A Sharp Lower Bound for the Spectrum of the Hodge Laplacian on K\"ahler Hyperbolic Manifolds and its Applications

TL;DR

This paper derives a precise lower bound for the spectrum of the Hodge Laplacian on Kähler hyperbolic manifolds, with applications to bounded symmetric domains, advancing understanding of geometric analysis in complex geometry.

Contribution

It provides a new explicit lower bound for the Hodge Laplacian spectrum on Kähler hyperbolic manifolds, linking it to the supremum norm of an associated 1-form.

Findings

Established a sharp spectral lower bound for Kähler hyperbolic manifolds.

Derived explicit bounds for bounded symmetric domains.

Connected spectral bounds to geometric structures via 1-forms.

Abstract

In this paper, we establish a sharp lower bound for the spectrum of the Hodge Laplacian on K\"ahler hyperbolic manifolds. This bound is expressed explicitly in terms of the supremum norm of the 1-form associated with the K\"ahler hyperbolic structure. As an application, we obtain explicit spectral lower bounds for bounded symmetric domains.