Rafael B. AndristFaculty of Mathematics and Physics,University of Ljubljana, Ljubljana, Sloveniarafael-benedikt.andrist@fmf.uni-lj.si

###### Abstract.

The boundary of every relatively compact Stein domain in a complex manifold of dimension at least two is connected. No assumptions on the boundary regularity are necessary. The same proofs hold also for $q$-complete domains, and in the context of almost complex manifolds as well.

###### Key words and phrases:

pseudoconvex domain, ends of manifolds, Morse theory, almost complex manifolds

###### 1991 Mathematics Subject Classification:

Primary 32F27, Secondary 32F10, 32Q60

## 1. Introduction

It seems to have been well-known to the experts in the 1980s that every bounded strictly pseudoconvex domain with $\mathcal{C}^{2}$-smooth boundary in $\mathbb{C}^{n},n\geq 2,$ has connected boundary. In fact, already in 1953 Serre pointed out that every Stein manifold of dimension at least $2$ has only one end (see Section 3). For a relatively compact Stein domain that admits a collar, this would already imply connectedness of the boundary.However, to the best knowledge of the author, the earliest publication mentioning that every bounded strictly pseudoconvex domain with $\mathcal{C}^{2}$-smooth boundary in $\mathbb{C}^{n},n\geq 2,$ has connected boundary, is due Rosay and Stout [MR0964461]*Corollary, p.1018 in 1989 where they actually prove a stronger result. Again, the connectedness of the boundary was noted by Balogh and Bonk [MR1793800]*p.513 under the same assumptions.In the monograph of Stout [MR2305474]*Corollary 2.4.7 it was established that every relatively compact, strictly pseudoconvex domain with $\mathcal{C}^{2}$-smooth boundary in a Stein manifold of dimension at least two has connected boundary. The proof given there follows from a theorem of ForstneriΔ [MR1278436] about complements of Runge domains.

The almost complex case, but with a $\mathcal{C}^{\infty}$-smooth defining $J$-plurisubharmonic function on a neighborhood of the closure of the relatively compact domain is treated by Bertrand and Gaussier [MR3359581].

For the special case of bounded pseudoconvex domains in $\mathbb{C}^{n}$, a proof without any assumptions on the boundary regularity can be found in the second edition of the textbook of Jarnicki and Pflug [MR4201928]*Corollary 2.6.10. It relies mainly on a topological argument that was provided by Czarnecki, Kulczycki and Lubawski [MR2855300] with an elementary proof: For a bounded domain in $\mathbb{R}^{n}$ the connectedness of its complement is equivalent to the connectedness of its boundary. A very similar proof is given by Izzo [Izzo] who uses an elegant hom*ology argument that relies only on $H_{1}(\mathbb{R}^{n},\mathbb{Z})=0$ to obtain the above-mentioned topological fact.

We will prove the natural generalization of these results without any assumption on boundary regularity and on the ambient space.

###### Definition 1 ([MR3700709]*Definition 3.1.3).

Let $X$ be a complex manifold. We say that a domain $\Omega\subset X$ is *$q$-complete*, if there exists a $\mathcal{C}^{2}$-smooth exhaustion function $\varphi\colon\Omega\to[0,+\infty)$ that is $q$-convex, i.e.if its Levi form has at most $q-1$ negative or zero eigenvalues at each point of $\Omega$.

###### Theorem 2.

Let $X$ be a complex manifold with $\dim_{\mathbb{C}}X=n$ and let $\Omega\subset X$ be a relatively compact $q$-complete domain with $n>q$. Then the boundary $\partial\Omega$ is connected.

By a result of Grauert [MR0098847] $\Omega$ is Stein if and only if it is $1$-complete, and thus we obtain the following corollary.

###### Corollary 3.

Let $X$ be a complex manifold with $\dim_{\mathbb{C}}X=n>1$ and let $\Omega\subset X$ be a relatively compact Stein domain. Then the boundary $\partial\Omega$ is connected.

Since a domain $\Omega\subset X$ in a Stein manifold $X$ is a domain of holomorphy if and only if it is pseudoconvex, we also arrive at the next corollary.

###### Corollary 4.

Let $X$ be a Stein manifold with $\dim_{\mathbb{C}}X=n>1$ and let $\Omega\subset X$ be a relatively compact domain of holomorphy. Then the boundary $\partial\Omega$ is connected.

The proof also extends to the almost complex situation.

###### Theorem 5.

Let $(X,J)$ be an almost complex manifold with $\dim_{\mathbb{R}}X\geq 4$, and let $\Omega\subset X$ be a relatively compact domain with a $\mathcal{C}^{2}$-smooth strictly $J$-plurisubharmonic exhaustion function. Then the boundary $\partial\Omega$ is connected.

###### Remark 6.

The result is sharp in the following sense:

- (1)
In one dimension, every bounded domain of $\mathbb{C}$ is pseudoconvex β and in fact a domain of holomorphy, but obviously does not need to have connected boundary. If we assume in addition that the domain is simply connected, its boundary will be connected, but in general not path-connected.

- (2)
Unbounded domains, even when simply connected, do not need to have a connected complement, e.g.an infinite strip in $\mathbb{C}$. By taking direct products, this yields also counterexamples in higher dimensions.

###### Remark 7.

For symplectic manifolds with contact type boundaries, the boundary does not need to be connected, see McDuff [MR1091622] and Geiges [MR1328705].

This short note is organized as follows:For completeness, in Section 2 we first give the proof for relatively compact, strictly pseudoconvex domains with $\mathcal{C}^{2}$-smooth boundary in a Stein manifold. We could not find a reference for this proof which was communicated by Franc ForstneriΔ.

In Section 3 we provide the topological background for the theory of ends and of continua. In Section 4 we give the proofs for the general situation.

## 2. A Classical Proof for Smooth Boundary

The following proof for the situation where $\Omega$ is a relatively compact, strongly pseudoconvex domain with $\mathcal{C}^{2}$-smooth boundary in a Stein manifold was communicated by Franc ForstneriΔ. This is likely the classical proof that was known at least since the 1980s:

Consider a relatively compact domain $\Omega=\{\rho<c\}$ in an $n$-dimensional Stein manifold $X$, where $n>1$ and $\rho$ is a strongly plurisubharmonic Morse function on a neighborhood of $\overline{\Omega}$, with $d\rho\neq 0$ on $\partial\Omega$. Let $\Omega_{t}=\{\rho\leq t\}$ for $t\leq c$. The topology of $\Omega_{t}$ and $\partial\Omega_{t}$ only changes when $t$ passes a critical level set of $\rho$. (The local normal form of a strongly plurisubharmonic Morse function at an isolated critical point is known in principle since 1924 due to a result of Takagi [zbMATH02595788] which was recovered again by Schur [zbMATH03097391] and by Harvey and Wells [MR0330510].) When passing a local minimum, a new connected component of $\Omega_{t}$ appears, which is not a concern. The only other type of points which can change connectivity of $\Omega_{t}$ is a critical point of index $1$. At such a point, we add a $1$-handle to $\Omega_{t}$. There are two possibilities β either this handle is attached with both ends to the same component of $\Omega_{t}$, or it joins two distinct components. In both cases we see by inspection that the boundary of any connected component remains connected.

Handles of higher index up to $2n-2$ do not change connectivity of the domain or its boundary:The core of the handle of index $k$ is a $k$-disc which is attached with its boundary $(k-1)$-sphere to $\partial\Omega_{t}$. Removing a submanifold of real codimension $\geq 2$ from a manifold does not disconnect the manifold: In our case, we apply this argument to the boundary sphere of the $k$-disc as a submanifold of $\partial\Omega_{t}$.

The connectivity of $\partial\Omega_{t}$ would change by attaching a handle of index $2n-1$ (i.e.of real codimension $1$), but such is not allowed since $\rho$ is plurisubharmonic and $n\geq 2$.

## 3. Ends of topological spaces

The definition of *ends* of a topological space goes back to Freudenthal [MR1545233] and has led to a well developed theory, see the textbook of Hughes and Ranicki [MR1410261].

###### Definition 8.

- (1)
A

*neighborhood of an end*in a non-compact topological space $X$ is a subspace $U\subset X$ which contains a component of $X\setminus K$ for a non-empty compact subspace $K\subset X$. - (2)
An

*end*$e$ of $X$ is an equivalence class of sequences of connected open neighborhoods $X\supset U_{1}\supset U_{2}\supset\dots$ such that$\bigcap_{i=1}^{\infty}U_{i}=\emptyset$ subject to the equivalence relation

$(X\supset U_{1}\supset U_{2}\supset\dots)\sim(X\supset V_{1}\supset V_{2}%\supset\dots)$ if for each $U_{i}$ there exists $j$ with $U_{i}\subseteq V_{j}$, and for each $V_{j}$ there exists $i$ with $V_{j}\subseteq U_{i}$.

###### Remark 9 ([MR1410261]*Example3(i)).

Let $X$ be a topological space with a proper map $\varphi\colon X\to[0,+\infty)$ which is onto, and such that the inverse images $U_{t}=\varphi^{-1}(t,+\infty)\subseteq X$, $t\geq 1$ are connected. Then $X$ has one end.

###### Lemma 10.

Let $X$ be a locally compact Hausdorff space with a countable basis of topology. Then $X$ has one end if and only if there exists an exhaustion by compacts $(K_{j})_{j\in\mathbb{N}}$ of $X$ such that $X\setminus K_{j}$ is connected for every $j\in\mathbb{N}$.

###### Proof.

The βifβ part is straightforward, see also Remark 9. We only need to provide a proof for the βonly ifβ part: Assume to get a contradiction that no exhaustion by compacts $(K_{j})_{j}$ of $X$ exists such that $X\setminus K_{j}$ is connected for every $j\in\mathbb{N}$, but that $X$ has only one end. We can always pass to a subsequence, and thus we may assume that $X\setminus K_{j}$ always has at least two connected components. Moreover, we can assume that none of these components is contained in a compact $K_{\ell(j)},\ell>j,$ for otherwise we could have included this component already in the compact $K_{j}$.Now we pick a connected component $U_{1}$ of $X\setminus K_{1}$. For every $j\geq 2$ we a pick a connected component of $X\setminus K_{j}$ that is contained in $U_{j-1}$. This is always possible, since we eliminated superficial connected components by our choice of compacts $K_{j}$. This sequence of neighborhoods $(U_{j})_{j}$ defines an end of $X$.Since $X\setminus K_{1}$ has at least two connected components, we choose another sequence of neighborhoods $(V_{j})_{j}$ such that $U_{j}$ and $V_{j}$ are different connected components of $X\setminus K_{j}$ for every $j\in\mathbb{N}$, and hence define two different ends.β

###### Definition 11.

A *continuum* is a non-empty, compact, connected metric space.

###### Lemma 12 ([MR1192552]*Theorem 1.8).

The intersection of a decreasing sequence of continua is a continuum.

###### Proposition 13.

Let $X$ be a manifold with countable basis of topology and let $\Omega\subset X$ be a relatively compact domain with one end. Then $\Omega$ has connected boundary.

###### Proof.

We apply Lemma 10 to obtain an exhaustion by compacts $K_{j}$ such that $\Omega\setminus K_{j}$ is connected for every $j\in\mathbb{N}$. Then $\Omega\setminus(K_{j})^{\circ}$ are a decreasing sequence of continua. Thus its limit, which is the boundary $\partial\Omega$, is also a continuum by Lemma 12.β

###### Remark 14.

If the boundary $\partial\Omega$ of the relatively compact domain $\Omega$ admits a collar, then we have a one-by-one correspondence between ends and boundary components.

## 4. Proofs

It was already noted by Serre that a Stein manifold of complex dimension at least two has only one end [MR0064155]*p.59, see also [MR3185220]. His short cohom*ological argument for this fact is given with more details by Gilligan and Huckleberry [MR0616269]*p.186.

Another way of seeing this is to consider a strictly plurisubharmonic Morse exhaustion function which is the approach taken by ForstneriΔ [MR1278436].

###### Proof of Theorem 2.

Let $\dim_{\mathbb{C}}X=n$. Since $\Omega$ is relatively compact, by a small perturbation we may assume that the $\mathcal{C}^{2}$-smooth exhaustion function $\varphi\colon\Omega\to[0,+\infty)$ is a Morse function and still $q$-complete. The Morse index at a critical point is at most $n+q-1$ (see the monograph of ForstneriΔ [MR3700709]*Sections 3.10 and 3.11 for more details). Note that $2n-1>n+q-1\Longleftrightarrow n>q$ is satisfied by assumption. Since we glue only handles of dimension $\leq n+q-1$, and $\Omega$ has real dimension $2n$, the complement of any sublevel set of $\varphi$ is connected. By Lemma 10 the domain $\Omega$ has only one end. Proposition 13 now gives the conclusion.β

###### Proof of Theorem 5.

Let $\dim_{\mathbb{R}}X=2n$. Let $\varphi\colon\Omega\to\mathbb{R}$ be a strictly $J$-plurisubharmonic exhaustion function. Since the domain $\Omega$ is relatively compact, we may slightly disturb $\varphi$ if necessary to obtain a strictly $J$-plurisubharmonic Morse exhaustion function. By [MR3012475]*Corollary 3.4 the Morse index of $\varphi$ at a critical point in an almost complex manifold is at most $n$. Now the conclusion is the same as in the proof of Theorem 2 above.β

###### Remark 15.

The proofs of these two theorems can also be given without using the theory of ends, by instead considering the exhaustion functions and their connected superlevel sets, and applying Lemma 12 directly to them. However, conceptually, it seems more natural to use Proposition 13 which reflects that the βreasonβ comes from the fact that these relatively compact domains have one end.

## Acknowledgments

The author would like to thank Matteo Fiacchi, Franc ForstneriΔ, Tobias Harz, Gerrit Hermann, Hendrik Hermann, Alexander Izzo, and Jaka Smrekar for interesting and helpful discussions and suggestions.

## Funding

The first author was supported by the European Union (ERC Advanced grant HPDR, 101053085 to Franc ForstneriΔ) and grant N1-0237 from ARRS, Republic of Slovenia.