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.

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.