References
T. Choi, W. H. Meeks III andB. White,A rigidity theorem for, properly embedded minimal surfaces in ℝ3. Journal of Differential Geometry, March, 1990.
M. do Carmo andC. K. Peng, Stable minimal surfaces in ℝ3 are planes. Bulletin of the AMS1, 903–906 (1979).
D. Fischer-Colbrie,On complete minimal surfaces with finite Morse index in 3-manifolds. Inventiones Math.82, 121–132 (1985).
D. Fischer-Colbrie andR. Schoen,The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature. Comm. Pure Appl. Math.33, 199–211 (1980).
C. Frohman andW. H. Meeks III,The topological uniqueness of complete one-ended minimal surfaces and Heegaard surfaces in ℝ3 (preprint).
R. Hardt andL. Simon,Boundary regularity and embedded minimal solutions for the oriented Plateau problem. Annals of Math.110, 439–486 (1979).
D. Hoffman andW. H. Meeks III,The strong halfspace theorem for minimal surfaces. Inventiones Math. (to appear).
D. Hoffman andW. H. Meeks III,The asymptotic behavior of properly embedded minimal surfaces of finite topology. Journal of the AMS2, 667–681 (1989).
R. Langevin andH. Rosenberg,A maximum principle at infinity for minimal surfaces and applications. Duke Math. Journal57, 819–828, (1988).
S. Lojasiewicz,Triangulation of semianalytic sets. Ann. Scuola Norm. Sup. Pisa18, 449–474 (1964).
W. H. Meeks III,The topological uniqueness of minimal surfaces in three-dimensional Euclidean space. Topology20, 389–410 (1981).
W. H. Meeks III andH. Rosenberg,The geometry of periodic minimal surfaces (preprint).
W. H. Meeks III andH. Rosenberg,The global theory of doubly periodic minimal surfaces. Inventiones Math.97, 351–379 (1989).
W. H. Meeks III, L. Simon andS. T. Yau,The existence of embedded minimal surfaces, exotic spheres and positive Ricci curvature. Annals of Math.116, 221–259 (1982).
W. H. Meeks III andS. T. Yau,The topological uniqueness theorem of complete minimal surfaces of finite topological type (preprint).
W. H. Meeks III andS. T. Yau,The existence of embedded minimal surfaces and the problem of uniqueness. Math. Z.179, 151–168 (1982).
R. Osserman,A Survey of Minimal Surfaces, 2nd edition. Dover Publications, New York, 1986.
M. Protter andH. Weinberger,Maximum Principles in Differential Equations, Prentice-Hall, Englewood 1967.
R. Schoen,Estimates for Stable Minimal Surfaces in Three Dimensional Manifolds, volume 103 ofAnnals of Math. Studies. Princeton University Press, 1983.
R. Schoen,Uniqueness, symmetry, and embeddedness of minimal surfaces. Journal of Differential Geometry18, 791–809 (1983).
L. Simon,Lectures on geometric measure theory. Inproceedings of the Center for Mathematical Analysis, volume 3. Australian National University, Canberra 1983.
J. A. Wolf,Spaces of Constant Curvature. McGraw-Hill, New York 1967.
Author information
Authors and Affiliations
Additional information
The research described in this paper was supported by research grant DE-FG02-86ER250125 of the Applied Mathematical Science subprogram of Office of Energy Research, U.S. Department of Energy, and National Science Foundation grant DMS-8611574.
Rights and permissions
About this article
Cite this article
Meeks, W.H., Rosenberg, H. The maximum principle at infinity for minimal surfaces in flat three manifolds. Commentarii Mathematici Helvetici 65, 255–270 (1990). https://doi.org/10.1007/BF02566606
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02566606