Tohoku Mathematical Journal
2004
December
SECOND SERIES VOL. 56, NO. 4

Tohoku Math. J.
56 (2004), 491-499

Title ON THE NONEXISTENCE OF STABLE CURRENTS IN SUBMANIFOLDS OF A EUCLIDEAN SPACE

Author Xueshan Zhang

(Received February 25, 2003, revised November 11, 2003)
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.

2000 Mathematics Subject Classification. Primary 53C20; Secondary 53C40, 49Q15.
Key words and phrases. Stable current, submanifold, shape operator, sectional curvature.

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