Canonical almost complex structures on ACH Einstein manifolds

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 $(0,1)$-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.

28 pages. The statement of Theorem 1.2 is modified; substantial corrections and clarifications are added throughout the text

References in corpus (2)

• Taming symplectic forms and the Calabi-Yau equation
• Deformation of Einstein metrics and $L^2$ cohomology on strictly pseudoconvex domains