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\title[CMC-1 surfaces]{
   Basic examples of mean curvature $1$ surfaces in \boldmath$H^3$
}
\author{Wayne Rossman}
\author{Masaaki Umehara}
\author{Kotaro Yamada}
\address[Rossman]{%
   Department of Mathematics, Faculty of Science,
   Kobe University,
   Rokko, Kobe 657-8501, Japan%
}
\email{wayne@math.kobe-u.ac.jp}
\address[Umehara]{%
   Department of Mathematics, Faculty of Science,
   Hiroshima University,
   Higashi-Hiroshima 739-8526, Japan%
}
\email{umehara@math.sci.hiroshima-u.ac.jp}
\address[Yamada]{%
   Faculty of Mathematics,
   Kyushu University 36, 6-10-1
   Hakozaki, Higashi-ku, Fukuoka 812-8185, Japan%
}
\email{kotaro@math.kyushu-u.ac.jp}
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\begin{document}
\begin{abstract}
 We describe the theory of complete 
 mean curvature one surfaces in $H^3$ and 
 give a detailed description of some basic examples.  This note 
 is intended to complement the three EG-Models 2001.01.048 and 
 2001.01.049 and 2001.01.050 by the authors on this subject.  
\end{abstract}
\maketitle
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% Text
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\section{Introduction}
There is a wide body of knowledge about minimal surfaces in Euclidean 
$3$-space $\R^3$, and there is a canonical local isometric
correspondence (sometimes called the Lawson correspondence, canonical
correspondence, or cousin correspondence)
between
minimal surfaces in $\R^3$ and \cmcone{} (constant mean curvature one)
surfaces in hyperbolic $3$-space $H^3$ (the complete simply-connected
$3$-manifold of constant sectional curvature $-1$).  
This has naturally led to the recent interest in and development of 
\cmcone{} surfaces in $H^3$ in the last decade.  
There are now many known examples, and here we give a detailed 
description of some of the most basic ones.  

By this canonical local isometric correspondence, 
minimal immersions in $\R^3$ are locally equivalent to \cmcone{}
immersions in $H^3$.  
But there are interesting differences between these two types of
immersions on the global level.  
There are period problems for an immersion of a 
non-simply-connected domain, which might be solved for the immersion into one 
$3$-space, but not the other.  
Solvability of the period problems is usually more likely in the $H^3$
case, leading to a wider variety of surfaces there.  
For example, a genus $0$ surface with finite total curvature and two embedded 
ends exists as a minimal surface in $\R^3$ only if it is a surface of 
revolution, but it may exist as a \cmcone{} surface in $H^3$ without being 
a surface of revolution (see Example~\ref{exa:catenoid}).  

\section{General comments on total absolute curvature}
The total absolute curvature of a minimal surface in $\R^3$ is 
equal to the area of the image (counted with multiplicity) of the Gauss map 
of the surface.  
Furthermore, as the Gauss map of a complete conformally parametrized 
finite-total-curvature minimal surface is meromorphic at each end, 
the area of the Gauss image must be an integer multiple of $4\pi$. 

However, unlike minimal surfaces in $\R^3$, we have a choice of
two different Gauss maps for \cmcone{} surfaces in $H^3$: 
the {\em hyperbolic Gauss map\/} $G$ and the {\em secondary Gauss map\/}
$g$.  The true total absolute curvature is the area of the image of $g$, but 
since $g$ might not be 
single-valued on the surface, the total curvature might not be an 
integer multiple of $4\pi$.  
The Osserman inequality does not hold for the true total 
absolute curvature.  
The weaker Cohn-Vossen inequality is the best general lower bound for
true absolute total curvature (with equality never holding \cite{uy1}).  

The area of the image of $G$, 
which we call the {\it dual\/} total absolute curvature, is the
true total curvature of the dual \cmcone{} surface (which we define in 
Section~\ref{sec:prelim}) in $H^3$.  
$G$ is single-valued on the surface, 
and so the dual total absolute curvature is always an integer multiple
of $4\pi$, like the case of minimal surfaces in $\R^3$.  
Furthermore, the dual total curvature satisfies not only the Cohn-Vossen
inequality, but also the Osserman inequality \cite{uy5,Yu2} 
(see also \eqref{eq:osserman} in Section~\ref{sec:prelim}).  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Basic preliminaries}
\label{sec:prelim}

Let $f\colon{}M\to H^3$ be a conformal \cmcone{} immersion of a Riemann 
surface $M$ into $H^3$.  
Let $ds^2$, $dA$ and $K$ denote the induced metric, induced area element
and Gaussian curvature, respectively.  
Then $K \leq 0$ and $d\sigma^2:=(-K)\,ds^2$ is a conformal pseudometric
of constant curvature $1$ on $M$.  
We call this pseudometric's developing map $g\colon{}\widetilde
M(:=\text{the universal cover of $M$})\to \CP^1=\C\cup\{\infty\}$ 
the {\em secondary Gauss map\/} of $f$.  
Namely, $g$ is a conformal map so that its pull-back of the 
Fubini-Study metric of $\CP^1$ equals $d\sigma^2$:
\begin{equation}\label{eq:pseudo}
    d\sigma^2 = (-K)\,ds^2 = \frac{4\,dg\,d\bar g}{(1+g\bar g)^2}\;.
\end{equation}
Such a map $g$ is determined by $d\sigma^2$ uniquely up to the change
\begin{equation}\label{eq:g-change}
   g\mapsto a\star g :=\frac{a_{11}g + a_{12}}{a_{21}g +a_{22}}~,\qquad
   a=\begin{pmatrix}
        a_{11} & a_{12} \\
        a_{21} & a_{22}
     \end{pmatrix}\in\SU(2)\;.
\end{equation}
Since $d\sigma^2$ is invariant under the deck transformation group
$\pi_1(M)$, there is a representation
\begin{equation}\label{eq:g-repr}
    \rho_g\colon{}\pi_1(M)\longrightarrow  \PSU(2)
    \quad\text{such that}\quad
    g\circ \tau^{-1}  = \rho_g(\tau)\star g \quad \bigl(\tau\in\pi_1(M)\bigr)\;,
\end{equation}
where $\PSU(2)=\SU(2)/\{\pm\id\}$.
The metric $d\sigma^2$ is called {\em reducible\/} if the image 
of $\rho_g$ can be diagonalized simultaneously, and is called 
{\em irreducible\/} otherwise.  
In the case $d\sigma^2$ is reducible,
we call it is {\em $\Hyp^3$-reducible\/} if the image of $\rho_g$ is the
identity, and is called 
{\em $\Hyp^1$-reducible\/} otherwise.
We call a \cmcone{} immersion $f\colon{}M\to H^3$ $\Hyp^1$-reducible
(resp.\ $\Hyp^3$-reducible) if the corresponding pseudometric
$d\sigma^2$ is $\Hyp^1$-reducible (resp.\ $\Hyp^3$-reducible).
For details on reducibility, see \cite{ruy1}, for example.  

In addition to $g$, two other holomorphic invariants $G$ and $Q$ are closely 
related to geometric properties of \cmcone{} surfaces.  
The {\em hyperbolic Gauss map\/} $G\colon{}M\to \CP^1$ is holomorphic
and is defined geometrically by identifying the ideal boundary of $H^3$
with $\CP^1$: 
$G(p)$ is the asymptotic class of the normal geodesic of $f(M)$ starting
at $f(p)$ and oriented in the mean curvature vector's direction.  
The {\em Hopf differential} $Q$ is a holomorphic symmetric
$2$-differential on $M$ such that $-Q$ is the $(2,0)$-part of the
complexified second fundamental form.
The Gauss equation implies 
\begin{equation}\label{eq:metrics-rel}
  ds^2 \cdot d\sigma^2 = 4 \, Q \cdot\overline Q\;,
\end{equation}
where $\cdot$ means the symmetric product.  
Moreover, these invariants are related by 
\begin{equation}\label{eq:schwarz}
   S(g)-S(G) = 2 Q\;,
\end{equation}
where $S(\cdot)$ denotes the Schwarzian derivative: 
\[
   S(h):=
    \left[
        \left(\frac{h''}{h'}\right)'-
        \frac{1}{2}\left(\frac{h''}{h'}\right)^2
    \right]\,dz^2\qquad \left ('=\frac{d}{dz}\right)
\]
with respect to a local complex coordinate $z$ on $M$.

In terms of $g$ and $Q$, the induced metric $ds^2$ and complexification
of the second fundamental form $h$ are 
\[
    ds^2 = (1+|g|^2)^2 \left| \frac{Q}{dg} \right|^2 \; , \qquad
    h= -Q-\overline{Q}+ds^2 \; . 
\]  

Since $K\leq 0$, we can define the {\em total absolute curvature} as 
\[
    \TA(f):=\int_M (-K)\,dA\in [0,+\infty] \;\; .
\]
Then $\TA(f)$ is the area of the image of $M$ in $\CP^1$ of the
secondary Gauss map $g$.  
$\TA(f)$ is generally not an integer multiple of $4\pi$ --- 
for catenoid cousins \cite[Example~2]{Bryant} and their $\delta$-fold 
covers, $\TA(f)$ admits {\em any\/} positive real number.  

For each conformal \cmcone{} immersion $f\colon{}M\to H^3$, there is a 
holomorphic null immersion $F\colon{}\widetilde M\to\SL(2,\C)$, 
the {\em lift\/} of $f$, satisfying the differential equation 
\begin{equation}\label{eq:bryant}
   dF = F \begin{pmatrix} 
             g & -g^2 \\ 1 & -g\hphantom{^2}
          \end{pmatrix}\omega \;\; ,\qquad \omega=\frac{Q}{dg}
\end{equation}
so that $f=FF^{*}$, where $F^{*}=\trans\overline{F}$ \cite{Bryant,uy1}.  
Here we consider 
\[
    H^3=\SL(2,\C)/\SU(2)=\{aa^{*}\,|\,a\in\SL(2,\C)\}\;.
\]
We call a pair $(g,\omega)$ the {\em Weierstrass data\/} of $f$.
The lift $F$ is said to be {\em null\/} because $\det F^{-1}dF$,
the pull-back of the Killing form of $\SL(2,\C)$ by $F$, vanishes
identically on $M$.
Conversely, for a holomorphic null immersion 
$F\colon{}\widetilde M\to\SL(2,\C)$,
$f:=FF^*$ is a conformal \cmcone{} immersion of $\widetilde M$ into
$H^3$.
If $F=(F_{ij})$, equation \eqref{eq:bryant} implies 
\begin{equation}\label{eq:g-F}
   g=-\frac{dF_{12}}{dF_{11}} =-\frac{dF_{22}}{dF_{21}} \; , 
\end{equation}
and it is shown in \cite{Bryant} that 
\begin{equation}\label{eq:G-F}
   G = \frac{dF_{11}}{dF_{21}} = \frac{dF_{12}}{dF_{22}} \; . 
\end{equation}
The inverse matrix $F^{-1}$ is also a holomorphic null immersion,
and produces a new \cmcone{} immersion 
$f^{\#}=F^{-1}(F^{-1}{})^{*}\colon{}\widetilde M\to H^3$, 
called the {\em dual\/} of $f$ \cite{uy5}.  
The induced metric $ds^2{}^{\#}$ and the Hopf differential $Q^{\#}$ of 
$f^{\#}$ are 
\begin{equation}\label{eq:fund-forms}
    ds^2{}^{\#}= (1+|G|^2)^2 \left|\frac{Q}{dG}\right|^2\;,
    \qquad
    Q^{\#}= - Q \; . 
\end{equation}
So $ds^2{}^{\#}$ and $Q^{\#}$ are well-defined on $M$ itself, 
even though $f^{\#}$ might be defined only on $\widetilde M$.
This duality between $f$ and $f^{\#}$ interchanges the roles of the 
hyperbolic Gauss map $G$ and secondary Gauss map $g$.  
In particular, one has 
\begin{equation} \label{eq:Fsharp}
  dF\, F^{-1}= -(F^{-1})^{-1}d(F^{-1})=
    \begin{pmatrix} G & -G^2 \\ 1 & -G\hphantom{^2} \end{pmatrix}
     \frac{Q}{dG}\;.
\end{equation} 
Hence $dFF^{-1}$ is single-valued on $M$, whereas $F^{-1}dF$ generally
is not.

Since  $ds^2{}^{\#}$ is single-valued on $M$,
we can define the {\em dual total absolute curvature\/} 
\[
    \TA(f^{\#}) := \int_M (-K^{\#})\,dA^{\#},
\]
where $K^{\#}$ ($\leq 0$) and $dA^{\#}$ are the Gaussian curvature and
area element of $ds^2{}^{\#}$, respectively.  As 
\begin{equation}\label{eq:pseudo-sharp}
   d\sigma^2{}^{\#}:=(-K^{\#})ds^2{}^{\#} = 
        \frac{4\,dG\,d\overline{G}}{(1+|G|^2)^2}
\end{equation}
is a pseudo-metric of constant curvature $1$ with developing map $G$,
$\TA(f^{\#})$ is the area of the image of $G$ on $\CP^1=S^2$.
The following assertion is important for us:  
\begin{lemma}[\cite{uy5,Yu2}] \label{lem:complete} 
  The Riemannian metric $ds^2{}^{\#}$ is complete
  $($resp.\ non\-degen\-erate$)$
  if and only if $ds^2$ is complete $($resp.~nondegenerate$)$.  
\end{lemma}

We now assume that the induced metric $ds^2$ 
(and consequently $ds^2{}^\#$) on $M$ is complete and that either
$\TA(f)<\infty$ or $\TA(f^\#)<\infty$, 
hence there exists a compact Riemann surface 
$\overline M_\gamma$ of genus $\gamma$ and a finite set of points 
$\{p_1,\dots,p_n\}\subset \overline M_\gamma$ ($n\geq 1$) so that 
$M$ is biholomorphic to $\overline M_\gamma\setminus\{p_1,\dots,p_n\}$ 
(see Theorem 9.1 of \cite{O}).  
We call the points $p_j$ the {\em ends\/} of $f$.  

Unlike the Gauss map for minimal surface with $\TA < \infty$ in $\R^3$,
the hyperbolic Gauss map $G$ of the surface might not extend to a
meromorphic function on $\overline M_\gamma$, as the Enneper cousin 
(Example~\ref{exa:enneper}) shows.  
However, the Hopf differential $Q$ does extend 
to a meromorphic differential on $\overline M_\gamma$ \cite{Bryant}.  
We say an end $p_j$ $(j=1,\dots,n)$ of a \cmcone{} immersion is 
{\em regular\/} if $G$ is meromorphic at $p_j$.  
When $\TA(f) < \infty$, an end $p_j$ is regular precisely when the order
of $Q$ at $p_j$ is at least $-2$, and otherwise $G$ has an essential
singularity at $p_j$ \cite{uy1}. 
Moreover, the pseudometric $d\sigma^2$ as in \eqref{eq:pseudo} has a
{\em conical singularity\/} at each end $p_j$ \cite{Bryant}.
For a definition of conical singularity, see \cite{uy3,uy7}.

\subsection*{Analogue of the Osserman inequality.} 

For a \cmcone{} surface of genus $\gamma$ with $n$ ends, 
the second and third authors showed that the equality of the Cohn-Vossen
inequality for the total absolute curvature never holds \cite{uy1}:
\begin{equation}\label{eq:cohn-vossen-general}
   \frac{1}{2\pi}\TA(f) > -\chi(M) = 2(\gamma-2)+n\;.
\end{equation}
The catenoid cousins (Example~\ref{exa:catenoid}) show that this 
inequality is the best possible.

On the other hand,  the dual total absolute curvature satisfies an
Osserman-type inequality \cite{uy5}: 
\begin{equation}\label{eq:osserman}
   \frac{1}{2\pi}\TA(f^{\#})\geq -\chi(M)+n = 2(\gamma+n-1) \; . 
\end{equation}   
Moreover, equality holds exactly when all the ends are embedded: 
This follows by noting that equality is equivalent to all ends being
regular and embedded (\cite{uy5}), 
and that any embedded end must be regular (proved recently by Collin,
Hauswirth and Rosenberg \cite{chr} and independently by Yu \cite{Yu3}).  

\subsection*{Effects of transforming the lift $\bmath{F}$.} 
Here we consider the change $\hat F=aFb^{-1}$ of the lift $F$,
where $a,b\in \SL(2,\C)$.  Then $\hat F$ is also a holomorphic null
immersion, and the hyperbolic Gauss map $\hat G$, the secondary Gauss
map $\hat g$ and the Hopf differential $\hat Q$ of $f=\hat F\hat F^*$ 
are given by (see \cite{uy3})
\begin{equation}\label{eq:changes}
  \hat G=a \star G, \quad \hat g=b\star g,\quad \hat Q=Q\;.
\end{equation}
In particular, the change $\hat F=aF$ moves the surface by a rigid
motion of $H^3$, 
and does not change $g$ and $Q$.  
Furthermore, for any $b \in \SU(2)$, the change $\hat F=Fb$ does not 
change the surface at all.  

By choosing a suitable rigid motion $a\in \SL(2,\C)$ of the surface in
$H^3$, the expression for $G$ can often be simplified, 
using 
\begin{equation} \label{eq:simplifyG}
  \hat G =a\star G=\frac{a_{11}G+a_{12}}{a_{21}G+a_{22}} \; , 
  \qquad (a_{ij})_{i,j=1,2} \in \SL(2,\C) \; . 
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Basic examples}
\label{sec:Added}

\begin{example}[Horosphere]\label{exa:horosphere}
 The horosphere is the only flat (and consequently totally umbilic) 
\cmcone{} surface in $H^3$.  It's Weierstrass data can be given by 
\[
    g=0,\qquad \omega=a\, dz\qquad (a\in \C\setminus\{0\}) \; .
\]
 The holomorphic lift $F\colon{}\C\to \SL(2,\C)$ of the surface
 with initial condition $F(0)=\id$ is given by
\[
 F=\begin{pmatrix}
    1  & 0 \\
    az & 1 
   \end{pmatrix}\;.
\]
 In particular the hyperbolic Gauss map is a constant function, 
 as well as the secondary Gauss map $g=0$.
 So the total curvature and the dual total 
 curvature of the horosphere are both equal to zero.
 Any flat or totally umbilic \cmcone{} surfaces are
 parts of this surface. Planes in $\R^3$ are 
 the corresponding minimal surfaces with the same
 Weierstrass data $(g,\omega)=(0,a\,dz)$.
 
 Since the horosphere is simply-connected, it is $\Hyp^3$-reducible.  
\end{example}

\begin{example}[Enneper cousin and dual of Enneper cousin]
\label{exa:enneper}
 The Enneper cousins are 
 given in \cite{Bryant}, with the same Weierstrass
 data as the Enneper surface in $\R^3$:
\[
      g=z,\qquad \omega=a\, dz\qquad \bigl(a\in \C\setminus\{0\}\bigr)\;.
\]
Hence the Enneper cousins are isometric to minimal Enneper surfaces.  

 The holomorphic lift $F\colon{}\C\to \SL(2,\C)$ of the surface
 with initial condition $F(0)=\id$ is given by
 \[
  F=
  \begin{pmatrix}
   \cosh(az) & a^{-1}\sinh(az)-z \cosh(az) \\
   a\sinh(a z) & \cosh(az)- az \sinh(az) 
  \end{pmatrix}\;.
 \]
 In particular the hyperbolic Gauss map $G$ is given by
 \[
   G=a^{-1}\tanh (az)\;,
 \]
 and hence the end at $z=\infty$ is irregular.
 So the Enneper cousins have complete induced 
 metrics of total absolute curvature $4\pi$ and infinite 
 dual total absolute curvature.  
 If one takes the inverse of $F$, one gets the duals of the Enneper
 cousins.  Since
 \[
   F d (F^{-1})=-dF F^{-1}=
   \begin{pmatrix}
     -a \cosh(az)\sinh(az) & \sinh^2(az) \\
     -a^2 \cosh^2(a z) & a \cosh(a z)\sinh(az) 
   \end{pmatrix}\;,
 \]
 the Weierstrass data $(g^\#,\omega^\#)$ of the dual of the Enneper cousin
 is given by
 \[
    g^\#=a^{-1}\tanh (az),\qquad \omega^\#=a^2 \cosh^2(az)\, dz\;.
 \]
 This dual surface also has a 
 complete induced metric, but now with infinite total absolute curvature
 (see Lemma~\ref{lem:complete}).  
 
  Since the Enneper cousins and their duals are simply-connected, they are 
  $\Hyp^3$-reducible.  
\end{example}

\begin{example}[Catenoid cousins and warped catenoid cousins]
\label{exa:catenoid}

Here we describe the catenoid cousins and the warped catenoid cousins.  
The catenoid cousins are the only \cmcone{} surfaces of revolution
\cite{Bryant}.  
The warped catenoid cousins \cite{uy1,ruy3} are less well known.  

 \cmcone{} surfaces of genus 0 with two regular ends are classified in 
 Theorem~6.2 in \cite{uy1}.
 Here we describe a slightly refined version of this classification, 
 which can also be found in \cite{ruy4}:  
 A complete conformal \cmcone{} immersion 
 $f\colon{}M=\C\setminus \{0\} \to H^3$ with regular ends has the
 following Weierstrass data
\begin{equation}
\label{eq:catcousdata}
    g = \frac{\delta^2-l^2}{4l}z^l+b \; ,\qquad
    \omega= z^{-l-1}dz \; , 
\end{equation}
 with $l>0$, $\delta\in\Z^+$, and $l \neq \delta$, and $b \geq 0$,
 where the case $b>0$ occurs only when $l\in \Z^+$. 
 When $b=0$ and $\delta=1$, the surface is called a 
 {\em catenoid cousin}, which is rotationally symmetric. 
 (The Weierstrass data of the catenoid cousin is often written as 
  $g=z^\mu$ and $\omega=(1-\mu^2)z^{-\mu-1}\,dz/(4\mu)$.  
 This is equivalent to \eqref{eq:catcousdata} for $b=0$ and $\delta=1$
 and $l=\mu$ by a coordinate change $z\mapsto ((1-\mu^2)/4\mu)^{(1/\mu)} z$.) 
 Catenoid cousins are embedded when
 $0<l<1$ and have one circle of self-intersection when $l>1$. 
 When $b=0$, $f$ is a $\delta$-fold cover of a catenoid cousin.  
 When $b>0$ (then automatically $l$ is a positive integer), 
 we call $f$ a {\em warped catenoid cousin}, 
 and its  discrete symmetry group is the natural $\Z_2$ extension of the
 dihedral group $D_l$.  
 Furthermore, the catenoid cousins and 
 warped catenoid cousins can be written explicitly as 
 \[ 
   f= F F^{*},\qquad F = F_0 B \; , 
 \]
 where 
 \[ 
    F_0 = \sqrt{\frac{\delta^2-l^2}{\delta}}
              \begin{pmatrix}
                 \dfrac{1}{l-\delta}    z^{(\delta-l)/2}  &
                 \dfrac{\delta-l}{4l}   z^{(l+\delta)/2} \\
                 \dfrac{1}{l+\delta}    z^{-(l+\delta)/2} &
                 \dfrac{-(l+\delta)}{4l}z^{(l-\delta)/2}
              \end{pmatrix} \quad 
          \text{and} \quad
        B =   \begin{pmatrix}
                    1 & -b \\
                    0 & \hphantom{-}1
              \end{pmatrix}   \,.
 \]
 In particular, the hyperbolic Gauss map and Hopf differential 
 are given by 
 \[
    G=z^\delta,\qquad Q=\frac{\delta^2-l^2}{4z^2}\,dz^2 \; , 
 \]
 which are equal to the Gauss map and Hopf differential of
 ($\delta$-fold covers of) the catenoids in $\R^3$.
 The dual total curvature of a catenoid cousin is $4\pi$, but
 its total curvature is $4\pi l$, which can take
 any value in $(0,4\pi) \cup (4\pi,\infty)$. 
 On the other hand, the total absolute curvature and the dual total
 absolute curvature of warped catenoid cousins are always integer
 multiples of $4\pi$.
 
  The catenoid cousins are generally $\Hyp^1$-reducible, except when 
  $l$ is an integer, in which case they are $\Hyp^3$-reducible.  
  The warped catenoid cousins are all $\Hyp^3$-reducible.  
\end{example}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{thebibliography}
\end{document}


