Flat minimal tori and Lu's second-gap conjecture

Lu’s first pinching theorem states that a closed minimal $n$-dimensional submanifold of the unit sphere satisfying $0\le S+{\lambda }_2\le n$ is one of the standard first-gap models; here $S$ is the squared norm of the second fundamental form and ${\lambda }_2$ is the second eigenvalue of Lu’s fundamental matrix. Lu’s second-gap conjecture asserts that, once $S+{\lambda }_2$ is constant and strictly larger than $n$, it is separated from $n$ by a positive gap depending only on the dimension and codimension. We construct closed embedded counterexamples for minimal surfaces in every codimension at least three. More precisely, in every odd codimension $q\ge 3$ the constant values of $S+{\lambda }_2$ realized by linearly full embedded flat minimal tori are dense in $(2,3)$. Thus the analogue of Chern’s discreteness statement fails for Lu’s refined quantity.