Construction of labyrinths in pseudoconvex domains

TL;DR

The paper constructs special convex sets within pseudoconvex domains in complex space, ensuring certain paths approaching the boundary have infinite length, thus solving an open problem in complex analysis.

Contribution

It introduces a method to build convex sets in pseudoconvex domains that influence the behavior of boundary-approaching paths, addressing a previously unresolved problem.

Findings

Constructed convex sets in pseudoconvex domains with specific boundary properties

Ensured paths approaching the boundary while avoiding these sets have infinite length

Solved an open problem from a recent research paper

Abstract

We build in a given pseudoconvex (Runge) domain $D$ of $\mathbb{C}^N$ a $\mathcal O(D)$ convex set $\Gamma$, every connected component of which is a holomorphically contractible (convex) compact set, enjoying the property that any continuous path $\gamma:[0,1)\rightarrow D$ with $\lim _{r\rightarrow 1}\gamma(r)\in \partial D$ and omitting $\Gamma$ has infinite length. This solves a problem left open in a recent paper by Alarc\'on and Forstneri\v{c}.

Full text

The first author is partly supported by the grant ANR-17-CE40-0021 of the French National Research Agency ANR (project Front). The second author is partly supported by the NCN grant SONATA BIS no. 2017/26/E/ST1/00723

1. Introduction

Alarcón and Forstnerič recently proved that the Euclidean ball ${\mathbb{B}}_{N}$ of ${\mathbb{C}}^{N}$, $N>1$, admits a nonsingular holomorphic foliation by complete properly embedded holomorphic discs [1, Theorem 1]. They asked the natural question whether their result extends to any Runge pseudoconvex domains. As explained in [1, Remark 1], the main obstruction that appears is how to construct a suitable labyrinth in such a domain. Here and in the sequel we call labyrinth of a given pseudoconvex domain $D$ in ${\mathbb{C}}^{N}$ a set $\Gamma$ in $D$ with the property that any continuous path $\gamma:[0,1)\rightarrow D$, with $\lim_{r\rightarrow\infty}\gamma(r)\in\partial D$, whose image does not intersect $\Gamma$, has infinite length. Such sets were already built in pseudoconvex domains by Globevnik, by properly embedding the pseudoconvex domain as a submanifold of ${\mathbb{C}}^{2N+1}$ [6], thus reducing the problem to a construction in ${\mathbb{B}}_{N}$ [5]. However Globevnik’s construction in [5, 6] did not provide with good topological properties of the connected components of the labyrinth, such as convexity or holomorphic contractibility. In [2] the authors simplified Globevnik’s construction building a labyrinth in ${\mathbb{B}}_{N}$ whose connected components are balls in suitably chosen affine real hyperplanes. Alarcón and Forstnerič used a slight modification of this construction to obtain [1, Theorem 1].

The main aim of this short note is to overcome the difficulty pointed out in [1, Theorem 1] and extend the construction made in [2] to pseudoconvex domains.

Theorem 1.1**.**

Let $D$ be a pseudoconvex domain in ${\mathbb{C}}^{N}$ and let $(D_{n})$ be an exhaustion of strictly pseudoconvex domains that are $\mathcal{O}(D)$-convex. Let also $(M_{n})$ be a sequence of positive numbers. Then there are holomorphically contractible compact sets $\Gamma_{n}\subset D_{n+1}\setminus\overline{D}_{n}$ such that $\overline{D}_{n}\cup\bigcup_{j=n}^{m}\Gamma_{j}$ is $\mathcal{O}(D)$-convex for every $m\geq n$, and any continuous path connecting $\partial D_{n}$ to $\partial D_{n+1}$ and omitting $\Gamma_{n}$ has length greater than $M_{n}$.

Moreover, if $D$ is Runge then it can be additionally assumed that each connected component of $\Gamma$ is the image of a $2N-1$-convex body under an automorphism of ${\mathbb{C}}^{N}$.

It will follow from the proof that the $2N-1$-convex bodies appearing in the theorem above are $\mathbb{R}$-linear transformations of $2N-1$-dimensional balls. Proceeding as in [1] one can use Theorem 1.1 to obtain the analogue of [1, Theorem 1] for Runge pseudoconvex domain, and thus answer the question posed by the authors.

The proof of Theorem 1.1 consists in three steps: first, to slightly adapt the main argument of [2] in order to build a suitable labyrinth in a strictly convex domain; second, to make use of a result by Diederich, Fornaess and Wold [3] to perform the construction in any strictly pseudoconvex domain, and three, to exhaust the given pseudoconvex domain by strictly pseudoconvex ones.

2. Proof of Theorem 1.1

Let $D$ be a pseudoconvex domain of ${\mathbb{C}}^{N}$ and let us denote by $\mathcal{O}(D)$ the space of holomorphic functions on $D$.

Let us first recall some classical notions and results about polynomial and holomorphic convexity. We refer the reader to Stout’s book [7]. Let $K$ and $L$ be compact subsets of a pseudoconvex domain $D$ of ${\mathbb{C}}^{N}$. We shall say that $K$ and $L$ are polynomially separated (respectively holomorphically separated with respect to $D$ - or simply $\mathcal{O}(D)$-separated) if there exists a holomorphic polynomial $P$ (resp. a function $f\in\mathcal{O}(D)$) such that $\widehat{p(K)}\cap\widehat{p(L)}$ is empty (resp. $\widehat{f(K)}\cap\widehat{f(L)}=\emptyset$). In particular if there exists a real hyperplane $\mathcal{H}$ such that $K$ is contained in one of the connected component of ${\mathbb{C}}^{N}\setminus\mathcal{H}$ and $L$ is contained in the other one, then $K$ and $L$ are polynomially separated. In this case, we will simply say that $K$ and $L$ are separated by a hyperplane. Note that if $K$ and $L$ are $\mathcal{O}(D)$-separated with $D={\mathbb{C}}^{N}$, then $K$ and $L$ are polynomially separated.

It follows from a classical result of Kallin [7, Theorem 1.6.19], that if $K$ and $L$ are both polynomially convex (resp. $\mathcal{O}(D)$-convex) and polynomially separated (resp. $\mathcal{O}(D)$-separated), then $K\cup L$ is polynomially convex (resp. $\mathcal{O}(D)$-convex).

Let us now proceed with the proof of Theorem 1.1. As a first step, we shall state an analogue of [2, Theorem 1.5] for strictly convex domains.

Lemma 2.1**.**

Let $D$ be a strictly convex domain in ${\mathbb{C}}^{n}$ and let $x\in\partial D$. There exists a neighborhood $U$ of $x$ such that for any $M>0$ and any compact set $K\subset D$ intersecting $U$, there exists a compact set $\Gamma\subset D$ with the following two properties:

(i) $\Gamma$ can be written as a finite union $\bigcup_{i}\Gamma_{i}$, where each $\Gamma_{i}$ is a convex body in a hyperplane, and any $\Gamma_{i}$ is separated from $\bigcup_{j<i}\Gamma_{j}$ by a hyperplane.

(ii) The length of any continuous path $\gamma:[0,1)\rightarrow D\setminus\Gamma$, with $\lim_{r\rightarrow}\gamma(r)\in\partial U\cap D$ and such that $\gamma(r_{0})\in K\cap U$ for some $r_{0}\in[0,1)$ and $\gamma(r)\in U$ for any $r_{0}<r<1$, is greater than $M$.

We only sketch the proof, as it is a simple modification of that of [2, Theorem 1.5]. It makes use of [2, Lemma 2.1], that we recall below for notational convenience.

Lemma 2.2**.**

There exist numbers $m\in{\mathbb{N}}$, $m\geq 2$, and $c\in{\mathbb{R}}$, $0<c<1/2$, depending only on $N$, such that for any $r>0$, there exist finitely many subsets $F_{1},\ldots,F_{m}$ of ${\mathbb{S}}_{N}$ which satisfy the following:

(i) $|p-q|\geq r$ for all $p,q\in F_{j}$, $p\neq q$, $j=1,\ldots,m$;

(ii) If $F:=\bigcup_{j=1}^{m}F_{j}$ then $F\neq\emptyset$ and $\text{dist}(p,F)\leq cr$ for all $p\in{\mathbb{S}}_{N}$.

Outline of the proof of Lemma 2.1.

Up to a translation and an $\mathbb{R}$-linear change of coordinates we can assume that $x=(1,0)\in{\mathbb{C}}\times{\mathbb{C}}^{n-1}$ and that near $x$ a defining function $r$ of $\partial D$ is of the form $r(z)=\|z\|^{2}-1+o(\|z-1\|^{2})$. Then there are two open balls $U_{1}$ and $U_{2}$ of $(1,0)$ and a diffeomorphism $\Phi:U_{1}\to U_{2}$ that maps $U_{1}\cap D$ onto $U_{2}\cap{\mathbb{B}}_{N}$. Upon shrinking the $U_{i}$’s, we can assume that $\Phi$ is arbitrarily close to the identity map in $\mathcal{C}^{2}$-topology. Let $U$ be any relatively compact ball in $U_{1}$, $(1,0)\in U$.

Let $m$ and $c$ be given by Lemma 2.2. Following [2], we fix a sequence $(s_{j})$ of positive numbers, increasing, tending to $1$ and such that $\sum_{j}\sqrt{s_{j}-s_{j-1}}=\infty$. We set $s_{j,k}:=s_{j-1}+k(s_{j}-s_{j-1})/(m+1)$. Let $S_{j,k}$ denote $\Phi^{-1}(U_{2}\cap s_{j,k}\mathbb{S}_{N})$. Then there is a uniform constant $a>0$ such that any tangent ball with origin in $S_{j,k}$ and radius $r_{j}:=a\sqrt{s_{j}-s_{j-1}}$ does not intersect $S_{j,k+1}$, and any ball centred in $S_{j,k}\setminus U_{1}$ does not intersect $U$. We fix $t>1$ such that $tc<1/2$. Let now $F_{1},\ldots,F_{m}$ be given by Lemma 2.2 applied to $r:=2tr_{j}$ and denote by $E_{j,k}$ the set $\Phi^{-1}(s_{j,k}F_{k}\cap U_{2})$. Let us denote by $\Gamma_{j,k,p}$ the tangent ball with origin $p\in E_{j,k}$ ans radius $r_{j}$. Observe that upon choosing $\Phi$ close enough to the identity map, in a way which depends only on $t>1$ - hence only on $N$, the sets $\Gamma_{j,k,p}$ can be separated by hyperplanes from $\bigcup_{(j^{\prime},k^{\prime},p^{\prime})\prec(j,k,p)}\Gamma_{j^{\prime},k^{\prime},p^{\prime}}$, where $\prec$ is the lexicographical order. Following the proof from [2] one easily checks that $\Gamma:=\bigcup_{j=1}^{J}\bigcup_{k,p}\Gamma_{j,k,p}$ satisfies the desired properties for some $J$ large enough. ∎

Remark 2.3**.**

(1) Observe that each connected component of $\Gamma$ is the image of a $2N-1$ dimensional ball under an $\mathbb{R}$-linear isomorphism.

(2) Note that setting $\Gamma:=\bigcup_{j=J}^{J^{\prime}}\bigcup_{k,p}\Gamma_{j,k,p}$ in the proof of Lemma 2.2, $J$ and $J^{\prime}$ can be chosen large enough so that $\Gamma$ is in any given $\epsilon$-neighbourhood of $\partial D$ and separated by a hyperplane from any given compact set in $D$.

The second step consists in extending [1, Theorem 1.5] to strictly pseudoconvex domains, using Lemma 2.1. This is the purpose of the next lemma.

Lemma 2.4**.**

Let $D$ be a strictly pseudoconvex domain in ${\mathbb{C}}^{N}$ and let $K$ be a compact subset of $D$. Then for any $M>0$ there is a compact set $\Gamma$ in $D\setminus K$ such that $\Gamma\cup K$ is $\mathcal{O}(D)$-convex, with the property that any continuous path $\gamma:[0,1)\rightarrow D\setminus\Gamma$, $\gamma(0)\in K$ and $\lim_{r\rightarrow 1}\gamma(r)\in\partial D$, has length greater than $M$.

$\Gamma$* can be chosen so that each of its connected components is holomorphically contractible. If additionally $D$ is Runge, the connected components of $\Gamma$ can even be chosen as the images of $2N-1$-dimensional balls under ${\mathbb{R}}$-linear isomorphisms and automorphisms of $\mathbb{C}^{N}$ .*

Proof.

Let $K$ be fixed as in the statement. By [3, Theorem 1.1], there exists an open ball $U$ centred at $x$ and a holomorphic embedding $\Phi_{x}:\overline{D}\to\overline{\mathbb{B}}_{N}$ such that $\Phi_{x}(U)$ and some $\Gamma^{\prime}\subset\Phi_{x}(D)$ satisfy the conclusion of Lemma 2.1 for some constant $M^{\prime}>0$. Upon choosing $M^{\prime}$ large enough - in a way which depends only on $\Phi_{x}$, the set $\Gamma:=\Phi_{x}^{-1}(\Gamma^{\prime})$ satisfies that any continuous path $\gamma$ in $D\setminus\Gamma$ connecting $K$ to $U\cap\partial D$ and satisfying $\gamma(r_{0})\in K^{\prime}\cap U$ for some $r_{0}\in[0,1)$ and $\gamma(r)\in U$, $r_{0}<r<1$, has length greater than $M$. Moreover, by Remark 2.3 (2) and Kallin’s theorem, $\Gamma$ can also be chosen so that $\Gamma\cup K$ is $\mathcal{O}(D)$-convex and contained in an $\epsilon$-neighbourhood of $\partial D$ for any given $\epsilon>0$.

Let us then consider a finite covering of $\partial D$ by such open balls $U_{x_{1}},\ldots,U_{x_{k}}$. Upon slightly shrinking the $U_{x_{j}}$’s, we may and shall assume that there exist $\delta,\eta>0$ and an open $\eta$-neighbourhood $V$ of $\partial D$ such that the distance between $V\cap\partial U_{x_{j}}$ and $V\cap\partial(\bigcup_{i\neq j}U_{x_{i}})$ is greater than $\delta$ for $j=1,\ldots,k$. From now on, let us fix $K^{\prime}=D\setminus V$. Observe that $K^{\prime}\cap U_{j}\neq\emptyset$ for any $j$ and that upon choosing $\eta$ small enough, we shall assume that $K\subset K^{\prime}$. Let us enumerate the sets $U_{x_{i}}$ as a sequence $(U_{j})$ in such a way that for any $i=1,\ldots,k$ there exist infinitely many $j$ such that $U_{j}=U_{x_{i}}$. We denote by $\Phi_{j}$ the mapping corresponding to $U_{j}$ and given by [3]. Following the above procedure with $K^{\prime}=D\setminus V$, we build a labyrinth $\Gamma_{1}$ in $D$ and such that:

1. (i)

$\Gamma_{1}$ is contained in $V$ and $\Gamma_{1}\cup K$ is $\mathcal{O}(D)$-convex; 2. (ii)

Any continuous path $\gamma$ in $D\setminus\Gamma_{1}$ connecting $K$ to $U_{1}\cap\partial D$ and satisfying $\gamma(r_{0})\in K^{\prime}\cap U_{1}$ for some $r_{0}\in[0,1)$ and $\gamma(r)\in U_{1}$, $r_{0}<r<1$, has length greater than $M$.

Assuming that $\Gamma_{1},\ldots,\Gamma_{j}$ have been built, we build $\Gamma_{j+1}$ in $D$ such that:

1. (i)

$\Gamma_{j+1}$ is contained in an $\eta_{j}$-neighborhood of $\partial D$, where $\eta_{j}$ is the distance from $\partial D$ to $\Gamma_{1}\cup\ldots\cup\Gamma_{j}$, and $\bigcup_{i=1}^{j+1}\Gamma_{i}\cup K$ is $\mathcal{O}(D)$-convex; 2. (ii)

Any continuous path $\gamma$ in $D\setminus\Gamma_{j+1}$ connecting $K$ to $U_{j+1}\cap\partial D$ and satisfying $\gamma(r_{0})\in K^{\prime}\cap U_{j+1}$ for some $r_{0}\in[0,1)$ and $\gamma(r)\in U_{j+1}$, $r_{0}<r<1$, has length greater than $M$.

It is now easily checked that there exists $J\in{\mathbb{N}}$ big enough so that $\Gamma:=\bigcup_{j=1}^{J}\Gamma_{j}$ has the desired property. Actually, it is enough to take $J$ such that each $U_{x_{j}}$ appears in a sequence $(U_{j})$ at least $n$ times, where $n\delta>M$. Indeed, let $\gamma$ be a path in $D$ with $\gamma(0)\in K$ and $\lim_{r\rightarrow 1}\gamma(r)\in\partial D$. Since $K\subset D\setminus V$ and $\gamma$ is continuous, without loss of generality, we may assume, up to re-parametrization, that $\gamma([0,1))\subset V$. If there exists $0<r_{1}<r_{2}<1$ and $j\leq J$ such that $\gamma(r)\in U_{j}$ for any $r_{1}\leq r\leq r_{2}$ and $\gamma(r_{1})\in V_{j-1}$ and $\gamma(r_{2})\in V_{j}$, where $V_{j}$ is the $\eta_{j}$-neighbourhood appearing in the construction, then the length of $\gamma$ is clearly greater than $M$. If not, it means that $\gamma$ has to escape from some $U_{j}$ as many times as it may have to pass through some $\Gamma_{j}$. With $J$ chosen as above, $\gamma$ would then have to pass at least $n$ times from a $U_{j}$ to another. Since the image of $\gamma$ is in $V$ and the distance between $V\cap\partial U_{x_{j}}$ and $V\cap\partial(\bigcup_{i\neq j}U_{x_{i}})$ is greater than $\delta$ for $j=1,\ldots,k$, the length of $\gamma$ has to bigger than $n\delta$.

The $\mathcal{O}(D)$-convexity of $\Gamma\cup K$ proceeds from the construction and Kallin’s theorem recalled above. Observe that the last assertion of the lemma directly follows from the construction.

∎

The third and last step is straightforward: Given $D$ a pseudoconvex domain, we consider an exhaustion $(D_{n})$ of $D$ by $\mathcal{O}(D)$-convex strictly pseudoconvex domains (Runge strictly pseudoconvex domains if $D$ is Runge) and inductively apply Lemma 2.4. For the existence of such an exhaustion, we refer to [4].