Rigidity results for Hermitian-Einstein manifolds
type: Item
instance of: scholarly article
title: Rigidity results for Hermitian-Einstein manifolds
zbMATH Open document ID: 1356.53051
DOI: 10.3318/PRIA.2016.116.03
published in: Mathematical Proceedings of the Royal Irish Academy
publication date: 2016-09-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/1311.6279
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 32Q15
zbMATH DE Number: 6629393
zbMATH Keywords: first eigenvalue
zbMATH Keywords: Kähler-Einstein
zbMATH Keywords: Gray's differential operator
zbMATH Keywords: rigidity theorems
MaRDI profile type: Publication
arXiv ID: 1311.6279
OpenAlex ID: W2963390956
author: Stuart J. Hall
author: T. Murphy
review text: 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.
reviewed by: Vamsi Pritham Pingali
description: scientific article; zbMATH DE number 6629393