#### Vol. 314, No. 2, 2021

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Canonical almost complex structures on ACH Einstein manifolds

### Yoshihiko Matsumoto

Vol. 314 (2021), No. 2, 375–410
##### Abstract

On asymptotically complex hyperbolic (ACH) Einstein manifolds, we consider a certain variational problem for almost complex structures compatible with the metric, for which the linearized Euler–Lagrange equation at Kähler–Einstein structures is given by the Dolbeault Laplacian acting on $\left(0,1\right)$-forms with values in the holomorphic tangent bundle. A deformation result of Einstein ACH metrics associated with critical almost complex structures for this variational problem is given. It is also shown that the asymptotic expansion of a critical almost complex structure is determined by the induced (possibly nonintegrable) CR structure on the boundary at infinity up to a certain order.

##### Keywords
asymptotically complex hyperbolic spaces, almost CR structures
##### Mathematical Subject Classification
Primary: 53C15
Secondary: 32T15, 32V15, 53B35, 53C25