A note on first eigenvalue estimates by coupling methods in K\"ahler and quaternion K\"ahler manifolds

TL;DR

This paper uses coupling methods to derive estimates for the first eigenvalue on Kähler and quaternion Kähler manifolds based on geometric parameters like dimension, diameter, and curvature bounds.

Contribution

It introduces a novel application of Kendall-Cranston coupling to obtain eigenvalue estimates in Kähler and quaternion Kähler geometries.

Findings

Eigenvalue estimates depend on dimension, diameter, and curvature bounds.

Coupling methods provide new tools for spectral geometry in complex and quaternionic manifolds.

Results extend previous bounds to specific geometric contexts.

Abstract

In this short note, using the Kendall-Cranston coupling, we study on K\"ahler (resp. quaternion K\"ahler) manifolds first eigenvalue estimates in terms of dimension, diameter, and lower bounds on the holomorphic (resp. quaternionic) sectional curvature.