Metric entropy of the space of holomorphic functions

Siarhei Finski

Metric entropy of the space of holomorphic functions

Complex Variables 2026-07-21 v1 Algebraic Geometry Classical Analysis and ODEs Functional Analysis

Authors: Siarhei Finski

Abstract

By Montel's theorem, the continuous functions on a compact subset of a complex manifold that admit uniformly bounded holomorphic extensions to the manifold form a compact set in the uniform topology. We render this compactness quantitative by determining the asymptotics of the associated metric entropy, thereby giving a new solution to a problem of Kolmogorov for domains in $\mathbb{C}^n$ and extending the solution to domains in arbitrary Stein manifolds.

Keywords

metric entropy holomorphic functions Montel theorem Kolmogorov problem Stein manifolds

Cite

@article{arxiv.2607.18890,
  title  = {Metric entropy of the space of holomorphic functions},
  author = {Siarhei Finski},
  journal= {arXiv preprint arXiv:2607.18890},
  year   = {2026}
}