Deformation of Einstein metrics and $L^2$ cohomology on strictly pseudoconvex domains

TL;DR

This paper develops new complete Einstein metrics on strictly pseudoconvex domains by deforming existing Kähler-Einstein metrics, linking geometric deformations with $L^2$ cohomology vanishing results in higher dimensions.

Contribution

It introduces a novel deformation method for Cheng-Yau's Kähler-Einstein metrics on pseudoconvex domains, extending previous work on complex hyperbolic metrics.

Findings

Existence of new complete Einstein metrics on pseudoconvex domains in Stein manifolds.

Deformation approach based on $L^2$ cohomology vanishing.

Applicability when the domain dimension is at least three.

Abstract

We construct new complete Einstein metrics on smoothly bounded strictly pseudoconvex domains in Stein manifolds. This is done by deforming the K\"ahler-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 vanishing of an $L^2$ 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.