Real submanifolds in complex space: new currents

The present project can be considered as a part of recently intensively studied branch of complex analysis named CR-geometry. CR-geometry goes back to Poincare and was deeply developed in the pioneering works of Cartan, Chern, Moser and Tanaka and in a large number of further papers. In this project we consider a number of open problems, where standard methods of CR-geometry are not applicable. These problems are concerned mainly with holomorphic equivalences and automorphism groups of real submanifolds in complex space, and with analytic continuation of holomorphic mappings between real submanifolds. We describe a number of new approaches, that can be used for the study of the above problems. Some of the new methods are successfully applied already in the upcoming papers of the applicant and the co-applicant, and enabled to achieve an essential progress in some of the open problems that we consider in the project. We corroborate the importance of the project by a historic outline and a list of relevant references.

A. We proved that there exist formally but not holomorphically equivalent real-analytic hypersur- faces in complex space. By that we have shown that a formal normal form is not sufficient anymore for solving the holomorphic equivalence problem. We applied this phenomenon to showing that there are more formal symmetries of real hypersurfaces than holomorphic ones in general.B. We obtained the optimal bound for the dimension of the authomorphism group of a real- analytic hypersurface in complex two-space. By that, we have answered a long-standing question of Poincare.C. We completed the study of the analytic continuation problem for a germ of a map of a real- analytic hypersurface in complex two-space into sphere. We found the optimal condition for extending a map to nonminimal points.D. We showed that there exist real-analytic hypersurfaces in complex space which are smoothly CR-equivalent but are inequivalent holomorphically. By that, we obtained the (negative) answer to the Conjecture of Ebenfelt and Huang.E. We constructed a convergent normal form for real-analytic hypersurfaces at a generic Levi degeneracy. This gave the first known convergent normal form at Levi degeneracy points in a real hypersurface.F. We found a general sufficient condition for the convergence of a Chern-Moser normal form of a real-analytic hypersurface in complex two-space.G. We discovered the sectorial extension phenomenon for maps of infinite type hypersurfaces in complex space. We also found an optimal condition for the analytic extension of a map to a full neighborhood of an infinite type point.