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Remarks on extremal Kähler metrics on ruled manifolds

Published online by Cambridge University Press:  22 January 2016

Akira Fujiki
Akira Fujiki*
Affiliation:
Kyoto University and Yoshida College, Kyoto 606, Japan
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Let X be a compact Kähler manifold and γ Kähler class. For a Kàhler metric g on X we denote by Rg the scalar curvature on X According to Calabi [3][4], consider the functional defined on the set of all the Kähler metrics g whose Kähler forms belong to γ, where dvg is the volume form associated to g. Such a Kähler metric is called extremal if it gives a critical point of Ф. In particular, if Rg is constant, g is extremal. The converse is also true if dim L(X) = 0, where L(X) is the maximal connected linear algebraic subgroup of AutoX (cf. [5]). Note also that any Kähler-Einstein metric is of constant scalar curvature.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 126 , June 1992 , pp. 89 - 101
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1992

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