On Lu's second gap conjecture in higher codimension

Let $M^n\to\Sph^{n+q}(1)$, $n\ge3$ and $q\ge2$, be a closed minimal immersion. Set $S=|h|^2$ and $Q=S+λ_2$, where $h$ is the second fundamental form and $λ_2$ is the second largest eigenvalue of Lu's fundamental matrix. We establish two complementary results concerning Lu's second-gap conjecture. Fi