Rigidity results for Hermitian-Einstein manifolds (Q2821813)

The first eigenvalue of the Laplacian of a Riemannian manifold is an important quantity to study. There are classical bounds on it due to Lichnerowicz if the Ricci curvature is positive. Theorem 1.2 of this paper bounds the first eigenvalue of a Sasaki metric on the sphere bundle of a Kahler-Einstein manifold.NEWLINENEWLINETheorem 1.4 (generalising a theorem of Berger, i.e., Theorem 1.3) computes the integral of the holomorphic sectional curvature \(H\) over any fibre of the sphere bundle. As a consequence (Corollary 1.5), an upper bound on \(H\) is provided on a closed Hermitian surface with equality holding if and only if the surface is a standard symmetric space.NEWLINENEWLINEFinally, Theorem 1.6 proves that \(\mathbb{CP}^1 \times \mathbb{CP}^1\) is the only closed positive scalar curvature Kähler-Einstein surface satisfying \(H_{av} = \frac{2}{3} H_{\max}\).NEWLINENEWLINEThe proofs use a differential operator originally defined by Gray who proved Theorem 1.1 about constant scalar curvature Kähler manifolds with non-negative sectional curvature being locally symmetric spaces of compact type.