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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On complete nonorientable minimal surfaces
with low total curvature


Author: Francisco J. Lopez
Journal: Trans. Amer. Math. Soc. 348 (1996), 2737-2758
MSC (1991): Primary 53A10; Secondary 53C42
DOI: https://doi.org/10.1090/S0002-9947-96-01618-2
MathSciNet review: 1351494
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Abstract: We classify complete nonorientable minimal surfaces in $\mathbb R^3 $ with total curvature $-8\pi $.


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  • [B] E. L. Barbanel, Complete minimal surfaces in $\mathbb R ^3$ of low total curvature, Thesis, Univ. of Massachusetts, 1987.
  • [BA] A. Alves de Barros, Complete nonorientable minimal surfaces in $\mathbb R ^3$ with finite total curvature, An. Acad. Bras. Cienc. 59 (1987), 141--143. MR 89h:53017
  • [BL] D. Bloss, Elliptische Funktionen und vollständige Minimalflächen, Thesis, Freie Universität Berlin, Berlin, 1989. Cf. MR 94j:53003
  • [CG] C. C. Chen and F. Gackstatter, Elliptische und Hyperelliptische Functionen und vollständige minimalflächen von Enneperschen Typ, Math. Ann. 259 (1982), 359--369. MR 84d:53005
  • [FK] H. M. Farkas and I. Kra, Riemann surfaces. Graduate Texts in Math., 71, Springer-Verlag, Berlin, 1980. MR 82c:30067
  • [HM] D. Hoffman and W. H. Meeks III, Embedded minimal surfaces of finite topology, Ann. of Math. 131 (1990), 1--34. MR 91i:53010
  • [I1] T. Ishihara, Complete Möbius strips minimally immersed in $\mathbb R ^3$, Proc. Amer. Math. Soc. 107 (1989), 803--806. MR 90c:53024
  • [I2] ------, Complete nonorientable minimal surfaces in $\mathbb R ^3$, Trans. Amer. Math. Soc. 333 (1992), 889--901. MR 92m:53013
  • [JM] L. Jorge and W. H. Meeks III, The topology of complete minimal surfaces of finite total Gaussian curvature, Topology 22 (1983), 203--221. MR 84d:53006
  • [K1] R. Kusner, Conformal geometry and complete minimal surfaces, Bull. Amer. Math. Soc. (N.S.) 17 (1987), 291--295. MR 88j:53008
  • [K2] ------, Global geometry of extremal surfaces, Dissertation, Univ. of California, Berkeley, 1988.
  • [L1] F. J. Lopez, The classification of complete minimal surfaces with total curvature greater than $-12\pi $, Trans. Amer. Math. Soc. 334 (1992), 49--74. MR 93a:53008
  • [L2] ------, A complete minimal Klein Bottle in $\mathbb R ^3$, Duke Math. J. 71 (1993), 23--30. MR 94e:53005
  • [LM] F. J. Lopez and F. Martin, Complete nonorientable minimal surfaces and symmetries, Duke Math. J. 79 (1995), 667--686.
  • [M] W. H. Meeks, The classification of complete minimal surfaces in $\mathbb R ^3$ with total curvature greater than $-8\pi $, Duke Math. J. 48 (1981), 523--535. MR 82k:53009
  • [OL] M. E. G. G. de Oliveira, Some new examples of nonorientable minimal surfaces, Proc. Amer. Math. Soc. 94 (1986), 629--636. MR 87m:53008
  • [OT] M. E. G. G. de Oliveira and E. Toubiana, Surfaces nonorientables de genre deux, Bol. Soc. Brasil. Mat. (N. S.) 24 (1993), 63--88. MR 94f:53011
  • [OS] R. Osserman, A survey of minimal surfaces, 2nd ed., Dover, New York, 1986. MR 87j:53012
  • [R] M. Ross, Complete nonorientable minimal surfaces in $\mathbb R ^3$, Comment. Math. Helv. 67 (1992), 64--76. MR 92k:53022
  • [S] N. Schmitt, Minimal surfaces with flat ends, Dissertation, Univ. of Massachusetts at Amherst, 1992-1993.
  • [SM] A. J. Small, Minimal surfaces in $\mathbb {R}^3$ and algebraic curves, Differential Geom. Appl. 2 (1992), 309--384. MR 94h:53014
  • [T] E. Toubiana, Surfaces minimales non orientables de genre quelconque, Bull. Soc. Math. France 121 (1993), 183--197. MR 94c:53013

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Additional Information

Francisco J. Lopez
Affiliation: Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de, Granada, 18071-Granada, Spain
Email: fjlopez@goliat.ugr.es

DOI: https://doi.org/10.1090/S0002-9947-96-01618-2
Received by editor(s): March 20, 1995
Additional Notes: Research partially supported by DGCYT grant No. PB91-0731
Article copyright: © Copyright 1996 American Mathematical Society

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