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Example of a complete minimal immersion in $${\mathbb{R}}^ 3$$ of genus one and three embedded ends. (English) Zbl 0613.53002
This paper contains parts of the author’s thesis written in 1982. T. Klotz and L. Sario have constructed complete minimal surfaces of finite total curvature with genus $$p\geq 1$$ and $$N\geq 4$$ ends or expanding tubes. There are also surfaces with $$p=0$$ and $$N\geq 1$$. In 1980, C. C. Chen and the reviewer [Math. Ann. 259, 359-369 (1982; Zbl 0468.53008)] have found surfaces of Enneper type with one and two handles $$(N=1$$, $$p=1$$ and 2). In his thesis the author has settled the two open cases $$p=1$$, $$N=2$$ and 3. Costa’s surface with $$N=3$$ tubes is very interesting, it is an embedded surface as D. Hoffman and W. Meeks III have shown 1984 [J. Differ. Geom. 21, 109-127 (1985; Zbl 0604.53002)]. The construction of this example is published in the present paper.
The above mentioned surface of Enneper type with $$p=1$$ and the two surfaces of Costa can be constructed in the following way. Take the Weierstraß $$\wp$$-function to the lattice with $$\tau =i$$ and $$set$$
N$$=1:$$ $$g(z)=a\wp '(z)/\wp (z)$$, $$f(z)=\wp (z),$$
N$$=2:$$ $$g(z)=b/\wp (z)\wp '(z)$$, $$f(z)=\wp^{'2}(z),$$
N$$=3:$$ $$g(z)=c/\wp '(z)$$, $$f(z)=\wp (z)$$
in the Weierstraß representation. If the constants a, b and c are suitable, the periods are pure imaginary. C. Costa uses our method developed two years earlier to construct the Enneper type surface. It would have been honest to mention this fact in the paper.
Reviewer: F.Gackstatter

MSC:
 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 30F30 Differentials on Riemann surfaces 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
Citations:
Zbl 0468.53008; Zbl 0604.53002
Full Text:
References:
  C.C. Chen and F. Gackstatter;Elliptic and Hiperelliptic functions and complete minimal surfaces with handles, IME-USP, no 27, 1981.  C.J. Costa;Complete minimal immersion in IR 3 of genus one and finite total curvature, Doctoral Thesis, IMPA, 1982.  G.H. Halphen;Traité des functions elliptiques et de leurs applications. Paris, Gauthier-Villars, Imprimeur-Libraire, 1886.  B. Lawson;Lectures on minimal surfaces, Monografias de Matemática, vol. 14, IMPA, Rio de Janeiro. · Zbl 0434.53006  W.H. Meeks III and L.P.M. Jorge;The topology of complete minimal surfaces of finite total Gaussian curvature, Topology, 22 (1983), 203–221. · Zbl 0517.53008  E.H. Neville;Elliptic Functions: A Primer Pergamon Press, First Edition, 1971.  R. Ossermann;A survey of minimal surfaces, Van Nostrand Reinhold Company, 1969.  J. Tannery and J. Molk;Éléments de la Théorie des fonctions elliptiques, Bronx, N.Y., Chelsea, 1972. · Zbl 0296.33003
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