The asymptotic behavior of properly embedded minimal surfaces of finite topology

Hoffman, David; Meeks, William H.

doi:10.1090/S0894-0347-1989-1002088-X

The asymptotic behavior of properly embedded minimal surfaces of finite topology

Authors: David Hoffman and William H. Meeks
Journal: J. Amer. Math. Soc. 2 (1989), 667-682
MSC: Primary 53A10; Secondary 49F10
DOI: https://doi.org/10.1090/S0894-0347-1989-1002088-X
MathSciNet review: 1002088
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Michael T. Anderson, Curvature estimates for minimal surfaces in $3$-manifolds, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 1, 89–105. MR 803196
J. L. Barbosa and M. do Carmo, On the size of a stable minimal surface in $R^{3}$, Amer. J. Math. 98 (1976), no. 2, 515–528. MR 413172, DOI https://doi.org/10.2307/2373899

M. Callahan, D. Hoffman, and W. H. Meeks III, Embedded minimal surfaces with four ends, 1989, preprint.

Michael Callahan, David Hoffman, and William H. Meeks III, The structure of singly-periodic minimal surfaces, Invent. Math. 99 (1990), no. 3, 455–481. MR 1032877, DOI https://doi.org/10.1007/BF01234428
Celso J. Costa, Example of a complete minimal immersion in ${\bf R}^3$ of genus one and three embedded ends, Bol. Soc. Brasil. Mat. 15 (1984), no. 1-2, 47–54. MR 794728, DOI https://doi.org/10.1007/BF02584707
Yi Fang and William H. Meeks III, Some global properties of complete minimal surfaces of finite topology in ${\bf R}^3$, Topology 30 (1991), no. 1, 9–20. MR 1081931, DOI https://doi.org/10.1016/0040-9383%2891%2990031-X
D. Fischer-Colbrie, On complete minimal surfaces with finite Morse index in three-manifolds, Invent. Math. 82 (1985), no. 1, 121–132. MR 808112, DOI https://doi.org/10.1007/BF01394782

D. Hoffman and W. H. Meeks III, One-parameter families of embedded minimal surfaces, 1989, preprint.

Antonio Ros, Properly embedded minimal surfaces with finite topology, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 907–926. MR 2275628

---, The strong halfspace theorem for minimal surfaces, 1987, preprint.

D. Hoffman and W. H. Meeks III, A variational approach to the existence of complete embedded minimal surfaces, Duke Math. J. 57 (1988), no. 3, 877–893. MR 975126, DOI https://doi.org/10.1215/S0012-7094-88-05739-0
David A. Hoffman and William Meeks III, A complete embedded minimal surface in ${\bf R}^3$ with genus one and three ends, J. Differential Geom. 21 (1985), no. 1, 109–127. MR 806705
David Hoffman and William H. Meeks III, Properties of properly embedded minimal surfaces of finite topology, Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 2, 296–300. MR 903736, DOI https://doi.org/10.1090/S0273-0979-1987-15566-2
Luquésio P. Jorge and William H. Meeks III, The topology of complete minimal surfaces of finite total Gaussian curvature, Topology 22 (1983), no. 2, 203–221. MR 683761, DOI https://doi.org/10.1016/0040-9383%2883%2990032-0
Tzee Char Kuo, On $C^{0}$-sufficiency of jets of potential functions, Topology 8 (1969), 167–171. MR 238338, DOI https://doi.org/10.1016/0040-9383%2869%2990007-X
William H. Meeks III and Harold Rosenberg, The global theory of doubly periodic minimal surfaces, Invent. Math. 97 (1989), no. 2, 351–379. MR 1001845, DOI https://doi.org/10.1007/BF01389046

---, The strong maximum principle for complete minimal surfaces in flat $3$-manifolds, 1988, preprint.

---, The geometry of periodic minimal surfaces, 1988, preprint.

W. H. Meeks III and S. T. Yau, The topological uniqueness theorem for minimal surfaces of finite type, 1987, preprint.

William H. Meeks III and Shing Tung Yau, The classical Plateau problem and the topology of three-dimensional manifolds. The embedding of the solution given by Douglas-Morrey and an analytic proof of Dehn’s lemma, Topology 21 (1982), no. 4, 409–442. MR 670745, DOI https://doi.org/10.1016/0040-9383%2882%2990021-0
William W. Meeks III and Shing Tung Yau, The existence of embedded minimal surfaces and the problem of uniqueness, Math. Z. 179 (1982), no. 2, 151–168. MR 645492, DOI https://doi.org/10.1007/BF01214308
Robert Osserman, A survey of minimal surfaces, 2nd ed., Dover Publications, Inc., New York, 1986. MR 852409

J. Pitts and H. Rubenstein, Existence of minimal surfaces of bounded topological type in three-manifolds, Proc. Centre Math. Anal. Austral. Nat. Univ., Vol. 10, Canberra, Australia, 1987.

Richard M. Schoen, Uniqueness, symmetry, and embeddedness of minimal surfaces, J. Differential Geom. 18 (1983), no. 4, 791–809 (1984). MR 730928
Leon Simon, Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. MR 756417
S. Smale, An infinite dimensional version of Sard’s theorem, Amer. J. Math. 87 (1965), 861–866. MR 185604, DOI https://doi.org/10.2307/2373250
Brian White, New applications of mapping degrees to minimal surface theory, J. Differential Geom. 29 (1989), no. 1, 143–162. MR 978083