Proceedings Ist International Meeting on Geometry and Topology Braga (Portugal)

Public. Centro de Matemática da Universidade do Minho p. 139–147, 1998

Huge Singly Periodic Minimal Surfaces in R^{3} and Symmetries

\underline{Francisco Martin} and Domingo Rodriguez

First of all, we are going to start by giving a brief outline of the development of SPMS ^{1}. During the last century or more the only known SPMS with infinite number of ends

were the Riemann examples.

At the begining of the 90’s, a large quantity of new examples appeared thanks to the work of Callahan, Hoffman, Meeks, Wolgemuth, among others. A problem of classification then arose due to the appearance of so many of these new examples. In this line of work we have obtained a uniqueness theorem for a discrete family of SPMS constructed by Callahan, Hoffman and Meeks in 1990.

But firstly we need to briefly summarize the principal results for complete embedded SPMS.

Let x̃ : M̃ ⟶ R^{3} be a proper minimal embedding of a surface M̃ in threedimensional Euclidean space, invariant under a cyclic group of translations T . Using isothermal parameters, M̃ has a natural complex structure in which the stereographic projection of the Gauss map of x̃, g̃, is a meromorphic function.

There is a holomorphic one form, ω̃, associated to this function, and both of them determine the minimal immersion as follows:

x̃ = Real (∫ (ϕ̃_{1}, ϕ̃_{2}, ϕ̃_{3}))

where

ϕ̃_{1} = \frac{1}{2}(1 − g̃^{2})ω̃, ϕ̃_{2} = \frac{i}{2}(1 + g̃^{2})ω̃, ϕ̃_{3} = g̃ω̃, (0.1)

are holomorphic one-forms on M̃ satisfying

∑_{i=1}^{3} |ϕ̃_{j}|^{2} ≠ 0.

1Singly Periodic Minimal Surfaces

140 Francisco Martin and Domingo Rodriguez

We label the pair (g̃, ω̃) as the Weierstrass representation of M̃.

We can work with the quotient M = M̃/T , and so we have a minimal immersion

x : M̃/T ⟶ R^{3}/T .

Since both the Gauss map and the holomorphic one form are translationinvariant, they can be induced to the quotient. If we label this new Weierstrass data as (g, ω), we can then reobtain the immersion x as follows:

x = Real (∫ (ϕ_{1}, ϕ_{2}, ϕ_{3}))

labeling

ϕ_{1} = \frac{1}{2}(1 − g^{2})ω, ϕ_{2} = \frac{i}{2}(1 + g^{2})ω, ϕ_{3} = gω. (0.2)

In particular, the period group

Λ = {Real (∫_{γ}(ϕ_{1}, ϕ_{2}, ϕ_{3})) : [γ] ∈ H_{1}(M, z)}

coincides with T . The total curvature of M is denoted by

C(M) = ∫_{M} K dA,

where K represents the Gauss curvature. Meeks and Rosenberg proved that a properly embedded minimal surface in a complete non-simply connected flat three-manifold has finite total curvature, if and only if it has finite topology. Then, M is conformally diffeomorphic to a compact Riemann surface punctured in a finite number of points,

M ≡ \overline{M} − \{P_{1}, …, P_{r}\}.

These points are called the ends of M.

Moreover, the Weierstrass data (g, ω) extends meromorphically to \overline{M}. Therefore, the degree of g is well defined and

C(M) = −4 π deg(g).

A surprising relationship exists between the topology of a properly embedded periodic minimal surface in R^{3} and the behaviour of its ends. This was studied by Callaham, Hoffman, and Meeks, who proved the following theorem:

Theorem 1 (Callahan-Hoffman-Meeks) Suppose M̃ is a properly embedded minimal surface in R^{3}, with more than one end, whose symmetry group is infinite. Then either M̃ is a catenoid, or:

Huge Singly Periodic Minimal Surfaces in R^{3} and Symmetries 141

(i) M̃ has an infinite number of ends;

(ii) M̃ is invariant under a screw motion S;

(iii) all annular ends of M̃ are flat ends;

(iv) the total curvature of M = M̃/S is finite if and only if M has finite topology,

in which case

C(M) = 2π (χ(M) − r(M))

where r(M) is the number of ends of M.

We write the isometry group of M̃ as Iso(M̃). The subgroup of Iso(M̃), which consists of isometries that are the rectriction of a rigid motion in R^{3}, leaving x̃(M̃) invariant, is denoted by Sym(M̃). Choi, Meeks, and White [4] proved the following:

Theorem 2 (Choi-Meeks-White) A properly embedded minimal surface M̃ in R^{3} which has more than one end is minimally rigid.

In particular Sym(M̃) = Iso(M̃).

The first examples of periodic minimal surfaces with more than one end were discovered by Riemann in 1867. He constructed a one-parametric family of properly embedded minimal surfaces {R_{λ} : λ > 0} which are invariant under a translation, T . Furthermore, the quotient R_{λ}/T has genus one, two ends, and total curvature −8π. Recently, Meeks, Pérez, and Ros have characterized Riemann’s examples as the only embedded minimal tori in R^{3}/T .

These surfaces were generalized by Callahan, Hoffman, and Meeks in 1989. The way in which they carried out this generalization was by increasing the genus and the symmetry of the surfaces at the same time. So they found examples of complete embedded minimal surfaces, with an infinite number of annular ends. This discrete family consists of a surface M_{k}, for each integer k > 0, which is invariant under a cyclic group of vertical translations, T . The quotient M_{k}/T has:

• genus 2k + 1;

• two flat ends;

• 8(k + 1) symmetries.

We have studied the surfaces M_{k} in depth, obtaining the following classification theorem:

Theorem 3 Let M̃ be a properly embedded minimal surface in R^{3} satisfying the following:

(i) M̃ has an infinite number of annular ends;

(ii) M̃ is invariant under a cyclic group of translations T ;

(iii) M̃/T has genus ϱ, ϱ ≥ 2, and 2 ends;

(iv) Iso(M̃/T ) has at least 4(ϱ + 1) elements.

Then ϱ is odd and M̃ is, up to homotheties and rigid motions, the Callahan-Hoffman-Meeks example of genus ϱ.

The main idea in the proof of this theorem is to use the great number of symmetries of the surface in order to determine the underlying complex structure for the quotient M̃/T . Furthermore, this abundance of symmetries permits the reduction of the number of possibilities for the Weierstrass representation for these kinds of surfaces.

We have included an outline of the proof of Theorem 3.

Let x̃ : M̃ ⟶ R^{3} be a properly embedded minimal surface in R^{3} satisfying:

(i) M̃ has an infinite number of annular ends;

(ii) M̃ is invariant under a cyclic group of translations, T ;

(iii) M̃/T has genus ϱ, ϱ ≥ 2, and two ends;

(iv) Iso (M̃/T ) has at least 4(ϱ + 1) elements.

From Theorem 1 we deduce that the ends of M̃ are flats and the total curvature of M = M̃/T is −4π(ϱ + 1).

As we have mentioned above, M has the conformal type of a compact Riemann surface, \overline{M}, punctured in two points, P_{1} and P_{2}, which correspond to the ends. Meeks and Rosenberg [12] proved that the Weierstrass data (g, η) extends meromorphically to \overline{M}. Since the ends of M are parallel we can suppose that, up to a rigid motion of R^{3}, g(P_{i}) ∈ {0, ∞}, i = 1, 2.

In the following, the generator of T is denoted by τ.

If Ã ∈ Sym(M̃), then a simple argument of Linear Algebra gives Ã∘τ ∘Ã^{−1} = τ^{ε}, where ε ∈ {1, −1}. In particular T is a normal subgroup of Sym(M̃).

If we write Δ = Sym(M̃)/T , then as a consequence of the rigidity of M̃ we obtain Iso(M) = Δ. A symmetry of M induces, in a natural way, a conformal automorphism of M which extends to \overline{M}, leaving the set \{P_{1}, P_{2}\} invariant. Since the subgroup of holomorphic transformations has index one or two in Δ, then Hurwitz’s Theorem (see [5]) implies that Δ is finite. For any Ã ∈ Sym(M̃) its class in Δ is denoted by A. In the same way, we write the class of Q̃ ∈ M̃ in M as Q. If we label H as the subgroup of the holomorphic transformations of Δ and G = {A ∈ H / A(P_{1}) = P_{1}}, it is obvious that

[Δ : H] ≤ 2, [H : G] ≤ 2 (0.3)

If we take D, a conformal disk centered at P_{1}, invariant under G, then {A_{|D} / A ∈ G} is a finite group of biholomorphisms of the disk fixing the origin, and so G is cyclic. Let J be a generator of G. As the normal vectors at the ends are vertical, then any J̃, in class J, maps a vertical line into another vertical line.

Under these conditions, and without loss of generality, we can assume that there exists a J̃ on J which preserves the x_{3}-axis and fixes the origin. Taking (0.3) and hypothesis (iv) into account, we have ord(J̃) ≥ ϱ + 1, ϱ ≥ 2. Then we can assume that either J̃ is a counterclockwise rotation around the x_{3}-axis by angle 2π/ord(J̃) or ord(J̃) is even and J̃is a counterclockwise rotation around the x_{3}-axis by angle 2π/ord(J̃) followed by a symmetry with respect to the (x_{1}, x_{2})- plane. Since the translation vector of τ, b⃗, is a eigenvector of J̃, we deduce that b⃗ has the direction of the x_{3}-axis. Up to an homothety, we can assume that τ(x) = x + (0, 0, 2).

For a point Q ∈ \overline{M} we define:

• the isotropy group of Q in G as ℑ(Q) = {A ∈ G / A(Q) = Q},

• μ(Q) = |ℑ(Q)|,

• the orbit of Q associated to G as orb(Q) = {Q, J(Q), …, J^{|G|−1}(Q)}.

Note that ♯(orb(Q)) = \frac{|G|}{μ(Q)}. Using the Riemann-Hurwitz formula for the branched covering ζ : \overline{M} → \overline{M}/⟨J⟩, we obtain

χ(\overline{M}) = ord(J) · χ(\overline{M}/⟨J⟩) − (2(ord(J) − 1) + ∑_{Q∈M}(μ(Q) − 1)) (0.4)

Let Q_{1}, …, Q_{s} be the points of M whose orbits are non-trivial. If we write m_{i} = ♯(orb(Q_{i})), i = 1, …, s, then equality (0.4) becomes:

2(ord(J) − ϱ) + ∑_{i=1}^{s}(ord(J) − m_{i}) = ord(J) · χ(\overline{M}/⟨J⟩) (0.5)

Taking into account that ord(J) ≥ ϱ + 1, one obtains χ(\overline{M}/⟨J⟩) > 0, that is to say, \overline{M}/⟨J⟩ is the Riemann sphere. Thus, substituting χ(\overline{M}/⟨J⟩) for 2 in (0.5), we have

∑_{i=1}^{s}(ord(J) − m_{i}) = 2ϱ (0.6)

As x̃ is an embedding, one has m_{i} ≤ 2, ∀i ∈ {1, …, s}.

We now consider the behaviour of the Gauss map at an end and at a point on the x_{3}-axis in the presence of a rotational symmetry R̃ of M̃. A catenoid end still produces a simple zero or a simple pole in g. However, a flat end must correspond to a zero or a pole of order jm + 1, where j is a positive integer and m is the order of R̃. On the other hand, if Q̃ ∈ M̃ is a point of the x_{3}-axis, then the order of Q̃ as zero or pole of g is jm − 1, for a positive integer j (for details see [2]).

Using this fact and the equality deg(g) = ϱ+1 (remember C(M) = −4π(ϱ+1)), we are able to obtain

(i) J̃ is a rotation followed by a symmetry, and so ord(J) is even.

(ii) The order of J is ϱ + 1. In particular, ϱ is odd.

(iii) s = 2 and m_{1} = m_{2} = 1.

For details see [9].

In the following, we have written ϱ = 2k + 1, where k is a positive integer. Note that this new notation leads to deg(g) = 2(k + 1) and ord(J) = 2(k + 1). Summarizing all the information and using the above ideas about the be-

haviour of the Gauss map in the presence of a rotational symmetry, it is possible to deduce that the divisor associated to g as a meromorphic function on \overline{M} is, up to relabelings,

[g] = \frac{P_{1}^{k+2} · Q_{1}^{k}}{P_{2}^{k+2} · Q_{2}^{k}} (0.7)

In the following we consider the fundamental piece of M̃ in the slab

{(x_{1}, x_{2}, x_{3}) ∈ R^{3} / − 1 < x_{3} ≤ 1}

Let Q̃_{1} and Q̃_{2} be the unique points in the fiber of Q_{1} and Q_{2}, respectively, which belong to the above slab. As J̃ is a rotation followed by a symmetry with respect to the plane x_{3} = 0 and J fixes the ends P_{1} and P_{2}, then the ends are asymptotic to the planes x_{3} = 0 and x_{3} = 1, respectively. In the same way, as J(Q_{i}) = Q_{i}, i = 1, 2, if we write x̃(Q̃_{i}) = (0, 0, q_{i}), q_{i} ∈] − 1, 1], i = 1, 2, then

(0, 0, q_{i}) = J̃(0, 0, q_{i}) + (0, 0, 2n_{i}) = (0, 0, 2n_{i} − q_{i}), n_{i} ∈ z, i = 1, 2.

which implies q_{i} = n_{i} ∈ z∩] − 1, 1] and so

{x̃(Q̃_{1}), x̃(Q̃_{2})} = {(0, 0, 0), (0, 0, 1)}.

This fact implies that any symmetry of \overline{M} which fixes the ends must fix the points Q_{1} and Q_{2}, and any symmetry of \overline{M} which interchanges the ends must interchange the points Q_{1} and Q_{2}.

Using similar techniques to those introduced by Hoffman and Meeks in [6], we obtain two possible conformal structures for \overline{M}:

(A) The surface \overline{M} is conformally diffeomorphic to

\overline{M}_{k a} = \{(z, w) ∈ (C ∪ \{∞\})^{2} : w^{2(k+1)} = \frac{z(z − a)}{az − 1}\}

Furthermore, P_{1} = (0, 0), P_{2} = (∞, ∞), Q_{1} = (a, 0), Q_{2} = (1/a, ∞) and

J(z, w) = (z, e^{\frac{πi}{k+1}} w). In this case the Weierstrass data are

M_{k a} = \overline{M}_{k a} − \{P_{1}, P_{2}\}, g = A\frac{w^{k}}{z} , gη = B \frac{dz}{(az − 1)w^{k+1}}

where A ∈ R − {0} and B ∈ C − {0}.

(B) The surface \overline{M} is conformally diffeomorphic to

\overline{N}_{k a} = \{(z, w) ∈ (C ∪ \{∞\})^{2} : w^{2(k+1)} = \frac{z(z − a)}{az + 1}\}

Furthermore, P_{1} = (0, 0), P_{2} = (∞, ∞), Q_{1} = (a, 0), Q_{2} = (−1/a, ∞) and

J(z, w) = (z, e^{\frac{πi}{k+1}} w). Now, the Weierstrass data are

N_{k a} = \overline{N}_{k a} − \{P_{1}, P_{2}\}, g = A\frac{w^{k}}{z} , gη = B \frac{dz}{(az + 1)w^{k+1}}

where A ∈ R − {0} and B ∈ C − {0}.

The first one corresponds to the Callahan-Hoffman-Meeks example and it is studied in [9, sec. 4], where we give a new approach to the solution of the period problem and we prove that this solution is unique. The second one does not give any example, because we have proved that the period problem has no solutions for this kind of surfaces (see [9, sec. 5]).

Finally, we would like to point out that the hypotheses of Theorem 3 are essential and cannot be changed, as there are examples (see [7] and [8]) which show this fact.

References

[1] M. CALLAHAN, D. HOFFMAN, H. KARCHER. A family of singly periodic minimal surfaces invariant under a screw motion. Experimental Mathematics, 2, No. 3 (1993), 157-182.

[2] M. CALLAHAN, D. HOFFMAN, W. H. MEEKS III. Embedded minimal surfaces with an infinite number of ends. Invent. math., 96 (1989), 459-505.

[3] M. CALLAHAN, D. HOFFMAN, W. H. MEEKS III. The structure of singlyperiodic minimal surfaces. Invent. math., 99 (1990), 455-481.

[4] H. I. CHOI, W. H. MEEKS III, B. WHITE. A rigidity theorem for properly embedded minimal surfaces in R^{3}. J. Differential Geometry, 32 (1990), 65-76.

[5] H. M. FARKAS, I. KRA. Riemann surfaces. Grad. Texts in Math., 72, Springer Verlag, Berlin, 1980.

[6] D. HOFFMAN, W. H. MEEKS III. Embedded minimal surfaces of finite topology. Annals of Mathematics, 131 (1990), 1-34.

[7] D. HOFFMAN, M. WOHLGEMUTH. New embedded periodic minimal surfaces of Riemann-type. In preparation.

[8] F. J. LOPEZ, F. MARTIN. Existence and uniqueness of properly embedded minimal surfaces bounded by straight lines. In preparation.

[9] F. MARTIN, D. RODRIGUEZ. A characterization of the periodic Callahan-Hoffman-Meeks surfaces in terms of their symmetries. Duke Math. J., 89, No. 3 (1997), 445-463.

[10] W. H. MEEKS III. “The geometry, topology, and existence of periodic minimal surfaces” in Differential Geometry: Partial Differential Equations on Manifolds (Los Angeles, Calif., 1990). Proceedings of Symposia in Pure Math., 54 (1993), 333-374.

[11] W. H. MEEKS III, J. PEREZ, A. ROS. Uniqueness of the Riemann minimal examples. To appear in Invent. math..

[12] W. H. MEEKS III, H. ROSENBERG. The geometry of periodic minimal surfaces. Comment. Math. Helvetici, 68 (1993), 538-578.

[13] R. OSSERMAN. A survey of minimal surfaces. Dover Publications, New York, second edition, 1986.

[14] B. RIEMANN. Über die Fläche wom kleinsten Inhalt bei gegebener Begrenzung. Abh. Königl. Ges. d. Wiss. Göttingen, Mathem. Cl. 13 (1867), 3-52.

[15] A. ROS. Embedded minimal surfaces: forces, topology and symmetries. Calc. Var. Partial Differential Equations 4 (1996), 469-496.

Francisco Martin Departamento de Geometria y Topologia Facultad de Ciencias Universidad de Granada 18071 Granada Spain

Email: fmartin@goliat.ugr.es