当前位置: X-MOL 学术manuscripta math. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
The rigidity on the second fundamental form of projective manifolds
manuscripta mathematica ( IF 0.6 ) Pub Date : 2019-09-16 , DOI: 10.1007/s00229-019-01151-8
Ping Li

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 $\underline{|\sigma|}^2\leq2n$. 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.

中文翻译:

射影流形的第二基本形式的刚性

令 $M$ 是 $\mathbb{P}^{n+r}$ 中的一个复杂的 $n$ 维射影流形,赋予恒定全纯截面曲率 $1$ 的 Fubini-Study 度量,$\sigma$ 是其第二基本$\underline{|\sigma|}^2$ 是 $\sigma$ 在 $M$ 上的平方长度的平均值。我们推导出 $\underline{|\sigma|}^2$ 的公式,并在 $\underline{|\sigma|}^2\leq2n$ 时对它们进行分类。我们介绍了这些结果的几种应用。第一个应用是确认 Loi 和 Zedda 的猜想,该猜想根据 $\sigma$ 的 $L^2$-范数表征线性子空间和二次曲面。第二个应用是改进Cheng 求解Oguie 旧猜想的结果,从逐点情况到均值情况。第三个应用是在 $\underline{|\sigma|}^2$ 上给出一个最优的第二间隙值,
更新日期:2019-09-16
down
wechat
bug