Maximal Normal Curvature and Veronese Rigidity

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 $F:(\mathrm{\Sigma },g)\looparrowright \bar{B}(1)$, define the maximal normal curvature by $$\kappa (F):=\underset{x\in \mathrm{\Sigma }}{\sup }\underset{\begin{array}{c}v\in T_x\mathrm{\Sigma }\\ \mathrm{\mid }v{\mathrm{\mid }}_g=1\end{array}}{\sup }\mathrm{\mid }{A}_x(v,v)\mathrm{\mid }\mathrm{.}$$ If ${\mathrm{\Sigma }}^{2n}$ is almost Hermitian with harmonic fundamental two-form, or ${\mathrm{\Sigma }}^{4n}$ is almost quaternion-Hermitian with harmonic fundamental four-form, $n\ge 2$, then $$\kappa (F)\ge \sqrt{\frac{2n}{n+1}}\mathrm{.}$$ 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.