Upper and lower bounds for the first eigenvalue and the volume entropy of noncompact K\"ahler manifolds

TL;DR

This paper establishes bounds for the first eigenvalue and volume entropy of noncompact Kähler manifolds using Calabi's diastasis function, with sharp results for complex hyperbolic spaces and applications to Hermitian symmetric spaces.

Contribution

It introduces bounds based on diastasis and diastatic entropy, providing explicit estimates for eigenvalues in noncompact Kähler and Hermitian symmetric spaces.

Findings

Bounds are sharp for complex hyperbolic space

Explicit lower bounds for eigenvalues of geodesic balls

New relations between eigenvalues, entropy, and diastasis functions

Abstract

We find upper and lower bounds for the first eigenvalue and the volume entropy of a noncompact real analytic K\"ahler manifold, in terms of Calabi's diastasis function and diastatic entropy, which are sharp in the case of the complex hyperbolic space. As a corollary we obtain explicit lower bounds for the first eigenvalue of the geodesic balls of an Hermitian symmetric space of noncompact type.