Compact complete minimal immersions in ℝ³

Alarcón, Antonio

doi:10.1090/S0002-9947-10-04741-0

Compact complete minimal immersions in $\mathbb {R}^3$
HTML articles powered by AMS MathViewer

by Antonio Alarcón PDF

Trans. Amer. Math. Soc. 362 (2010), 4063-4076 Request permission

Abstract:

In this paper we find, for any arbitrary finite topological type, a compact Riemann surface $\mathcal {M},$ an open domain $M\subset \mathcal {M}$ with the fixed topological type, and a conformal complete minimal immersion $X:M\to \mathbb {R}^3$ which can be extended to a continuous map $X:\overline {M}\to \mathbb {R}^3,$ such that $X_{|\partial M}$ is an embedding and the Hausdorff dimension of $X(\partial M)$ is $1.$

We also prove that complete minimal surfaces are dense in the space of minimal surfaces spanning a finite set of closed curves in $\mathbb {R}^3$, endowed with the topology of the Hausdorff distance.

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
Antonio Alarcón, Leonor Ferrer, and Francisco Martín, Density theorems for complete minimal surfaces in $\Bbb R^3$, Geom. Funct. Anal. 18 (2008), no. 1, 1–49. MR 2399094, DOI 10.1007/s00039-008-0650-2
Antonio Alarcón and Nikolai Nadirashvili, Limit sets for complete minimal immersions, Math. Z. 258 (2008), no. 1, 107–113. MR 2350037, DOI 10.1007/s00209-007-0161-0
Eugenio Calabi, Quasi-surjective mappings and a generalization of Morse theory, Proc. U.S.-Japan Seminar in Differential Geometry (Kyoto, 1965) Nippon Hyoronsha, Tokyo, 1966, pp. 13–16. MR 0216513
Tobias H. Colding and William P. Minicozzi II, The Calabi-Yau conjectures for embedded surfaces, Ann. of Math. (2) 167 (2008), no. 1, 211–243. MR 2373154, DOI 10.4007/annals.2008.167.211
Jesse Douglas, Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931), no. 1, 263–321. MR 1501590, DOI 10.1090/S0002-9947-1931-1501590-9
L. Ferrer, F. Martín and W. H. Meeks III, Existence of proper minimal surfaces of arbitrary topological type. Preprint.
F. J. López, F. Martín, and S. Morales, Adding handles to Nadirashvili’s surfaces, J. Differential Geom. 60 (2002), no. 1, 155–175. MR 1924594, DOI 10.4310/jdg/1090351086
Francisco Martín, William H. Meeks III, and Nikolai Nadirashvili, Bounded domains which are universal for minimal surfaces, Amer. J. Math. 129 (2007), no. 2, 455–461. MR 2306042, DOI 10.1353/ajm.2007.0013
Francisco Martín and Santiago Morales, On the asymptotic behavior of a complete bounded minimal surface in $\Bbb R^3$, Trans. Amer. Math. Soc. 356 (2004), no. 10, 3985–3994. MR 2058515, DOI 10.1090/S0002-9947-04-03451-8
Francisco Martín and Santiago Morales, Complete proper minimal surfaces in convex bodies of $\Bbb R^3$. II. The behavior of the limit set, Comment. Math. Helv. 81 (2006), no. 3, 699–725. MR 2250860, DOI 10.4171/CMH/70
Francisco Martín and Nikolai Nadirashvili, A Jordan curve spanned by a complete minimal surface, Arch. Ration. Mech. Anal. 184 (2007), no. 2, 285–301. MR 2299764, DOI 10.1007/s00205-006-0023-7
Frank Morgan, Geometric measure theory, 3rd ed., Academic Press, Inc., San Diego, CA, 2000. A beginner’s guide. MR 1775760
Nikolai Nadirashvili, Hadamard’s and Calabi-Yau’s conjectures on negatively curved and minimal surfaces, Invent. Math. 126 (1996), no. 3, 457–465. MR 1419004, DOI 10.1007/s002220050106
Nikolai Nadirashvili, An application of potential analysis to minimal surfaces, Mosc. Math. J. 1 (2001), no. 4, 601–604, 645 (English, with English and Russian summaries). Dedicated to the memory of I. G. Petrovskii on the occasion of his 100th anniversary. MR 1901078, DOI 10.17323/1609-4514-2001-1-4-601-604
Tibor Radó, On Plateau’s problem, Ann. of Math. (2) 31 (1930), no. 3, 457–469. MR 1502955, DOI 10.2307/1968237