CR functions on Subanalytic Hypersurfaces

TL;DR

This paper explores CR functions on subanalytic hypersurfaces in complex space, revealing geometric conditions that influence the extendability of these functions and challenging existing assumptions about extension criteria in singular cases.

Contribution

It introduces a framework for CR functions on subanalytic hypersurfaces and demonstrates that classical extension conditions are not sufficient or necessary in the singular setting.

Findings

Certain geometric conditions allow non-extendable CR functions at singular points

The classical absence of complex hypersurfaces does not guarantee holomorphic extension in singular cases

Extension properties differ significantly between smooth and singular hypersurfaces

Abstract

A general class of singular real hypersurfaces, called subanalytic, is defined. For a subanalytic hypersurface M in C^n, Cauchy-Riemann (or simply CR) functions on M are defined, and certain properties of CR functions discussed. In particular, sufficient geometric conditions are given for a point p on a subanalytic hypersurface M to admit a germ at p of a smooth CR function f that cannot be holomorphically extended to either side of M. As a consequence it is shown that a well-known condition of the absence of complex hypersurfaces contained in a smooth real hypersurface M, which guarantees one-sided holomorphic extension of CR functions on M, is neither a necessary nor a sufficient condition for one-sided holomorphic extension in the singular case.