A family of genus one minimal surfaces with two catenoid ends and one Enneper end
Introduction
During the 80's and 90's, the Weierstrass representation and the theory of elliptic functions were fundamental tools for finding a large quantity of new examples of minimal surfaces: Chen-Gackstatter, Costa, Hoffman, Meeks, Nelli and Karcher (see [3], [5], [11], [12], [18]), among others.
The Chen-Gackstatter surface (C–G) (see [3]) was the first example of a complete minimal surface of genus one with one Enneper-type end and total curvature . The construction of this example was obtained using the elliptic function ℘ and the Weierstrass data In the same paper, Chen and Gackstatter have proven that there is a complete minimal surface with genus two and one Enneper-type end. The genus one C-G surface was generalized by Karcher [15] and the genus two C-G surface was generalized by Thayer [20]. These generalizations are similar to C-G surfaces, but with a higher winding order at the end. Other generalizations of the Chen-Gackstatter surface also can be found in [6], [7] and [14]. Among many other surfaces of genus one with three ends, we can mention the beautiful Costa surface (see [4]) given by Weierstrass data that is an embedded minimal surface with one planar end and two catenoid ends (see [12]).
Examples of genus one minimal surfaces in with three ends have a long history and are very important for the theory of complete minimal surfaces, especially those that are embedded. Many generalizations about the genus and symmetries of minimal surfaces with three ends have already been obtained (see [11], [16], [17]). The main goal of this paper will be to prove that there exists a one parameter family, , of complete immersed minimal surfaces of genus one with two parallel catenoid ends and one Enneper end and finite total curvature . Moreover, we will be proving that the symmetry group of is the dihedral group G with 8 elements, while the symmetry group of is generated by two orthogonal vertical planes of reflectional symmetry. Using the theory of elliptic function, we will also give a different approach to the Weber's and Fujimori - Shoda's minimal surface (see [8], [21]) and we explicitly solve the period problem that surface.
Section snippets
The Weierstrass representation
The main tool we will use to prove the existence of the surfaces described in the Theorem 1, Theorem 2 and Proposition 5 is the Weierstrass representation formula (see [7], [19]).
Proposition 1 Let be a compact Riemann surface and . Suppose is a meromorphic function and η is a meromorphic 1-form such that whenever has a pole of order k, then η has a zero of order 2k and η has no other zeros on M. Let If for any closed curve γ in M,
Classical approach to the Weber's and Fujimori - Shoda's minimal surface example
The Professor Matthias Weber [21] has constructed numerically an example of a minimal surface of genus one, with one catenoid-type end and one Enneper-type end. That surface was also studied by S. Fujimori and T. Shoda, [8]. In the Proposition 5 below, we will use a different approach which uses the theory of elliptic functions and we will give explicit proof of the solution of the period problem.
Proposition 5 There exists a complete genus one immersed minimal surface S in with two ends and the following
An example of genus one minimal surface S with two catenoid ends and one Enneper end
In order to find a new example of a minimal surface with three ends, being two catenoid-type ends and one Ennerper-type end (See Fig. 2.), we will add one more catenoid-type end to minimal surface described in Proposition 5, by symmetry.
Theorem 1 There exists a complete genus one immersed minimal surface S in with three ends and the following properties: The total curvature of S is ; S has two catenoid-type ends and one Enneper-type end; The symmetry group of S is the dihedral group G with 8 elements
A family of genus one minimal surfaces with two catenoid ends and one Enneper end
In the above section, we obtained a minimal surface S described in Theorem 1 by symmetry of the minimal surface given in Propositon 5. In [22], Matthias Weber suggested numerically that the surface S belongs to a one-parameter family of minimal surfaces with two catenoid ends and one Enneper end. In this section we will give an analytically prove that S is indeed a lie inside the one-parameter sub-family of genus one minimal surfaces with two catenoid ends and one Enneper end. Moreover, we
Acknowledgements
We would like to express our gratitude to the Editors and Reviewers for their corrections and suggestions, and also to give our thanks to Felipe Castro Vilhena for reading the paper.
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