On the volume of positively curved Kähler manifolds
type: Item
instance of: scholarly article
title: On the volume of positively curved Kähler manifolds
zbMATH Open document ID: 0647.53050
author: Shin Nayatani
published in: Osaka Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: Let M be a complete Kähler manifold of complex dimension n and assume that Ric\(\geq 2(n+1)\delta^ 2\) and \(K_ H\geq 4\delta^ 2\), where Ric and \(K_ H\) denote the Ricci and holomorphic sectional curvature of M, respectively. Here the author's concern is to prove a Bishop-Gromov comparison theorem for M. Let B(p,r) and S(p,r) be the geodesic ball and the geodesic sphere of radius r at \(p\in M\). Then, by computing the mean curvature of S(p,r) with respect to the unit normal of S(p,r), it is proved here that \[ vol(B(p,r))/vol(B(p,r'))\leq V_{\delta,n}(r')/V_{\delta,n}(r),\quad r'\geq r\geq 0, \] where \(V_{\delta,n}(r)\) is the volume of the geodesic ball of radius r in the projective space \(P^ n_{\delta}(C)\). This theorem yields the final result which says that \(vol(M) \leq vol(S_\delta^{2n}\).
reviewed by: W.Miesner
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 4056493
zbMATH Keywords: Kähler manifold
zbMATH Keywords: holomorphic sectional curvature
zbMATH Keywords: Bishop-Gromov comparison theorem
zbMATH Keywords: geodesic ball
zbMATH Keywords: volume
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4056493