Metric entropy of the space of holomorphic functions

Siarhei Finski

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arXiv · 2607.18890

Metric entropy of the space of holomorphic functions

Siarhei Finski

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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.

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Siarhei Finski. 2026-07-21. Metric entropy of the space of holomorphic functions. https://arxiv.org/abs/2607.18890

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