Coupled reciprocal-sum inequalities for Neumann eigenvalues in rank-one symmetric spaces

Abstract

Let $M$ be a rank-one symmetric space of compact or noncompact type and real dimension $m=kn$, where $k\in \{1,2,4,8\}$ is the real dimension of the underlying division algebra. Meng and Wang proved sharp isoperimetric inequalities for reciprocal sums of the first $p$ or $\ell$ nonzero Neumann eigenvalues, with indices strictly below $m$ in the nonconstant-curvature cases. We keep the whole $m$-dimensional transplanted first eigenspace of an equimeasurable geodesic ball and treat its mass and stiffness forms as a coupled generalized eigenvalue problem. For every bounded connected smooth domain in a noncompact rank-one symmetricspace we prove the full sharp estimate \[ \sum_{j=1}^{m}\frac{1}{μ_j(Ω)} \ge \frac{m}{μ_1(B)}, \] where $B$ is a geodesic ball with $|B|=|\Omega |$; equality characterizes geodesic balls. The Hopf-direction contribution is encoded by a positive, unital, trace-preserving projection on symmetric matrices, and the finite-dimensional closure is supplied by a positive-projection Jensen inequality. In compact type, under the localization hypothesis of Meng--Wang, the same full estimate is obtained whenever ${\cot }^{2}R\ge k-1$, where $R$ is the radius of the comparison ball, as well as in the isotropic endpoint cases and for Hopf-balanced angular second moment. Consequently the complete $m$-term inequality holds for all complex and quaternionic projective spaces under the original localization assumption, and for the Cayley plane when $R\le \arctan (1/\sqrt{7})$. In the remaining admissible Cayley interval we exhibit a finite-dimensional obstruction showing that the affine tangent-matrix bounds alone cannot yield the full trace estimate. This obstruction concerns the proof mechanism and is not a counterexample to the unresolved spectral inequality.

22 pages