The first gap for minimal submanifolds in spheres

Let f:Mⁿ→n+q(1) be a minimal immersion of a closed connected manifold with n≥3 and q≥2, and let S be the squared length of its second fundamental form. We prove an integral inequality that gives maxM S≥ n if the immersion is not totally geodesic. If n≠4 or $q=2$, equality maxM S=n holds if and only if the immersion is a Clifford minimal hypersurface; if $n=4$ and q≥3, equality holds if and only if the immersion is a Clifford minimal hypersurface or the standard minimal embedding of CP² in ⁷(1). Both classifications are understood up to an ambient isometry, a totally geodesic inclusion, and a finite covering. We construct the integral inequalities using polynomial weights and divergence identities. The required algebraic estimates are proved by sums of squared tensor norms.