Journal of the American Mathematical Society

The topological uniqueness of complete one-ended minimal surfaces and Heegaard surfaces in $\mathbb {R}^3$

Author(s): Charles Frohman; William H. Meeks III
Journal: J. Amer. Math. Soc. 10 (1997), 495-512.
MSC (1991): Primary 57N12, 53A10
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1.: M. Boileau and J. P. Otal, Sur les scindements de Heegaard du tore ${\mathbb {T}^3}$ , Journal of Differential Geometry, 32, (1990), pp. 209-233. MR 91i:57006
2.: F. Bonahon and J. P. Otal, Scindements de Heegaard des espaces Lenticulares, Ann. Scient. Éc. Norm. Sup. 16 (1983), no. 4, 451-466. MR 85c:57010
3.: M. Callahan, D. Hoffman, and W. H. Meeks III, The structure of singly-periodic minimal surfaces, Inventiones Math. 99 (1990), 455-481. MR 92a:53005
4.: A. J. Casson and C. McA. Gordon, Reducing Heegaard splittings, Topology and Its Applications 27 (1987), 275-283. MR 89c:57020
5.: M. do Carmo and C. K. Peng, Stable minimal surfaces in $\mathbb {R}^3$ are planes, Bulletin of the AMS 1 (1979), 903-906.
6.: D. Fischer-Colbrie, On complete minimal surfaces with finite Morse index in 3-manifolds, Inventiones Math. 82 (1985), 121-132. MR 87b:53090
7.: D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature, Comm. on Pure and Appl. Math. 33 (1980), 199-211. MR 81i:53044
8.: M. Freedman, J. Hass, and P. Scott, Least area incompressible surfaces in 3-manifolds, Inventiones Math. 71 (1983), 609-642. MR 85e:57012
9.: C. Frohman, Minimal surfaces and Heegaard splittings of the three-torus, Pacific Journal of Math. 124 (1986), 119-130. MR 87j:57011
10.: -, The topological uniqueness of triply-periodic minimal surfaces in $\mathbb {R}^{3}$ , Journal of Differential Geometry 31 (1990), 277-283. MR 92e:53008
11.: -, Heegaard splittings of the three-ball, Topology '90, Proceedings of the Special Semester on Low Dimensional Topology, Ohio State University, De Gruyter, Berlin, 1992. MR 94b:57015
12.: C. Frohman and J. Hass, Unstable minimal surfaces and Heegaard splittings, Inventiones Math. 95 (1989), 529-540. MR 90e:57028
13.: R. Hardt and L. Simon, Boundary regularity and embedded minimal solutions for the oriented Plateau problem, Annals of Math. 110 (1979), 439-486. MR 81i:49031
14.: J. Hempel, Three manifolds, Annals of Math Studies, no. 72, Princeton University Press, Princeton, NJ, 1976. MR 54:3702
15.: S. Hildebrandt, Boundary behavior of minimal surfaces, Archive Rational Mech. Anal. 35 (1969), 47-81. MR 40:1901
16.: D. Hoffman, H. Karcher, and F. Wei, Adding handles to the helicoid, Bulletin of the AMS, New Series 29 (1993), no. 1, 77-84. MR 94g:53004
17.: D. Hoffman and W. H. Meeks III, The strong halfspace theorem for minimal surfaces, Inventiones Math. 101 (1990), 373-377. MR 92e:53010
18.: H. B. Lawson, The unknottedness of minimal embeddings, Inventiones Math. 11 (1970), 183-187. MR 44:4651
19.: W. H. Meeks III, The topological uniqueness of minimal surfaces in three-dimensional Euclidean space, Topology 20 (1981), 389-410. MR 82h:53010
20.: -, The theory of triply-periodic minimal surfaces, Indiana University Math. Journal 39 (1990), no. 3, 877-936. MR 92e:53012
21.: W. H. Meeks III and H. Rosenberg, The maximum principle at infinity for minimal surfaces in flat three-manifolds, Commentari Mathematici Helvetici 65 (1990), 255-270. MR 91d:53011
22.: W. H. Meeks III, L. Simon, and S. T. Yau, The existence of embedded minimal surfaces, exotic spheres and positive Ricci curvature, Annals of Math. 116 (1982), 221-259.
23.: W. H. Meeks III and S. T. Yau, The classical Plateau problem and the topology of three-dimensional manifolds, Topology 21 (1982), no. 4, 409-442. MR 84g:53016
24.: -, The existence of embedded minimal surfaces and the problem of uniqueness, Math. Z. 179 (1982), 151-168. MR 83j:53060
25.: -, The topological uniqueness of complete minimal surfaces of finite topological type, Topology 31 (1992), no. 2, 305-316. MR 93g:57037
26.: R. Schoen, Estimates for stable minimal surfaces in three dimensional manifolds, Annals of Math. Studies, vol. 103, Princeton University Press, 1983. MR 86j:53094
27.: -, Uniqueness, symmetry, and embeddedness of minimal surfaces, Journal of Differential Geometry 18 (1983), 791-809. MR 85f:53011
28.: G. P. Scott and T. Wall, Homological group theory, London Math Society Lecture Notes, no. 36, Cambridge University Press, 1979. MR 81m:57002
29.: L. Simon, Lectures on geometric measure theory, Proceedings of the Center for Mathematical Analysis (Canberra, Australia), vol. 3, Australian National University, 1983. MR 87a:49001
30.: J. Singer, Three-dimensional manifolds and their Heegaard diagrams, Transactions of the AMS 35 (1933), 88-111.
31.: F. Waldhausen, Heegaard-zerlegungen der 3-sphäre, Topology 7 (1968), 195-203. MR 37:3576

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 57N12, 53A10

Charles Frohman
Affiliation: Mathematics Department, University of Iowa, Iowa City, Iowa 52242
Email: frohman@math.uiowa.edu

William H. Meeks III
Affiliation: Department of Mathematics, University of Massachusetts, Amherst, Massachusetts 01003
Email: bill@gang.umass.edu

DOI: 10.1090/S0894-0347-97-00215-4
PII: S 0894-0347(97)00215-4
Received by editor(s): February 1, 1991
Received by editor(s) in revised form: June 1, 1995
Additional Notes: This research was supported by the National Science Foundation grant DMS-8701736
The research described in this paper was supported by research grant DE-FG02-86ER250125 of the Applied Mathematical Science subprogram of the Office of Energy Research, U.S. Department of Energy, and National Science Foundation grants DMS-8900285 and DMS-9505101
Copyright of article: Copyright 1997, American Mathematical Society

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ISSN: 1088-6834(e) ISSN: 0894-0347(p)

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