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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On Calabi's extremal metric and properness

Author: Weiyong He
Journal: Trans. Amer. Math. Soc. 372 (2019), 5595-5619
MSC (2010): Primary 53C55; Secondary 35J60
Published electronically: December 10, 2018
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Abstract: In this paper we extend a recent breakthrough of Chen and Cheng on the existence of a constant scalar Kähler metric on a compact Kähler manifold to Calabi's extremal metric. There are no new a priori estimates needed, but rather there are necessary modifications adapted to the extremal case. We prove that there exists an extremal metric with extremal vector $ V$ if and only if the modified Mabuchi energy is proper, modulo the action of the subgroup in the identity component of the automorphism group which commutes with the flow of $ V$. We introduce two essentially equivalent notions, called reductive properness and reduced properness. We observe that one can test reductive properness/reduced properness only for invariant metrics. We prove that existence of an extremal metric is equivalent to reductive properness/reduced properness of the modified Mabuchi energy.

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Weiyong He
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403

Received by editor(s): February 7, 2018
Received by editor(s) in revised form: October 11, 2018
Published electronically: December 10, 2018
Additional Notes: The author is partly supported by an NSF grant, award no. 1611797.
Article copyright: © Copyright 2018 American Mathematical Society