Abstract
The last ten years have seen an intense activity on certain questions that arise in connection with the study of minimal surfaces. Among such questions one should mention those of regularity, embeddability, stability and finiteness of the number of minimal surfaces spanning a given boundary. In this lecture I would like to describe a few ideas, results and problems related to the questions of stability and finiteness. For simplicity, I will restrict myself to minimal surfaces of the topological type of the disk in a Riemannian manifold.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J. L. Barbosa and M. do Carmo, On the size of a stable minimal surface in R 3 Amer. J. Math. 98 (1976), 515—526.
Stability of minimal surfaces and eigenvalues of the Laplacian (preprint).
R. Böhme and F. Tomi, Zür Struktur der Lösungmenge Plateauproblems, Math. Z. 193 (1973), 1—29.
R. Böhme and A. Tromba, The index theorem/or classical minimal surfaces (preprint, Bonn 1977).
R. Gulliver, The Plateau problem for surfaces of prescribed mean curvature in a Riemannian manifold, J. Differential Geometry 8 (1972), 317—330.
R. Gulliver and J. Spruck, On embedded minimal surfaces, Ann. of Math. 103 (1976), 331—347.
H. Kaul, lsoperimetrische Ungleichung und Gauss—Bonnet Formel für H—F/iichen in Riemannschen Mannigfa/tigkeiten, Arch. Rational Mech. Anal. 45 (1972), 194—221.
H. B. Lawson, Lectures on minimal submanifolds, IMPA, Rio de Janeiro, 1973.
W. H. Meeks Ill, Lectures on Plateau's problem, IMPA, Rio de Janeiro, 1978.
L. P. de Melo Jorge, Estabilidade C2 das cur vas com soluroes niio degeneradas do problema de Plateau, D. Sc. Thesis, IMPA, 1978.
C. B. Morrey, Jr., The problem of Plateau on a Riemannian manifold, Ann. of Math. 49 (1948), 807—851.
J. C. C. Nitsche, A new uniqueness theorem for minimal surfaces, Arch. Rational Mech. Anal. 52 (1973), 319—329.
Contours bounding at most finitely many solutions of Plateau's problem (pre print).
M. Shiffman, Unstable minimal surfaces with any rectifiable boundary, Proc. Nat. Acad. Sci. U.S.A. 28 (1942), 103—108.
F. Tomi, On the local uniqueness of the problem of least area. Arch. Rational Mech. Anal. 52 (1973), 312—318.
On the finite solvability of Plateau's problem, Lecture Notes in Math. Vol. 597, Springer—Verlag, Berlin and New York, 1977, pp. 679—695.
A. Tromba, On the number of simply—connected minimal surfaces spanning a curve. Mem. Amer. Math. Soc. No. 194 (1977).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Carmo, M.P.d. (2012). Minimal Surfaces: Stability and Finiteness. In: Tenenblat, K. (eds) Manfredo P. do Carmo – Selected Papers. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25588-5_12
Download citation
DOI: https://doi.org/10.1007/978-3-642-25588-5_12
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-25587-8
Online ISBN: 978-3-642-25588-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)