INTRODUCTION

Complete minimal surfaces immersed or embedded in ℝ^{3}constitute a honorable field of research. In particular, many geometers studied with special emphasis the Gauss map properties of such surfaces. Several important results have been shown and many other problems have emerged, among them the following one: let * X : M *→ ℝ^{3} be a complete non-planar minimal immersion and let *N : M* → 𝕊^{2} ≅ ℂ ∪ {∞} be its Gauss map. How many points can be omitted by *N*?

In 1961 R. Osserman (^{Osserman 1961}) proved that *N* does not omit more than one set of logarithmic capacity zero. In 1981 F. Xavier (^{Xavier 1981}) proved that *N* can omit at most six points. In 1988, H. Fujimoto (^{Fujimoto 1988}) refined Xavier's result showing that *N*omits at most four points. This is the best possible result since there are complete immersed minimal surfaces in ℝ^{3} whose Gauss map omits exactly four values. An example of such a surface is the Scherk's surface and other examples can be found in the book of R. Osserman (^{Osserman 1986}). However, these examples have infinite total curvature. This is an expected fact since R. Osserman (^{Osserman 1964}) proved that if*M* has finite total curvature, then *N* omits at most three points. Until now examples of complete minimal surfaces with finite total curvature whose Gauss map omits three points of 𝕊^{2} are not known. Hence, the maximum number of points that can be omitted by *N* is either two or three. By trying to answer this question we naturally ask what are the conditions for the existence of such a surface. However little progress has been made in this direction since the question was posed by R. Osserman. We mention that a result of R. Osserman (^{Osserman 1961}) states that if *M* is a complete, orientable minimal surface in ℝ^{3}; then its total curvature is a multiple of −4*π*: We recall that even if a minimal surface*M* is not orientable, its oriented double covering
is an orientable minimal surface. In that way we can study the Gauss map of
. Here we consider only the orientable case. In 1987, Weitsman and Xavier (^{Weitsman and Xavier 1987}) proved that if *N* omits three values, the total curvature of*M* is less than or equal to −16*π*:

In this paper we give a new proof of the Osserman's theorem in the case of finite total curvature (^{Osserman 1964}). We believe that a simpler proof, using fewer arguments of complex variables, could lead to a better understanding and even to the possibility of improving the result. Our proof uses the Weierstrass representation, the fact that *M* is a compact surface minus a finite number of points and the Poincaré's theorem about the sum of indices of a differentiable vector field with isolated singularities on a compact surface.

We consider certain particular functions on the surface
and we apply the Poincaré's theorem to the gradient vector field of these functions. This permits us to obtain relations between the Euler characteristic of
; the number of ends, the index of each end, the total curvature of*M* and the number of points omitted by the spherical image of*M*: Finally, under additional hypothesis on the type of ends of the surface, we obtain results that improve the Osserman's theorem.

Let *X : M*→ ℝ^{3} be an isometric minimal immersion of a Riemann surface *M* into ℝ^{3}: Let *N* be Gauss map and C (M) := ∫_{M}
*KdM* the total curvature of *M*. If*M* has finite total curvature (ie |*C*(*M*)| < ∞), then we know the following results (see ^{Osserman 1986}).

**1.1.**
*The Riemann surface M is diffeomorphic to a compact surface
of genus γ minus a finite set points, that is,*

**1.2.**
*The Gauss mapping N extends continuously to a function
: So,
is a branched covering of 𝕊 ^{2}(1);*

**1.3.**
*If m represents the number of times that
covers each point of 𝕊 ^{2} then the total curvature of M is given by −4πm:*

Let π: 𝕊^{2}→ ℂ ∪ {∞} be any identication of 𝕊^{2} withℂ ∪ {∞}. Represent by *g* the composition*g*:=π ο *N* and set
.

For each choice of *π* there is a meromorphic one-form*ω* on
satisfying:

**1.4.**
*at each point of M where g has a pole of order v the form ω has a zero of order* 2*v*;

**1.5.**
*the form
has no real periods on M*;

**1.6.**
*the Weierstrass representation gives the following integral expression* X = 2ℜ (∫^{z}α).Since*M* is complete with the induced metric, then*α* must have poles at each point of
. We know that (see, for example, ^{Osserman 1986}):

**1.7.**
*the poles of α have order greater than or equal to two.*

The points of
, or punctured neighborhoods of such points, are called*ends of M*. L. Jorge and W. Meeks (^{Jorge and Meeks 1983}) have shown that:

**1.8.** for *ρ* > 0 *sufficiently large, the set M _{ρ} :*= {

*p*∈

*M*, |

*X*(

*p*)| ≥

*ρ*}

*is a disjoint union of closed sets E*

_{1},

*E*

_{2}, ...,

*E*,

_{n}*where each E*:=

_{i}is an end of M and V_{i}*E*?{

_{i}*p*}

_{i}*is a disk neighborhood of p*

_{i}in*for i*= 1, ...,

*n*;

**1.9.**
*if γ ^{i}ρ: 𝕊^{1} → M parametrizes ∂V_{i} and
, then C^{i}ρ converges, in the C^{∞} topology, to the great circle of 𝕊^{2}(1) perpendicular to
covered I(p_{i}) times;*

**1.10.**
*the number I(p _{i}) is an integer greater than or equal to one.*

*If I(p _{i}) = 1 then, for ρ sufficiently large, X (E_{i}) is graph of a function f : 𝒰_{i}
→ ℝ, where 𝒰_{i} is some neighborhood of infinite in the subspace perpendicular to
:*

HEIGHT FUNCTIONS

For each ξ ∈ 𝕊^{2}(1)we define:

*ξ*(

^{T}*p*) means the orthogonal projection of

*ξ*into

*T*. The singularities of this vector fields are the points in

_{p}M*N*

^{–1}{

*ξ*, −

*ξ*}.

**Proposition 2.1.**
*If p ∈ N ^{–1} {ξ, − ξ} then grad(f_{ξ})(p) = 0 and the index of this singularity is −v(p), where v(p) is the order of p as a zero of g − g(p).*

**Proof.** By a change of coordinates in ℝ^{3}; if necessary, we may suppose that *ξ* = ±(0, 0, 1). Consider the identication AQUI π : 𝕊^{2}(1) → ℂ ∪ {0}, for which it holds that *π*(0, 0, 1) = ∞ and *π* (0, 0, −1)=0. As *N*(*p*) = ±*ξ* we have*g*(*p*) = 0 or*g*(*p*) = ∞. We may suppose that *g*(*p*) = 0. Let*z* = *u* + *iv* be a local complex parameter of *M* such that *p*corresponds to *z* = 0. Then, in a neighborhood of*p*, we have:

*h*is a holomorphic function such that

*h*(0) = 0 and

*h*′(0) ≠ 0: Hence, we get:andwhat implies that the index of

*grad*(

*f*) at

_{ξ}*p*is −

*v*(

*p*).

In order to determine the behavior of the vector field*grad* (*f _{ξ}
*) at the points
we consider its restriction to the curves γ

^{i}ρ defined in (1.9).

**Proposition 2.2.**
*For ρ sufficiently large, the index of grad*(*f _{ξ}
*) along γ

^{i}ρ is

**Proof.** Choose coordinates on ℝ^{3} so that*ξ* = ± (0,0,1) and ĝ (Pi) ≠ ∞. With respect to the local parameter *z* =*u* + *iv* around *p _{i}
* we have:

*µ*(

*p*) ≥ 2. As in the proof of the Proposition 2.1 we have:It follows that one of the following cases can occur:

_{i}These cases correspond respectively to:

As before, consider the gradient vector field

Then, take large enough so that γ^{i}ρ(𝕊^{1)} is contained in the neighborhood of *p _{i}
* we are considering. It follows that the index of

*grad*(

*f*) along γ

_{ξ}^{i}ρ is

*µ*in the case (A) and

*v*−

*µ*in the case (B).

The following lemma completes the proof of Proposition 2.2.

**Lemma 2.3.**
*If*
, ĝ(p) ≠ ∞ *and ω has a pole of order µ at p, then I*(*p*) =*µ* − 1.

**Proof.** We may choose coordinates in 𝕊^{2}(1) in such a way that ĝ(p) = 0. By using (1.6) we have:

*f*

_{1}and

*f*

_{2}are power series with at most terms in .

Setting *z* = *re ^{iθ}
* and
we obtain

Therefore, for *ρ* sufficiently large,*X*
_{1} + *iX*
_{2} takes the circle of radius *r* centered at*p* into a closed curve that rotates (*µ* − 1)-times around the origin, that is*I*(*p*) =*µ* − 1.

**Corollary 2.4.**
*Let
be a complete minimal isometric immersion with finite total curvature. If m is the number of times that
covers 𝕊 ^{2}(1) then:*

*ξ*such that {−

*ξ*,

*ξ*} does not intersect . We have:If the genus of is

*γ*, then . Hence, we obtain . Since

*I*(

*p*) ≥ 1 for all

_{i}*i*∈ {1,2, ...,

*n*} it follows that:Note that the equality holds if and only if

*I*(

*p*) = 1 for all

_{i}*i*∈ {1,2, ...,

*n*}. In this case every end of

*M*are embedded. Moreover, (5) is equivalent to:

γ ≤ *m - n*+1.

SUPPORT TYPE FUNCTIONS

Now we consider, for each ξ ∈ 𝕊^{2}, the support type function

A simple computation shows that the singular points of grad(𝔰_{ξ}) are the points where*ξ ^{T}
*= 0 or

*K*= 0 (branch points of

*g*). By a change of coordinates in ℝ

^{3}we may always assume that

*ξ*= (0, 0, 1).

Since:

(see, for example,^{Barbosa and Colares 1986}), we have:

Consider a point *p* such that
. In terms of local coordinates *z* =*u*+*iv*; with *p*corresponding to *z* = 0; we have
;
; when *v* is a nonzero integer.

If *v* > 0 then 𝔰_{ξ}(0) = -1 and

*p*is then a local minimum for 𝔰

_{ξ}with index + 1.

If *v* < 0 then 𝔰_{ξ}(0) = 1 and

*p*is then a local maximum and so its index is also +1.

When
, then
,
, where *v* ≥ 1. It follows that
and

Considering *z* = *re ^{iθ}
* one obtains

_{0}b

_{0}= Re

^{iα}, we have:

From this expression it follows that the difference 𝔰_{ξ} (Z) ― 𝔰_{ξ} (0) changes sign in any neighborhood of *z* = 0. It follows easily that the index of grad(𝔰_{ξ}) at *p* is 1 − *v*(*p*).

Therefore we have proved:

**Proposition 3.1.**
*If p is a critical point of 𝔰 _{ξ} bof index 1. If
and p is a critical point of 𝔰_{ξ}then g has a branch point of order v at p and the index of grad(𝔰_{ξ}) at p is 1 − v.*

THE OSSERMAN'S THEOREM

The following theorem was proved by Osserman (^{Osserman 1964}) using a different method.

**Theorem 4.1.**
*Let
be a complete minimal isometric immersion with finite total curvature, − 4πm. Suppose that { ξ _{1}, ξ_{2, ..., ξk
}} = 𝕊^{2} (1) \ g(M). If k ≥ 4, then M is flat.*

**Proof.** Set:

*g*. Counting indices of the singularities of grad(𝔰

_{ξ}) we obtain:Observe that and . Hence,Now using 2.4, we have . Using (9) it follows that:Thus (4 −

*k)m*> 0. Therefore

*k*< 4.

**Corollary 4.2.**
*Let*
*be a complete minimal isometric immersion with finite total curvature* −4*πm. If* N : M → 𝕊^{2} (1) *omits 3 points then*
*. Moreover, if*
*we have:*

**Proof.** It follows from equation (9) that if*k* = 3 then:

*n*−

*m*≤ 0, where . Moreover, if then

*m*=

*n*and

*B*=

*C*=

*ϕ*. It is consequence of (5) that the ends

*M*are embedded.

**Corollary 4.3.**
*Under the same hypothesis of the previous corollary it holds that if N : M → 𝕊^{2}omits 3 points then m ≥ 3:*

**Proof.** From the proof of the previous corollary we have
. Hence *m* ≥ *n*: However, for each point q ∈ 𝕊^{2} \ M (M)there is at least one point
such that
. Hence, we have *n* ≥ 3.

RESULTS

Now we study the behavior of *M* at infinity. For this purpose we consider
as before and let (*g*,*ω*) be the the corresponding Weierstrass data. After a rotation we may assume, for *j* fixed, that ĝp(_{j}) = 0 . Let *z* be a local parameter in an open subset
with p_{j}∈ D such that (*p _{j}
*) = 0. Hence, locally we have:

Moreover, as the immersion is complete it follows that *f*has pole at *p _{j}
* whose order is greater than or equal to 2. Hence we have near

*p*that:

_{j}By Lemma (2.3) we have*I*(*p _{j}
*) =

*µ*− 1: Thus,

*p*is an embedded end if and only if

_{j}*µ*(

*p*) = 2. Suppose that

_{j}*p*is an embedded end. Then

_{j}*α*

_{3}:=

*gω*=

*gfdz*and

Hence, if *v* > 1 then *gf* is holomorphic and X_{3}(p) = ℜ(∫^{P}⨍ω) satisfies

*F*is asymptotic to the plane {

_{j}*X*

_{3}=

*L*}. In this case, we say that

*F*is a planar end of order

_{j}*v*− 1:

If *v* = 1, then*gf*(*z*) = *a*
_{0}
*b*
_{0}
*z*
^{–1}+ (holomorphic function). Since: ℜ(∮_{c}gfdz) = 0 for any closed curve C ⊂ M, we have ℜ(2πia_{0}b_{0}) = 0 and then a_{0}b_{0} ∈ ℝ. Therefore: X_{3} = a_{0}b_{0} |z| +R(z), where*R* is an harmonic function in a neighborhood of 0 ∈ ℂ and *R*(0) = 0. The number*a*
_{0}
*b*
_{0} is the logarithmic growth of the end *F _{j}
*. In this case the end

*F*is of catenoidal type. This means geometrically that

_{j}*F*is asymptotic to a catenoid in ℝ

_{j}^{3}.

In the general case we have
. Now consider *N*
_{0} := *v* − *µ*. If*N*
_{0} ≥ 0 then *fg* is holomorphic. Otherwise, if*N*
_{0} < 0 then
holomorphic function.

**Denition 5.1.**
*Let*
*be a end of M. We say that*
*is a non-degenerate end if, and only if, N*
_{0}(*p*) :=*v*(*p*) −*µ*(*p*) ≤ 0.

For non-degenerate ends we have:

**Theorem 5.2.**
*Let
be a complete non-flat minimal immersion with finite total curvature and non-degenerate ends. If the Gauss map omit k points, then k ≤ 2:*

**Proof.** By the Osserman's theorem (4.1) we know that*k* ≤ 3: Suppose that *k* = 3 and let*ξ*
_{1}, *ξ*
_{2} and *ξ*
_{3} be the omitted points. As in the proof of the Theorem 4.1 consider the sets:

Counting the indices of the singularities of*grad*(*f _{ξ}
*) we obtain:

Moreover, since *N*
_{0} ≤ 0 we have:

*k*= 2.

In the case *k* = 2 with non-degenerate ends we have:

**Theorem 5.3.**
*Let
be a complete non-at minimal immersion with finite total curvature and non-degenerate ends. Suppose that the Gauss map omits two points. Then M is topologically a sphere minus two points.*

**Proof.** Let ξ1, ξ2 ∈ 𝕊^{2} be the two points omitted. Similarly to the proof of Theorem 5.2 we have:

In both cases,
what implies that
is homeomorphic to 𝕊^{2} and therefore: M = 𝕊^{2} \ {p_{1}; p^{2}, ...., p_{n}}

We claim that *n* ≤ 3.

By contradiction, suppose that *n* > 3 and let*p*
_{1}, *p*
_{2}, *p*
_{3}, *p*
_{4} four distinct points in
. Consider two simple and differentiable curves*α*
_{1} and *α*
_{2} joining *p*
_{1} to *p*
_{2} and *p*
_{3} to *p*
_{4} respectively. Assume also that the curves *α*
_{1} and *α*
_{2} do not intersect each other and are contained in *M*except for its endpoints.

Cutting *M* along the curves *α*
_{1} and *α*
_{2} we obtain another surface topologically equivalent to a cylinder. Joining two copies of the later surface along the curves *α*
_{1} and *α*
_{2} we obtain another compact surface of genus 1 and 2(*n* − 4) + 4 points removed.

Denote by *M*′ this new surface and let*R* : *M*′ → *M* the canonical covering map. Now *X ο R : M'*→ ℝ^{3} is a minimal immersion with the same image of*X*. The Gauss map of X ○ R is given by N' = N ○ R and has the same image as the Gauss map of *X*. For the ends of *M*′ we still have *N*
_{0} < 0. In fact, the ends that are not endpoints of the curves*α _{i}
*,

*i*= 1, 2, remain with the same local behavior and hence the same

*I*(

*p*) and

*v*(

*p*). Thus the value

*N*

_{0}=

*v*−

*µ*=

*v*−

*I*−1 remains unaltereted. However when the ends are endpoints of the curves

*α*,

_{i}*i*= 1, 2, some changes occur. For instance, a neighborhood of

*p*

_{1}now covers twice a old neighborhood. It follows that

*I*(

*p*

_{1}) is changed for 2

*I*(

*p*

_{1}) and

*g*(

*z*) is changed for . Therefore,

*v*(

*p*

_{1}) becomes 2

*v*(

*p*

_{1}). Since

*I*(

*p*

_{1}) −

*v*(

*p*

_{1}) is changed for 2

*I*(

*p*

_{1}) − 2

*v*(

*p*

_{1}) the inequality

*I*(

*p*

_{1}) −

*µ*(

*p*

_{1}) + 1 > 0. is preserved. Thus X ○ R : M' → ℝ

^{3}is an immersion whose ends are not degenerate. Moreover, this immersion satisfies the hypothesis of the theorem. From the first part of the proof we have . This is a contradiction because . Hence

*n*≤ 3 as we claimed.

Now we claim that *n* = 2.

Clearly *n* ≥ 2 (*n* ≥*k* = 2). Assume *n* = 2 and let *p*
_{1}, *p*
_{2} and *p*
_{3} be the ends of *M*. Consider a curve*α* joining *p*
_{1} to *p*
_{2} as before.

The previous construction of *M*′ now leads to a surface with *n* = 4 which contradicts *n* ≤ 3. Therefore, *n* = 2.