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Polynomial convexity of $\bar\partial$-flat perturbations of totally real sets

Leandro Arosio, Håkan Samuelsson Kalm, Erlend F. Wold

TL;DR

The paper addresses when the union of a polynomially convex compact set $K$ and a totally real submanifold $X$ can be made polynomially convex by a small perturbation. It constructs near-identity, $\bar\partial$-flat diffeomorphisms of $\mathbb{C}^n$ that fix $K$ and annihilate higher-order derivatives along $K\cup X$, forcing $DF$ to be $\mathbb{C}$-linear on $K\cup X$ and producing a polynomially convex image. The main contribution is extending the AW framework by making the perturbation of $X$ arise from a global diffeomorphism of $\mathbb{C}^n$ with controlled flatness, and providing a technique to ensure polynomial convexity of $K\cup X'$ while keeping the perturbation arbitrarily small. This yields hull-controlled embeddings and perturbation results for submanifolds, with potential applications to perturbing lower-dimensional manifolds to be polynomially convex under dimension constraints $d<n$.

Abstract

We show that if $X$ is a totally real $d$-dimensional manifold attached to a polynomially convex compact set $K$ in $\mathbb{C}^n$, $d<n$, then there are arbitrarily small perturbations $X'$ of $X$ such that $K\cup X'$ is polynomially convex. The perturbations are induced by diffeomorphisms of $\mathbb{C}^n$ fixing $K$, which are $\bar\partial$-flat on $K\cup X$, and which are arbitrarily $C^k$-close to the identity.

Polynomial convexity of $\bar\partial$-flat perturbations of totally real sets

TL;DR

The paper addresses when the union of a polynomially convex compact set and a totally real submanifold can be made polynomially convex by a small perturbation. It constructs near-identity, -flat diffeomorphisms of that fix and annihilate higher-order derivatives along , forcing to be -linear on and producing a polynomially convex image. The main contribution is extending the AW framework by making the perturbation of arise from a global diffeomorphism of with controlled flatness, and providing a technique to ensure polynomial convexity of while keeping the perturbation arbitrarily small. This yields hull-controlled embeddings and perturbation results for submanifolds, with potential applications to perturbing lower-dimensional manifolds to be polynomially convex under dimension constraints .

Abstract

We show that if is a totally real -dimensional manifold attached to a polynomially convex compact set in , , then there are arbitrarily small perturbations of such that is polynomially convex. The perturbations are induced by diffeomorphisms of fixing , which are -flat on , and which are arbitrarily -close to the identity.

Paper Structure

This paper contains 3 sections, 6 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['glass']}

Key Result

Theorem 1.1

Let $K\subset \mathbb C^n$ be a polynomially convex compact set, and let $X\subset\mathbb C^n\setminus K$ be a closed, bounded, totally real $C^\infty$-smooth submanifold with $\mathrm{dim}(X)< n$. Then for any $k\in\mathbb{N}$, $\delta>0$, and $\kappa\in\mathbb{N}$ there exists a $C^\infty$-smooth

Theorems & Definitions (10)

  • Theorem 1.1
  • Proposition 2.1
  • Corollary 2.2
  • Proposition 2.3
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • proof : Proof of Theorem \ref{['glass']}