Complete minimal Möbius strips in \(\mathbb R^n\) and the Björling problem. (English) Zbl 1107.53007
The main goal of the author is to study the minimal Möbius strips of \(\mathbb{R}^n\) that are complete and of finite total curvature. For this the author formulates at first in a global way the general solution of the Björling problem in \(\mathbb{R}^n\), and uses it to provide a general description free of the period problem for the minimal Möbius strips of \(\mathbb{R}^n\). Then he uses the Björling problem to investigate the complete minimal Möbius strips of finite total curvature in \(\mathbb{R}^n\), showing that any such surface is the solution to a particular Björling problem in which the initial data are expressed as vector trigonometric polynomials. Finally in the excellently written paper the author gives a geometric construction of all complete minimal Möbius strips of finite total curvature in \(\mathbb{R}^n\) that attain equality in the fundamental Gackstatter inequality of the theory.
Reviewer: Hans Sachs (Leoben)
MSC:
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
References:
| [1] | J.A. Aledo, R.M.B. Chaves, J.A. Glvez, The Cauchy problem for improper affine spheres and the Hessian one equation, preprint.; J.A. Aledo, R.M.B. Chaves, J.A. Glvez, The Cauchy problem for improper affine spheres and the Hessian one equation, preprint. |
| [2] | Alías, L. J.; Mira, P., A Schwarz-type formula for minimal surfaces in Euclidean space \(R^n\), C.R. Acad. Sci. Paris, Ser. I, 334, 389-394 (2002) · Zbl 1013.53004 |
| [3] | Dierkes, U.; Hildebrant, S.; Kster, A.; Wohlrab, O., Minimal Surfaces I, A Series of Comprehensive Studies in Mathematics, vol. 295 (1992), Springer-Verlag · Zbl 0777.53012 |
| [4] | Gackstatter, F., ber die Dimension eine minimalflche und zur Ungleichung von St. Cohn-Vossen, Arch. Rat. Mech. Anal., 61, 141-152 (1976) · Zbl 0328.53002 |
| [5] | Glvez, J. A.; Mira, P., Dense solutions to the Cauchy problem for minimal surfaces, Bull. Braz. Math. Soc., 35, 387-394 (2004) · Zbl 1103.53029 |
| [6] | Glvez, J. A.; Mira, P., The Cauchy problem for the Liouville equation and Bryant surfaces, Adv. Math., 195, 456-490 (2005) · Zbl 1083.53010 |
| [7] | Glvez, J. A.; Mira,, P., Embedded isolated singularities of flat surfaces in hyperbolic 3-space, Calc. Var. Partial Differential Equations, 24, 239-260 (2005) · Zbl 1080.53008 |
| [8] | Hoffman, D.; Osserman, R., The geometry of the generalized Gauss map, Mem. Am. Math. Soc., 28 (1980) · Zbl 0469.53004 |
| [9] | Kokubu, M.; Umehara, M.; Yamada, K., Minimal surfaces that attain equality in the Chern-Osserman inequality. Differential geometry and integrable systems, Contemp. Math., 223-228 (2002) · Zbl 1029.53013 |
| [10] | Meeks, W. H., The classification of complete minimal surfaces in \(R^3\) with total curvature greater than \(- 8 \pi \), Duke Math. J., 48, 523-535 (1981) · Zbl 0472.53010 |
| [11] | Nitsche, J. C.C., Lectures on Minimal Surfaces, vol. I (1989), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0202.20601 |
| [12] | de Oliveira, M. E., Some new examples of nonorientable minimal surfaces, Proc. Am. Math. Soc., 98, 629-636 (1986) · Zbl 0607.53003 |
| [13] | Osserman, R., A Survey of Minimal Surfaces (1986), Dover Publications: Dover Publications New York, (Enlarged republication of the 1969 original) · Zbl 0209.52901 |
| [14] | Yang, K., Complete Minimal Surfaces of Finite Total Curvature. Mathematics and its Applications (1994), Kluwer Academic Publishers Group: Kluwer Academic Publishers Group Dordrecht · Zbl 0813.53003 |
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