Deformation of Einstein Metrics and L Cohomology on Strictly Pseudoconvex Domains

We construct new complete Einstein metrics on a smoothly bounded strictly pseudoconvex domain of a Stein manifold. The approach that we take here is to deform the KählerEinstein metric constructed by Cheng and Yau, which generalizes a work of Biquard on the deformations of the complex hyperbolic metric on the unit ball. Recasting the problem into the question of vanishing of an L cohomology and taking advantage of the asymptotic complex hyperbolicity of the Cheng–Yau metric, we establish the possibility of such a deformation when the dimension is at least three.