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Kobayashi complete domains in complex manifolds

Rumpa Masanta

TL;DR

The paper develops sufficient conditions for Cauchy-completeness of the Kobayashi metric on Kobayashi hyperbolic domains $\Omega$ embedded in ambient complex manifolds $X$. It introduces local holomorphic weak peak functions and a notion of hyperbolic imbedding, proving that if $X$ belongs to a broad class (including Kobayashi hyperbolic manifolds, Stein manifolds, holomorphic fiber bundles with hyperbolic fibers, or holomorphic coverings over such bases) and $\Omega\Subset X$ has boundary points with local weak peaks, then $(\Omega, K_\Omega)$ is Cauchy-complete. A second theorem extends the result to the non-compact ambient setting by employing a peak-at-infinity condition at infinity on $X^\infty$ together with hyperbolic imbedding, yielding a purely metric-geometry proof. The work extends Gaussier’s results to manifolds and clarifies the role of peak functions in ensuring Kobayashi completeness beyond the Stein setting, with implications for the geometric structure of Kobayashi hyperbolic domains.

Abstract

In this paper, we give sufficient conditions for Cauchy-completeness of Kobayashi hyperbolic domains in complex manifolds. The first result gives a sufficient condition for completeness for relatively compact domains in several large classes of manifolds. This follows from our second result, which may be of independent interest, in a much more general setting. This extends a result of Gaussier to the setting of manifolds.

Kobayashi complete domains in complex manifolds

TL;DR

The paper develops sufficient conditions for Cauchy-completeness of the Kobayashi metric on Kobayashi hyperbolic domains embedded in ambient complex manifolds . It introduces local holomorphic weak peak functions and a notion of hyperbolic imbedding, proving that if belongs to a broad class (including Kobayashi hyperbolic manifolds, Stein manifolds, holomorphic fiber bundles with hyperbolic fibers, or holomorphic coverings over such bases) and has boundary points with local weak peaks, then is Cauchy-complete. A second theorem extends the result to the non-compact ambient setting by employing a peak-at-infinity condition at infinity on together with hyperbolic imbedding, yielding a purely metric-geometry proof. The work extends Gaussier’s results to manifolds and clarifies the role of peak functions in ensuring Kobayashi completeness beyond the Stein setting, with implications for the geometric structure of Kobayashi hyperbolic domains.

Abstract

In this paper, we give sufficient conditions for Cauchy-completeness of Kobayashi hyperbolic domains in complex manifolds. The first result gives a sufficient condition for completeness for relatively compact domains in several large classes of manifolds. This follows from our second result, which may be of independent interest, in a much more general setting. This extends a result of Gaussier to the setting of manifolds.
Paper Structure (5 sections, 5 theorems, 19 equations)

This paper contains 5 sections, 5 theorems, 19 equations.

Table of Contents

  1. Introduction and statement of results
  2. A remark on our notation
  3. Preliminary lemmas
  4. The proof of Theorem \ref{['T:weak_hyperbolic']}
  5. The proof of Theorem \ref{['T:k-complete']}

Key Result

Theorem 1.2

Let $X$ be any one of the following: Let $\Omega\varsubsetneq X$ be a relatively compact domain. If each point of $\partial\Omega$ admits a local holomorphic weak peak function, then $(\Omega, K_\Omega)$ is Cauchy-complete.

Theorems & Definitions (15)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Definition 1.4
  • Remark 1.5
  • Theorem 1.6
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • ...and 5 more