Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-10T21:03:45.478Z Has data issue: false hasContentIssue false
  • English
  • Français

A uniqueness theorem for properly embedded minimal surfaces bounded by straight lines

Part of: Classical differential geometry Global differential geometry

Published online by Cambridge University Press:  09 April 2009

Francisco J. Lopez  and
Francisco Martin
Francisco J. Lopez
Affiliation:
Departamento de Geometría y Topología Universidad de Granada18071 GranadaSpain e-mail: fjlopez@ugr.es, fmartin@ugr.es
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 prove a uniqueness theorem for minimal discs in R3 spanning a polygonal boundary.

Keywords

properly embedded minimal surfacesboundarystraight lines

MSC classification

Secondary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature 53C42: Immersions (minimal, prescribed curvature, tight, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 69 , Issue 3 , December 2000 , pp. 362 - 402
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Alexandrov, A. D., ‘Uniqueness theorems for surfaces in the large I’, Vestnik Leningrad. Univ. 11 (1956), 5–17;Google Scholar
English translation: Amer. Math. Soc. Transl. (2) 21 (1962), 341–354.Google Scholar
[2]Burckel, R. B., An introduction to classical complex analysis vol. 1 (Birkhäuser, Basel, 1979).CrossRefGoogle Scholar
[3]Dierkes, U., Hildebrandt, S., Küster, A. and Wohlrab, O., Minimal Surfaces I, Grundlehren Math. Wiss. 295 (Springer, Berlin, 1992).Google Scholar
[4]Fang, Y., ‘Minimal annuli in R 3 bounded by non-compact complete convex curves in parallel planes’, J. Austral. Math. Soc. (Series A) 60 (1996), 369–388.Google Scholar
[5]Hoffman, D. and Ill, W. H. Meeks, ‘The asymptotic behavior of properly embedded minimal surfaces of finite topology’, J. Amer. Math. Soc. (4) 2 (1989), 667–682.CrossRefGoogle Scholar
[6]Jenkins, H. and Serrin, J., ‘Variational problems of minimal surface type. II. Boundary value problems for the minimal surface equation’, Arch. Rat. Mech. Anal. 21 (1966), 321–342.CrossRefGoogle Scholar
[7]Karcher, H., ‘Construction of minimal surfaces’, in: Surveys in Geomentry 1989/90 (University of Tokyo, Tokyo, 1989). Also in: Vorlesungsreihe Nr. 1, SFB 256, Bonn, 1989.Google Scholar
[8]López, F. J., López, R. and Souam, R., ‘Maximal surfaces of Riemann type in Lorentz-Minkowski space L 3’, Michigan Math. J., to appear.Google Scholar
[9]López, F. J. and Martín, F., ‘Minimal surfaces in a wedge of a slab’, Comm. Anal. Geom., to appear.Google Scholar
[10]López, F. J. and Wei, F., ‘Properly immersed minimal discs bounded by straight lines’, Math. Ann., to appear.Google Scholar
[11]Meeks, W. H. III and Rosenberg, H., ‘The geometry and conformal structure of properly embedded minimal surfaces of finite topology in R 3’, Invent. Math. 114 (1993), 625–639.Google Scholar
[12]Nitsche, J. C. C., ‘A supplement to the condition of J. Douglas’, Rend. Circ. Mat. Palermo (2) 13 (1964), 192–198.CrossRefGoogle Scholar
[13]Nitsche, J. C. C., Lectures on minimal surfaces, vol. 1 (Cambridge University Press, Cambridge, 1989).Google Scholar
[14]Osserman, R., A survey of minimal surfaces, 2nd edition (Dover Publications, New York, 1986).Google Scholar
[15]Rossman, W., ‘Minimal surfaces with planar boundary curves’, Kyushu J. Math. 52 (1998), 209–225.CrossRefGoogle Scholar
[16]Schoen, R., ‘Uniqueness, symmetry and embeddedness of minimal surfaces’, J. Differential Geom. 18 (1983), 791–809.CrossRefGoogle Scholar