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Abstract We consider the obstruction flatness problem for small deformations of the standard CR 3-sphere. That rigidity holds for the CR sphere was previously known (in all dimensions) for the case of embeddable CR structures, where it also holds at the infinitesimal level. In the 3-dimensional case, however, a CR structure need not be embeddable.Unlike in the embeddable case, it turns out that in the nonembeddable case there is an infinite-dimensional space of solutions to the linearized obstruction flatness equation on the standard CR 3-sphere and this space defines a natural complement to the tangent space of the embeddable deformations. In spite of this, we show that the CR 3-sphere does not admit nontrivial obstruction flat deformations, embeddable or nonembeddable.more » « less
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On the Classification of Normal Stein Spaces and Finite Ball Quotients With Bergman–Einstein Metricsnull (Ed.)Abstract We study the Bergman metric of a finite ball quotient $$\mathbb{B}^n/\Gamma $$, where $$n \geq 2$$ and $$\Gamma \subseteq{\operatorname{Aut}}({\mathbb{B}}^n)$$ is a finite, fixed point free, abelian group. We prove that this metric is Kähler–Einstein if and only if $$\Gamma $$ is trivial, that is, when the ball quotient $$\mathbb{B}^n/\Gamma $$ is the unit ball $${\mathbb{B}}^n$$ itself. As a consequence, we characterize the unit ball among normal Stein spaces with isolated singularities and abelian fundamental groups in terms of the existence of a Bergman–Einstein metric.more » « less
