A variational problem for almost complex structures compatible with a given asymptotically complex hyperbolic (ACH) Einstein metric is proposed. Then the known locally unique smooth assignment of an Einstein ACH metric to a given conformal infinity is enhanced to that of a pair of such a metric and a critical almost complex structure. It is also shown that the asymptotic expansion of a critical almost complex structure is determined locally by the conformal infinity up to a certain order.
Almost Complex Structure on $S^{2n}$
Results concerning almost complex structures on the six-sphere
Stable almost complex structures on certain $10$-manifolds
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