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The Schwarz reflection principle states that a minimal surface

During the

“A

With this terminology, he proved that such symmetries induce symmetries in the

By classifying the symmetries this way, we sort out the space groups that might admit one, both, or none of them. Since minimal surfaces may model some natural structures, like crystals and copolymers, an example within a given symmetry group might fit an already existing compound, or even hint at nonexisting ones. However, several symmetry groups are not yet represented by any minimal surface (see [

Restricted to symmetries given by reflections in the plane of principal geodesics and by 180°-rotations about straight lines contained in the surface, outside the triply periodic class it is easy to find complete embedded minimal surfaces in

In the class of triply periodic minimal surfaces almost all known examples have either both or none of such symmetries, after suitable motion in

The “TT-surfaces” are generated by an annulus, of which the boundary consists of two twisted equilateral triangles. For edge length

In the present work, we give existence proofs for examples that are probably the first triply periodic minimal surfaces with only horizontal symmetries, of which the translation group is given by an orthogonal lattice. They are constructed by Karcher’s method [

Regarding examples with only vertical symmetries, we believe they have not been found yet.

The examples presented herein are inspired in the surfaces

(a) The surfaces

(a) Schwarz’s D-surfaces; (b) the surfaces

The reader will notice that the surfaces

We are going to prove the following results:

There exists a one-parameter family of triply periodic minimal surfaces in

The quotient by its translation group has genus 9.

The whole surface is generated by a fundamental piece, which is a surface with boundary in

By successive reflections with respect to planes bounding the fundamental domain, and successive vertical translations, one obtains the triply periodic surface.

For

The quotient by its translation group has genus

The whole surface is generated by a fundamental piece, which is a surface with boundary in

By successive rotations about the boundary of the fundamental piece, and successive vertical translations, one obtains the triply periodic surface.

Sections

In this section we state some basic definitions and theorems. Throughout this work, surfaces are considered connected and regular. Details can be found in [

Let

Let

Let

The pair

Under the assumptions of Theorems

The function

Consider the surface indicated in Figure

(a) The fundamental domain of

The surface

Let us define

First of all, observe that Jorge-Meeks’ formula gives

By following Karcher’s method in [

Since

Now we analyse what happens to (

At this point we are ready to prove that

Now we are ready to find some relations that the parameters

Consider the curves

On

Therefore,

Equation (

Figure

Definition domain of the

It is not difficult to prove that (

Of course, the right-hand side of (

Regarding our remaining restriction, namely,

Symmetry | |||
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Notice that the points

Since the surface

If we had a well-defined square root of the function at the right-hand side of (

Now we apply (

At (

Symmetry | |||
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From Table

The analysis of the period problems can be reduced to the analysis of the fundamental domain of our minimal immersion. If this fundamental domain is contained in a rectangular prism of

In order to obtain such a prism, a little reflection will show us that the following two conditions will be enough.

The symmetry

After an orthonormal projection of the fundamental domain in the direction

The first condition is easy to prove. Take a path

The path

Therefore, our minimal surface is

Now we are ready to deal with the second condition. Consider Figure

On

On

Now define

The next proposition will solve the period problem given by (

For any fixed positive value of

By recalling (

Since

By fixing

The integrand of (

Since

Proposition

From now on we will denote our triply periodic surfaces by

A possible

It is not difficult to prove that the contour of the shaded region in Figure

Hence, the stretch

But even so, it can happen that the expanded triply periodic surface will not be embedded. We do not know whether the curve

By using the maximum principle, if we find an embedded member of our family in the curve

There is an

We will prove that

By recalling (

Equation (

We have fixed

But

But

Now we use

Together with (

In order to prove Theorem

(a) The fundamental domain of

Since

For the surfaces

(a) The fundamental domain of

From this point on we redefine the following:

Based on Figure

Integrals

For any fixed positive value of

The proof of Proposition

For this present paper, V. R. Batista was supported by the Grants “Bolsa de Produtividade Científica” from CNPq—Conselho Nacional de Desenvolvimento Científico e Tecnológico, and “Bolsa de Pós-Doutorado” FAPESP 2000/07090-5.