Tohoku Mathematical Journal

Noncongruent minimal surfaces with the same symmetries and conformal structure

Valério Ramos Batista

Full-text: Open access

Abstract

Concerning complete orientable minimal surfaces with finite total curvature in Euclidean three-space, we show for any positive genus the existence of noncongruent examples having the same symmetry group and conformal type.

Article information

Source
Tohoku Math. J. (2), Volume 56, Number 2 (2004), 237-254.

Dates
First available in Project Euclid: 11 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1113246552

Digital Object Identifier
doi:10.2748/tmj/1113246552

Mathematical Reviews number (MathSciNet)
MR2053320

Zentralblatt MATH identifier
1063.53009

Subjects
Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]

Citation

Ramos Batista, Valério. Noncongruent minimal surfaces with the same symmetries and conformal structure. Tohoku Math. J. (2) 56 (2004), no. 2, 237--254. doi:10.2748/tmj/1113246552. https://projecteuclid.org/euclid.tmj/1113246552


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References

  • C. C. Chen and F. Gackstatter, Elliptische und Hyperelliptische Funktionen und vollständige Minimalflächen von Enneperschen Typ, Math. Ann. 259 (1982), 359--369.
  • C. Costa, Example of a complete minimal immersion in $\real^3$ of genus one and three embedded ends, Bol. Soc. Brasil. Mat. 15 (1984), 41--54.
  • C. Costa, Uniqueness of minimal surfaces embedded in $\real^3$ with total curvature $12\pi$, J. Differential Geom. 30 (1989), 597--618.
  • D. Hoffman and H. Karcher, Complete embedded minimal surfaces of finite total curvature, Encyclopaedia of Math. Sci. 90, Ed. R. Osserman, 5--93, Springer-Verlag, Berlin, Heidelberg, 1997.
  • D. Hoffman and W. Meeks, A complete embedded minimal surface in $\real^3$ with genus one and three ends, J. Differential. Geom. 21 (1985), 109--127.
  • D. Hoffman and W. Meeks, Embedded minimal surfaces of finite topology, Ann. of Math. 131 (1990), 1--34.
  • A. Huber, On subharmonic functions and differential geometry in the large, Comment. Math. Helv. 32 (1957), 13--72.
  • H. Karcher, Construction of minimal surfaces, Surveys in Geometry, University of Tokyo (1989), 1--96, and Lecture Notes 12, SFB256, Bonn, 1989.
  • F. J. López, The classification of complete minimal surfaces with total curvature greater than $-12\pi$, Trans. Amer. Math. Soc. 334 (1992), 49--74.
  • F. J. López and F. Martín, Complete nonorientable minimal surfaces with the highest symmetry group, Amer. J. Math. 119 (1997), 55--81.
  • F. J. López and A. Ros, On embedded complete minimal surfaces of genus zero, J. Differential Geom. 33 (1991), 293--300.
  • J. C. C. Nitsche, Lectures on minimal surfaces, Second edition, Dover, New York, 1986.
  • R. Osserman, Global properties of minimal surfaces in $E^3$ and $E^n$, Ann. of Math. (2) 80 (1964), 340--364.
  • R. Osserman, A survey of minimal surfaces, Second edition, Dover Publications, New York, 1986.
  • V. Ramos Batista, Construction of new complete minimal surfaces in $\real^3$ based on the Costa surface, Doctoral thesis, University of Bonn, 2000.
  • V. Ramos Batista, The use of unitary functions in the behaviour analysis of elliptic integrals, Technical Report 03/03, University of Campinas, 2003. http://www.ime.unicamp.br/rel_pesq/2003/rp03-03.html.
  • R. Schoen, Uniqueness, symmetry and embeddedness of minimal surfaces, J. Differential Geom. 18 (1983), 791--809.
  • M. Wohlgemuth, Higher genus minimal surfaces by growing handles out of a catenoid, Manuscripta Math. 70 (1991), 397--428.