Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-13T09:13:29.220Z Has data issue: false hasContentIssue false
  • English
  • Français

Minimal annuli in R3 bounded by non-compact complete convex curves in parallel planes

Part of: Partial differential equations

Published online by Cambridge University Press:  09 April 2009

Yi Fang
Yi Fang
Affiliation:
Centre for Mathematics and its Applications School of Mathematical SciencesThe Australian National UniversityCanberra ACT 0200Australia e-mail: yi@maths.anu.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we consider the Plateau problem for surfaces of annular type bounded by a pair of convex, non-compact curves in parallel planes. We prove that for certain symmetric boundaries there are solutions to the non-compact Plateau problems (Theorem B). Except for boundaries consisting of a pair of parallel straight lines, these are the first known examples.

MSC classification

Secondary: 35A10: Cauchy-Kovalevskaya theorems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 60 , Issue 3 , June 1996 , pp. 369 - 388
Copyright
Copyright © Australian Mathematical Society 1996

References

[1]Barbosa, L. and do Carmo, M., ‘On the size of a stable minimal surface in R3’, Amer. J. Math. 98 (1976), 515–528.CrossRefGoogle Scholar
[2]Douglas, J., ‘The problem of Plateau for two contours’, J. Math. Phys. Massachusetts Inst. of Tech. 10 (1930–1931), 315–359.Google Scholar
[3]Fang, Y., ‘On minimal annuli in a slab’, Comment. Math. Helv. (3) 69 (1994), 417–430.CrossRefGoogle Scholar
[4]Hoffman, D., Karcher, H. and Rosenberg, H., ‘Embedded minimal annuli in R3 bounded by a pair of straight lines’, Comment. Math. Helv. 66 (1991), 599–617.CrossRefGoogle Scholar
[5]Hoffman, D. and Meeks, W. III, ‘A variational approach to the existence of complete embedded minimal surfaces’, Duke Math. J. (3) 57 (1989), 877–893.Google Scholar
[6]Meeks, W. III, and White, B., ‘Minimal surfaces bounded by convex curves in paralles planes’, Comment. Math. Helv. 66 (1991), 263–278.CrossRefGoogle Scholar
[7]Nitsche, J. C. C., Lectures on minimal surfaces, vol. I (Cambridge University Press, 1989).Google Scholar
[8]Osserman, R., ‘Minimal surfaces, Gauss maps, total curvature, eigenvalue estimates, and stability’, in: The Chern Symposium, 1979 Proc. Internal. Sympos., Berkeley, Calif. (1979) pp. 199–227.Google Scholar
[9]Osserman, R. and Schiffer, M., ‘Doubly-connected minimal surfaces’, Arch. Rational Mech. Anal. 58 (1975), 285–307.CrossRefGoogle Scholar
[10]Schoen, R., ‘Estimates for stable minimal surfaces in three dimensional manifolds’, in: Seminar on minimal submanifolds, Annals of Mathematics Studies 103 (Princeton University Press, 1983).Google Scholar
[11]Shiffman, M., ‘On surfaces of stationary area bounded by two circles, or convex curves, in paralles planes’, Ann. Math. 63 (1956), 77–90.CrossRefGoogle Scholar
[12]Toubiana, E., ‘On the minimal surfaces of Riemann’, Comment. Math. Helv. 67 (1992), 546–570.CrossRefGoogle Scholar