On the regularity of mappings, arising from solutions of tangent Cauchy-Riemann PDEs

Ilya Kossovskiy
University of Vienna, Austria

Date(s) : 31/03/2014   iCal
10 h 00 min - 11 h 00 min

One can define functions holomorphic on open sets in Cn as smooth functions, annihilated by the Cauchy-Riemann operator ¯∂ . Similarly, one can consider smooth functions on a real submanifold M of $\CC{n}$, annihilated by the naturally defined on M tangent Cauchy-Riemann operator. The corresponding objects are called CR-functions on M. CR-functions naturally occur in complex analysis as restrictions of holomorphic functions onto a real submanifold M, and also as boundary values of functions, holomorphic in a domain. A remarkable property of CR-functions is their wedge holomorphic extension, provided the CR-manifold M satisfies certain nondegeneracy conditions (Tumanov, 1980’s). Another remarkable property is the analyticicty of CR-diffeomorphisms between nondegenerate real-analytic submanifolds M,Min Cn (i.e., CR-mappings in this case appear to be simply restrictions of holomorphic mappings). It was an open problem for a while whether one can drop the nondegeneracy conditions, posed on M and M. In this talk we provide a negative answer for this question. Our construction is based on a connection between geometry of degenerate (more precisely, nonminimal) real hypersurfaces in C2 and that of second order singular holomorphic differential equations.


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