##### Department of Mathematics,

University of California San Diego

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### Analysis Seminar

## Ilya Kossovskiy

#### University of Western Ontario

## Analytic Continuation of Holomorphic Mappings From Nonminimal Hypersurfaces

##### Abstract:

The classical result of H.Poincare states that a local biholomorphic mapping of an open piece of the 3-sphere in $\mathbb{C}^2$ onto another open piece extends analytically to a global holomorphic automorphism of the sphere. A big stream of further publications was dedicated to the possibility to extend local biholomorphic mapping between real hypersurfaces in complex space. The most general results were obtained by D.Hill, R.Shafikov and K.Verma who generalized Poincare's extension phenomenon for the case of an essentially finite hypersurface in the preimage and a quadric in the image, and also for the case of a minimal hypersurface (in the sense of Tumanov) in the preimage and a sphere in the image. In this joint work with R.Shafikov we consider the - essentially new - case where a hypersurface $M$ in the preimage contains a complex hypersurface, i.e. where $M$ is nonminimal. We demonstrate that the above extension results fail in this case, and prove the following analytic continuation phenomenon: a local biholomorphic mapping of $M$ onto a non-degenerate hyperquadric in $\mathbb{CP}^n$ extends to a punctured neighborhood of the complex hypersurface $X$, lying in $M$, as a multiple-valued locally biholomorphic mapping. The extension phenomenon is based on the properties of Segre sets introduced by Baouendi, Ebenfelt and Rothschild near the complex hypersurface $X$. We also establish an interesting interaction between nonminimal spherical real hypersurfaces and linear differential equations with an isolated singular point.

Host: Peter Ebenfelt

### May 15, 2012

### 10:30 AM

### AP&M 7321

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