On the first eigenvalue of invariant K\"ahler metrics

TL;DR

This paper studies the first eigenvalue of the Laplacian on invariant K"ahler Einstein metrics on compact flag manifolds, identifying critical points and characterizing maxima, especially on full flag manifolds.

Contribution

It provides necessary and sufficient conditions for invariant K"ahler Einstein metrics to be critical points of the first eigenvalue functional, with a complete characterization on full flag manifolds.

Findings

g is a critical point iff certain conditions are met

g is a maximum only when M=SU(3)/T^2

On full flag manifolds, critical points are characterized explicitly

Abstract

Given a simply connected compact generalized flag manifold M together with its invariant K\"ahler Einstein metric g, we investigate the functional given by the first eigenvalue of the Hodge Laplacian on smooth functions restricted to the space of invariant K\"ahler metrics. We give sufficient and necessary conditions so that the metric g is a critical point for this functional. Moreover we prove that when M is a full flag manifold, the metric g is critical if and only if M= SU(3)/T^2 and in this case g is a maximum.