The purpose of this article is twofold. The first aim is to characterize an n-dimensional Kobayashi hyperbolic complex manifold M exhausted by a sequence <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\{\Omega _j\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi mathvariant="normal">Ω</mi> <mi>j</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> of domains in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\mathbb {C}}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> via an exhausting sequence <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\{f_j:\Omega _j\rightarrow M\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>f</mi> <mi>j</mi> </msub> <mo>:</mo> <msub> <mi mathvariant="normal">Ω</mi> <mi>j</mi> </msub> <mo stretchy="false">→</mo> <mi>M</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$f_j^{-1}(a)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>f</mi> <mi>j</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> converges to a boundary point <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\xi _0 \in \partial \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ξ</mi> <mn>0</mn> </msub> <mo>∈</mo> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Omega = \bigcup _{j=1}^{\infty } \Omega _j$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <msubsup> <mo>⋃</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </msubsup> <msub> <mi mathvariant="normal">Ω</mi> <mi>j</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$a\in M$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>∈</mo> <mi>M</mi> </mrow> </math></EquationSource> </InlineEquation>. Then, our second aim is to show that any spherically extreme boundary point must be strongly pseudoconvex.

Exhaustion of Hyperbolic Complex Manifolds and Relations to the Squeezing Function

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发表日期：

2026年3月24日

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Van Thu Ninh,
Huy Vu Trinh,
Quang Dieu Nguyen

摘要

The purpose of this article is twofold. The first aim is to characterize an n-dimensional Kobayashi hyperbolic complex manifold M exhausted by a sequence $\{\Omega _j\}$ ${Ω_{j}}$ of domains in ${\mathbb {C}}^n$ $C^{n}$ via an exhausting sequence $\{f_j:\Omega _j\rightarrow M\}$ ${f_{j} : Ω_{j} \to M}$ such that $f_j^{-1}(a)$ $f_{j}^{- 1} (a)$ converges to a boundary point $\xi _0 \in \partial \Omega $ $ξ_{0} \in \partial Ω$ , where $\Omega = \bigcup _{j=1}^{\infty } \Omega _j$ $Ω = ⋃_{j = 1}^{\infty} Ω_{j}$ and $a\in M$ $a \in M$ . Then, our second aim is to show that any spherically extreme boundary point must be strongly pseudoconvex.