30 research outputs found
A regularity result for CR mappings between infinite type hypersurfaces
The Schwarz reflection principle in one complex variable can be stated as follows. Let M and M' be two real analytic curves in ℂ and f a holomorphic function defined on one side of M, extending continuously through M, and mapping M into M'. Then f has a holomorphic extension across M. In this paper, we extend this classical theorem to higher complex dimensions for a class of hypersurfaces of infinite type
Approximation and convergence of formal CR-mappings
Let be a minimal real-analytic CR-submanifold and a real-algebraic subset through points and . We show
that that any formal (holomorphic) mapping ,
sending into , can be approximated up to any given order at by a
convergent map sending into . If is furthermore generic, we also
show that any such map , that is not convergent, must send (in an
appropriate sense) into the set of points of D'Angelo
infinite type. Therefore, if does not contain any nontrivial
complex-analytic subvariety through , any formal map as above is
necessarily convergent
Higher order symmetries of real hypersurfaces in ℂ³
We study nonlinear automorphisms of Levi degenerate hypersurfaces of finite multitype. By results of Kolar, Meylan, and Zaitsev in 2014, the Lie algebra of infinitesimal CR automorphisms may contain a graded component consisting of nonlinear vector fields of arbitrarily high degree, which has no analog in the classical Levi nondegenerate case, or in the case of finite type hypersurfaces in . We analyze this phenomenon for hypersurfaces of finite Catlin multitype with holomorphically nondegenerate models in complex dimension three. The results provide a complete classification of such manifolds. As a consequence, we show on which hypersurfaces 2-jets are not sufficient to determine an automorphism. The results also confirm a conjecture about the origin of nonlinear automorphisms of Levi degenerate hypersurfaces, formulated by the first author for an AIM workshop in 2010
Chern–Moser operators and polynomial models in CR geometry
We consider the fundamental invariant of a real hypersurface in CN – its holomorphic symmetry group – and analyze its structure at a point of degenerate Levi form. Generalizing the Chern–Moser operator to hypersurfaces of finite multitype, we compute the Lie algebra of infinitesimal symmetries of the model and provide explicit description for each graded component. Compared with a hyperquadric, it may contain additional components consisting of nonlinear vector fields defined in terms of complex tangential variables.As a consequence, we obtain exact results on jet determination for hypersurfaces with such models. The results generalize directly the fundamental result of Chern and Moser from quadratic models to polynomials of higher degre
Stationary discs and finite jet determination for non-degenerate generic real submanifolds
In case M is Levi non-degenerate in the sense Tumanov, we construct
stationary discs for . If furthermore M satisfies an additional
non-degeneracy condition, we apply the method of stationary discs to obtain
2-jet determination of CR automorphisms of M