Rigidity Results for Hermitian-Einstein manifolds

Abstract

A differential operator introduced by A. Gray on the unit sphere bundle of a Kähler-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 Kähler-Einstein manifold is derived. Some rigidity theorems classifying complex space forms amongst compact Hermitian surfaces and the product of two projective lines amongst all Kähler-Einstein surfaces are then derived.

V4. Some typos corrected. To appear in Math. Proc. Roy. Irish. Acad