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
-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
