The rigidity on the second fundamental form of projective manifolds

Abstract

Let $M$ be a complex $n$-dimensional projective manifold in ${\mathbb{P}}^{n+r}$ endowed with the Fubini-Study metric of constant holomorphic sectional curvature $1$, $\sigma$ its second fundamental form, and ${\underline{|\sigma |}}^2$ the mean value of the squared length of $\sigma$ on $M$. We derive a formula for ${\underline{|\sigma |}}^2$ and classify them when ${\underline{|\sigma |}}^2\le 2n$. We present several applications to these results. The first application is to confirm a conjecture of Loi and Zedda, which characterizes the linear subspace and the quadric in terms of the $L^2$-norm of $\sigma$. The second application is to improve a result of Cheng solving an old conjecture of Oguie from pointwise case to mean case. The third application is to give an optimal second gap value on ${\underline{|\sigma |}}^2$, which can be viewed as a complex analog to those on minimal submanifolds in the unit spheres.

10 pages, final version to appear in Manuscripta Mathematica