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Free, publicly-accessible full text available August 1, 2024
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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
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We investigate regularity of CR-mappings between real-analytic infinite type hypersurfaces in $\mathbb{C}^2$. We show that, under the Fuchsian type condition, all (respectively formal or smooth) CR-diffeomorphisms between them are automatically analytic. The Fuchsian condition appears to be in a certain sense optimal for the regularity problem.more » « less