A conformal invariant and its application to the nonexistence of minimal submanifolds

Abstract

Let $(M^m,g)$ be an $m$-dimensional closed Riemannian manifold with non-negative sectional curvatures, $m\ge 3$. We define a conformal invariant and prove that, if the conformal invariant is bounded from above by a constant depending only on $m$, then there are no closed $n$-dimensional stable minimal submanifolds in $M$ for all $\xi(m)\le n\le m-2$, where $\xi(m)=1$ when $3\le m\le 5$ and $\xi(m)=2$ when $m\ge 6$. In particular, a conformal $m$-sphere with non-negative sectional curvatures does not admit any closed $n$-dimensional stable minimal submanifold for all $\xi(m)\le n\le m-2$.