Josai Mathematical Monographs vol 3 (2001), pp. 145-150

EXISTENCE OF A FAMILY OF COMPLETE MINIMAL SURFACES OF GENUS ONE WITH ONE END AND

FINITE TOTAL CURVATURE

KATSUHIRO MORIYA

Abstract. We will report our recent result on existence of a complex one-parameter family of complete minimal surfaces of genus one with one end and finite total curvature. The family connects a minimal surface with total curvature −12π and that with total curvature less than −12π.

1. Introduction

The purpose of this paper is to report our recent result [8] on existence of a complete minimal surface with finite total curvature in Euclidean space in terms of Weierstrass representation.

Let M be a two-dimensional oriented manifold. If X: M → ℝ^{3} is a complete minimal surface with finite total curvature, then we may assume M an open Riemann surface biholomorphic to a compact Riemann surface M̄ with finitely many points removed. The genus of M̄ is called the genus of X and each removed point a puncture point. The image of an neighborhood of each puncture point by X is called an end of X. The integral ∫_{M} K dA for the Gaussian curvature K and the area element dA is called the total curvature of X. The total curvature of an oriented complete minimal surface is either −4πm for some nonnegative integer m or infinite.

We will focus the case where a Riemann surface M is a square torus M̄ with one puncture point. By applying Jorge-Meeks’s formula ([7, Theorem 4]), the total curvature is less than −4π in this case. We can see existence of the minimal surfaces from examples constructed by applying Weierstrass representation and theory of elliptic functions, for examples, Chen and Gakstatter [3], Bloß [2], Fang [4], and Abi-Khuzam [1].

In [8], we discussed existence of the minimal surfaces in terms of Weierstrass representation without constructing examples by elliptic functions. We will describe the outline of the proof of the following theorem:

Figure 1. A canonical basis {a, b}.

Theorem 1.1 ([8]). There exists a complex one-parameter family of complete minimal surfaces of genus one with one end and total curvature less than −12π and greater than −36π.

2. Elliptic curves

We will start with preliminaries on the theory of elliptic curves. For more details, see [5] or [6].

Let us denote by M̄ the elliptic curve which is the zero locus of a cubic polynomial

w^{2} = z(z + 1)(z − 1).

The Riemann surface M̄ is conformally equivalent to a square torus. We will introduce a representation of the elliptic curve M̄ as a two sheeted covering of the sphere ℂ ∪ {∞} branched at −1, 0, 1, and ∞. The covering is obtained by cutting the sphere ℂ ∪ {∞} along two intervals [−1, 0] and [1, ∞] and by pasting two copies of this slit sphere along the slits. We will denote by S_{I} and S_{II} these two sheets. We will take the branches of w as follows:

w = \begin{cases}\sqrt{x(x^{2} − 1)} & x ∈ [−1, 0] ⊂ S_{I}, \\ −\sqrt{x(x^{2} − 1)} & x ∈ [−1, 0] ⊂ S_{II}. \end{cases}

Let a and b be two oriented one-cycles on M̄ indicated in Figure 1, where the parts of the cycles that lie on the sheet S_{I} are indicated by solid lines and those on the sheet S_{II} by broken lines. The set {a, b} of cycles forms a canonical basis of the first homology group of M̄, that is, the cycle a intersects the cycle b once positively.

We will recall that all holomorphic differentials on an elliptic curve form a one-dimensional complex vector space. Since dz/w is a holomorphic one-form on M̄, any holomorphic one-form on M̄ is a constant multiple of dz/w. We can see that the integral ∫_{a} dz/w is not equal to 0. Hence, the holomorphic differential

ω: = (1 /∫_{a} \frac{dz}{w} ) \frac{dz}{w}

on M̄ becomes the dual of a.

For a divisor D on M̄, let D_{+} and D_{−} be two nonnegative divisors satisfying D = D_{+} − D_{−}. For a meromorphic function f on M̄ and a meromorphic one-forms η on M̄, we will denote by (f) and (η) the divisors of f and η respectively. we will denote by P the point (w, z) = (0, 0) on M̄. Let L be the complex vector space of meromorphic functions on M̄ whose divisors are equal to or greater than −8P:

L = {f | (f) ≥ −8P }.

We can see dim L = 8 from the Riemann-Roch theorem. Since

(z)_{−} = 2P, (ω)_{−} = 3P, (2.1)

the vector space L is spanned by the following basis:

1, \frac{1}{z}, w, \frac{1}{z^{2}} , \frac{w}{z}, \frac{1}{z^{3}} , \frac{w}{z^{2}} , \frac{1}{z^{4}} .

3. An example of a minimal surface with total curvature −12π

In this section, we will recall the theory of complete minimal surfaces of genus one with one end and finite total curvature. For more details, see [9] or [10].

We will denote by M̄, a, b, P, and ω as in Section 2 and M the open Riemann surface M̄ \ {P }. We can see that {a, b} is a basis for the first homology group of M, too.

We can see the following holds:

Lemma 3.1. A pair (g, f) of meromorphic functions on M̄ satisfying

Re ∫_{γ} (\frac{1}{2}(1 − g^{2})fω, \frac{\sqrt{−1}}{2}(1 + g^{2})fω, gfω) = (0, 0, 0) (γ = a, b), (3.1)

− 2(g)_{−} + (f) = −nP, n ∈ ℤ, n ≥ 2 (3.2)

produces a complete minimal surface with finite total curvature X: M → ℝ^{3} by

X(p) = Re ∫ ^{p} (\frac{1}{2}(1 − g^{2})fω, \frac{\sqrt{−1}}{2}(1 + g^{2})fω, gfω) .

The meromorphic function g of a pair (g, f) in Lemma 3.1 is the stereo-graphic projection of the normal Gauss map of the corresponding minimal surface X. Hence, the total curvature of the minimal surface corresponding to (g, f) is equal to −4π deg g.

For the convenience, we will denote by Φ the the triplet of integrands in (3.1) corresponding to a pair (g, f):

Φ = (\frac{1}{2}(1 − g^{2})fω, \frac{\sqrt{−1}}{2}(1 + g^{2})fω, gfω) . (3.3)

Let us denote by F, N, and r three positive real numbers such that

F = ∫_{−1}^{0} \sqrt{x(x^{2} − 1)} dx, N = ∫_{−1}^{0} \frac{dx}{\sqrt{x(x^{2} − 1)}}, r = \sqrt{\frac{2F}{N}} .

We will denote by G the meromorphic function r/w on M̄. Then, we can show the following:

Lemma 3.2. The pair (G, 1/G^{2}) produces a complete minimal surfaces of genus one with one end and total curvature −12π by applying Lemma 3.1.

Proof. It is easy to see that the pair (G, 1/G^{2}) satisfies the condition (3.2) by (2.1).

The triplet

(Φ_{1}, Φ_{2}, Φ_{3}) = (\frac{1}{2} (\frac{w dz}{Nr^{2}} − ω) , \frac{\sqrt{−1}}{2} (\frac{w dz}{Nr^{2}} + ω) , \frac{dz}{Nr})

of meromorphic differentials on M̄ is corresponding to (G, 1/G^{2}) by (3.3). Since Φ_{3} is exact, we see

∫_{a} Φ_{3} = ∫_{b} Φ_{3} = 0.

We can see the real periods of Φ_{1} and Φ_{2} along a and b vanish. For example, the period of Φ_{1} along a is calculated as follows:

∫_{a} Φ_{1} =\frac{1}{2Nr^{2}} ∫_{−1}^{0} \sqrt{x(x^{2} − 1)} dx

+ \frac{1}{2Nr^{2}} ∫_{0}^{−1} −\sqrt{x(x^{2} − 1)} dx − \frac{1}{2} ∫_{a} ω

=\frac{F}{Nr^{2}} − \frac{1}{2} = 0.

Hence, the pair (G, 1/G^{2}) satisfies (3.1).

Thus the pair (G, 1/G^{2}) produces a complete minimal surface of genus one with one end by applying Lemma 3.1. Since deg g is equal to 3, the total curvature of the corresponding minimal surface is equal to −12π.

4. Existence of a family of minimal surfaces

In this section, we will show Theorem 1.1.

Proof of Theorem 1.1. We will denote by L the complex vector subspace of L spanned by the following basis:

\frac{1}{z^{2}} , \frac{w}{z}, \frac{1}{z^{3}} , \frac{w}{z^{2}} , \frac{1}{z^{4}} .

Let us define a set M of meromorphic functions on M̄ by

M := \{g | \frac{1}{g} − \frac{w}{r} ∈ L\} .

We will consider M as ℂ^{5} by identifying an element g of M such that

\frac{1}{g} − \frac{w}{r} = c_{1} \frac{1}{z^{2}} + c_{2} \frac{w}{z} + c_{3} \frac{1}{z^{3}} + c_{4} \frac{w}{z^{2}} + c_{5} \frac{1}{z^{4}}

with an element (c_{1}, c_{2}, c_{3}, c_{4}, c_{5}) ∈ ℂ^{5}. It is easy to see that each pair (g, 1/g^{2}) (g ∈ M) satisfies the condition (3.2).

We can see that the tuple F = (F_{1}, …, F_{6}) such that

F_{i}(g) = ∫_{a} Φ_{i}, F_{i+3}(g) = ∫_{b} Φ_{i} (i = 1, 2, 3)

defines a holomorphic map from M to ℂ^{6} where (Φ_{1}, Φ_{2}, Φ_{3}) is the triplet of meromorphic one-forms corresponding to (g, 1/g^{2}) by (3.3). Then, each element (g, 1/g^{2}) (g ∈ (Re F)^{−1}(0, …, 0)) produces a complete minimal surface with finite total curvature by applying Lemma 3.1. In Section 3, we have already seen that G which corresponds to (c_{1}, …, c_{5}) is an element of (Re F)^{−1}(0, …, 0).

We can see that the following holds:

\frac{∂F_{i}}{∂c_{j}} (G) = ∫_{a} \frac{∂ϕ_{i}}{∂c_{j}} (G) dz, \frac{∂F_{i+3}}{∂c_{j}} (G) = ∫_{b} \frac{∂ϕ_{i}}{∂c_{j}} (G) dz,

where ϕ_{i} = Φ_{i}/dz (i = 1, 2, 3, j = 1, …, 5). Since we obtain

(\frac{∂ϕ_{1}}{∂c_{1}} (G), …, \frac{∂ϕ_{1}}{∂c_{5}} (G)) = (\frac{1}{rz^{2}} , \frac{(z^{2} − 1)}{rw}, \frac{1}{rz^{3}} , \frac{a(z^{2} − 1)}{rzw}, \frac{1}{rz^{4}} ) ,

(\frac{∂ϕ_{2}}{∂c_{1}} (G), …, \frac{∂ϕ_{2}}{∂c_{5}} (G)) = \sqrt{−1} (\frac{∂ϕ_{1}}{∂c_{1}} (G), …, \frac{∂ϕ_{1}}{∂c_{5}} (G)) ,

(\frac{∂ϕ_{3}}{∂c_{1}} (G), …, \frac{∂ϕ_{3}}{∂c_{5}} (G)) = (\frac{1}{z^{2}w}, \frac{1}{z}, \frac{1}{z^{3}w}, \frac{1}{z^{2}} , \frac{1}{z^{4}w}) ,

the Jacobian matrix of F at G becomes as follows:

(\begin{matrix}0 & C_{1} & 0 & C_{2} & 0 \\ 0 & \sqrt{−1}C_{1} & 0 & \sqrt{−1}C_{2} & 0 \\ C_{3} & 0 & C_{4} & 0 & C_{5} \\ 0 & −\sqrt{−1}C_{1} & 0 & \sqrt{−1}C_{2} & 0 \\ 0 & C_{1} & 0 & −C_{2} & 0 \\ −\sqrt{−1}C_{3} & 0 & \sqrt{−1}C_{4} & 0 & −\sqrt{−1}C_{5} \end{matrix}) ,

where

C_{1} = 2 ∫_{−1}^{0} \frac{(x^{2} − 1)}{r\sqrt{x(x^{2} − 1)}} dx, C_{2} = 2 ∫_{−1}^{0} \frac{(x^{2} − 1)}{rx\sqrt{x(x^{2} − 1)}} dx,

C_{3} = 2 ∫_{−1}^{0} \frac{dx}{x^{2}\sqrt{x(x^{2} − 1)}}, C_{4} = 2 ∫_{−1}^{0} \frac{dx}{x^{3}\sqrt{x(x^{2} − 1)}},

C_{5} = 2 ∫_{−1}^{0} \frac{dx}{x^{4}\sqrt{x(x^{2} − 1)}}.

The rank of the Jacobian matrix of F is equal to 4 since the number C_{i} does not vanish (i = 1, …, 5). Thus, F^{−1}(F(G)) is a complex submanifold of ℂ^{5} of dimension one. The degree of any element of M except G is greater than 3 and less than 9. Thus, each pair (g, 1/g^{2}) (g ∈ F^{−1}(F(G))) except (G, 1/G^{2}) produces a complete minimal surfaces of genus one with one end and total curvature less than −12π and greater than −36π

References

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Institue of Mathematics University of Tsukuba Ibaraki 305-8571 Japan moriya@math.tsukuba.ac.jp