Cartan uniqueness theorem on nonopen sets

TL;DR

This paper extends Cartan's uniqueness theorem to a broad class of nonopen sets in complex space, including real-analytic subvarieties, by establishing conditions for the extendibility of CR functions and automorphisms.

Contribution

It introduces conditions under which CR functions extend to neighborhoods for nonopen sets, generalizes the concept of CR automorphisms, and characterizes automorphisms of certain subvarieties.

Findings

CR functions extend uniquely under specified conditions

Automorphisms of certain subvarieties are linear

Generalization of infinitesimal CR automorphisms

Abstract

Cartan's uniqueness theorem does not hold in general for CR mappings, but it does hold under certain conditions guaranteeing extendibility of CR functions to a fixed neighborhood. These conditions can be defined naturally for a wide class of sets such as local real-analytic subvarieties or subanalytic sets, not just submanifolds. Suppose that $V$ is a locally connected and locally closed subset of ${\mathbb{C}}^n$ such that the hull constructed by contracting analytic discs close to arbitrarily small neighborhoods of a point always contains the point in the interior. Then restrictions of holomorphic functions uniquely extend to a fixed neighborhood of the point. Using this extension, we obtain a version of Cartan's uniqueness theorem for such sets. When $V$ is a real-analytic subvariety, we can generalize the concept of infinitesimal CR automorphism and also prove an analogue of the theorem. As an application of these two results we show that, for circular subvarieties satisfying the condition, the only automorphisms, CR or infinitesimal, are linear.