Second Eigenvalue Estimates and Spectral Rigidity for Submanifolds of Compact Rank-One Symmetric Spaces

In this paper, we establish upper bounds for the second eigenvalue of the Schr"odinger operator $L=\mathrm{\Delta }+\mathrm{\mid }\sigma {\mathrm{\mid }}^2+k$ 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.