A form of Bernstein theorem states that a complete stable minimal surface in euclidean space is a plane. A generalization of this statement is that there exists no complete stable hypersurface of an $n$-euclidean space with vanishing $(n-1)$-mean curvature and nowhere zero Gauss-Kronecker curvature. We show that this is the case, provided the immersion is proper and the total curvature is finite.
Publié le : 2004-06-14
Classification:
Stability,
r-mean curvature,
complete,
finite total curvature,
53C42
@article{1113246548,
author = {do Carmo, Manfredo and Elbert, Maria F.},
title = {On stable complete hypersurfaces with vanishing {$r$}-mean curvature},
journal = {Tohoku Math. J. (2)},
volume = {56},
number = {1},
year = {2004},
pages = { 155-162},
language = {en},
url = {http://dml.mathdoc.fr/item/1113246548}
}
do Carmo, Manfredo; Elbert, Maria F. On stable complete hypersurfaces with vanishing {$r$}-mean curvature. Tohoku Math. J. (2), Tome 56 (2004) no. 1, pp. 155-162. http://gdmltest.u-ga.fr/item/1113246548/