On the nonexistence of stable currents in submanifolds of a Euclidean space

Abstract

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.