Some Picard theorems for minimal surfaces

López, Francisco

doi:10.1090/S0002-9947-03-03213-6

Some Picard theorems for minimal surfaces
HTML articles powered by AMS MathViewer

by Francisco J. López PDF

Trans. Amer. Math. Soc. 356 (2004), 703-733 Request permission

Abstract:

This paper deals with the study of those closed subsets $F \subset \mathbb {R}^3$ for which the following statement holds: If $S$ is a properly immersed minimal surface in $\mathbb {R}^3$ of finite topology that is eventually disjoint from $F,$ then $S$ has finite total curvature. The same question is also considered when the conclusion is finite type or parabolicity.

References

Lars V. Ahlfors and Leo Sario, Riemann surfaces, Princeton Mathematical Series, No. 26, Princeton University Press, Princeton, N.J., 1960. MR 0114911, DOI 10.1515/9781400874538
Robert B. Burckel, An introduction to classical complex analysis. Vol. 1, Pure and Applied Mathematics, vol. 82, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR 555733, DOI 10.1007/978-3-0348-9374-9
Jaigyoung Choe and Marc Soret, Nonexistence of certain complete minimal surfaces with planar ends, Comment. Math. Helv. 75 (2000), no. 1, 189–199. MR 1760502, DOI 10.1007/s000140050119
Pascal Collin, Topologie et courbure des surfaces minimales proprement plongées de $\mathbf R^3$, Ann. of Math. (2) 145 (1997), no. 1, 1–31 (French). MR 1432035, DOI 10.2307/2951822
Tobias H. Colding and William P. Minicozzi II, Complete properly embedded minimal surfaces in $\mathbf R^3$, Duke Math. J. 107 (2001), no. 2, 421–426. MR 1823052, DOI 10.1215/S0012-7094-01-10726-6
Collin, P., Kusner, R., Meeks, W. H. III, Rosenberg, H.: The topology, geometry and conformal structures of properly embedded minimal surfaces. Preprint.
Yi Fang and William H. Meeks III, Some global properties of complete minimal surfaces of finite topology in $\textbf {R}^3$, Topology 30 (1991), no. 1, 9–20. MR 1081931, DOI 10.1016/0040-9383(91)90031-X
G. M. Goluzin, Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, Vol. 26, American Mathematical Society, Providence, R.I., 1969. MR 0247039, DOI 10.1090/mmono/026
David Hoffman and William H. Meeks III, The asymptotic behavior of properly embedded minimal surfaces of finite topology, J. Amer. Math. Soc. 2 (1989), no. 4, 667–682. MR 1002088, DOI 10.1090/S0894-0347-1989-1002088-X
D. Hoffman and W. H. Meeks III, The strong halfspace theorem for minimal surfaces, Invent. Math. 101 (1990), no. 2, 373–377. MR 1062966, DOI 10.1007/BF01231506
Dietrich Suschowk, Verallgemeinerte Ausstrahlungslösungen der Wellengleichung bei beliebiger Dimensionszahl, Bayer. Akad. Wiss. Math.-Nat. Kl. S.-B. 1959 (1960), 387–413 (1960) (German). MR 0117442
N. N. Ural′tseva, Estimates of the maximum moduli of gradients for solutions of capillarity problems, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 115 (1982), 274–284, 312 (Russian). Boundary value problems of mathematical physics and related questions in the theory of functions, 14. MR 660089
Nicholas J. Korevaar, Rob Kusner, and Bruce Solomon, The structure of complete embedded surfaces with constant mean curvature, J. Differential Geom. 30 (1989), no. 2, 465–503. MR 1010168
Francisco J. López and Francisco Martín, Minimal surfaces in a wedge of a slab, Comm. Anal. Geom. 9 (2001), no. 4, 683–723. MR 1868918, DOI 10.4310/CAG.2001.v9.n4.a2
Francisco J. López, Minimal surfaces in a cone, Ann. Global Anal. Geom. 20 (2001), no. 3, 253–299. MR 1866418, DOI 10.1023/A:1012451110396
López, F. J., Pérez, J.: Parabolicity and Gauss map of minimal surfaces. To appear in Indiana J. Math.
William H. Meeks III and Harold Rosenberg, The geometry and conformal structure of properly embedded minimal surfaces of finite topology in $\textbf {R}^3$, Invent. Math. 114 (1993), no. 3, 625–639. MR 1244914, DOI 10.1007/BF01232681
Meeks, W. H., III, Rosenberg, H.: Maximum principles at infinity with applications to minimal and constant mean curvature surfaces. Preprint.
Meeks, W. H., III, Rosenberg, H.: The uniqueness of the helicoid and the asymptotic geometry of properly embedded minimal surfaces with finite topology. Preprint.
Morales, S.: On the existence of a proper minimal surface in $\mathbb {R}^3$ with the conformal type of a disk. To appear in GAFA.
Robert Osserman, A survey of minimal surfaces, Van Nostrand Reinhold Co., New York-London-Melbourne, 1969. MR 0256278
Xiaokang Mo and Robert Osserman, On the Gauss map and total curvature of complete minimal surfaces and an extension of Fujimoto’s theorem, J. Differential Geom. 31 (1990), no. 2, 343–355. MR 1037404
V. S. Malakhovsky, Differential geometry of manifolds of figures, J. Math. Sci. (New York) 89 (1998), no. 3, 1261–1278. Geometry, 3. MR 1619704, DOI 10.1007/BF02414871
Harold Rosenberg, Minimal surfaces of finite type, Bull. Soc. Math. France 123 (1995), no. 3, 351–359 (English, with English and French summaries). MR 1373739, DOI 10.24033/bsmf.2263
Xavier, F.: Why no new complete simply-connected embedded minimal surfaces have been found since 1776. Preprint.