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MAXIMAL ANNULI WITH PARALLEL PLANAR BOUNDARIES IN THE THREE-DIMENSIONAL LORENTZ–MINKOWSKI SPACE

Part of: Global differential geometry Classical differential geometry

Published online by Cambridge University Press:  27 January 2010

JUNCHEOL PYO
JUNCHEOL PYO*
Affiliation:
Department of Mathematics, Seoul National University, Seoul 151-742, Korea (email: jcpyo@snu.ac.kr)
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Abstract

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We prove that maximal annuli in 𝕃3 bounded by circles, straight lines or cone points in a pair of parallel spacelike planes are part of either a Lorentzian catenoid or a Lorentzian Riemann’s example. We show that under the same boundary condition, the same conclusion holds even when the maximal annuli have a planar end. Moreover, we extend Shiffman’s convexity result to maximal annuli; but by using Perron’s method we construct a maximal annulus with a planar end where a Shiffman-type result fails.

Keywords

Lorentzian catenoidmaximal annulus surfaceLorentzian Riemann’s example

MSC classification

Secondary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) 53A10: Minimal surfaces, surfaces with prescribed mean curvature
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 81 , Issue 2 , April 2010 , pp. 208 - 222
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

[1]Calabi, E, ‘Examples of the Bernstein problem for some nonlinear equation’, Proc. Sympos. Pure Math. 15 (1970), 223–230.CrossRefGoogle Scholar
[2]Cheng, S. Y. and Yau, S. T., ‘Maximal space-like hypersurfaces in the Lorentz–Minkowski spaces’, Ann. of Math. (2) 104 (1976), 407–419.CrossRefGoogle Scholar
[3]Fang, Y., ‘On minimal annuli in a slab’, Comment. Math. Helv. 69 (1994), 417–430.CrossRefGoogle Scholar
[4]Fang, Y., ‘Lectures on minimal surfaces in ℝ3’, Proc. Centre Math. Appl. Austral. Nat. Univ. 35 (1996).Google Scholar
[5]Fang, Y. and Hwang, J. F., ‘A note on Shiffman’s theorems’, Geom. Dedicata 81 (2000), 167–171.CrossRefGoogle Scholar
[6]Fang, Y. and Wei, F., ‘On uniqueness of Riemann’s examples’, Proc. Amer. Math. Soc. 126 (1998), 1531–1539.CrossRefGoogle Scholar
[7]Fernandez, I., Lopez, F. J. and Souam, R., ‘The space of complete embedded maximal surfaces with isolated singularities in the three-dimensional Lorentz–Minkowski space’, Math. Ann. 332 (2005), 605–643.CrossRefGoogle Scholar
[8]Fernandez, I., Lopez, F. J. and Souam, R., ‘The moduli space of embedded singly periodic maximal surfaces with isolated singularities in the Lorentz–Minkowski space 𝕃3’, Manuscripta Math. 122 (2007), 439–463.CrossRefGoogle Scholar
[9]Gilbarg, D. and Trudinger, N., Elliptic Partial Differential Equations of Second Order, 3rd edn Grundlehren der Mathematischen Wissenschaften, 224 (Springer, Berlin, 2001).CrossRefGoogle Scholar
[10]Huber, A., ‘On subharmonic functions and differential geometry in the large’, Comment. Math. Helv. 32 (1957), 13–72.Google Scholar
[11]Imaizumi, T., ‘Maximal surfaces with simple ends’, Kyushu J. Math. 58 (2004), 59–70.CrossRefGoogle Scholar
[12]Kim, Y. W. and Yang, S. D., ‘A family of maximal surfaces in Lorentz–Minkowski three-space’, Proc. Amer. Math. Soc. 134 (2006), 3379–3390.CrossRefGoogle Scholar
[13]Klyachin, A. A., ‘Description of the set of singular entire solutions of the maximal surface equation’, Sb. Math. 194 (2003), 1035–1054.CrossRefGoogle Scholar
[14]Kobayashi, O., ‘Maximal surface in the three-dimensional Minkowski space 𝕃3’, Tokyo J. Math. 6 (1983), 297–309.CrossRefGoogle Scholar
[15]Kobayashi, O., ‘Maximal surface with conelike singularities’, J. Math. Soc. Japan 36 (1984), 609–617.Google Scholar
[16]López, F. J., López, R. and Souam, R., ‘Maximal surface of Riemann type in Lorentz–Minkowski space 𝕃3’, Michigan Math. J. 47 (2000), 469–497.CrossRefGoogle Scholar
[17]Nitsche, J. C. C., Lectures on Minimal Surfaces, Vol. 1 (Cambridge University Press, Cambridge, 1989).Google Scholar
[18]Shiffman, M., ‘On surfaces of stationary area bounded by two circles, or convex curves, in parallel planes’, Ann. of Math. (2) 63 (1956), 77–90.CrossRefGoogle Scholar
[19]Umehara, M. and Yamada, K., ‘Maximal surfaces with singularities in Minkowski space’, Hokkaido Math. J. 35 (2006), 13–40.CrossRefGoogle Scholar