arXiv subjects
Publications and source records attributed to D. Zaitsev.
We consider a Cauchy problem for an overdetermined system of PDEs, and give necessary and sufficient conditions for solvability of this Cauchy problem for all data. As an application, we find all real tube hypersurfaces in complex space whose Levi number is maximal.
This paper shows that an arbitrary generic submanifold in a complex manifold can be deformed into a 1-parameter family of generic submanifolds satisfying strong nondegeneracy conditions. The proofs use a careful analysis of the jet spaces of embeddings satisfying certain nondegeneracy properties, and also make use of the Thom transversality theorem, as well as the stratification of real-algebraic sets. Optimal results on the order of nondegeneracy are given.
We introduce new biholomorphic invariants for real-analytic hypersurfaces in 2-dimensional complex space and show how they can be used to show that a hypersurface possesses few automorphisms. We give conditions, in terms of the new invariants, guaranteeing that the stability group is finite, and give (sharp) bounds on the cardinality of the stability group in this case. We also give a sufficient condition for the stability group to be trivial. The main technical tool developed in this paper is a complete (formal) normal form for a certain class of hypersurfaces. As a byproduct, a complete classification, up to biholomorphic equivalence, of the finite type hypersurfaces in this class is obtained.
The goal of this paper is twofold. First, to give purely local boundary uniqueness results for maps defined only on one side as germs at a boundary point and hence not necessarily sending any domain to itself and also under the weaker assumption that $f(z)=z+o(|z-p|^3)$ holds only for $z$ in a proper cone in $D$ with vertex $p$. Such results have no analogues in one complex variable in contrast to the situation when a domain is preserved. And second, to extend the above results from boundaries of domains to submanifolds of higher codimension.
We consider the class of Levi nondegenerate hypersurfaces $M$ in $\bC^{n+1}$ that admit a local (CR transversal) embedding, near a point $p\in M$, into a standard nondegenerate hyperquadric in $\Bbb C^{N+1}$ with codimension $k:=N-n$ small compared to the CR dimension $n$ of $M$. We show that, for hypersurfaces in this class, there is a normal form (which is closely related to the embedding) such that any local equivalence between two hypersurfaces in normal form must be an automorphism of the associated tangent hyperquadric. We also show that if the signature of $M$ and that of the standard hyperquadric in $\bC^{N+1}$ are the same, then the embedding is rigid in the sense that any other embedding must be the original embedding composed with an automorphism of the quadric.
We establish results on holomorphic extension of CR-mappings of class $C^\infty$ between a real-analytic CR-submanifold of $\C^N$ and a real-algebraic CR-submanifold of $\C^{N'}$.
We show that germs of local real-analytic CR automorphisms of a real-analytic hypersurface $M$ in $\C^2$ at a point $p\in M$ are uniquely determined by their jets of some finite order at $p$ if and only if $M$ is not Levi-flat near $p$. This seems to be the first necessary and sufficient result on finite jet determination and the first result of this kind in the infinite type case. If $M$ is of finite type at $p$, we prove a stronger assertion: the local real-analytic CR automorphisms of $M$ fixing $p$ are analytically parametrized (and hence uniquely determined) by their 2-jets at $p$. This result is optimal since the automorphisms of the unit sphere are not determined by their 1-jets at a point of the sphere.