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Zalcman's lemma, Pinchuk's rescaling method, and Catlin's estimates revisited

François Berteloot

Abstract

We present a renormalization lemma for certain maps defined on the unit disc of C and taking values in some metric space. We show that the classical renormalization lemmas of Zalcman and Miniowitz can be deduced from our lemma. We also use it to establish a general normality statement for the Pinchuk's scaling method in C^2 and, incidentally, reprove the Catlin's estimates for the Kobayashi metric in finite type domains.

Zalcman's lemma, Pinchuk's rescaling method, and Catlin's estimates revisited

Abstract

We present a renormalization lemma for certain maps defined on the unit disc of C and taking values in some metric space. We show that the classical renormalization lemmas of Zalcman and Miniowitz can be deduced from our lemma. We also use it to establish a general normality statement for the Pinchuk's scaling method in C^2 and, incidentally, reprove the Catlin's estimates for the Kobayashi metric in finite type domains.

Paper Structure

This paper contains 4 sections, 11 theorems, 47 equations.

Table of Contents

  1. Introduction and main result
  2. Zalcman and Miniowitz lemmas
  3. A Bloch principle for the Bedford-Pinchuk rescaling method in $\mathbb{C}^2$
  4. Estimates for the Kobayashi pseudo-metric near finite type boundaries in $\mathbb{C}^2$

Key Result

Lemma 1.1

Let $(f_n)_n$ be a sequence of holomorphic maps from the unit disc $\mathbb{D}$ of $\mathbb{C}$ to the Riemann sphere which is not normal at $0$. Then there exists a sequence of affine contractions $(r_n)_n$, converging to $0$ and such that the renormalized sequence $(f_n\circ r_n)_n$ is converging

Theorems & Definitions (12)

  • Lemma 1.1
  • Definition 1.2
  • Lemma 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 3.1
  • Proposition 3.2
  • Theorem 3.3
  • Lemma 3.4
  • Lemma 3.5
  • ...and 2 more