Holomorphic extension of CR functions from quadratic cones

Abstract

It is proved that CR functions on a quadratic cone M in C n , n > 1, admit one-sided holomorphic extension if and only if M does not have two-sided support, a geometric condition on M which generalizes minimality in the sense of Tumanov. A biholomorphic classification of quadratic cones in C 2 is also given.

CR functions on subanalytic hypersurfaces

“…First of all, without an additional topological assumption, non-minimality by itself does not suffice. Further, there is another geometric condition, first introduced in [9], called two-sided support, that also gives rise to non-extendable CR functions. We say that M has proper two-sided support at p ∈ M if there is an open neighbourhood Ω of p such that M divides Ω into two connected components Ω + and Ω − , and there exist germs at p of distinct complex analytic hypersurfaces A ± ⊂ Ω ± such that A + ∩ M = A − ∩ M , see §4.3 for details.…”

“…This has been generalized to graphs of continuous real-valued functions by Chirka [8]. It was shown in [9] that for singular hypersurfaces, minimality is no longer a sufficient condition for local one-sided holomorphic extension (see §4.3). Combining this with Theorem 3 above we conclude that for singular hypersurfaces minimality is neither a necessary nor a sufficient condition for one-sided holomorphic extension of CR functions.…”

“…We wish to generalize the notion of CR functions to singular hypersurfaces. In [9], we considered the class of real-analytic hypersurfaces. The smooth part of such hypersurfaces is always locally orientable, and this allows us to define CR functions using the adjoint form of the Cauchy-Riemann equations.…”

Holomorphic extension of CR functions from quadratic cones

“…We write (7) simply means that f is CR whenever the current f [M ] 0,1 is ∂-closed in Ω. This leads to the following Definition 3.1.…”

“…Note on this version: An earlier version of this article was published as [6], and subsequently an erratum [7] correcting some mistakes was published. This version incorporates the corrections from [7], as well as corrections of a few minor typos, and is posted in ArXiv.…”