Homogeneous bundles and the first eigenvalue of symmetric spaces

TL;DR

This paper establishes the stability of certain homogeneous bundles and derives a precise upper bound for the first Laplacian eigenvalue on compact Hermitian symmetric spaces, advancing understanding in geometric analysis.

Contribution

It introduces a stability result for homogeneous bundles and provides a sharp eigenvalue estimate for Kähler metrics on symmetric spaces.

Findings

Proves stability of the Gieseker point for homogeneous bundles

Provides a sharp upper bound for the first Laplacian eigenvalue

Applies results to compact Hermitian symmetric spaces of ABCD-type

Abstract

We prove the stability of the Gieseker point of an irreducible homogeneous bundle over a rational homogeneous space. As an application we get a sharp upper estimate for the first eigenvalue of the Laplacian of an arbitrary Kaehler metric on a compact Hermitian symmetric spaces of ABCD--type.