Uniformization and Steinness

TL;DR

This paper proves that the unit ball in complex n-space uniquely covers both Stein and non-Stein strictly pseudoconvex domains, highlighting its special role in complex geometry.

Contribution

It establishes the uniqueness of the unit ball as the universal cover for certain classes of complex domains, linking uniformization and Steinness.

Findings

The unit ball is the only universal cover for both Stein and non-Stein strictly pseudoconvex domains.

This result characterizes the special status of the unit ball in complex manifold theory.

The paper connects uniformization properties with Steinness in complex analysis.

Abstract

It is shown that the unit ball in ${\mathbb C}^n$ is the only complex manifold that can universally cover both Stein and non-Stein strictly pseudoconvex domains.

Figures

Equations

$$\underset{\pi (A)}{\sup }\phi =\underset{n\to \infty }{\lim }\phi (x_n)=\phi (x).$$

$$\tilde{\phi }(\xi )<\phi (x)-\epsilon \text{ for all }\xi \in \partial U\cap \{\phi <\phi (x)\}$$

$$\psi (x):={\left(\underset{{\pi }^{\prime }(y)=x}{\sup }\phi \circ \pi (y)\right)}^{\ast },$$

$${\psi |}_B\equiv const.<0.$$

$$\phi \circ \pi (y)\le const.<0\text{ for all }y\in {\pi }^{\prime -1}(B),$$

Full text

The first author was partly supported by SFB/TR 191 of the DFG and RFBR grant №17-01-00592-a. The second author was supported by a grant from the Natural Sciences and Engineering Research Council of Canada.

In this note we use methods from [5] to show that the unit ball in $\mathbb{C}^{n}$ is the only simply connected complex manifold that can cover both Stein and non-Stein strictly pseudoconvex domains.

Here a strictly pseudoconvex domain is a relatively compact domain in a complex manifold such that its boundary admits a $C^{2}$-smooth strictly plurisubharmonic defining function.

Examples of strictly pseudoconvex domains covered by the ball in $\mathbb{C}^{2}$ which contain compact complex curves (and hence are not Stein) may be found in [2]. It is well-known that the ball covers compact complex manifolds as well.

Recall also from [4, 5] that a Stein strictly pseudoconvex domain is covered by the unit ball if and only if its boundary is everywhere locally CR-diffeomorphic to the unit sphere.

The theorem will follow immediately from the two lemmas below.

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Choose a convex coordinate neighbourhood $U\ni x$ and a strictly plurisubharmonic function $\widetilde{\varphi}$ on $\overline{U}$ such that $\widetilde{\varphi}(x)=\varphi(x)$ and

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for some $\varepsilon>0$, see Fig. 1.

For each $n\gg 1$, there is a local inverse $\psi_{n}$ to $\pi$ defined on $\overline{U}$ so that $\psi_{n}(x_{n})=y_{n}$. Set $A_{n}:=\psi_{n}^{-1}(A)$. This is a complex analytic subset of $U$ with boundary in $\partial U$ and ${\varphi|}_{A_{n}}\leq\varphi(x)$ by the choice of $x$. Thus, ${\widetilde{\varphi}|}_{\partial A_{n}}<\varphi(x)-\varepsilon$ for all $n$ by construction, whereas $\widetilde{\varphi}(x_{n})\to\varphi(x)$ as $n\to\infty$. This contradicts the maximum principle for plurisubharmonic functions on complex analytic sets (see e.g. [1, §6.3]), which proves the lemma. ∎

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where $*$ denotes the upper semicontinuous regularisation. As shown in [5, §2.3], it follows from [5, Corollary 2.3] that $\psi$ is plurisubharmonic and strictly negative on $M$. (It is explained in [5, §3.2] how to modify the proof of [5, Corollary 2.3] for non-Stein domains.) By the maximum principle,

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Hence,

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and therefore $\pi\left(\pi^{\prime-1}(B)\right)$ is relatively compact in $D$. ∎