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

2024/10/23 by François Berteloot, Berteloot, François · 2 citations
Mathematics · #Complex Variables (math.CV) #FOS: Mathematics #Fuzzy Systems and Optimization

paper · pdf · doi:10.48550/arxiv.2410.17633

openalex publication_date 2024/10/23 · openalex created_date 2024/10/27 · openalex updated_date 2026/07/28

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 C² and, incidentally, reprove the Catlin's estimates for the Kobayashi metric in finite type domains.

Cited by

• Properties and examples of Kobayashi hyperbolic Riemannian manifolds
• Asymptotic behavior of the Bergman kernel and associated invariants in weakly pseudoconvex domains

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