A Sharp Lower Bound for the Spectrum of the Hodge Laplacian on Kähler Hyperbolic Manifolds and its Applications
Ye-Won Luke Cho, Young-Jun Choi, Kang-Hyurk Lee
TL;DR
We address sharp lower bounds for the spectrum of the Hodge Laplacian on complete Kähler hyperbolic manifolds by exploiting a $d$-bounded Kähler form $\omega=d\eta$ and the Lefschetz framework. The main technical result provides explicit Rayleigh-quotient bounds $\lambda_0^k(X) \ge \frac{c_k}{\|\eta\|_{L^{\infty}}^2}$ for $k=p+q\neq n$, with detailed constants $c_k$ and $c_{p,q}$, derived via a Lefschetz-based reduction and primitive decomposition. In the middle-dimension case, the paper shows a bound on the orthogonal complement of $\mathcal{H}^{n}_{(2)}(X)$, while for $k=0$ a sharper bound is obtained, with optimality demonstrated in the unit-ball model. As an application, the authors obtain explicit lower bounds for the bottom of the spectrum on irreducible bounded symmetric domains via the Kähler-hyperbolicity length $\mathsf{L}_{\Omega}$ and holomorphic sectional curvature control, recovering a universal bound $\lambda_0(\Omega) \ge \frac{n^2}{4}K$ in terms of $K$, and providing precise bounds for all classical types I–IV and the exceptional V, VI. Overall, the work extends Lefschetz-based vanishing techniques to quantitative spectral estimates and yields sharp, domain-specific spectral data for symmetric spaces.
Abstract
In this paper, we establish a sharp lower bound for the spectrum of the Hodge Laplacian on Kähler hyperbolic manifolds. This bound is expressed explicitly in terms of the supremum norm of the 1-form associated with the Kähler hyperbolic structure. As an application, we obtain explicit spectral lower bounds for bounded symmetric domains.