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 whenever 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.…”

“…In fact, if A ± are distinct complex hypersurfaces in an open set Ω in C n such that E = A + ∩ A − is non-empty with p ∈ E, and each of A ± \ E is connected (this happens when A ± are irreducible), then M = {z ∈ Ω : dist(z, A + ) = dist(z, A − )} has proper two-sided support at p. It is easy to verify that Ω ± = {z ∈ Ω : ±(dist(z, A + )−dist(z, A − )) > 0} are connected, and it follows from [4, Remarks 3.11] that M is subanalytic (and real-analytic if A ± are smooth.) If M is not minimal at p, after a small perturbation, we get a hypersurfaceM which has two-sided support at p by A ± , and which is minimal at p. In [9], the quadratic cones (zero-sets of real quadratic forms in C n ) with two-sided support were classified.…”