Duke Mathematical Journal

Complete properly embedded minimal surfaces in ℝ3

Tobias H. Colding and William P. Minicozzi

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Abstract

In this short paper, we apply estimates and ideas from [CM4] to study the ends of a properly embedded complete minimal surface Σ2⊂ℝ3 with finite topology. The main result is that any complete properly embedded minimal annulus that lies above a sufficiently narrow downward sloping cone must have finite total curvature.

Article information

Source
Duke Math. J. Volume 107, Number 2 (2001), 421-426.

Dates
First available in Project Euclid: 5 August 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1091736761

Digital Object Identifier
doi:10.1215/S0012-7094-01-10726-6

Mathematical Reviews number (MathSciNet)
MR1823052

Zentralblatt MATH identifier
1010.49025

Subjects
Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]
Secondary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]

Citation

Colding, Tobias H.; Minicozzi, William P. Complete properly embedded minimal surfaces in ℝ 3 . Duke Math. J. 107 (2001), no. 2, 421--426. doi:10.1215/S0012-7094-01-10726-6. http://projecteuclid.org/euclid.dmj/1091736761.


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References

  • S. Y. Cheng and S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975), 333--354.
  • T. H. Colding and W. P. Minicozzi II, Convergence of embedded minimal surfaces without area bounds in three-manifolds, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), 765--770.
  • --. --. --. --., ``Embedded minimal surfaces without area bounds in $3$-manifolds'' in Geometry and Topology (Aarhus, 1998), Contemp. Math. 258, Amer. Math. Soc., Providence, 2000, 107--120.
  • --------, Minimal Surfaces, Courant Lect. Notes Math. 4, New York Univ., New York, 1999.
  • --------, Convergence and compactness of minimal surfaces without density bounds, I: Partial regularity and convergence, preprint, 2000.
  • --------, Convergence and compactness of minimal surfaces without density bounds, II: Compactness and removable singularities, in preparation.
  • --------, Convergence and compactness of minimal surfaces without density bounds, III: Morse index bounds and applications to topology, in preparation.
  • P. Collin, Topologie et courbure des surfaces minimales proprement plongées de $\RR^3$, Ann. of Math. (2) 145 (1997), 1--31.
  • D. Hoffman and W. H. Meeks III, The asymptotic behavior of properly embedded minimal surfaces of finite topology, J. Amer. Math. Soc. 2 (1989), 667--682.
  • W. H. Meeks III, ``The geometry, topology, and existence of periodic minimal surfaces'' in Differential Geometry: Partial Differential Equations on Manifolds (Los Angeles, 1990), Proc. Sympos. Pure Math. 54, Part 1, ed. R. Greene and S. T. Yau, Amer. Math. Soc., Providence, 1993, 333--374.
  • W. H. Meeks III and H. Rosenberg, The geometry and conformal structure of properly embedded minimal surfaces of finite topology in $\RR^3$, Invent. Math. 114 (1993), 625--639.
  • R. Osserman and M. Schiffer, Doubly-connected minimal surfaces, Arch. Rational Mech. Anal. 58 (1975), 285--307.
  • H. Rosenberg and E. Toubiana, A cylindrical type complete minimal surface in a slab of $\RR^3$, Bull. Sci. Math. (2) 111 (1987), 241--245.