Rigidity Results for Hermitian-Einstein manifolds

TL;DR

This paper investigates a differential operator on the sphere bundle of Kähler-Einstein manifolds, establishing eigenvalue bounds and rigidity theorems that classify certain complex space forms and product surfaces.

Contribution

It introduces bounds for the Laplacian's first eigenvalue and derives rigidity theorems that classify specific Hermitian-Einstein manifolds.

Findings

Lower bound for the first Laplacian eigenvalue on sphere bundles

Rigidity theorems classifying complex space forms

Classification of product of two projective lines as Kähler-Einstein surfaces

Abstract

A differential operator introduced by A. Gray on the unit sphere bundle of a K\"ahler-Einstein manifold is studied. A lower bound for the first eigenvalue of the Laplacian for the Sasaki metric on the unit sphere bundle of a K\"ahler-Einstein manifold is derived. Some rigidity theorems classifying complex space forms amongst compact Hermitian surfaces and the product of two projective lines amongst all K\"ahler-Einstein surfaces are then derived.