The problem of stability of minimal submanifolds in Riemannian and pseudo-Riemannian spaces. (English) Zbl 1073.53072

Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 2nd international conference on geometry, integrability and quantization, Varna, Bulgaria, June 7–15, 2000. Sofia: Coral Press Scientific Publishing (ISBN 954-90618-2-5/pbk). 7-32 (2001).
This is a survey on certain aspects of minimal submanifolds, with emphasis on questions of stability. Among other things, the proof is described, given by Pogorelov in the simply-connected case, of a theorem of do Carmo-Peng, and the interesting fact that the substance of Pogorelov’s proof has been generalized in 1996 to a hypersurface \(M^n\) of Euclidean space by Klyachin and Miklukov. The early result of the author that a two-dimensional minimal sphere in a simply-connected Riemannian manifold with sectional curvature \(1/4<K_\sigma\leq 1\) is unstable is presented with a sketch of proof. Some more recent results of the author on the expression of the curvature tensor of a submanifold of Euclidean space defined by a system of equations are also presented
For the entire collection see [Zbl 0957.00038].


53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)