Maximal Normal Curvature and Veronese Rigidity

Abstract

We prove a sharp Veronese rigidity theorem for closed immersed submanifolds of the Euclidean unit ball under intrinsic harmonic-structure assumptions. For an isometric immersion $F:(\Sigma,g)\looparrowright\overline B(1)$, define the maximal normal curvature by \[ \kappa(F):= \sup_{x\in\Sigma} \sup_{\substack{v\in T_x\Sigma\\ |v|_g=1}} |A_x(v,v)|. \] If $\Sigma^{2n}$ is almost Hermitian with harmonic fundamental two-form, or $\Sigma^{4n}$ is almost quaternion-Hermitian with harmonic fundamental four-form, $n\ge2$, then \[ \kappa(F)\ge \sqrt{\frac{2n}{n+1}} . \] In the equality case the harmonic form is parallel and the immersion is, up to a totally geodesic inclusion, the standard complex or quaternionic Veronese embedding of projective spaces. The key input is a Bochner--Gauss mechanism that turns the Bochner curvature term of the harmonic form into a sharp algebraic estimate for the shape operators.