Ilya Kossovskiy

Divergent CR-equivalences and meromorphic differential equations

Using the analytic theory of differential equations, we construct examples of formally but not holomorphically equivalent real-analytic Levi nonflat hypersurfaces in

New extension phenomena for solutions of tangential Cauchy–Riemann equations

\CC{n}

Normal Form for Second Order Differential Equations

together with examples of such hypersurfaces with divergent formal CR-automorphisms.

Sphericity of a real hypersurface via projective geometry

ABSTRACT In our recent work, we showed that C∞ CR-diffeomorphisms of real-analytic Levi-nonflat hypersurfaces in ℂ2 are not analytic in general. This result raised again the question on the nature of CR-maps of a real-analytic hypersurfaces. In this paper, we give a complete picture of what CR-maps actually are. First, we discover an analytic continuation phenomenon for CR-diffeomorphisms which we call the sectorial analyticity property. It appears to be the optimal regularity property for CR-diffeomorphisms in general. We emphasize that such type of extension never appeared previously in the literature. Second, we introduce the class of Fuchsian type hypersurfaces and prove that (infinitesimal generators of) CR-automorphisms of a Fuchsian type hypersurface are still analytic. In particular, this solves a problem formulated earlier by Shafikov and the first author. Finally, we prove a regularity result for formal CR-automorphisms of Fuchsian type hypersurfaces.

Classification of homogeneous CR-manifolds in dimension 4

Applying methods of CR-geometry, we give a solution to the local equivalence problem for second order (smooth or analytic) ordinary differential equations. We do so by presenting a complete normal form (which is smooth or analytic respectively) for this class of ordinary differential equations (ODEs). The normal form is optimal in the sense that it is defined up to the automorphism group of the model (flat) ODE y″ = 0. For a generic ODE, we also provide a unique (up to a discrete group action) normal form. By doing so, we give a solution to a problem which remained unsolved since the work of Arnold (1988). As another application of the normal form, we obtain distinguished curves associated with a differential equation that we call chains due to their analogy with the chains defined by Chern and Moser (Acta Math. 7;133:219–271).

Analytic differential equations and spherical real hypersurfaces

In this work, we obtain an unexpected geometric characterization of sphericity of a real-analytic Levi-nondegenerate hypersurface

Analytic continuation of holomorphic mappings from nonminimal hypersurfaces

M\subset\mathbb C^{2}

On the analyticity of CR-diffeomorphisms

. We prove that

Convergent normal form for real hypersurfaces at a generic Levi-degeneracy

M

The envelope of holomorphy of a model surface of the third degree and the ``rigidity'' phenomenon

is spherical if and only if its Segre\,(-Webster) varieties satisfy an elementary combinatorial property, identical to a property of straight lines on the plane and known in Projective Geometry as the {\em Desargues Theorem}.