Lawson–Simons conjecture on stable minimal submanifolds in strictly quarter-pinched manifolds

Let $M$ be a compact, simply connected Riemannian manifold whose sectional curvature lies in $(1/4,1]$. A closed minimal submanifold of $M$ is stable when its second variation satisfies

$$I(V,V):=\int_\Sigma \langle \mathfrak{J}V,V\rangle\geq 0$$

for every section $V$ of the normal bundle. Lawson–Simons conjecture. There are no closed stable minimal submanifolds in $M$. This conjecture extends the known nonexistence results for stable minimal submanifolds in the standard sphere and the classification in compact rank-one symmetric spaces. It remains open, although known results support it.