A solution to Lu's second gap conjecture

Let $M^n\to\mathbb{S}^{n+q}(1)$ be a closed connected minimal immersion, where $n\ge3$, and set $Q=S+λ_2$, with $S=|h|^2$ and $λ_2$ the second largest eigenvalue of Lu's fundamental matrix. We determine the sharp codimension range for Lu's second-gap conjecture. For every $2\le q\le n$, there exists