xe^{131}$ with(ii) and without symmetry energy(i).
The top(a), middle(b) and bottom (c) panels are, respectively, for scaled
impact parameters $\hat{b}$ = 0, 0.3, 0.6. Different symbols are for free
nucleons, LCP’s and IMF’s.

In Fig.1, we display the final phase space of a single event of
${}_{54}Xe^{131}~{}+~{}_{54}Xe^{131}$ at incident energy of 400 MeV/nucleon,
with and without symmetry energy. The top, middle and bottom panels are at
$\hat{b}$ = 0, 0.3 and 0.6, respectively. Here phase space of free particles
[A =1], light charged particles (LCP’s) [2$\leq$A$\leq$4] and intermediate
mass fragments(IMF’s)[5$\leq$ A $\leq$ 44] is displayed. We note that
irrespective of the symmetry energy, central collisions lead to complete
spherical distribution of particles, indicating, spreading of the nucleons in
all directions. It means that breaking of initial correlations among nucleons
is maximal in this region and, as a result, more randomization and stopping in
the hot and compressed nuclear matter occurs. This effect seems to decrease
with impact parameter. Since free particles as well as LCP’s originate from
the mid-rapidity region, they are better suited for studying the degree of
stopping reached in a heavy-ion collision. On the other hand, IMF’s seems to
originate either from the target or from the projectile region, therefore, are
the remnant/residue of the spectator matter. This observation is in agreement
with many other studies Sing00 ; Dhaw06 ; Dhaw07 .

Figure 2:  The rapidity distribution $\frac{dN}{dY}$ as a function of reduced
rapidity for free nucleons(a), LCP’s (b) and IMF’s(c) at different impact
parameters. The reaction under study is ${}_{54}Xe^{131}~{}+~{}_{54}Xe^{131}$
at incident energy E = 400 MeV/nucleon. The left and right panels are with
(ii) and without symmetry energy (i). Figure 3:  The anisotropy ratio $R$ (i)
and quadrupole moment $\left({Q_{zz}/nucleon}\right)$ (ii) as a function of
normalized impact parameter with and without symmetry energy. In panel (b),
the results are also displayed with isospin-dependent cross-section (55mb).
The panels from top to bottom are at incident energy of 50 (a), 400 (b), 600
(c) and 1000 MeV/nucleon (d), respectively.

To further quantify this observation, we display in Fig.2, the rapidity
distribution $\frac{dN}{dY}$ for the emission of free nucleons as well as
LCP’s and IMF’s. We see that free particles and LCP’s emitted in the central
collisions form a single narrow Gaussian shape, whereas, IMF’s have broader
Gaussian, indicating less thermalization. As we increase impact parameter,
single Gaussian distribution splits into two Gaussian (at target and
projectile rapidities), indicating correlated matter. From the shape of the
Gaussian, one sees that free particles and LCP’s are better indicator of
thermal source. Obviously, this condition is necessary, but, not a sufficient
one.
From the figure, it is also evident that the symmetry energy does not plays
significant role for the rapidity distribution. The peak value of the Gaussian
for LCP’s is altered by about 10$\%$, whereas, nearly no effect is seen in the
case of intermediate mass fragments. The reason is that LCP’s can feel the
role of mean field directly, while, the heavy fragments have weak sensitivity
Yan07 . From the figure, one sees a one to one relation between the degree of
stopping and emission of LCP’s. These conclusions match with the findings of
Fig.1 and Ref.Dhaw06 .
In Fig.3, we display impact parameter dependence of global variables ($R$ and
$Q_{zz}/nucleon$), whereas, the multiplicity dependence of free nucleons and
LCP’s is displayed in Fig.4. The displayed results are at $E_{Sym}$= 0 and
$E_{Sym}$ = 32 MeV in each panel, while, in panel (b) the results are also
displayed with isospin-dependent cross-section. The value of cross-section is
denoted in the superscript. From Fig.3, We observe that $R$ and
${Q_{zz}/nucleon}$ behave in opposite fashion i.e. $R$ and
$\frac{1}{Q_{zz}/nucleon}$ will behave in a similar fashion. For $R>1$ and
$Q_{zz}/nucleon<0$, it can be explained by the preponderance of momentum flow
perpendicular to the beam directionRenf84 . The maximum stopping is observed
around 400 MeV/nucleon, which is in supportive nature with the findings of W.
Reisdorf et al.Reis04a . In their work, they measured the nuclear stopping
from 0.090 to 1.93 GeV/nucleon and maximal stopping was observed around 400
MeV/nucleon. It is clear that if the reaction reaches the maximal stopping
around certain energies, the matter formed in the reaction should reach
minimum transparency and thus most of the particles are preferentially out-of-
plane. On the other hand, no visible effect is seen for symmetry energy term.
We see both quantities are nearly independent of the symmetry energy, while,
strongly depends on the isospin-dependent cross-section.

Figure 4:  Same as in Fig.3, but for the multiplicity of free
particles/nucleon and LCP’s/nucleon.

As we know, major contribution for the stopping of nuclear matter is from the
hot and compressed region, where symmetry energy does not play any role. Some
small spikes can be seen at lower beam energies, however, outcome is
independent of the symmetry energy at higher incident energies. This is due to
the fact that above the Fermi energy, incident energy itself is sufficient to
break the initial correlations among the nucleons. On the other hand, isospin-
dependent cross-section will lead to violent N-N collisions, which further
cause the transformation of the initial longitudinal motion in other
directions and hence thermalization of the system. This dominant role played
by the isospin-dependent cross-section gradually disappears with increase in
the impact parameter. As discussed earlier, stopping is the phenomena which
originates from the participant zone and this zone goes on decreasing with
increase in the impact parameter and hence the effect of cross-section on
nuclear stopping. These findings are also in supportive nature with the
findings of Liu et al., Liu01 .

Figure 5:  Impact parameter dependence of (a) R, (b) ${Q_{zz}/nucleon}$, (c)
free particles/nucleon, (d) LCP’s/nucleon, with hard (H) and soft (S)
equations of state. The results are displayed at $E_{Sym}=0$ (i) and 32 MeV
(ii).

To correlate the degree of stopping with the multiplicity of fragments, we
display in Fig.4, the impact parameter dependence of the multiplicity of free
nucleons as well as of LCP’s. The behavior of all curves is similar to that of
nuclear stopping parameters $R$ and $\frac{1}{Q_{zz}/nucleon}$, as discussed
in Fig.3. In addition, LCP’s are more sensitive towards symmetry energy
compared to free particles. Due to pairing nature of LCP’s, symmetry energy
term $\propto$ $(N-Z)^{2}$ contributes considerably. The effect of isospin-
dependent cross-section is more visible for the LCP’s as compared to free
particles. This also gives us clue that LCP’s production can act as a
indicator for the nuclear stopping. Moreover, free particles/nucleon are found
to increase monotonically with the incident energy, while LCP’s/nucleon behave
in similar fashion as that of nuclear stopping i.e. maximum around 400
MeV/nucleon and then decreases. It is also evident from Ref.Dhaw06 , LCP’s
production act as a barometer for nuclear stopping compared to the free
particles.

Figure 6:  System size dependence of (a) R, (b) $\frac{1}{Q_{zz}/nucleon}$,
(c) free particles, (d) LCP’s, in the presence of symmetry energy. All the
curves are fitted with power law.

In Fig.5, we have checked the sensitivity of nuclear stopping as well as
fragment production with the nuclear equation of state (EOS). For this
purpose, a hard (H) and soft (S) equations of state with compressibility
$\kappa$ = 380, 200 MeV are employed, respectively. The nuclear stopping is
found to be weakly dependent on the equations of state, while, the fragments
production is sensitive to different equations of state. It means that the
fragment production with different equations of state can act as a global
indicator for the nuclear stopping as it is weakly dependent on equations of
state.
It also becomes important to study the system size dependence and isospin
asymmetry of R, $\frac{1}{Q_{zz}/nucleon}$, free particles and LCP’s. For
this, in Fig.6, we have displayed the results for the reactions of
${}_{20}Ca^{40}~{}+~{}_{20}Ca^{40}$, ${}_{28}Ni^{58}~{}+~{}_{28}Ni^{58}$,
${}_{41}Nb^{93}~{}+~{}_{41}Nb^{93}$, ${}_{54}Xe^{131}~{}+~{}_{54}Xe^{131}$ and
${}_{79}Au^{197}~{}+~{}_{79}Au^{197}$, in which Z as well as A is varied. On
the other hand, results are displayed, in Fig.7, for the reactions of
${}_{20}Ca^{34}~{}+~{}_{20}Ca^{34}$ (N/Z = 0.7),
${}_{20}Ca^{40}~{}+~{}_{20}Ca^{40}$ (N/Z = 1),
${}_{20}Ca^{48}~{}+~{}_{20}Ca^{48}$ (N/Z = 1.4) and
${}_{20}Ca^{57}~{}+~{}_{20}Ca^{57}$ (N/Z = 1.85), having same Z and different
A, in the presence of symmetry energy and isospin-dependent cross-section. The
curves in Figs.6 and 7 are parametrized with the power law $Y=CX^{\tau}$,
where C and $\tau$ are constants, while X and Y are the respective parameters
on X and Y axis.

Figure 7:  Isospin asymmetry of (a) R, (b) $\frac{1}{Q_{zz}/nucleon}$, (c)
free particles, (d) LCP’s, in the presence of symmetry energy and isospin-
dependent cross-section (55 mb).

From the Fig.6, it is observed that the parameters R,
$\frac{1}{Q_{zz}/nucleon}$, free particles as well as LCP’s are in similar
trend with the composite mass of the system. All the parameters are found to
increase with the composite mass of the system. For a fixed geometry (semi-
central here), more heavier is the composite system, more hot is the
compressed zone, which further results in more thermalization or global
stopping. Looking the parallel side, the free particles and LCP’s will always
originate from the participant zone. With an increase in the composite mass of
the system, the participant zone goes on increasing for a fixed geometry
(semi-central here) and hence the production of free particles and LCP,s.
Similar findings are also published in the Ref.Liu01 ; Reis04 .
The dependence of these parameters on the isospin asymmetry (N/Z dependence )
displayed, in Fig.7, is also found to be in supportive nature with the
findings in Fig.6. An increase in the number of neutrons will increase the
number of collisions and hence dominance of R, $\frac{1}{Q_{zz}/nucleon}$,
free particles as well as LCP’s is observed with increase in N/Z ratio.
Nuclear stopping as well as LCP’s are observed to be strongly dependent on the
isospin-dependent cross-section. Similar results with isospin dependent cross-
section are observed in Figs.3 and 4. From here, one may conclude that the
nuclear stopping and LCP’s can also be used as a tool to investigate the
isospin-dependent cross-section.

Figure 8:  The scaled free particles/nucleon and LCP’s/nucleon along with
anisotropy ratio $R$ (a) and $\frac{1}{Q_{zz}/nucleon}$(b) as a function of
normalized impact parameter in the presence of symmetry energy. The reaction
is at incident energy 400 MeV/nucleon.

To further elaborate this point, we display in Fig.8, Multiplicity/nucleon
(free and LCP’s) as well as $R$ and $\frac{1}{Q_{zz}/nucleon}$. Once free
nucleons and LCP’s are normalized with $R$ at the starting point of impact
parameter, we see that their behavior with respect to impact parameter is
similar to that of anisotropy ratio, whereas, visible difference occurs with
reference to quadrupole moment. This similarity in all three quantities in the
presence of symmetry energy makes LCP’s good indicator of global stopping in
heavy-ion collisions.

##  IV Conclusion

In summary, using the isospin-dependent quantum molecular dynamics
(IQMD)model, we investigate the emission of free particles, LCP’s, and degree
of stopping reached in a heavy-ion collisions in the presence of symmetry
energy and isospin dependent cross-section. We observed that nuclear stopping
in term of anisotropy ratio and quadrupole moment depends weakly on the
symmetry energy and strongly on the isospin-dependent cross-section. On the
other hand, the symmetry energy and isospin-dependent cross-section has an
effect of $10\%$ on the LCP’s production. It means nuclear stopping and LCP’s
production can be used as a tool to investigate the isospin-dependent cross-
section. The LCP’s production is found to be highly correlated with the global
stopping.

###### Acknowledgements.

This work has been supported by the Grant no. 03(1062)06/ EMR-II, from the
Council of Scientific and Industrial Research (CSIR) New Delhi, Govt. of
India.

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