A splitting theorem for extremal Kaehler metrics

TL;DR

This paper proves that extremal Kaehler metrics on product manifolds split into products of extremal metrics on each factor under certain conditions, extending Yau's splitting result for Kaehler-Einstein metrics.

Contribution

It generalizes Yau's splitting theorem to extremal Kaehler metrics, under conditions on Futaki invariants and automorphism groups.

Findings

Extremal Kaehler metrics on product manifolds split into factors.

The splitting holds when Futaki invariants vanish or automorphism groups satisfy constraints.

Extends Yau's splitting theorem from Kaehler-Einstein to extremal metrics.

Abstract

Based on recent work of S. K. Donaldson and T. Mabuchi, we prove that any extremal Kaehler metric in the sense of E. Calabi, defined on the product of polarized compact complex projective manifolds is the product of extremal Kaehler metrics on each factor, provided that the integral Futaki invariants of the polarized manifold vanish or its automorphism group satisfies a constraint. This extends a result of S.-T. Yau about the splitting of a Kaehler-Einstein metric on the product of compact complex manifolds to the more general setting of extremal Kaehler metrics.