On the nonexistence of stable minimal submanifolds and the Lawson–Simons conjecture
Volume 96 / 2003
Colloquium Mathematicum 96 (2003), 213-223
MSC: 53C20, 53C40, 53C42.
DOI: 10.4064/cm96-2-6
Abstract
Let $\, \overline {\! M}$ be a compact Riemannian manifold with sectional curvature $K_{\, \overline {\! M}}$ satisfying $1/5< K_{\, \overline {\! M}}\le 1$ (resp. $2\le K_{\, \overline {\! M}}<10$), 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 $\, \overline {\! M}$. This extends Shen and Xu's result for ${1\over 4}$-pinched Riemannian manifolds and also suggests a modified version of the well-known Lawson–Simons conjecture.
