Uniqueness of Kähler-Ricci solitons.
type: Item
instance of: scholarly article
title: Uniqueness of Kähler-Ricci solitons.
zbMATH Open document ID: 1036.53052
DOI: 10.1007/BF02392630
author: Xiaohua Zhu
author: Gang Tian
published in: Acta Mathematica
publication date: 2000-12-06T00:00:00Z
review text: Let \(X\) be a holomorphic vector field on a compact Kähler manifold \((M,\omega)\). Then \(\omega\) is a Kähler-Ricci soliton with respect to \(X\) if \(\text{Ric} (\omega)-\omega=L_X\omega\) where \(\text{Ric} (\omega)\) is the Ricci form and \(L_X\) is the Lie derivative with respect to \(X\). It is easily seen that in this case \(c_1(M)>0\) and that \(c_1(M)\) is represented by \(\omega\). On the other hand, by computing the Futaki invariant one finds that if a Kähler-Ricci soliton exists, then the manifold cannot admit any Kähler-Einstein metrics. The paper under review proves the uniqueness (modulo a certain subgroup of Kähler automorphisms) of a Kähler-Ricci soliton on a fixed compact Kähler manifold. The proof consists of discussing a Monge-Ampère equation depending on a parameter \(t\in [0,1]\) such that for \(t=1\) one obtains the Kähler-Ricci soliton equation.
reviewed by: Liviu Ornea
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 35Q51
Mathematics Subject Classification ID: 32Q20
Mathematics Subject Classification ID: 32W20
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C44
zbMATH DE Number: 1541224
zbMATH Keywords: Kähler manifold
zbMATH Keywords: Kähler-Ricci soliton
zbMATH Keywords: Monge-Ampère equation
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1541224