Stability under deformations of Hermitian-Einstein almost-Kähler metrics

Details Stability under deformations of Hermitian-Einstein almost-Kähler metrics Stability under deformations of Hermitian-Einstein almost-Kähler metrics

2012-04-24

arxiv.org/abs/1204.5438v1

Mathematics

Differential Geometry

17 pages. typos and quotations corrected

Scientific paper

In this paper, we continue to study stability under deformations of extremal almost-K\"ahler metrics on a compact symplectic manifold. In dimension 4, we consider a smooth family of compatible almost-complex structures such that at time zero the induced metric is Hermitian-Einstein almost-K\"ahler metric with zero or negative hermitian scalar curvature. We prove, under certain hypothesis, the existence of a smooth family of compatible almost-complex structures, diffeomorphic at each time to the initial family, and inducing constant hermitian scalar curvature metrics. Furthermore, in any dimension, we show that any family of compatible almost-complex structures invariant under a maximal torus in the hamiltonian group of symplectomorphisms, such that at time zero the induced almost-K\"ahler metric is extremal, can be deformed to a smooth family inducing extremal almost-K\"ahler metrics.

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