Metric entropy of the space of holomorphic functions

Technical Summary: Metric Entropy of the Space of Holomorphic Functions

Problem Statement
The paper addresses a classical problem posed by Kolmogorov concerning the asymptotic behavior of the metric entropy (or $\varepsilon$-entropy) of spaces of holomorphic functions. Specifically, given a domain $\Omega$ in a Stein manifold $X$ of dimension $n$ and a compact non-pluripolar subset $K \subset \Omega$, the author investigates the growth rate of $H_\varepsilon(A^\Omega_K, C(K))$ as $\varepsilon \to 0$. Here, $A^\Omega_K$ denotes the set of continuous functions on $K$ that admit holomorphic extensions to $\Omega$ bounded by 1. While the compactness of this set is guaranteed by Montel's theorem, the quantitative rate of this compactness—measured by the number of $\varepsilon$-balls required to cover the set—remains a central question in complex analysis.

Previous work had established the asymptotics for domains in $\mathbb{C}$ (Babenko, Erokhin) and $\mathbb{C}^n$ (Zakharyuta, Nivoche, Bandtlow-Nivoche), often relying on explicit formulas like the Bergman-Weil formula or specific pluripotential-theoretic conjectures. However, extending these results to arbitrary Stein manifolds, where global coordinate systems and explicit integral formulas are not available, remained an open challenge.

Methodology
The author employs a multi-step reduction strategy to generalize the problem from arbitrary Stein manifolds to settings where algebraic and geometric tools are more accessible:

1. Reduction to Affine Manifolds and Runge Domains:
   Using results by Stout and properties of strictly hyperconvex domains, the problem is reduced to the case where the ambient space $X$ is an affine manifold (a submanifold of $\mathbb{C}^m$) and $\Omega$ is a strictly Runge domain. This allows the replacement of the space of holomorphic functions $H^0(\Omega)$ with finite-dimensional approximations.

2. Polynomial Approximation:
   The space of holomorphic functions is approximated by spaces of polynomials of bounded degree, $A^k[X]$. Leveraging Siciak's results on optimal polynomial approximation (extended by Zeriahi to affine varieties), the author establishes that the $\varepsilon$-entropy of the holomorphic space is asymptotically equivalent to that of these polynomial subspaces, provided the degree $k$ scales logarithmically with $\varepsilon^{-1}$.

3. Volume Ratios and Unit Balls:
   The metric entropy of the finite-dimensional polynomial spaces is related to the ratio of volumes of unit balls defined by two different sup-norms: one on $K$ and one on $\Omega$. This step utilizes estimates on the entropy of finite-dimensional vector spaces (Proposition 2.1) and the John ellipsoid theorem to relate general norms to Hermitian norms.

4. Connection to Monge-Ampère Energy:
   The ratio of volumes of these unit balls is linked to the relative Monge-Ampère energy via a theorem by Berman and Boucksom. This energy is defined using the relative extremal functions (Siciak-Zaharuta functions) $V_\Omega$ and $V_{cK, \Omega}$.

5. Relating Energy to Capacity:
   A crucial technical contribution involves proving that the relative Monge-Ampère energy, when normalized by $c^{n+1}$ as $c \to 0$, converges to the relative capacity $C(K, \Omega)$. This bridges the gap between the global geometric energy (dependent on the embedding in $\mathbb{C}^m$) and the intrinsic relative capacity (which is independent of the ambient manifold).

Key Results
The main theorem (Theorem 1.1) provides sharp bounds for the asymptotic behavior of the metric entropy:

$$\frac{2C(K, \Omega)}{(n+1)!} \le \liminf_{\varepsilon \to 0} \frac{H_\varepsilon(A^\Omega_K, C(K))}{(\log \varepsilon^{-1})^{n+1}} \le \limsup_{\varepsilon \to 0} \frac{H_\varepsilon(A^\Omega_K, C(K))}{(\log \varepsilon^{-1})^{n+1}} \le \frac{2C(K, \hat{\Omega})}{(n+1)!}$$

where:

• $C(K, \Omega)$ is the relative capacity defined via an exhaustion by hyperconvex domains.
• $\hat{\Omega}$ is the envelope of holomorphy of $\Omega$.
• $C(K, \hat{\Omega})$ is the relative capacity with respect to the envelope.

Corollaries and Specific Cases:

• Strictly Hyperconvex Domains: If $\Omega$ is strictly hyperconvex, the envelope of holomorphy coincides with $\Omega$ (up to biholomorphism), and the bounds collapse to a precise limit:
  $$\lim_{\varepsilon \to 0} \frac{H_\varepsilon(A^\Omega_K, C(K))}{(\log \varepsilon^{-1})^{n+1}} = \frac{2C(K, \Omega)}{(n+1)!}$$
  This result extends previous findings in $\mathbb{C}^n$ to arbitrary Stein manifolds.
• Quantitative Runge Property: The paper also derives a quantitative version of the Runge property (Theorem 8.2), relating the discrepancy between the entropy of functions on $\Omega$ and those extendable to the ambient manifold $X$ to the capacity of the holomorphic hull of $\Omega$.

Significance and Claims
The paper claims to provide a new, independent solution to Kolmogorov's problem for domains in $\mathbb{C}^n$ and, more significantly, to extend this solution to arbitrary Stein manifolds. The significance lies in:

1. Generalization: Unlike previous approaches that relied on explicit integral formulas (e.g., Bergman-Weil) specific to $\mathbb{C}^n$, this method works for general Stein manifolds by reducing the problem to affine varieties and utilizing pluripotential theory.
2. Conceptual Clarity: The work offers a conceptual explanation for the appearance of relative capacity in entropy estimates. Previously, this connection was established through explicit computations in special cases (e.g., analytic polyhedra). Here, the link is derived through the relative Monge-Ampère energy, showing that the capacity emerges naturally as the limit of this energy.
3. Robustness: The proof handles the subtleties of the envelope of holomorphy, which may not be hyperconvex even if the original domain is, by utilizing a limiting process over hyperconvex exhaustions.

The author notes that the result is consistent with known cases in $\mathbb{C}^n$ but provides a unified framework that does not require the ambient space to be $\mathbb{C}^n$, thereby addressing domains that embed into Stein manifolds but not necessarily into Euclidean space.