Deformation of Einstein metrics and 𝐿² cohomology on strictly pseudoconvex domains

Abstract

We construct new complete Einstein metrics on smoothly bounded strictly pseudoconvex domains in Stein manifolds. This is done by deforming the Kähler–Einstein metric of Cheng and Yau, the approach that generalizes the works of Roth and Biquard on the deformations of the complex hyperbolic metric on the unit ball. Recasting the problem into the question of the vanishing of an L 2 L^2 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 of the domain is larger than or equal to three.