# Canonical almost complex structures on ACH Einstein manifolds

## TL;DR

This paper studies a variational problem for almost complex structures on ACH Einstein manifolds, revealing deformation results and how boundary CR structures influence the asymptotic behavior of solutions.

## Contribution

It introduces a new variational framework for almost complex structures on ACH Einstein manifolds and analyzes their deformations and boundary asymptotics.

## Key findings

- Deformation results for Einstein ACH metrics with critical almost complex structures.
- Asymptotic expansion of critical structures determined by boundary CR structures.
- Linearized equation given by the Dolbeault Laplacian on (0,1)-forms.

## 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 K\"ahler-Einstein structures is given by the Dolbeault Laplacian acting on $(0,1)$-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 non-integrable) CR structure on the boundary at infinity up to a certain order.

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Source: https://tomesphere.com/paper/1812.09633