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On the nonexistence of stable minimal submanifolds and the Lawson-Simons conjecture

Ze-Jun Hu; Guo-Xin Wei — 2003

Colloquium Mathematicae

Let M̅ be a compact Riemannian manifold with sectional curvature $K_{M ̅}$ satisfying $1 / 5 < K_{M ̅} \leq 1$ (resp. $2 \leq K_{M ̅} < 10$ ), which can be isometrically immersed as a hypersurface in the Euclidean space (resp. the unit Euclidean sphere). Then there exist no stable compact minimal submanifolds in M̅. This extends Shen and Xu’s result for 1/4-pinched Riemannian manifolds and also suggests a modified version of the well-known Lawson-Simons conjecture.

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