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Cartan uniqueness theorem on nonopen sets

Jiri Lebl, Alan Noell, Sivaguru Ravisankar

TL;DR

The paper establishes a Cartan-type uniqueness theorem for CR mappings on broad classes of sets $V\subset\mathbb{C}^n$ that are locally connected and locally closed, under a contracting disc hull condition at a point $p$ and a generic tangent cone $C_p V$. By proving that holomorphic functions on $V$ extend to a fixed neighborhood $\Omega$ of $p$, it shows that a $f\in\mathcal{O}(V)$ with $f(p)=p$ and $Df(p)$ acting as the identity on $C_p V$ must be the identity on $V$, yielding a finite jet-type determination. The results are specialized to generic real-analytic submanifolds, yielding a CR-analytic analogue of Cartan uniqueness, and to infinitesimal CR automorphisms, where flows with suitable growth imply triviality. For circular subvarieties satisfying the hull condition, automorphisms (and infinitesimal automorphisms) are shown to be linear, linking geometric symmetry to linear structure and enabling a classification of such maps.

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.

Cartan uniqueness theorem on nonopen sets

TL;DR

The paper establishes a Cartan-type uniqueness theorem for CR mappings on broad classes of sets that are locally connected and locally closed, under a contracting disc hull condition at a point and a generic tangent cone . By proving that holomorphic functions on extend to a fixed neighborhood of , it shows that a with and acting as the identity on must be the identity on , yielding a finite jet-type determination. The results are specialized to generic real-analytic submanifolds, yielding a CR-analytic analogue of Cartan uniqueness, and to infinitesimal CR automorphisms, where flows with suitable growth imply triviality. For circular subvarieties satisfying the hull condition, automorphisms (and infinitesimal automorphisms) are shown to be linear, linking geometric symmetry to linear structure and enabling a classification of such maps.

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 is a locally connected and locally closed subset of 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 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.
Paper Structure (6 sections, 9 theorems, 11 equations, 1 figure)

This paper contains 6 sections, 9 theorems, 11 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The theorem for mappings
  3. Generic real-analytic submanifolds
  4. The theorem for infinitesimal automorphisms
  5. Automorphisms of circular subvarieties
  6. On the erratum in the published version

Key Result

Lemma 2.1

Let $K \subset {\mathbb{C}}^n$ be a compact and connected subset and $p \in K$. Suppose that $\widehat{K}_{CD,p}$ has $p \in K$ in its interior, and let $B \subset \widehat{K}_{CD,p}$ be a ball centered at $p$ such that $B \cap K$ is connected. Let $f\in{\mathcal{O}}(K)$. Then there exists a holomor

Figures (1)

  • Figure 1: Diagram of the set $K$ as seen in the $z$-plane with the point $(1,0)$ marked. Note that the smaller circle represents an entire sphere minus a cap (a "fishbowl"), while the larger arc is simply an arc in the $z$-plane.

Theorems & Definitions (25)

  • Definition 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Theorem 2.4
  • proof
  • Example 2.5
  • Example 2.6
  • ...and 15 more