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Polynomially convex embeddings and CR singularities of real manifolds

Purvi Gupta, Rasul Shafikov

Abstract

It is proved that any smooth manifold $\mathcal M$ of dimension $m$ admits a smooth polynomially convex embedding into $\mathbb C^n$ when $n\geq \lfloor 5m/4\rfloor$. Further, such embeddings are dense in the space of smooth maps from $\mathcal M$ into $\mathbb C^n$ in the $\mathcal C^3$-topology. The components of any such embedding give smooth generators of the algebra of complex-valued continuous functions on $\mathcal M$. A key ingredient of the proof is a coordinate-free description of certain notions of (non)degeneracy, as defined by Webster and Coffman, for CR-singularities of order one of an embedded real manifold in $\mathbb C^n$. The main result is obtained by inductively perturbing each stratum of degeneracy to produce a global polynomially convex embedding.

Polynomially convex embeddings and CR singularities of real manifolds

Abstract

It is proved that any smooth manifold of dimension admits a smooth polynomially convex embedding into when . Further, such embeddings are dense in the space of smooth maps from into in the -topology. The components of any such embedding give smooth generators of the algebra of complex-valued continuous functions on . A key ingredient of the proof is a coordinate-free description of certain notions of (non)degeneracy, as defined by Webster and Coffman, for CR-singularities of order one of an embedded real manifold in . The main result is obtained by inductively perturbing each stratum of degeneracy to produce a global polynomially convex embedding.

Paper Structure

This paper contains 7 sections, 20 theorems, 110 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A stratification of the set of CR singular points
  3. Some notions of degeneracy of order-one CR singularities
  4. Some stratifications of the set of order-one CR singularities
  5. A perturbation result for totally real points
  6. A perturbation result for Coffman nondegenerate points
  7. Proof of the main result

Key Result

Theorem 1

Let ${\bf M}_m$ the class of all smooth closed (i.e., compact without boundary) manifolds of dimension $m>0$. Let $\mathcal{M} \in {\bf M}_{m}$. Let $n \ge \lfloor 5m/4\rfloor$. Then, given any $\mathcal{C}^\infty$-smooth embedding $f: \mathcal{M}\hookrightarrow \mathbb C^n$, there exists a $\mathca

Figures (1)

  • Figure 1: The grey region is $K$; the dashed and dotted lines bound $V_0$ and $V_0'$, respectively.

Theorems & Definitions (39)

  • Theorem 1
  • Proposition 2
  • proof
  • Definition 3
  • Remark 4
  • Lemma 5
  • proof
  • Definition 6
  • Remark 7
  • Lemma 8
  • ...and 29 more