zbMATH — the first resource for mathematics

On complete minimal surfaces with finite Morse index in three manifolds. (English) Zbl 0573.53038
The author obtains several good results on the Morse index (for area) of an oriented complete minimal surface (M,g) in a Riemannian 3-manifold N. Amongst them: 1) If index M\(<\infty\), there is a compact \(C\subset M\) so that \(M\setminus C\) is stable and there is a positive function \(u: M\to {\mathbb{R}}\) with \(Lu=0\) on \(M\setminus C\), where L is the second variation operator. If N has scalar curvature \(\geq 0\), then \(u^ 2g\) is a complete metric on M with Gaussian curvature \(\geq 0\) on \(M\setminus C\). In particular, M is conformally equivalent to a Riemann surface with a finite number of punctures. 2) If \(N={\mathbb{R}}^ 3\), then index M\(<\infty\) iff M has finite total curvature.
Reviewer: J.Eells

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
PDF BibTeX Cite
Full Text: DOI EuDML
[1] [CY] Cheng, S.Y., Yau, S.I.: Differential equations on Riemannian manifolds and their geometric applications. Comment. Pure Appl. Math.28, 333-354 (1975) · Zbl 0312.53031
[2] [CV] Cohn-Vossen, S.: Kürzeste Wege und Totalkrümmung auf Flächen. Compos. Math.2, 69-133 (1935) · JFM 61.0789.01
[3] [CH] Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. 1, N.Y.: Interscence 1937 · JFM 57.0245.01
[4] [FS] Fischer-Colbrie, D., Schoen, R.: The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature. Comment. Pure Appl. Math.33, 199-211 (1980) · Zbl 0439.53060
[5] [GT] Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Berlin-Heidelberg-New York: Springer 1977 · Zbl 0361.35003
[6] [MT] Morse, M., Tompkins, C.B.: On the existence of minimal surfaces of critical types. Ann. Math.40, 443-472 (1939) · JFM 65.0456.04
[7] [O] Osserman, R.: A Survey of Minimal Surfaces. New York: Van Nostrand Reinhard 1969 · Zbl 0209.52901
[8] [S] Shiffman, M.: The plateau problem for non-relative minima. Ann. Math.40, 834-854 (1939) · Zbl 0023.39802
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.