We derive an explicit formula for the well-known Chern-Moser-Weyl tensor for nondegenerate real hypersurfaces in complex space in terms of their defining functions. The formula is considerably simplified when applying to "pluriharmonic perturbations" of the sphere or to a Fefferman approximate solution to the complex Monge-Ampère equation. As an application, we show that the CR invariant one-form Xα constructed recently by Case and Gover is nontrivial on each real ellipsoid of revolution in C 3 , unless it is equivalent to the sphere. This resolves affirmatively a question posed by these two authors in 2017 regarding the (non-) local CR invariance of the I ′ -pseudohermitian invariant in dimension five and provides a counterexample to a recent conjecture by Hirachi.
We give a new proof of Faran's and Lebl's results by means of a new CR-geometric approach and classify all holomorphic mappings from the sphere in C 2 to Levi-nondegenerate hyperquadrics in C 3 . We use the tools developed by Lamel, which allow us to isolate and study the most interesting class of holomorphic mappings. This family of so-called nondegenerate and transversal maps we denote by F . For F we introduce a subclass N of maps which are normalized with respect to the group G of automorphisms fixing a given point. With the techniques introduced by Baouendi-Ebenfelt-Rothschild and Lamel we classify all maps in N . This intermediate result is crucial to obtain a complete classification of F by considering the transitive part of the automorphism group of the hyperquadrics.
Based on the results in [Rei14a] we deduce some topological results concerning holomorphic mappings of Levi-nondegenerate hyperquadrics under biholomorphic equivalence. We study the class F of so-called nondegenerate and transversal holomorphic mappings sending locally the sphere in C 2 to a Levi-nondegenerate hyperquadric in C 3 , which contains the most interesting mappings. We show that from a topological point of view there is a major difference when the target is the sphere or the hyperquadric with signature (2, 1). In the first case F modulo the group of automorphisms is discrete in contrast to the second case where this property fails to hold. Furthermore we study some basic properties such as freeness and properness of the action of automorphisms fixing a given point on F to obtain a structural result for a particularly interesting subset of F .
In this paper we continue our study of local rigidity for maps of CR submanifolds of the complex space. We provide a linear sufficient condition for local rigidity of finitely nondegenerate maps between minimal CR manifolds. Furthermore, we show higher order infinitesimal conditions can be used to give a characterization of local rigidity.
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