AI 中文总结
该研究针对紧射影空间及凯莱射影平面上的闭子流形，建立薛定谔算子第二特征值的上界，推导普适估计，证明等式对应全脐子流形，确定达界的全测地例子。
AI 中文摘要
本文针对紧射影空间$\boldsymbol{\text{F}}P^m$（其中$\boldsymbol{\text{F}}\boldsymbol{\boldsymbol{\text{R}},\text{C},\text{H}}$）以及凯莱射影平面上的$k$维闭子流形，建立了薛定谔算子$L=\boldsymbol{\boldsymbol{\text{\rm \text{Δ}}}}+|\boldsymbol{\boldsymbol{\text{σ}}}|^2+k$的第二特征值的上界。该估计通过结合这些空间的标准球嵌入与共形检验函数论证得到。在复、四元数和凯莱情形下，所得界包含修正项，用于记录子流形切空间相对于对应几何结构的位置。我们还推导了仅依赖子流形维数和基础可除代数的普适估计，证明尖锐估计中的等式成立时子流形必为全脐的。最后，利用已知的全脐子流形分类结果，计算了标准实、复、四元数和凯莱模型的第二特征值，并确定了达到尖锐上界的全测地例子。
英文摘要
In this paper, we establish upper bounds for the second eigenvalue of the Schrödinger operator $L=Δ+|σ|^2+k$ on $k$-dimensional closed submanifolds of the compact projective spaces $\mathbb F P^m$, where $\mathbb F\in\{\mathbb R,\mathbb C,\mathbb H\}$, as well as the Cayley projective plane. The estimates are obtained by combining the standard spherical embeddings of these spaces with a conformal test-function argument. In the complex, quaternionic, and Cayley cases, the resulting bounds involve correction terms that record the position of the tangent spaces of the submanifold relative to the corresponding geometric structures. We also derive universal estimates depending only on the dimension of the submanifold and the underlying division algebra. We show that equality in the sharp estimates forces the submanifold to be totally umbilical. Finally, using known classifications of totally umbilical submanifolds, we compute the second eigenvalue for the standard real, complex, quaternionic, and Cayley models and identify the totally geodesic examples that attain the sharp upper bounds.
Comments33 pages. Comments welcome
