A Picard type theorem for Hyperbolic Gauss Map of CMC-1 Surfaces in Hyperbolic 3-space and de Sitter 3-space

TL;DR

This paper extends Picard type theorems to the hyperbolic Gauss map of CMC-1 surfaces in hyperbolic and de Sitter spaces, showing limitations on the image omissions similar to classical minimal surface results.

Contribution

It establishes a Picard type theorem for the hyperbolic Gauss map of CMC-1 surfaces in hyperbolic and de Sitter spaces, generalizing known minimal surface theorems.

Findings

The hyperbolic Gauss map omits at most 3 points for certain CMC-1 surfaces.

The results apply to surfaces with finite total curvature and regular ends.

The approach adapts techniques from minimal surface theory to CMC-1 contexts.

Abstract

In a recent paper Jorge and Mercuri proved that the image of Gauss map of a complete non flat minimal surfaces in R3 with finite total curvature omits at most 2 points. In this work we follow their idea and prove 3a similar result for CMC-1 with finite total curvature in H and CMC-1 faces with finite type and regular ends in S31.

Equations

$$⟨(x_0,x_1,x_2,x_3),(y_0,y_1,y_2,y_3)⟩=-x_0{y}_0+x_1{y}_1+x_2{y}_2+x_3{y}_3$$

$${\mathbb{H}}^3=\{v\in {\mathbb{L}}^4|⟨v,v⟩=-1,x_0(v)>0\}$$

$$(x_{0},x_{1},x_{2},x_{3})\mapsto\left[\begin{array}[]{c c }x_{0}+x_{3}&x_{1}+ix_{2}\\ x_{1}-ix_{2}&x_{0}-x_{3}\end{array}\right]$$

$${\mathbb{H}}^3=\{X{X}^{\ast }|X\in SL(2,\mathbb{C})\}{X}^{\ast }={\overline{X}}^t$$

$$⟨X,Y⟩=-\frac{1}{2}tr(X,\tilde{Y}),⟨X,X⟩=-det(X)$$

$$X\mapsto YX{Y}^{\ast },X\in {\mathbb{H}}^{3}Y\in PSL(2,\mathbb{C})$$

$$F^{-1}dF=\left[\begin{array}[]{c c }g&-g^{2}\\ 1&-g\end{array}\right]\omega$$

$$F(z)=\left[\begin{array}[]{c c }A(z)&B(z)\\ C(z)&D(z)\end{array}\right]\in SL(2,\mathbb{C})$$

$$g=-\frac{dB}{dA},\omega =AdC-CdA$$

$$F^{-1}dF=\left[\begin{array}[]{c c }g&-g^{2}\\ 1&g\end{array}\right]\omega$$

$$d{s}^2=(1+|g{|}^2{)}^2|f{|}^{2}d{z}^2$$

$$\kappa (z)=\frac{-4|g^{\prime }{|}^2}{|f{|}^2(1+|g{|}^2{)}^4}$$

$$G(z)=\underset{t\to \infty }{\lim }\gamma (t)$$

$$\displaystyle F(z)=\left[\begin{array}[]{c c}A(z)&B(z)\\ B(z)&D(z)\end{array}\right]$$

$$G(z)=\frac{dA}{dC}=\frac{dB}{dD}$$

$$S(g)-S(G)=2Q$$

$$S(g)$$

$$S_z(g)$$

$$Q$$

$$\frac{1}{2\pi }{\int }_{M}KdA\le (\chi (M)-n)$$

$$\frac{1}{2\pi }{\int }_{M}KdA<\chi (M)$$

$${\varphi }^{♯}=(F^{♯})(F^{♯}{)}^{\ast }$$

$$(F^{\sharp})^{-1}dF^{\sharp}=\left[\begin{array}[]{c c }g^{\sharp}&-(g^{\sharp})^{2}\\ 1&-g^{\sharp}\end{array}\right]\omega^{\sharp}$$

$$G^{♯}=g,g^{♯}=G,{\omega }^{♯}=-\frac{Q}{dG},Q^{♯}=-Q$$

$$d{s}^{♯2}$$

$$=$$

$$K_T^{♯}={\int }_M-K^{♯}d{A}^{♯}.$$

$$\frac{1}{2\pi }{\int }_M{K}^{♯}d{A}^{♯}\le (\chi (M)-n)$$

$$⟨(x_0,x_1,x_2,x_3),(y_0,y_1,y_2,y_3)⟩=-x_0{y}_0+x_1{y}_1+x_2{y}_2+x_3{y}_3$$

$${\mathbb{S}}_1^3=\{v\in {\mathbb{L}}^4|⟨v,v⟩=1\}$$

$$(x_{0},x_{1},x_{2},x_{3})\mapsto\left[\begin{array}[]{c c }x_{0}+x_{3}&x_{1}+ix_{2}\\ x_{1}-ix_{2}&x_{0}-x_{3}\end{array}\right]$$

$$e_{0}=\left[\begin{array}[]{c c }1&0\\ 0&1\end{array}\right]\ ,\ e_{1}=\left[\begin{array}[]{c c }0&1\\ 1&0\end{array}\right]\ ,\ e_{2}=\left[\begin{array}[]{c c }0&i\\ -i&0\par\end{array}\right]\ ,\ e_{3}=\left[\begin{array}[]{c c }1&0\\ 0&-1\par\end{array}\right]$$

Full text

1 Introduction

A classical problem in theory of minimal surfaces of $\mathbb{R}^{3}$ is to determine the number of the missing points on the image of the Gauss map, in other words, if $M$ is a complete minimal surface and $G:M\longrightarrow\mathbb{S}^{2}$ is the Gauss map of $M$, how many points in $\mathbb{S}^{2}\backslash G(M)$ are there? Fujimoto showed that if $M$ is not the plane, then $\sharp(\mathbb{S}^{2}\backslash G(M))\leqslant 4$ and is exactly $4$ for the Scherk’s surface. Osserman, studied this problem under the hypothesis that the total curvature is finite. In this case, he showed that $\sharp(\mathbb{S}^{2}\backslash G(M))\leqslant 3$. Recently, Jorge and Mercuri [13] improved Osserman’s estimate in the following theorem

Theorem 1.1** (Jorge, Mercuri).**

If the Gauss map of a complete minimal surface of $\mathbb{R}^{3}$ with finite total curvature misses $3$ or more points of the sphere then it is a plane.

This estimative is sharp since catenoid’s Gauss map omits exactly 2 points. So the problem is fully answered for this class of surfaces. Theorem 1.1 follows from analysis of immersions with Gauss map missing three points.

A natural question arises: Which other class of surfaces have the same conformal type of the complex plane and Gauss map as meromorphic functions?

As first choice we have the Bryant’s Surfaces. Those surfaces share many properties with minimal surfaces of $\mathbb{R}^{3}$ and act like an analogue of minimal surfaces in hyperbolic 3-space. Indeed, Collin, Hauswirth and Rosenberg [4] proved an analogue of Osserman’s result for Bryant’s surfaces, that is, the hyperbolic Gauss map of a complete Bryant’s surface with finite total curvature misses at most 3 points.

Kawakami [14] proved partial results for other classes of surfaces. He studied the hyperbolic Gauss map of algebraic Bryant Surfaces and CMC-1 faces in de Sitter 3-space. Algebraic Bryant Surfaces are Bryant Surfaces with finite total curvature dual (dual in the Umehara and Yamada’s sense). CMC-1 faces are spacelike CMC-1 surfaces in de Sitter space with a specific kind of singularities. Kawakami showed that hyperbolic Gauss map of a complete algebraic Bryant’s Surfaces and of a complete CMC-1 face can miss 3 points at most, otherwise it is constant.

In this paper, we use Jorge and Mercuri’s ideas to prove the following theorem:

Theorem 1**.**

Let $M$ be one of the following:

1. (1)

a complete Bryant surface with finite total curvature or, 2. (2)

a complete algebraic Bryant surface, 3. (3)

a complete CMC-1 face of finite type with elliptic ends.

Then the hyperbolic Gauss map $G$ of $M$ omit at most 2 points, otherwise $G$ is constant. Further, in case of constant hyperbolic Gauss map $M$ is a horosphere for items (1) and (2) and a horosphere space like type surface for item (3).

All results in this theorem are sharp.

A surface is parabolic if positive harmonic functions are constant. By [12] the volume growth of geodesics ball $B_{r}(x_{o})$ of a complete minimal surface into $\mathbb{R}^{3}$ with finite total curvature is of type $r^{2}$. In §5 we study the decay of curvature of those surfaces of theorem 1 and concluded that all of them have $\text{vol}_{M}(B_{r})\leq cr^{2},\ r\geq r_{0}$ form some constant $c_{0}$ and fixed $r_{0}>0.$ This implies next result.

Theorem 2**.**

Let $M$ be one complete surface of type:

1. (1)

minimal surface with finite total curvature into $\mathbb{R}^{3}$; 2. (2)

Algebraic Bryant surface with finite total curvature; 3. (3)

Algebraic Bryant surface endowed with the dual metric; 4. (4)

CMC-1 Face surface of finite type, elliptic ends endowed with the lift metric.

Then $M$ is parabolic. If $M$ has a Kähler structure then all bounded holomorphic map $g\colon M\to\mathbb{C}$ is constant.

Remark 1.1**.**

All surfaces in items (1), (2), (3), (4), are isometric to a complete minimal surface $M$ into $\mathbb{R}^{3}$ with finite total curvature. Then they have a natural Kähler structure. If the Gauss map $G$ of $M$ miss at least one point $y\in\mathbb{S}^{2}$ we can rotated $M$ inside $\mathbb{R}^{3}$ to making $y=(0,\ 0,\ 1).$ If $g\colon M\to\mathbb{C}$ is the stereographic projection of $G$ then $g$ is holomorphic. Hence all surfaces described in the theorem 2 are Kähler. Unfortunately, if $g(M)=\mathbb{C}\setminus\{a,\ b\}$ we can not do a lifting of $g$ to the unit disk $\mathbb{D}$ by a modular function ot get a easy proof of theorem 1.1 unless $M$ is simply connected.

2 Hyperbolic Space

Let $\mathbb{L}^{4}$ be Lorentz-Minkowski 4-space with Lorentzian metric

[TABLE]

Define the hyperbolic $3$-space

[TABLE]

with the induced metric from $\mathbb{L}^{4}$. Give to $\mathbb{H}^{3}$ the orientation which $v_{1},v_{2},v_{3}$ forms a oriented base of $T_{v}\mathbb{H}^{3}$ if, and only if, $v,v_{1},v_{2},v_{3}$ forms a base of $\mathbb{L}^{4}$. We have that $\mathbb{H}^{3}$ is a Riemannian manifold 3-dimensional simply connected with constant mean curvature $-1$.

The space $\mathbb{H}^{3}$ is not compact, but can be compacted by adding a ”sphere at infinity” $\mathbb{S}_{\infty}^{2}$ in such way that the rigid motions of $\mathbb{H}^{3}$ extend to homeomorphisms of $\overline{\mathbb{H}^{3}}=\mathbb{S}_{\infty}^{2}\cup\mathbb{H}^{3}$.

Identify $\mathbb{L}^{4}$ with the set $Herm(2)=\{X^{*}=X\}$ of $2\times 2$ hermitian matrices in the follow way

[TABLE]

where $i=\sqrt{-1}$.

With this identification, we can see $\mathbb{H}^{3}$ as

[TABLE]

with the metric

[TABLE]

where $\tilde{Y}$ is the cofactor of $Y$. $PSL(2,\mathbb{C})=SL(2,\mathbb{C})/\{\pm id\}$ acts isometrically on $\mathbb{H}^{3}$ by

[TABLE]

Note that the map $PSL(2,\mathbb{C})\longrightarrow Herm(2)$ such that $F\mapsto FF^{*}$ take values in $\mathbb{H}^{3}$.

2.1 Bryant Surfaces

Surfaces of constant mean curvature (CMC) are objects of great interest because they are critic points of functional area with respect to volume preserving variations that fix their boundaries. Minimal surfaces are a special class of CMC surfaces that are critical for all variations, not just volume preserving ones. Beside that, minimal surfaces can be described by a pair of holomorphic functions, called Weierstrass data. Many properties of minimal surfaces can be obtained using this Weierstrass data.

In 1970, Lawson described a correspondence between minimal surfaces and CMC-1 in $\mathbb{H}^{3}$, the ”Lawson’s correspondence”. So, as the minimal surfaces are described by a pair of holomorphic function, the Weierstrass data, one may ask if CMC-1 surfaces in $\mathbb{H}^{3}$ can be described in a similar way. Such question was answered by Robert Bryant [3] in 1987. He showed that this class of surfaces can be described by a pair of holomorphic functions, known as the Bryant data.

Theorem 2.1** (Bryant Representation).**

Let $M$ be a Riemann’s surface and $F:M\longrightarrow SL(2,\mathbb{C})$ conformal immersion such that$det(F^{-1}dF)=0$. Let $\phi:M\longrightarrow\mathbb{H}^{3}$ be such that $\phi=FF^{*}$. So $\phi$ is a immersion from $M$ to $\mathbb{H}^{3}$ with constant mean curvature 1. Conversely, if $\phi:M\longrightarrow\mathbb{H}^{3}$ is a immersion with constant mean curvature 1, there is a holomorphic lift $\phi$ to universal recovery $\tilde{F}:\tilde{M}\longrightarrow SL(2,\mathbb{C})$, such that $det(F^{-1}dF)=0$ and $\phi=\tilde{F}\tilde{F}^{*}$.

Because of this work, the CMC-1 surfaces in hyperbolic 3-space became known as Bryant Surfaces.

Remark 2.1**.**

A holomorphic map $F:M\longrightarrow SL(2,\mathbb{C})$, such that $det(F^{-1}dF)=0$ is called holomorphic null immersion.

Remark 2.2**.**

Let $F:M\longrightarrow SL(2,\mathbb{C})$ be a map such that $det(F^{-1}dF)=0$. Locally, we can write:

[TABLE]

Where $g$ is a meromorphic function and $\omega$ is a holomorphic 1-form. Indeed, write

[TABLE]

Where $z$ is conformal local coordinate and $A,B,C,D$ are holomorphic. So, take

[TABLE]

As $F$ is holomorphic, the forms $\omega$ e $g^{2}\omega$ are also holomorphic, and $F^{-1}dF$ never vanish, by the fact that $F$ is a immersion. Therefore poles of $g$ are zeros of $\omega$, and a pole of order $k$ of $g$is a zero of order $2k$ of $\omega$. The pair $(g,\omega)$ is the Bryant data associated to $F$.

Remark 2.3**.**

We can write the holomorphic 1-form $\omega$, in local coordinates, as $\omega=f(z)dz$. So, we can write the Bryant data as $(g,f)$.

Let $F$ be a immersion of $M$ in $SL(2,\mathbb{C})$ such that

[TABLE]

By Bryant’s representation theorem and Lawson’s correspondence, if $(g,f)$ is a Bryant data of a Bryant surface so the metric and Gaussian curvature can be described using $g$ and $F$ by.

[TABLE]

[TABLE]

Note that these are the same expressions for the Weierstrass data and minimal surfaces.

Remark 2.4**.**

Note That the Weierstrass representation can be obtained as the limit of Bryant’s theorem by collapsing the Lie group $SL(2,\mathbb{C})$ in the abelian group $\mathbb{C}^{3}$.

2.2 Hyperbolic Gauss Map

In 1986, Epstein introduced the hyperbolic Gauss map while he studied immersed surfaces in the hyperbolic 3-space in Poincaré model. Later, Bryant [3] reintroduced the concept in the hyperboloid of Minkowski space model, in the following way:

Definition 2.1**.**

Let $M$ be a immersed surface in the hyperbolic 3-space. Define the hyperbolic Gauss Map $G:M\longrightarrow\mathbb{S}_{\infty}^{2}$ in the following way: Let $z\in M$, take $\gamma$ normal oriented geodesic starting from $z$. Define $G(z)$ as the point where $\gamma$ intercepts the ideal boundary $\mathbb{S}_{\infty}^{2}$ of $\mathbb{H}^{3}$. In other words

[TABLE]

For Bryant surfaces we have the follow characterization. Let $M$ be a Bryant surface and $F:M\longrightarrow SL(2,\mathbb{C})$ the conformal immersion given by theorem $2.1$. Denote

[TABLE]

then,

[TABLE]

Following the terminology introduced by Umehara and Yamada, $g(z)=-\frac{dB}{dA}$ is called Secondary Gauss Map. Note that the secondary Gauss map is the map $g$ from Bryant data $(g,\omega)$. the motivation for the name comes from the fact that in a minimal surface immersed in $\mathbb{R}^{3}$ with Weierstrass data $(\tilde{g},\tilde{f})$ the map $\tilde{g}$ can be seen as the composition of Gauss map with stereographic projection. So, the map $\tilde{g}$ has 2 fundamentals roles:

1. (i)

Describe the metric. 2. (ii)

Describe the stereographic projection of unit normal vector.

But in the case of Bryant data $(g,f)$, the roles $(i)$ e $(ii)$ are played by 2 different maps, $g$ plays the role $(i)$ and $G$ plays the role $(ii)$. Therefore, we can think that the Bryant representation has 2 Gauss Maps, the Hyperbolic Gauss Map and the Secondary Gauss Map.

Umehara e Yamada showed the following relationship between the Hyperbolic Gauss Map and Secondary Gauss Map.

[TABLE]

where

[TABLE]

$S_{z}(g)$ is the Schwarzian derivate and $Q$ is the Hopf’s differential.

2.3 Finite Total Curvature

The basic tools in the modern theory of minimal immersions $\phi:M^{2}\longrightarrow\mathbb{R}^{3}$ was establish by Osserman [15]. In this paper he shows the following results about complete minimal surfaces $M$ of $\mathbb{R}^{3}$ with finite total curvature:

There is a compact Riemann surface $\overline{M}$ of finite genus $\mu$ and a finite set $E=\{p_{1},\cdots,p_{k}\}\subset\overline{M}$ such that $M$ is conformal to $\overline{M}\setminus E.$ 2. 2.

The Gauss map $G\colon M\to\mathbb{S}^{2}$ extends to a conformal branched covering map $G\colon\overline{M}\to\mathbb{S}^{2}$. 3. 3.

\sharp\big{(}\mathbb{S}^{2}\setminus G(M)\big{)}\leq 3 unless $M$ is a flat plane.

The points $p_{i}\in E$ or some times one neighborhood of them are known as ends of $M$. In the hyperbolic context those items have similar results with fill exception.

Theorem 2.2** (Bryant, [3]).**

Let $M$ a complete Bryant surface with finite total curvature immersed in $\mathbb{H}^{3}$. Then there is a compact Riemann surface $\overline{M}$ and a finite set $E=\{p_{1},\ ...\ ,p_{n}\}\subset\overline{M}$ such that $M$ is conformal to $\overline{M}\setminus E$.

Theorem 2.3** (Collin, Hauswirth, Rosenberg, [4]).**

Let $M$ be a immersed complete Bryant surface with finite total curvature in $\mathbb{H}^{3}$. Then the hyperbolic Gauss map $G$ of $M$ omit at most $3$ points unless $G$ is constant and $M$ is a horosphere.

The hyperbolic Gauss map is holomorphic but does not necessarily has meromorphic extension to the ends. In fact some ends could be an essential singularity of the Gauss map and motivated the next definition.

Definition 2.2**.**

Let $M$ be a complete Bryant surface with finite total curvature and $E=\{p_{1},\ ...\ ,p_{n}\}$ the ends of $M$. We say that an end $p_{i}$ is regular if the hyperbolic Gauss map does extend meromorphically to $p_{i}$. Otherwise the end $p_{i}$ is called irregular.

The existence of irregular ends is the first big difference between minimal surfaces and Bryant surfaces.

2.4 Dual Surfaces

It is well known that a minimal surface $M$ with finite total curvature immersed in $\mathbb{R}^{3}$ satisfies the Osserman’s inequality

[TABLE]

where $K$ is the Gaussian curvature of $M$ and $n$ is the number of ends of the surface. For Bryant surfaces with finite total curvature, there is no analogue relation. In fact, Umehara and Yamada [19] showed that Bryant surfaces satisfy only the Cohn-Vossen inequality

[TABLE]

Motivated by this fact, Umehara and Yamada [19] introduced the concept of dual surface, in a way that if $\phi:M^{2}\longrightarrow\mathbb{H}^{3}$ is a CMC-1 immersion and $\phi^{\sharp}:\tilde{M}^{2}\longrightarrow\mathbb{H}^{3}$ is the dual immersion, then $M$ satisfies an analogue of Osserman’s inequality, but in terms of dual surface.

Let $\phi:M\longrightarrow\mathbb{H}^{3}$ be a complete CMC-1 immersion with finite total curvature. Let $G$ be the hyperbolic Gauss map, $(g,\omega)$ its Bryant data and $Q=\omega dg$ the Hopf differential of $\phi$.

Definition 2.3**.**

Define the dual CMC-1 immersion $\phi^{\sharp}:\tilde{M}\longrightarrow\mathbb{H}^{3}$ associated to Bryant data $(g,\omega)$ of $\phi$ by

[TABLE]

where $F$ satisfies $(4)$, $F^{\sharp}$ is the inverse of the matrix $F$ and $\tilde{M}$ is the universal cover of $M$.

Remark 2.5**.**

Observe that $\phi^{\sharp}$ is not necessarily single-valued in $M$. But is easy see that $\phi^{\sharp}$ is single-valued in $M$ if, and only if, $g$ is single-valued in $M$.

Let $(g^{\sharp},\omega^{\sharp})$ be the pair defined by

[TABLE]

thus, taking $(\phi^{\sharp})^{\sharp}$, we get a map $(F^{\sharp})^{\sharp}:\tilde{\tilde{M}}\longrightarrow\mathbb{H}^{3}$ such that, $(F^{\sharp})^{\sharp}=(F^{-1})^{-1}=F$, this shows that $(\phi^{\sharp})^{\sharp}=\phi$.

Now we show a important result due to Umehara and Yamada [19] that shows the relationship between the Bryant data and its dual data.

Proposition 2.4** (Umehara, Yamada).**

Let $\phi^{\sharp}$ be the dual immersion of the CMC-1 immersion $\phi$ of a surface $M$ in a hyperbolic 3-space. Let $(g,\omega)$ be the Bryant data of $\phi$. So, $\phi^{\sharp}$ is a CMC-1 immersion of $M$ in the hyperbolic 3-space, with hyperbolic Gauss map $G^{\sharp}$, Bryant data $(g^{\sharp},\omega^{\sharp})$ and Hopf differential $Q^{\sharp}$ $\phi^{\sharp}$ given by:

[TABLE]

Remark 2.6**.**

Observe that the dual surface changes the hyperbolic Gauss map with the secondary Gauss map of $M$.

If $\phi$ is a CMC-1 immersion in hyperbolic 3-space and $\phi^{\sharp}$ its dual immersion. By the above theorem, $\phi^{\sharp}$ is a CMC-1 immersion also, so it admits a Bryant data $(g^{\sharp},\omega^{\sharp})$. Thus the metric of the dual immersion is given by

[TABLE]

Such metric is called dual metric.

Remark 2.7**.**

Observe that the metric $ds^{\sharp 2}$ is well defined and single valued in $M$. So, make sense take $M$ with the metric $ds^{\sharp 2}$, such surface is a Bryant surface with Bryant data $(G,\frac{-Q}{dG})$.

There is a relation between the metric $ds^{2}$ and the dual metric $ds^{\sharp 2}$ [20]

Proposition 2.5** (Yu).**

The dual metric $ds^{\sharp 2}$ is complete (resp. non degenerated) if, and only if, the metric $ds^{2}$ is complete (resp. non degenerated).

2.4.1 Algebraic Bryant Surfaces

As the dual metric is well defined in $M$, we have the following definition:

Definition 2.4**.**

Let $M$ be a Bryant surface. Define the dual total curvature of $M$, as

[TABLE]

Where $K^{\sharp}$ and $dA^{\sharp}$ are the Gaussian curvature and the area element of the surface $M$ with the dual metric.

Observe that the dual total curvature is the area of $M$ with respect to the (singular) metric induced by Fubini-Study metric in $\mathbb{C}\mathbb{P}^{1}$.

Definition 2.5**.**

Let $M$ be a Bryant surface. We say that $M$ is algebraic if $M$ has finite dual total curvature.

The algebraic Bryant surfaces satisfy the following theorem

Theorem 2.6** (Bryant, Huber, Z. Yu).**

Let $Mbe$ a algebraic Bryant surface. So:

1. (i)

$M$* is biholomorphic to $\overline{M_{\gamma}}\setminus E$, where $\overline{M_{\gamma}}$ is a closed surface of genus $\gamma$ and $E\subset\overline{M_{\gamma}}$ is a finite set $E=\{p_{1},...,p_{m}\}$.* 2. (ii)

the dual Bryant data $(G,\omega^{\sharp})$ extends meromorphically to $\overline{M_{\gamma}}$.

The points of set $E$ are the ends of $M$.

Using this, Umehara and Yamada [19] deduced an analogue to Osserman’s inequality to dual surfaces. Explicitly they proved that

Theorem 2.7** (Umehara, Yamada).**

Let $M$ be a Riemann surface and $\phi:M\longrightarrow\mathbb{H}^{3}$ a CMC-1 complete conformal immersion with finite dual total curvature. Let $\phi^{\sharp}:{M}^{2}\longrightarrow\mathbb{H}^{3}$ be the dual immersion. then,

[TABLE]

Where $\tilde{K}$ and $dA^{\sharp}$ the dual Gaussian curvature and the dual area element, and $n$ is the number of ends of the original surface $M$.

Due o this properties, one may ask if there is a Picard Type theorem for the hyperbolic Gauss map of algebraic Bryant surfaces too. Indeed, some partial results were obtained by Kawakami [14].

3 De Sitter 3-Space

In this section we study the CMC-1 faces. This CMC-1 faces are spacelike CMC-1 surfaces in de Sitter 3-space with some kind of singularities. Such surfaces share a many properties with Bryant surfaces, in particular, they have an analogue for the Bryant representation. For CMC-1 faces, is possible to define a hyperbolic Gauss map in a similar way we did in surfaces immersed in hyperbolic 3-space, so we can estimate the number of omitted points in the image of this map. In this work we show a sharp estimative for this number.

Lets give a brief description of de Sitter space.

Let $\mathbb{L}^{4}$ be the Lorentz-Minkowski 4-space with Lorentz metric

[TABLE]

Define the de Sitter 3-space $\mathbb{S}_{1}^{3}$ as

[TABLE]

with the induced metric $\mathbb{L}^{4}$. $\mathbb{S}_{1}^{3}$ is a 3-dimensional Lorentz manifold simply connected with constant sectional curvature 1.

We identify $\mathbb{L}^{4}$ with the set of hermitian matrices $2\times 2$ $Herm(2)=\{X^{*}=X\}$ by

[TABLE]

where $i=\sqrt{-1}$.

With this identification, we can define

[TABLE]

Thus,

[TABLE]

with the metric

[TABLE]

Where $X^{*}=\overline{X^{t}}$.

Remark 3.1**.**

Observe that with above definition $\langle X,X\rangle=-detX$.

3.1 CMC-1 Face

In this section, we define CMC-1 faces, enumerate some important results and at last we prove a Picard type theorem for this surfaces.

Definition 3.1**.**

A immersion $f:M^{2}\longrightarrow\mathbb{S}_{1}^{3}$ of a surface $M$ is called spacelike if the induced metric in $M$ is positive definite.

As an analogue for the Bryant surfaces, we have the following theorem [1]:

Theorem 3.1** (Aiyama-Akutagawa).**

Let $D$ be a simply connected domain in $\mathbb{C}$ and $z_{0}\in D$ a base point. Let

[TABLE]

be a meromorphic function and $\omega$ a holomorphic 1-form in $D$ such that

[TABLE]

is a Riemannian metric in $D$.

Take $F=(F_{ij}):D\longrightarrow SL(2,\mathbb{C})$ holomorphic immersion such that $F(z_{0})=e_{0}$ and

[TABLE]

then $f:D\longrightarrow\mathbb{S}_{1}^{3}$ defined by

[TABLE]

is a conformal spacelike immersion, with constant mean curvature 1.

The induced metric $ds^{2}$ in $D$, satisfies

[TABLE]

Conversely, every CMC-1 immersion of a simply connected surface has this form.

Remark 3.2**.**

The pair $(g,\omega)$ is called Aiyama data.

Following Umehara and Yamada definitions for CMC-1 immersions in $\mathbb{H}^{3}$, we define

Definition 3.2**.**

Let $f:D\longrightarrow\mathbb{S}_{1}^{3}$ and $F=(F_{ij})$ as above. Define the hyperbolic Gauss Map $G$ of $f$ as

[TABLE]

We define also the Hopf differential $Q$ of immersion by

[TABLE]

Remark 3.3**.**

The hyperbolic Gauss map has the following meaning. Let $\mathbb{S}_{\infty}^{2}$ be the pointing future ideal boundary ideal of $\mathbb{S}_{1}^{3}$. $\mathbb{S}_{\infty}^{2}$ can be identified with $\mathbb{C}\cup\{\infty\}$. So, given $z\in D$, take $\gamma$ geodesic in $\mathbb{S}_{1}^{3}$ with initial velocity as the unity normal vector of $f(D)$ in $f(z)$. Then $G(z)$ is the point where $\gamma$ intercepts the ideal boundary $\mathbb{S}_{\infty}^{2}$

Remark 3.4**.**

Observe that given $f:D\longrightarrow\mathbb{S}_{1}^{3}$ with $f=Fe_{3}F^{*}$, then $\hat{f}:D\longrightarrow\mathbb{H}^{3}$ given by $\hat{f}=FF^{*}$ is a conformal CMC-1 immersion, with metric $d\hat{s}^{2}$ and the same hyperbolic Gauss map $G$ and Hopf differential $Q$ of $f$.

Definition 3.3**.**

Let $M$ be a oriented surface. A smooth map $f:M\longrightarrow\mathbb{S}_{1}^{3}$ is called CMC-1 map if there is an open dense set $W\subset M$ such that $f|_{W}$ is a spacelike immersion. A point $p\in M$ is called a singular point of $f$ if the indexed metric $ds^{2}$ is degenerated in $p$.

Definition 3.4**.**

Let $f:M\longrightarrow\mathbb{S}_{1}^{3}$ be a CMC-1 map and $W\subset M$ an open dense set such that $f|_{W}$ is a CMC-1 immersion. A point $p\in M\setminus W$ is an admissible singular point if:

1. (1)

There is a map $\beta:U\cap W\longrightarrow\mathbb{R}^{+}$ of class $C^{1}$, where $U$ is a neighborhood of $p$, such that $\beta ds^{2}$ extends to a Riemannian metric in $U$ of class $C^{1}$. 2. (2)

$df(p)\neq 0$, that is, $df$ has rank 1 in $p$.

We say that a CMC-1 map $f$ is a CMC-1 face if all its singular points are admissible.

Proposition 3.2**.**

Let $M$ be a oriented surface and $f:M\longrightarrow\mathbb{S}_{1}^{3}$ 0a CMC-1 face where $W\subset M$ is the open dense such that $f|_{W}$ is a CMC-1 immersion. Then there is a only one complex structure $J$ on $M$ such that

1. (1)

$f|_{W}$* is conformal with respect to $J$.* 2. (2)

There is a immersion $F:\tilde{M}\longrightarrow SL(2,\mathbb{C})$ which is holomorphic with respect to $J$, such that

[TABLE]

Where $\varrho:\tilde{M}\longrightarrow M$ is the universal cover of $M$.

$F$* is called holomorphic null lift.*

Remark 3.5**.**

The holomorphic null lift is unique except of right multiplication for a constant matrix in $SU(1,1)$.

By the above proposition, given a CMC-1 face $f:M\longrightarrow\mathbb{S}_{1}^{3}$ , always exists a complex structure $J$ in $M$. Henceforward, $M$ will be treated as a Riemann surface with this complex structure.

Proposition 3.3**.**

Let $M$ be a Riemann surface and $F:M\longrightarrow SL(2,\mathbb{C})$ a holomorphic null immersion. Assume that the symmetric $(0,2)$-tensor $det[d(Fe_{3}F^{*})]$ is not identically zero. Then

[TABLE]

is a CMC-1 face, and a point $p\in M$ is a singular point of $M$ if, and only if, $det[d(Fe_{3}F^{*})]_{p}=0$. Beside that, $-det[d(FF^{*})]$ is positive definite on $M$.

Remark 3.6**.**

The hypothesis of $det[d(Fe_{3}F^{*})]\neq 0$ is essential. In fact, take $F:M\longrightarrow\mathbb{S}_{1}^{3}$ holomorphic null immersion such that

[TABLE]

So, $f=Fe_{3}F^{*}$ degenerates in every point of $\mathbb{C}$, therefore does not provides a immersion. It comes from the fact that $det[d(Fe_{3}F^{*})]$ is identically zero.

Using the above theorems, is possible to extend Aiyama-Akutagawa representation to CMC-1 faces which domain is not simply connected.

Remark 3.7**.**

Let $F$ be a holomorphic null lift a CMC-1 face $f$ with Aiyama data $(g,\omega)$. Let $B\in SU(1,1)$ be a constant matrix, that is

[TABLE]

Thus, $FB$ is a holomorphic null lift of $f$. The Aiyama data $(\hat{g},\hat{\omega})$ corresponding to $(FB)^{-1}d(FB)$ is given by

[TABLE]

2 Aiyama datas $(g,\omega)$ and $(\hat{g},\hat{\omega})$ are called equivalents if satisfy the above equality for some $B\in SU(1,1)$. We call the equivalence class of Aiyama data $(g,\omega)$ as Aiyama data associated to $f$.

Remark 3.8**.**

Observe that, if $(g,\omega)$ e $(\hat{g},\hat{\omega})$ are 2 equivalents Aiyama data, and if $G$ and $\hat{G}$ are the hyperbolic Gauss maps associated with this data, then

[TABLE]

Therefore, $G=\hat{G}$, that is, the hyperbolic Gauss map does not depend of the choice of $F$ in the class.

3.2 CMC-1 Faces With Elliptics Ends

Definition 3.5**.**

Let $M$ be a Riemann surface, and $f:M\longrightarrow\mathbb{S}_{1}^{3}$ a CMC-1 face. Let $ds^{2}=f^{*}(ds_{\mathbb{S}_{1}^{3}}^{2})$. $f$ is complete (resp, of finite type) if there is a compact set $C$ and a symmetric $(0,2)$-tensor $T$ in $M$ such that $T$ vanishes in $M\setminus C$ and $ds^{2}+T$ is a complete Riemannian metric (resp. has finite total curvature).

Remark 3.9**.**

For CMC-1 immersions in $\mathbb{S}_{1}^{2}$, the Gaussian curvature $K$ is not negative, So the total curvature is the same as the absolute total curvature. However, for CMC-1 faces with singular points, the total curvature never is finite.

Remark 3.10**.**

The universal cover of a complete CMC-1 face (resp finite type) is not necessarily complete (resp finite type), Because the singular set may not be compact the universal cover.

Let $f:M\longrightarrow\mathbb{S}_{1}^{3}$ be a complete CMC-1 face of finite type. So $(M,ds^{2}+T)$ is a complete Riemannian surface with finite total curvature. Therefore, $M$ has finite topological type.

Let $\phi:\tilde{M}\longrightarrow M$ be the universal cover of $M$, and $F:\tilde{M}\longrightarrow SL(2,\mathbb{C})$ a holomorphic null lift of a CMC-1 face $f:M\longrightarrow\mathbb{S}_{1}^{3}$. Fix a point $z_{0}\in M$. Let $\gamma:[0,1]\longrightarrow M$ a loop such that $\gamma(0)=\gamma(1)=z_{0}$. So there is a unique deck transformation $\tau$ of $\tilde{M}$ associated to the homotopy class of $\gamma$. Define the monodromy representation $\Phi_{\gamma}$ of $F$ by

[TABLE]

As $f$ is well defined in $M$, $\Phi_{\gamma}\in SU(1,1)$ for every loop $\gamma$. Therefore, $\Phi_{\gamma}$ is conjugated to one of the following matrices

[TABLE]

De Sitter 3-Space for $\theta\in[0,2\pi)$, $s\in\mathbb{R}\setminus\{0\}$.

Definition 3.6**.**

Let $f:M\longrightarrow\mathbb{S}_{1}^{3}$ be a complete CMC-1 face of finite type with holomorphic null lift $F$. A end of $f$ is called elliptic, hyperbolic or parabolic if tis monodromy representation is conjugated to $E_{1}$, $E_{2}$ or $E_{3}$ in $SU(1,1)$ respectively.

Remark 3.11**.**

As every matrix in $SU(2,\mathbb{C})$ is conjugated to $E_{1}$ in $SU(2,\mathbb{C})$, CMC-1 immersions in $\mathbb{H}^{3}$ have similar properties to CMC-1 faces with elliptic ends in $\mathbb{S}_{1}^{3}$.

Proposition 3.4**.**

Let $V$ be a neighborhood of a end of $f$ and $f|_{V}$ a spacelike CMC-1 immersion of finite total curvature, which is complete in the end. Suppose that the end is elliptic. So there is a holomorphic null lift $F:\tilde{V}\longrightarrow SL(2,\mathbb{C})$ of $f$ with associated Aiyama data $(g,\omega)$ such that

[TABLE]

is single valued in $V$. Besides, $d\hat{s}^{2}$ has complete total curvature and is complete at the end.

Proposition 3.5**.**

Let $f:M\longrightarrow\mathbb{S}_{1}^{3}$ be a complete CMC-1 face of finite type with elliptic ends. So, there is a compact Riemann surface $\overline{M}$ and a finite number of points $p_{1},...,p_{n}\in\overline{M}$, such that $M$ é biholomorphic to $\overline{M}\setminus\{p_{1},...,p_{n}\}$. Besides, the Hopf differential $Q$ of $f$ extends meromorphically to $\overline{M}$.

Similarly to immersed Bryant surfaces in hyperbolic 3-space, the hyperbolic Gauss map does not extends necessarily to the ends. Because of this, we have the following definition

Definition 3.7**.**

Let $f:M\longrightarrow\mathbb{S}_{1}^{3}$ a CMC-1 face. A end $p_{j}$ of $M$ is regular if the hyperbolic Gauss map does extend meromorphically to $p_{j}$, otherwise is called irregular.

Let $f:M\longrightarrow\mathbb{S}_{1}^{3}$ be a CMC-1 face of finite type with elliptic ends. Thus $M=\overline{M}\setminus\{p_{1},...,p_{n}\}$. Let $G$ and $Q$ the hyperbolic Gauss map and the Hopf differential of $f$ respectively.

Definition 3.8**.**

We call the metric

[TABLE]

of lift metric of CMC-1 face $f$. Besides

[TABLE]

Remark 3.12**.**

Observe that $d\hat{s}^{\sharp 2}$ and $-(K_{d\hat{s}^{\sharp 2}})d\hat{s}^{\sharp 2}$ are given in terms of $G$ and $Q$, and such functions are defined in $M$, thus $d\hat{s}^{\sharp 2}$ and $-(K_{d\hat{s}^{\sharp 2}})d\hat{s}^{\sharp 2}$ are defined in $M$ also.

Fujimori [6] showed that

Proposition 3.6**.**

Let $f:M\longrightarrow\mathbb{S}_{1}^{3}$ be a CMC-1 face. Assume that each end of $f$ elliptic and regular. Thus, if $f$ is complete and of finite type, then the lift metric $d\hat{s}^{\sharp 2}$ is complete and of finite total curvature in $M$.

4 The proof of theorem 1

Proof.

Suppose that $\phi:M\longrightarrow\mathbb{H}^{3}$ is a complete algebraic CMC-1 immersion with hyperbolic Gauss map $G$ and Bryant data $(g,\omega)$. Then, take $\phi^{\sharp}:M\longrightarrow\mathbb{H}^{3}$ its dual immersion with dual data $(G,\omega^{\sharp})$ and let $\psi:M\longrightarrow\mathbb{R}^{3}$ be the minimal immersion cousin of the $\phi^{\sharp}$, with Weierstrass data $(G,\omega^{\sharp})$. Note that $\psi$ is a complete minimal surface with finite total curvature, since $\phi^{\sharp}$ is complete and the total curvature of $\psi$ coincides with total dual curvature of $\psi^{\sharp}$, witch is finite since $\phi$ is algebraic. Besides, the Gauss map of $\psi$ coincides with the hyperbolic Gauss map $G$ of $\phi$. Then by theorem 1.1, $G$ omits 2 points at most.

Suppose now that $M$ is a complete Bryant surface with finite total curvature. Then $M$ is conformal to $\overline{M}\setminus E$, where $\overline{M}$ is a compact surface and $E$ is a finite set, the ends of the surface. Suppose that $M$ has at least one irregular end $p\in E$. Then $p$ is a essential singularity of $G$, so by Big Picard’s theorem, $G$ omits 2 points at most. Therefore we can suppose that all ends of $M$ are regular. If all ends are regular, then $M$ is a algebraic Bryant surface, and therefore, $G$ can omits 2 points at most.

Suppose now that $M$ is a a complete CMC-1 face of finite type with elliptic ends. Let $(g,\omega)$ be its Aiyama data. Define $\phi:M\longrightarrow\mathbb{H}^{3}$ the immersion wich has $(g,\omega)$ as its Bryant data. Then, $\phi:M\longrightarrow\mathbb{H}^{3}$ is a complete CMC-1 immersion. Note that the lift metric of the CMC-1 face $M$ coincides with the dual metric of $\phi$. Then $\phi(M)$ is a algebraic Bryant surface, because $M$ has finite type. Observe that the hyperbolic Gauss map $G$ of $M$ coincides with the hyperbolic Gauss map of $\phi:M\longrightarrow\mathbb{H}^{3}$. Therefore, $G$ can miss 2 points at most.

∎

Remark 4.1**.**

This estimative is sharp since:

1. (i)

the catenoid cousin is a complete Bryant surface with finite total curvature witch hyperbolic Gauss map that omits exactly 2 points. 2. (ii)

$\mathbb{C}\setminus\{0,1\}$* is a complete algebraic Bryant surface witch hyperbolic Gauss map that omits exactly 2 points.* 3. (iii)

the elliptical catenoid is a complete CMC-1 face of finite type with elliptic ends with hyperbolic Gauss map that omits exactly 2 points.

Corolary 4.1**.**

Let $M$ be a properly embedded Bryant surface of finite topological type. Then, the hyperbolic Gauss map is constant and $M$ is a horosphere or the image of hyperbolic Gauss map omits 2 points at most.

Proof.

Collin, Hauswirth and Rosenberg showed in [5] that a properly embedded Bryant surface of finite topological type has finite total curvature, thus the result follows from the previous theorem. ∎

5 Curvature estimate and volume growth.

We begin with the following theorem

Theorem 5.1**.**

Let $M$ be one complete surface of type:

1. (1)

minimal surface with finite total curvature into $\mathbb{R}^{3}$; 2. (2)

Algebraic Bryant surface with finite total curvature; 3. (3)

Algebraic Bryant surface endowed with the dual metric; 4. (4)

CMC-1 Face surface of finite type, elliptic ends endowed with the lift metric.

If $\kappa\colon M\to\mathbb{R}$ is the curvature of $M$, there is a function $k:[0,\infty)\longrightarrow[0,\infty)$, at least $C^{2}$ such that,

1. (i)

$0\geq\kappa(p)\geq-k(r(p)),\quad\forall\ p\in M,\ r(p)=\text{dist}^{M}(p,p_{0}),\ p_{0}\ \text{fixed},$** 2. (ii)

$\int_{0}^{\infty}sk(s)ds<\infty.$**

Proof.

The Bryant surface with finite total curvature and regular ends is isometric to its cousin, that is, to one complete minimal surface into $\mathbb{R}^{3}$ with finite total curvature. The CMC-1 Face dual surface of finite type and elliptic ends is isometric to the dual of one algebraic Bryant surface (with finite total dual curvature) and both are isometric to one complete minimal surface into $\mathbb{R}^{3}$ with finite total curvature.

We know that there is a compact Riemann surface $\overline{M}$ and a finite set $E=\{w_{1},\ ...\ ,w_{m}\}$ such that $M$ is conformal to $\overline{M}\smallsetminus E$, where $w_{i}$ are the ends of the surface. The Gauss map, for a minimal surface, and the hyperbolic Gauss map, for the Bryant surface, $G$ of $M$ does extend meromorphically to the ends, and $G:\overline{M}\longrightarrow\mathbb{S}_{\infty}^{2}$ is a branched covering. Take $R_{0}>0$ such that $\overline{M}\setminus B_{R_{0}}^{M}(o)$ is the finite union of the subends $E_{1},...,E_{m}$ . Bryant Proposition 4 [3] showed that is possible parametrize each subend $E_{i}$ using the Bryant data $(g,f)$, in$\{z\in\mathbb{C}/0<|z|<r_{j}\ ,\ \ r_{j}>0\}$ by

[TABLE]

Where $\hat{g}$ and $\hat{f}$ are analytic, never vanish and extend meromorphically to $z=0$. Is well known that is possible to get a similar parametrization for minimal surfaces with finite total curvature.

Let $\gamma_{j}=\partial E_{j}.$ For each $|z|<r_{j}$ let $\alpha_{j}$ be the path $re^{i\theta}$, $|z|\leqslant r\leqslant r_{j}$. We have

[TABLE]

Observe that $\hat{f}$ e $\hat{g}$ are bounded in $0<|z|<r_{j}$. So, in this set, we have

[TABLE]

Therefore

[TABLE]

Beside that, the Gaussian curvature $\kappa$ satisfies

[TABLE]

Observe that $1\leq|1+|g(z)|^{2}|$, therefore $\frac{1}{|1+|g(z)|^{2}|}\leq 1$. Besides, $\hat{f}$, $\hat{g}$ and $\hat{g}^{\prime}$ are bounded in $0<|z|<r_{j}$. So,

[TABLE]

Where $|\hat{g}(z)|\leq\hat{c}_{1}$, $|\hat{g}^{\prime}(z)|\leq\hat{c}_{2}$ and $|\hat{3}(z)|\geq\hat{c}_{3}$ for $|z|\leq r_{j}$, and $\hat{c}_{4}=4.max\{\hat{c}_{1},\hat{c}_{2}\}$.

Now, let $\rho(z)$ be the geodesic distance from $o\in M$ to $z$. Note that $\rho(z)\leq\frac{c_{3}}{|z|^{I(w_{j}}}$, so

[TABLE]

therefore,

[TABLE]

Replacing this in the Gaussian curvature inequality above

[TABLE]

Thus, we have

[TABLE]

Take $c_{0}\geq max_{j}\{c_{6}\}$, and define

[TABLE]

and

[TABLE]

Take a cut off function $k:[0,\infty)\longrightarrow[0,\infty)$ of class $C^{\infty}$ such that $k(s)\geq|\kappa(z)|$ for all $\rho(p)\leq R_{o}$ and $k(s)=\frac{c_{o}}{s^{2+\epsilon}}$ , for all $s\geq R_{o}$.

So we have

[TABLE]

∎

Lemma 5.2**.**

Let $M$ be a complete surface with curvature $\kappa,\ 0\geq\kappa\geq k,$ where $k(r)$ is the function of last theorem. If $B_{r}$ is the geodesic ball of $M$ with fixed center then there are constant $c$ and $r_{0}>0$ such that $\text{vol}(B_{r})\leq cr^{2}$ for all $r\geq r_{0}.$

Proof.

Let $M_{o}$ be a model with metric $dr^{2}+h^{2}dt^{2}$ where $h^{\prime\prime}/h=k$ the function $k(s)$ defined in theorem 5.1 item (ii). By lemma 4.5 of [8] there is a constant $c_{0}>1$ such that $r\leq h(r)\leq c_{0}r$.

Set $\exp=\exp_{p_{0}}\colon\mathbb{R}^{2}\to M,\ \mathbb{R}^{2}=T_{p_{0}}M.$ Let $F$ be the set $M\setminus\mathcal{C}(p_{0})$ where $\mathcal{C}(p_{0})$ is the cut locus of $p_{0}.$ Let $U$ be the closure of the connected component of $\exp^{-1}(F)$ passing at the origin. The metric into $U$ endowed by $\exp$ has the expression $dr^{2}+g^{2}dt^{2}.$ If $B_{r}$ is the geodesic ball with center $p_{0}$ then

[TABLE]

By the Rauch comparison theorem we have $g(r)\leq h(r)\leq c_{0}r$ for all $r_{0}\leq r\leq r^{\star},$ where $[r_{0},\ r^{\star}]$ is the interval where the radial geodesic $\gamma(r)$ is into $U$. Hence

[TABLE]

∎

6 Proof of theorem 2

It is well know that $\text{vol}(B_{r})\leq c_{0}r^{2}$ of geodesic balls imply those surfaces are parabolic. For example

[TABLE]

and by [9] theorem 11.14 $M$ is parabolic. Then all surfaces of theorem 2 are parabolic.