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Local Polynomial Convexity at Hyperbolic CR-singularities in $M \subset \mathbb{C}^n$

Harshith Alagandala

TL;DR

The work analyzes local polynomial convexity of real $M^n\subset\mathbb{C}^n$ at order-1 CR-singularities, focusing on hyperbolic points. It employs Bishop's normal form, a decomposition into perturbations of totally real planes, and tools like Kallin's lemma and fibre theorems to lift 2D convexity results to higher dimensions. A key finding is that if the normal-form perturbation $F$ vanishes to order two in $w$, then $M$ is locally polynomially convex at a hyperbolic point, and similarly for flat hyperbolic points; in the $M^2\subset\mathbb{C}^2$ case, a quantitative neighborhood size is obtained depending on the hyperbolicity parameter $\gamma>\tfrac12$. The paper thereby extends known 2D results to higher dimensions via slicing and induction, providing explicit conditions and constructive proofs for local polynomial convexity in several complex variables.

Abstract

Let $M$ be a smooth manifold of dimension $n$ embedded in $\mathbb{C}^n$. If $T_pM \subset T_p\mathbb{C}^n$ is a totally real subspace for $p\in M$, then $M$ is locally polynomially convex at $p$. For a generic embedding $M$, we are interested in assessing polynomial convexity of $M$ at a CR-singularity, i.e., at a point $p\in M$ where $T_pM$ is not totally real. An order one CR-singularity in $M$ can be broadly classified as elliptic and hyperbolic. It is known that elliptic points give obstruction to polynomial convexity. In the case $n=2$, $M^2 \subset \mathbb{C}^2$ is locally polynomially convex at a hyperbolic complex point. We investigate local polynomial convexity of $M^n \subset \mathbb{C}^n$ at hyperbolic points in higher dimension.

Local Polynomial Convexity at Hyperbolic CR-singularities in $M \subset \mathbb{C}^n$

TL;DR

The work analyzes local polynomial convexity of real at order-1 CR-singularities, focusing on hyperbolic points. It employs Bishop's normal form, a decomposition into perturbations of totally real planes, and tools like Kallin's lemma and fibre theorems to lift 2D convexity results to higher dimensions. A key finding is that if the normal-form perturbation vanishes to order two in , then is locally polynomially convex at a hyperbolic point, and similarly for flat hyperbolic points; in the case, a quantitative neighborhood size is obtained depending on the hyperbolicity parameter . The paper thereby extends known 2D results to higher dimensions via slicing and induction, providing explicit conditions and constructive proofs for local polynomial convexity in several complex variables.

Abstract

Let be a smooth manifold of dimension embedded in . If is a totally real subspace for , then is locally polynomially convex at . For a generic embedding , we are interested in assessing polynomial convexity of at a CR-singularity, i.e., at a point where is not totally real. An order one CR-singularity in can be broadly classified as elliptic and hyperbolic. It is known that elliptic points give obstruction to polynomial convexity. In the case , is locally polynomially convex at a hyperbolic complex point. We investigate local polynomial convexity of at hyperbolic points in higher dimension.

Paper Structure

This paper contains 15 sections, 13 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Polynomial Convexity
  4. CR-singularities
  5. Preliminary inequalities
  6. Local polynomially convexity of $M^k \subset \mathbb{C}^n$ for $k<n$
  7. Proof of Theorem \ref{['thm_hyperbolic_higher_dimension_vanish_order_two']}
  8. Hyperbolic points in $M^2 \subset \mathbb{C}^2$
  9. Polynomial convexity of $S_1$ and $S_2$
  10. Polynomial convexity of $S_1 \cup S_2$
  11. Flat hyperbolic point
  12. The set of CR-singularities
  13. Slices of $M$
  14. Normal form representation of two-dimensional slices
  15. Local polynomial convexity at a flat hyperbolic point

Key Result

Theorem 1.1

Let $p$ be a hyperbolic point of $M^n \subset \mathbb{C}^n$. Let equation (eqn_bishop_normal_form) be the normal form of $M$ centered at $p$ ($p=0$). If $F(t,w,\bar{w})$ vanishes to order two in $w$ (Definition defn:vanishes_order_two_in_w), then $M$ is locally polynomial convex at $p$. Further, $\m

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1: Polynomial Convexity
  • Theorem 2.2: stout2007polynomial
  • Definition 2.3: Local Polynomial Convexity
  • Lemma 2.4
  • proof
  • Remark
  • Theorem 2.5: stout2007polynomial
  • ...and 22 more