In 1973, Lawson and Simons conjectured that there are no stable currents in any compact, simply connected Riemannian manifold $M^m$ which is $1/4$-pinched. In this paper, we regard $M^m$ as a submanifold immersed in a Euclidean space and prove the conjecture under some pinched conditions about the sectional curvatures and the principal curvatures of $M^m$. We also show that there is no stable $p$-current in a submanifold of $M^m$ and the $p$-th homology group vanishes when the shape operator of the submanifold satisfies certain conditions.
@article{1113246746,
author = {Zhang, Xueshan},
title = {On the nonexistence of stable currents in submanifolds of a Euclidean space},
journal = {Tohoku Math. J. (2)},
volume = {56},
number = {1},
year = {2004},
pages = { 491-499},
language = {en},
url = {http://dml.mathdoc.fr/item/1113246746}
}
Zhang, Xueshan. On the nonexistence of stable currents in submanifolds of a Euclidean space. Tohoku Math. J. (2), Tome 56 (2004) no. 1, pp. 491-499. http://gdmltest.u-ga.fr/item/1113246746/