We construct new complete Einstein metrics on smoothly bounded strictly pseudoconvex domains in Stein manifolds. This is done by deforming the Kahler-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.

中文翻译：

---

爱因斯坦度量的变形和严格伪凸域上的 $L^2$ 上同调

我们在 Stein 流形中的平滑有界严格赝凸域上构建新的完整爱因斯坦度量。这是通过对 Cheng 和 Yau 的 Kahler-Einstein 度量进行变形来完成的，这种方法概括了 Roth 和 Biquard 在单位球上的复双曲度量变形方面的工作。将问题重新转化为$L^2$上同调消失的问题，并利用Cheng-Yau度量的渐近复双曲性，我们建立了当域的维数大于或等于时这种变形的可能性到三个。