Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Complete minimal surfaces and the puncture number problem

Author: Kichoon Yang
Journal: Proc. Amer. Math. Soc. 119 (1993), 261-265
MSC: Primary 53A10; Secondary 30F10, 53A30
MathSciNet review: 1181179
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Given a nonnegative integer $ g$, let $ \mathcal{P}(g)$ denote the set of integers $ N$ such that an arbitrary compact Riemann surface with genus $ g$ can be completely conformally and minimally immersed in $ {\mathbb{R}^3}$ (with finite total curvature) with exactly $ N$ punctures. We prove that the infimum of $ \mathcal{P}(g)$ is at most $ 4g$ and that the set $ \mathcal{P}(g)$ may not miss any $ 3g$ consecutive integers larger than the infimum of $ \mathcal{P}(g)$.

References [Enhancements On Off] (What's this?)

  • [B] D. Bloss, Elliptische funktionen und vollstandige minimalflachen, doctoral dissertation, Freien Universitat, Berlin, 1989.
  • [CO] S. Chern and R. Osserman, Complete minimal surfaces in Euclidean $ n$-space, J. Analyse Math. 19 (1967), 15-34. MR 0226514 (37:2103)
  • [GK] F. Gackstatter and R. Kunert, Konstruktion vollstänger Minimalflächen von endlicher Gesamtkrümmung, Arch. Rational Mech. Anal. 65 (1977), 289-297. MR 0448264 (56:6573)
  • [JM] L. Jorge and W. Meeks, The topology of complete minimal surfaces of finite total Gaussian curvature, Topology 22 (1983), 203-221. MR 683761 (84d:53006)
  • [KS] T. Klotz and L. Sario, Existence of complete minimal surfaces of arbitrary connectivity and genus, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 42-44. MR 0178408 (31:2665)
  • [M] X. Mo, Gauss map and moduli space of minimal surfaces in Euclidean spaces, Doctoral dissertation, Stanford University, Stanford, CA, 1990.
  • [O] R. Osserman, A survey of minimal surfaces, Van Nostrand, New York, 1969. MR 0256278 (41:934)
  • [Y1] K. Yang, Meromorphic functions on a compact Riemann surface and associated complete minimal surfaces, Proc. Amer. Math. Soc. 105 (1989), 706-711. MR 953749 (89h:53029)
  • [Y2] K. Yang, Complete and compact minimal surfaces, Kluwer Academic Publishers, Boston, MA, 1989. MR 1020302 (91h:53058)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 53A10, 30F10, 53A30

Retrieve articles in all journals with MSC: 53A10, 30F10, 53A30

Additional Information

Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society