# Complete nonsingular holomorphic foliations
on Stein manifolds

Antonio Alarcón and Franc Forstnerič  Antonio Alarcón, Departamento de
Geometría y Topología e Instituto de Matemáticas (IMAG), Universidad de
Granada, Campus de Fuentenueva s/n, E–18071 Granada, Spain<EMAIL_ADDRESS>Franc Forstnerič, Faculty of Mathematics and Physics, University of Ljubljana,
and Institute of Mathematics, Physics, and Mechanics, Jadranska 19, 1000
Ljubljana, Slovenia<EMAIL_ADDRESS>
(Date: 26 December 2023)

###### Abstract.

Let $X$ be a Stein manifold of complex dimension $n>1$ endowed with a
Riemannian metric $\mathfrak{g}$. We show that for every integer $k$ with
$\left[\frac{n}{2}\right]\leq k\leq n-1$ there is a nonsingular holomorphic
foliation of dimension $k$ on $X$ all of whose leaves are closed and
$\mathfrak{g}$-complete. The same is true if $1\leq
k<\left[\frac{n}{2}\right]$ provided that there is a complex vector bundle
epimorphism $TX\to X\times\mathbb{C}^{n-k}$. We also show that if
$\mathcal{F}$ is a proper holomorphic foliation on $\mathbb{C}^{n}$ $(n>1)$
then for any Riemannian metric $\mathfrak{g}$ on $\mathbb{C}^{n}$ there is a
holomorphic automorphism $\Phi$ of $\mathbb{C}^{n}$ such that the image
foliation $\Phi_{*}\mathcal{F}$ is $\mathfrak{g}$-complete. The analogous
result is obtained on every Stein manifold with Varolin’s density property.

###### Key words and phrases:

Stein manifold; complete holomorphic foliation; density property

###### 2010 Mathematics Subject Classification:

Primary 32M17, 32M25; secondary 32H02, 37F75

Dedicated to Josip Globevnik

##  1\. Introduction

Let $X$ be a complex manifold endowed with a Riemannian metric $\mathfrak{g}$.
A locally closed subvariety $Y$ of $X$ is said to be $\mathfrak{g}$-complete
if every smooth divergent path $\gamma:[0,1)\to Y$ (i.e., one leaving every
compact subset of $Y$) has infinite $\mathfrak{g}$-length:
$\int_{0}^{1}\mathfrak{g}(\gamma(t),\dot{\gamma}(t))\,dt=+\infty.$ If
$\mathfrak{g}$ is a complete metric on $X$ then clearly every closed connected
complex subvariety of $X$ of positive dimension is $\mathfrak{g}$-complete. A
holomorphic foliation on $X$ is said to be $\mathfrak{g}$-complete if every
leaf is $\mathfrak{g}$-complete. For the theory of holomorphic foliations, see
Scárdua [26].

The results of this paper concern the problem, posed by Yang in 1977 [31, 30],
whether there exist bounded complex submanifolds of a Euclidean space
$\mathbb{C}^{n}$ for $n>1$ which are complete in the Euclidean metric. We
begin with a brief overview of the main developments on this subject,
referring the interested reader to the more complete recent survey by Alarcón
[3]. We also point out that Yang’s problem is related to the Calabi–Yau
problem in the theory of minimal surfaces; see [7, Chapter 7] for a recent
survey of the latter subject.

The first affirmative result on Yang’s problem was obtained by Jones [23], who
constructed a holomorphic immersion of the unit disc into $\mathbb{C}^{2}$ and
an embedding into $\mathbb{C}^{3}$ with bounded image and complete induced
metric. His method is based on the BMO duality theorem. Much later, Martín,
Umehara, and Yamada [25] used a geometric method to construct complete bounded
complex curves in $\mathbb{C}^{2}$ with arbitrary finite genus and finitely
many ends. This was followed by Alarcón and López [10] who showed that any
topological type is possible. The methods used in these constructions do not
provide any control of the complex structure on the curve, except in the
simply connected case when any such curve is biholomorphic to the disc. In
[4], the authors used the Riemann–Hilbert boundary value problem and gluing
methods to prove that every bordered Riemann surface admits a complete proper
holomorphic immersion into the ball of $\mathbb{C}^{2}$ and a complete proper
holomorphic embedding into the ball of $\mathbb{C}^{3}$. A related result for
Riemann surfaces with finite genus and countably many ends was obtained by the
authors in [6, Theorem 1.8]. These are still the only results on Yang’s
problem with a complete control of the conformal structure on the underlying
Riemann surface.

All results mentioned so far pertain to complex curves. This is how things
stood until the seminal work of Globevnik [21] in 2015. In his landmark
construction, Globevnik showed that for every pair of integers $1\leq k<n$
there exists a holomorphic foliation of the unit ball $\mathbb{B}^{n}$ in
$\mathbb{C}^{n}$ by closed complete complex subvarieties of complex dimension
$k$, most of which are smooth (without singularities). The leaves of
foliations in his construction are connected components of the level sets of
holomorphic maps $f:\mathbb{B}^{n}\to\mathbb{C}^{n-k}$. Completeness of the
leaves is ensured by choosing the map $f$ to grow sufficiently fast on
components of a suitable labyrinth $\Gamma\subset\mathbb{B}^{n}$ having the
property that any divergent path in $\mathbb{B}^{n}$ avoiding all but finitely
many components of $\Gamma$ has infinite Euclidean length (see Lemma 2.1). The
construction of such labyrinths was one of the main new results of Globevnik’s
paper. In the sequel [22], Globevnik extended his construction to any
pseudoconvex domain in $\mathbb{C}^{n}$. Further improvements and
generalizations of his results were made by Alarcón [1, 2].

This is a suitable point to state our first main result. It generalizes
Globevnik’s theorem to an arbitrary Riemannian Stein manifold, and the
foliations that we find are nonsingular.

######  Theorem 1.1.

Let $X$ be a Stein manifold of complex dimension $n>1$ endowed with a
Riemannian metric $\mathfrak{g}$. For every integer $k$ with
$\left[\frac{n}{2}\right]\leq k\leq n-1$ there exists a nonsingular
holomorphic foliation of dimension $k$ on $X$ all of whose leaves are closed
and $\mathfrak{g}$-complete. The same holds if $1\leq
k<\left[\frac{n}{2}\right]$ provided that there is a complex vector bundle
epimorphism $TX\to X\times\mathbb{C}^{n-k}$. In particular, if $X$ is
parallelizable then it admits a nonsingular holomorphic foliation of any
dimension $k\in\\{1,\ldots,n-1\\}$ with closed $\mathfrak{g}$-complete leaves.

Foliations in Theorem 1.1 are given by holomorphic submersions
$X\to\mathbb{C}^{n-k}$. The proof (see Section 2) combines the methods of
Globevnik [21, 22] with those of Forstnerič [16]; the latter provide the
h-principle for holomorphic submersions from any Stein manifold $X$ to
Euclidean spaces of dimension $<\dim X$.

The construction methods used in the proof of Theorem 1.1 (like those in [21,
22, 1, 2]) do not provide any control of the topology or the complex structure
of the leaves. Soon after Globevnik’s work, Alarcón, Globevnik, and López [8,
9] combined the use of labyrinths with the approximation theory for
holomorphic automorphisms of complex Euclidean spaces to construct complete
properly embedded complex hypersurfaces in the ball $\mathbb{B}^{n}$ of
$\mathbb{C}^{n}$ $(n>1)$ with partial control on their topology, and with
prescribed topology if $n=2$. The main idea is to inductively deform a given
closed complex submanifold $Y\subset\mathbb{C}^{n}$ by holomorphic
automorphisms of $\mathbb{C}^{n}$ such that the images avoid more and more
pieces of a given labyrinth $\Gamma\subset\mathbb{B}^{n}$, and they converge
to a properly embedded complete complex submanifold of $\mathbb{B}^{n}$ which
is biholomorphic to a Runge domain in $Y$. This construction is possible if
$\Gamma$ is polynomially convex and the connected components of $\Gamma$ are
holomorphically contractible. The labyrinths used in the aforementioned papers
consist of balls in suitably placed affine real hyperplanes in
$\mathbb{C}^{n}$. A bit later, the authors proved in [5] that the ball
$\mathbb{B}^{n}$ for $n>1$ admits a nonsingular holomorphic foliation by
complete holomorphic discs. By using labyrinths in pseudoconvex shells in
$\mathbb{C}^{n}$, constructed by Charpentier and Kosiński in [13], one obtains
the analogous result with the ball replaced by an arbitrary Kobayashi
hyperbolic pseudoconvex Runge domain in $\mathbb{C}^{n}$ with $n>1$ (see [5,
Remark 1]). If the domain fails to be hyperbolic then some leaves of the
foliation may be complex lines. See the survey [3] for more information.

In order to extend this technique to more general Stein manifolds, we must
assume that the manifold has many holomorphic automorphisms. The suitable
class are Stein manifolds with the density property; see Definition 1.6.
Another technical issue is to find suitable labyrinths in $X$ with
holomorphically contractible components. We do this in Section 3 in the course
of proof of Theorem 1.7 (see Lemma 3.2, which is an important new tool).

We shall consider foliations satisfying the following condition.

######  Definition 1.2.

Let $X$ be a connected Stein manifold of dimension $>1$. A (possibly singular)
holomorphic foliation $\mathcal{F}$ on $X$ is proper if every leaf
$\mathcal{F}_{x}$ $(x\in X)$ is closed and satisfies $\dim\mathcal{F}_{x}\geq
1$, and for every compact subset $K\subset X$ the set

(1.1) $\mathcal{F}(K):=\overline{\bigcup_{x\in K}\mathcal{F}_{x}}$

is such that $X\setminus\mathcal{F}(K)$ is nonempty and not relatively compact
in $X$.

A biholomorphic map clearly takes a proper foliation to another such
foliation.

######  Proposition 1.3.

Every holomorphic foliation $\mathcal{F}$ on a connected Stein manifold $X$
given by a nonconstant holomorphic map
$f=(f_{1},\ldots,f_{q}):X\to\mathbb{C}^{q}$ with $1\leq q<\dim X$ is proper.

###### Proof.

Every irreducible component of a nonempty fibre $f^{-1}(z)$
$(z\in\mathbb{C}^{q})$ is a closed complex subvariety of $X$ of some dimension
$d\in\\{1,\ldots,\dim X-1\\}$. In particular, the foliation does not have any
zero dimensional leaves. Given a compact set $K\subset X$ we clearly have that

(1.2) $\mathcal{F}(K)\subset f^{-1}(f(K))\subset f_{j}^{-1}(f_{j}(K))\ \
\text{for every $j=1,\ldots,q$.}$

Since $f$ is nonconstant, the function $f_{j}:X\to\mathbb{C}$ is nonconstant
for some $j$, hence an open map. Thus, $f_{j}(X)$ is an open subset of
$\mathbb{C}$ containing the compact subset $f_{j}(K)$. Choose a nonempty open
subset $U\subset f_{j}(X)\setminus f_{j}(K)$. The set $X\setminus
f_{j}^{-1}(f_{j}(K))$ then contains $f_{j}^{-1}(U)$, hence is nonempty and not
relatively compact. By (1.2) the same is true for $X\setminus\mathcal{F}(K)$.
∎

The proof of Proposition 1.3 also gives the following criterium for
properness.

######  Corollary 1.4.

Assume that $X$ is a connected Stein manifold of dimension $>1$ and
$\mathcal{F}$ is a holomorphic foliation on $X$ with all leaves closed and of
positive dimension. If there is a nonconstant holomorphic function on $X$
which is constant on every leaf of $\mathcal{F}$, then $\mathcal{F}$ is a
proper foliation.

An example of a nonproper (singular) holomorphic foliation is given by
punctured complex lines through the origin in $\mathbb{C}^{2}$, with the
origin a leaf of dimension zero. Taking a Cartesian product with $\mathbb{C}$
gives a nonproper holomorphic foliation of $\mathbb{C}^{3}$ with a closed leaf
of dimension $1$ and nonclosed leaves of dimension $2$. Proposition 1.3 and
Corollary 1.4 show that there are no simple examples of nonproper holomorphic
foliations having closed leaves.

Our second main result is the following. It is proved in Section 3.

######  Theorem 1.5.

Let $\mathfrak{g}$ be a Riemannian metric on $\mathbb{C}^{n}$, $n>1$. For
every proper holomorphic foliation $\mathcal{F}$ on $\mathbb{C}^{n}$ (see
Definition 1.2) there is a holomorphic automorphism
$\Phi\in\mathrm{Aut}(\mathbb{C}^{n})$ such that the image foliation
$\Phi_{*}\mathcal{F}$ with leaves $\Phi(\mathcal{F}_{z})$
$(z\in\mathbb{C}^{n})$ is $\mathfrak{g}$-complete.

This theorem is nontrivial if the metric $\mathfrak{g}$ is not complete on
$\mathbb{C}^{n}$, and its main interest is when $\mathfrak{g}$ decays fast at
infinity. Note that the foliation $\Phi_{*}\mathcal{F}$ has exactly the same
leaves as the original foliation $\mathcal{F}$ up to biholomorphisms, but they
are now sufficiently twisted in $\mathbb{C}^{n}$ to become
$\mathfrak{g}$-complete.

By using labyrinths provided by Lemma 3.2, we shall also prove the analogue of
Theorem 1.5 for every Stein manifold with the density property, a notion
introduced by Varolin [29].

Recall that a holomorphic vector field on a complex manifold $X$ is called
complete if its flow exists for all complex values of time, so it forms a
complex one-parameter group of holomorphic automorphisms of $X$.

######  Definition 1.6 (See Varolin [29] or Definition 4.10.1 in [18]).

A complex manifold $X$ has the density property if every holomorphic vector
field on $X$ can be approximated uniformly on compacts by sums and commutators
of complete holomorphic vector fields on $X$.

The fact that the Euclidean space $\mathbb{C}^{n}$ for $n>1$ has the density
property was discovered by Andersén and Lempert [11], thereby giving birth to
this theory. It is known that most complex Lie groups and complex homogeneous
manifolds have the density property. Surveys of this subject can be found in
[18, Chapter 4], [19], and [24]. An important point is that, on any Stein
manifold $X$ with the density property, one can approximate isotopies of
biholomorphic maps between Stein Runge domains in $X$ by isotopies of
holomorphic automorphisms of $X$; see Forstnerič and Rosay [20] for the case
$X=\mathbb{C}^{n}$ and [18, Theorem 4.10.5] for the general case. This result
plays an essential role in the proofs of Theorems 1.5, 1.7, and 1.8.

The following result generalizes Theorem 1.5; it is proved in Section 3. As
above, it is nontrivial only if the given metric $\mathfrak{g}$ on $X$ is not
complete.

######  Theorem 1.7.

Let $X$ be a Stein manifold with the density property, endowed with a
Riemannian metric $\mathfrak{g}$. For every proper holomorphic foliation
$\mathcal{F}$ on $X$ there is a holomorphic automorphism
$\Phi\in\mathrm{Aut}(X)$ such that the image foliation $\Phi_{*}\mathcal{F}$
is $\mathfrak{g}$-complete.

The analogous construction applies on any pseudoconvex Runge domain in a Stein
manifold with the density property. This gives the following result,
generalizing the aforementioned results in [5, 8, 9] for domains in
$\mathbb{C}^{n}$ with $n>1$. See also Theorem 3.3.

######  Theorem 1.8.

Let $X$ be a Stein manifold of dimension $>1$ with the density property, and
let $\Omega$ be a pseudoconvex Runge domain in $X$ endowed with a Riemannian
metric $\mathfrak{g}$. For every proper holomorphic foliation
$\mathcal{F}_{0}$ on $X$ there is a $\mathfrak{g}$-complete holomorphic
foliation $\mathcal{F}$ on $\Omega$ such that every leaf of $\mathcal{F}$ is
biholomorphic to a pseudoconvex Runge domain in a leaf of $\mathcal{F}_{0}$.

Theorems 1.5, 1.7, and 1.8 suggest that the main obstruction to finding
complete holomorphic foliations with specific types of leaves on a given Stein
Riemannian manifold with the density property lies in the topological and the
complex structure of the manifold, and not in the choice of the Riemannian
metric.

##  2\. Proof of Theorem 1.1

We begin by sketching the proof of Globevnik’s main theorem in [21]. The
essential ingredient are labyrinths $\Gamma$ as in the following lemma from
[21]. A simpler construction can be found in [9], where the connected
components of $\Gamma$ are balls in affine real hyperplanes.

######  Lemma 2.1.

Given numbers $0<r<s$ and $M>0$, there is a compact set $\Gamma\subset
s\mathbb{B}^{n}\setminus r\overline{\mathbb{B}}^{n}$ with connected complement
satisfying the following two conditions:

  1. (a)

the compact set $r\overline{\mathbb{B}}^{n}\cup\Gamma$ is polynomially convex,
and

  2. (b)

every piecewise smooth path $\gamma:[0,1]\to\mathbb{C}^{n}\setminus\Gamma$
with $\gamma(0)\in r\overline{\mathbb{B}}^{n}$ and
$\gamma(1)\in\mathbb{C}^{n}\setminus s\mathbb{B}^{n}$ has Euclidean length
$\geq M$.

######  Remark 2.2.

Given a Riemannian metric $\mathfrak{g}$ on $\mathbb{C}^{n}$ and a constant
$M>0$, there is a labyrinth $\Gamma$ as in the lemma such that any path
crossing the shell $s\mathbb{B}^{n}\setminus r\overline{\mathbb{B}}^{n}$ and
avoiding $\Gamma$ has $\mathfrak{g}$-length $\geq M$. The reason is that any
two Riemannian metrics are comparable on a compact set. When condition (b)
holds for a metric $\mathfrak{g}$, we say that $\Gamma$ enlarges the
$\mathfrak{g}$-distance by $M$.

Granted the lemma, the proof in [21] proceeds as follows. The main case is to
find a complete foliation by hypersurfaces given as level sets of a
holomorphic function $f:\mathbb{B}^{n}\to\mathbb{C}$. (By Sard’s theorem, most
level sets of $f$ are nonsingular.) Pick a sequence $0<r_{1}<r_{2}<\cdots<1$
with $\lim_{i\to\infty}{r_{i}}=1$. In each spherical shell
$S_{i}=\\{z\in\mathbb{C}^{n}:r_{i}<|z|<r_{i+1}\\}$ we choose a labyrinth
$\Gamma_{i}$ satisfying Lemma 2.1 with a constant $M_{i}>0$, chosen such that

(2.1) $\sum_{i=1}^{\infty}M_{i}=+\infty.$

Since the compact set $K_{i}=r_{i}\overline{\mathbb{B}}^{n}\cup\Gamma_{i}$ is
polynomially convex for every $i\in\mathbb{N}$, a standard construction using
the Oka–Weil theorem gives a holomorphic function $f$ on $\mathbb{B}^{n}$
satisfying

(2.2) $\lim_{i\to\infty}\min_{z\in\Gamma_{i}}|f(z)|=+\infty.$

It follows that any divergent path $\gamma:[0,1)\to\mathbb{B}^{n}$ on which
$f$ is bounded avoids $\Gamma_{i}$ for all sufficiently big $i\in\mathbb{N}$,
and hence $\gamma$ has infinite Euclidean length by Lemma 2.1 (b) and (2.1).
Thus, every nonempty level set $\\{f=c\\}$ is a complete closed (possibly
singular) complex hypersurface in $\mathbb{B}^{n}$. Foliations of lower
dimensions are obtained by adding generically chosen additional component
functions, noting that the leaves are automatically complete.

The construction in the proof of Theorem 1.1 is similar to the one of
Globevnik, except that we also use the results of Forstnerič [16] on the
existence of holomorphic submersions from Stein manifolds to Euclidean spaces.
His main result (see [16, Theorem 2.5]) is that a holomorphic submersion
$f:X\to\mathbb{C}^{q}$ always exists if $1\leq
q\leq\left[\frac{n+1}{2}\right]$ where $n=\dim X$, so we obtain a nonsingular
holomorphic foliation of $X$ of any dimension $k=n-q$ with
$\left[\frac{n}{2}\right]\leq k\leq n-1$. If on the other hand
$\left[\frac{n+1}{2}\right]<q\leq n-1$ then a holomorphic submersion
$X^{n}\to\mathbb{C}^{q}$ exists if and only if the holomorphic tangent bundle
of $X$ admits a surjective complex vector bundle map $\theta:TX\to
X\times\mathbb{C}^{q}$ onto the trivial bundle of rank $q$. In fact, the
h-principle holds: every surjective complex vector bundle map $\theta$ as
above is homotopic through maps of the same kind to the tangent map of a
holomorphic submersion $f:X\to\mathbb{C}^{q}$. In [16, Theorem 2.5] it is also
shown that the analogous results hold with interpolation on closed complex
subvarieties and approximation on compact $\mathscr{O}(X)$-convex subsets of
$X$, in analogy to the constructions of holomorphic functions in the
Oka–Cartan theory and of holomorphic maps in Oka theory. Noncritical
holomorphic functions also exist on reduced Stein spaces with singularities,
see [17].

###### Proof of Theorem 1.1.

We embed the Stein manifold $X$ as a closed complex submanifold in a Euclidean
space $\mathbb{C}^{N}$; see [18, Theorems 2.4.1 and 9.3.1] for a survey of the
classical embedding theorems. For every $i\in\mathbb{N}$ the set $X_{i}=X\cap
i\overline{\mathbb{B}}^{N}$ is compact and $\mathscr{O}(X)$-convex, i.e.,
holomorphically convex in $X$. Pick a number $c_{i}>0$ such that

(2.3) $\mathfrak{g}(x,v)\geq c_{i}|v|$ holds for every $x\in X_{i+1}$ and
$v\in T_{x}X$.

Here, $|v|$ denotes the Euclidean length of a tangent vector $v\in
T_{x}X\subset T_{x}\mathbb{C}^{N}\cong\mathbb{C}^{N}$. In every spherical
shell $S_{i}=\\{z\in\mathbb{C}^{N}:i<|z|<i+1\\}$ $(i\in\mathbb{N})$ we choose
a labyrinth $\Gamma_{i}$ satisfying Lemma 2.1 with a constant $M_{i}>0$,
chosen such that

(2.4) $\sum_{i=1}^{\infty}c_{i}M_{i}=+\infty.$

We explain the basic case with $k=n-1$. Using the Oka–Weil type approximation
theorem for noncritical holomorphic functions (see [16, Theorem 2.1])
inductively, we find a holomorphic function $f:X\to\mathbb{C}$ without
critical points such that condition (2.2) holds for all $i\in\mathbb{N}$ with
$\Gamma_{i}$ replaced by $\Gamma^{\prime}_{i}:=\Gamma_{i}\cap X$. To be
precise, choose constants $C_{i}>0$ $(i\in\mathbb{N})$ with
$\lim_{i\to\infty}C_{i}=+\infty$. In the induction step, we are given a
noncritical holomorphic function $f_{i}$ on a neighbourhood of $X_{i}$ in $X$.
Next, we define $f_{i}$ on a neighbourhood of $\Gamma^{\prime}_{i}$ in $X$ to
be an arbitrary noncritical holomorphic function satisfying
$\min_{\Gamma^{\prime}_{i}}|f_{i}|>C_{i}$. Since
$X_{i}\cup\Gamma^{\prime}_{i}$ is a compact $\mathscr{O}(X)$-convex set, we
can apply [16, Theorem 2.1] to find a noncritical holomorphic function
$f_{i+1}$ on a neighbourhood of $X_{i+1}$ in $X$ which approximates $f_{i}$ on
$X_{i}\cup\Gamma^{\prime}_{i}$. Assuming that the approximation is close
enough at every step, the sequence $f_{i}$ converges uniformly on compacts in
$X$ to a noncritical holomorphic function $f:X\to\mathbb{C}$ satisfying
$\min_{\Gamma^{\prime}_{i}}|f|>C_{i}$ for every $i\in\mathbb{N}$. Pick a
divergent path $\gamma:[0,1)\to X$ contained in a level set of $f$ and choose
$i_{0}\in\mathbb{N}$ such that $\gamma(0)\in i_{0}\mathbb{B}^{N}$. Note that
$\gamma$ also diverges in $\mathbb{C}^{N}$ since $X$ is a closed noncompact
submanifold. We see as before that there is an integer $i_{1}\geq i_{0}$ such
that $\gamma$ avoids the labyrinth $\Gamma^{\prime}_{i}$, and hence also the
labyrinth $\Gamma_{i}\subset\mathbb{C}^{N}$ for all $i\geq i_{1}$. From (2.3)
and (2.4) it follows that

$\int_{0}^{1}\mathfrak{g}(\gamma(t),\dot{\gamma}(t))dt\ \geq\
\sum_{i=i_{1}}^{\infty}c_{i}M_{i}=+\infty.$

Hence, every nonempty level set $f^{-1}(c)$ is a closed complete complex
hypersurface in $X$, which is smooth since $f$ has no critical points.

The same argument gives nonsingular $\mathfrak{g}$-complete holomorphic
foliations of $X$ of any dimension $k\geq\left[\frac{n}{2}\right]$. In this
case, one inductively uses the Oka-Weil type approximation theorem for
holomorphic submersions (see [16, Theorem 2.5]) in the above proof; we leave
out the obvious details. The same holds for $1\leq k<\left[\frac{n}{2}\right]$
if we assume the existence of a surjective complex vector bundle map $TX\to
X\times\mathbb{C}^{q}$ with $q=n-k>\left[\frac{n+1}{2}\right]$. ∎

By using the full power of [16, Theorem 2.5] we obtain the following stronger
statement.

######  Theorem 2.3.

Let $X$ be a closed complex submanifold of dimension $n>1$ in $\mathbb{C}^{N}$
endowed with a Riemannian metric $\mathfrak{g}$. For every integer $k$ with
$N-\left[\frac{n+1}{2}\right]\leq k\leq N-1$ there is a nonsingular
holomorphic foliation $\mathcal{F}$ of dimension $k$ on $\mathbb{C}^{N}$,
given by a holomorphic submersion $f:\mathbb{C}^{N}\to\mathbb{C}^{N-k}$, such
that every leaf $\mathcal{F}_{z}$ is transverse to $X$ and the induced
foliation on $X$ is $\mathfrak{g}$-complete.

###### Proof.

Theorem 1.1 furnishes a holomorphic submersion $f_{0}:X\to\mathbb{C}^{q}$ with
$\mathfrak{g}$-complete leaves, where $q=N-k\leq\left[\frac{n+1}{2}\right]$.
Clearly, $f_{0}$ extends to a holomorphic submersion $U\to\mathbb{C}^{q}$ from
an open neighbourhood $U$ of $X$ in $\mathbb{C}^{N}$. By [16, Theorem 2.5]
there is a holomorphic submersion $f:\mathbb{C}^{N}\to\mathbb{C}^{q}$ which
agrees with $f_{0}$ to the second order along $X$. The foliation on
$\mathbb{C}^{N}$ determined by $f$ then clearly satisfies Theorem 2.3.
Furthermore, given a Riemannian metric $\mathfrak{h}$ on $\mathbb{C}^{N}$, one
can choose $\mathcal{F}$ to be $\mathfrak{h}$-complete, transverse to $X$, and
such that its trace on $X$ is $\mathfrak{g}$-complete. We leave further
details to the reader. ∎

######  Remark 2.4.

The proof of Theorem 1.1 actually gives for any $1\leq
q\leq\left[\frac{n+1}{2}\right]$ (where $n=\dim X$) a holomorphic submersion
$f:X\to\mathbb{C}^{q}$ which is unbounded on every divergent path of finite
$\mathfrak{g}$-length in $X$. The same holds also for
$q\in\\{\left[\frac{n+1}{2}\right]+1,\ldots,n-1\\}$ if there is a surjective
complex vector bundle map $TX\to X\times\mathbb{C}^{q}$. This is in the spirit
of Globevnik’s main theorem in [21] when $X$ is the ball in $\mathbb{C}^{n}$
and $f$ is a single holomorphic function, possibly with critical points.
Furthermore, one can strengthen the statement in the spirit of those obtained
by Charpentier and Kosiński [14] for holomorphic functions on pseudoconvex
domains in $\mathbb{C}^{n}$. In particular, under the above conditions, a
holomorphic submersion $f:X\to\mathbb{C}^{q}$ can be chosen such that the
image of any divergent path of finite $\mathfrak{g}$-length in $X$ is
everywhere dense in $\mathbb{C}^{q}$. It follows that the foliation of $X$
defined by $f$ is nonsingular and $\mathfrak{g}$-complete.

##  3\. Proofs of Theorems 1.5, 1.7, and 1.8

We shall need the following result generalizing [5, Lemma 2].

######  Lemma 3.1.

Let $B$ be a compact polynomially convex set in $\mathbb{C}^{n}$ $(n>1)$, and
let $\Gamma=\bigcup_{j=1}^{m}\Gamma_{j}\subset\mathbb{C}^{n}\setminus B$ be a
union of finitely many pairwise disjoint compact convex sets $\Gamma_{j}$ such
that the set $B\cup\Gamma$ is polynomially convex. If $E$ is a closed subset
of $\mathbb{C}^{n}$ with unbounded complement, then for any $\epsilon>0$ there
exists an automorphism $\Theta\in\mathrm{Aut}(\mathbb{C}^{n})$ such that

  1. (a)

$\Theta(E)\cap\Gamma=\varnothing$, and

  2. (b)

$|\Theta(z)-z|<\epsilon$ for all $z\in B$.

###### Proof.

Pick a compact neighbourhood $K_{0}$ of $B$ and compact convex neighbourhoods
$K_{j}$ of $\Gamma_{j}$ for $j=1,\ldots,m$ such that the sets
$K_{0},\ldots,K_{m}$ are pairwise disjoint and $K=\bigcup_{j=0}^{m}K_{j}$ is
polynomially convex. (The existence of such sets follows from standard results
on polynomial convexity; see Stout [28].) Let
$\Psi_{0}=\mathrm{Id}\in\mathrm{Aut}(\mathbb{C}^{n})$ be the identity and set
$K^{\prime}_{0}=K_{0}$. For every $j=1,\ldots,m$ we choose an automorphism
$\Psi_{j}\in\mathrm{Aut}(\mathbb{C}^{n})$ such that the compact sets
$K^{\prime}_{j}:=\Psi_{j}(K_{j})$ are pairwise disjoint, we have that

(3.1) $K^{\prime}_{j}\cap(E\cup K^{\prime}_{0})=\varnothing\quad\text{for
$j=1,\ldots,m$},$

and the union $\bigcup_{j=0}^{m}K^{\prime}_{j}$ is polynomially convex. Such
$\Psi_{j}$ are obtained by squeezing $K_{j}$ $(j=1,\ldots,m)$ by a dilation
into a small neighbourhood of an interior point, then translating the images
into small pairwise disjoint balls around some points in
$\mathbb{C}^{n}\setminus(E\cup K^{\prime}_{0})$ (this set is open and nonempty
by the assumption on $E$), and applying [20, Theorem 1.1] to approximate the
final map by an automorphism of $\mathbb{C}^{n}$. By [20, Theorem 2.3] (see
also [18, Corollary 4.12.4]), given $\delta>0$ there is an automorphism
$\Psi\in\mathrm{Aut}(\mathbb{C}^{n})$ satisfying

(3.2) $|\Psi(z)-\Psi_{j}(z)|<\delta\quad\text{for all $z\in K_{j}$, \
$j=0,1,\ldots,m$.}$

Let $\Theta=\Psi^{-1}$. If $\delta>0$ is small enough then condition (b)
holds, and we have that $\Psi(\Gamma_{j})\subset\mathring{K}^{\prime}_{j}$ and
hence $\Gamma_{j}\subset\Theta(\mathring{K}^{\prime}_{j})$ for every
$j=1,\ldots,m$, which by (3.1) also yields (a). ∎

###### Proof of Theorem 1.5.

Let $\mathbb{B}$ denote the open unit ball of $\mathbb{C}^{n}$. Set $r_{1}=1$,
$B_{1}=r_{1}\overline{\mathbb{B}}=\overline{\mathbb{B}}$,
$\mathcal{F}_{1}=\mathcal{F}$, and $E_{1}=\mathcal{F}_{1}(B_{1})$; see (1.1).
Choose a sequence $M_{j}>0$ such that

(3.3) $\sum_{j=1}^{\infty}M_{j}=+\infty.$

Pick a closed ball $B^{\prime}_{1}\subset\mathbb{C}^{n}$ centred at $0$ and
containing $B_{1}$ in its interior. By Lemma 2.1 and Remark 2.2 there is a
labyrinth $\Gamma_{1}\subset\mathring{B}^{\prime}_{1}\setminus B_{1}$ which
enlarges the $\mathfrak{g}$-distance by $M_{1}>0$ such that
$B_{1}\cup\Gamma_{1}$ is polynomially convex. Fix a number $0<\epsilon_{1}<1$.
Lemma 3.1 furnishes an automorphism $\phi_{1}\in\mathrm{Aut}(\mathbb{C}^{n})$
such that

  1. (a1)

$\phi_{1}(E_{1})\cap\Gamma_{1}=\varnothing$, and

  2. (b1)

$|\phi_{1}(z)-z|<\epsilon_{1}$ for all $z\in B_{1}$.

In the second step, we choose a number $r_{2}>2$ such that
$\phi_{1}(\overline{\mathbb{B}})\subset(r_{2}-1)\mathbb{B}$ and set

$B_{2}=r_{2}\overline{\mathbb{B}},\quad\mathcal{F}_{2}=(\phi_{1})_{*}\mathcal{F}_{1},\quad
E_{2}=\mathcal{F}_{2}(B_{2}).$

Pick a slightly bigger ball $B^{\prime}_{2}\supset B_{2}$ containing $B_{2}$
in its interior. By Lemma 2.1 there is a labyrinth
$\Gamma_{2}\subset\mathring{B}^{\prime}_{2}\setminus B_{2}$ which enlarges the
$\mathfrak{g}$-distance by $M_{2}>0$. Given $\epsilon_{2}>0$, Lemma 3.1
furnishes an automorphism $\phi_{2}\in\mathrm{Aut}(\mathbb{C}^{n})$ such that

  1. (a2)

$\phi_{2}(E_{2})\cap\Gamma_{2}=\varnothing$, and

  2. (b2)

$|\phi_{2}(z)-z|<\epsilon_{2}$ for all $z\in B_{2}$.

Continuing inductively, we obtain an increasing sequence of numbers $r_{j}>0$,
balls

(3.4) $B_{j}=r_{j}\overline{\mathbb{B}}\subset B^{\prime}_{j}\subset
B_{j+1}=r_{j+1}\overline{\mathbb{B}},$

labyrinths $\Gamma_{j}\subset\mathring{B}_{j+1}\setminus B_{j}$, constants
$\epsilon_{j}>0$, automorphisms $\phi_{j}\in\mathrm{Aut}(\mathbb{C}^{n})$, and
foliations $\mathcal{F}_{j}$ of $\mathbb{C}^{n}$ such that, setting

(3.5) $\text{$E_{j}=\mathcal{F}_{j}(B_{j})$, \ \
$\Phi_{j}=\phi_{j}\circ\cdots\circ\phi_{1}\in\mathrm{Aut}(\mathbb{C}^{n})$, \
\ and\ \ $\mathcal{F}_{j+1}=(\Phi_{j})_{*}\mathcal{F}$},$

the following conditions hold for every for $j=1,2,\ldots$:

  1. (ij)

$\phi_{j}(E_{j})\cap\Gamma_{j}=\varnothing$.

  2. (iij)

$|\phi_{j}(z)-z|<\epsilon_{j}$ for $z\in B_{j}$.

  3. (iiij)

$0<\epsilon_{j+1}<\frac{1}{2}\min\\{\epsilon_{j},\mathrm{dist}(\phi_{j}(E_{j}),\Gamma_{j})\\}$.
(Note that the set $\phi_{j}(E_{j})$ is closed and $\Gamma_{j}$ is compact, so
these sets are at positive distance by (iij).)

  4. (ivj)

$\Phi_{j}(j\overline{\mathbb{B}})\cup
B^{\prime}_{j}\subset(r_{j+1}-1)\mathbb{B}$.

  5. (vj)

The labyrinth $\Gamma_{j}$ enlarges the $\mathfrak{g}$-distance by $M_{j}$ and
$B_{j}\cup\Gamma_{j}$ is polynomially convex.

Assuming that we have obtained these quantities for indices $\leq j$, the
induction step goes as follows. Let $\mathcal{F}_{j+1}$ and $E_{j+1}$ be given
by (3.5). Pick a number $\epsilon_{j+1}$ satisfying (iiij) and a number
$r_{j+1}>r_{j}$ satisfying (ivj), and set
$B_{j+1}=r_{j+1}\overline{\mathbb{B}}$. Choose a ball
$B^{\prime}_{j+1}\supsetneq B_{j+1}$. Lemma 2.1 gives a labyrinth
$\Gamma_{j+1}\subset\mathring{B}^{\prime}_{j+1}\setminus B_{j+1}$ satisfying
(vj+1). Then, Lemma 3.1 gives an automorphism
$\phi_{j+1}\in\mathrm{Aut}(\mathbb{C}^{n})$ satisfying (ij+1) and (iij+1),
closing the induction.

We claim that the sequence $\Phi_{j}\in\mathrm{Aut}(\mathbb{C}^{n})$ converges
uniformly on compacts in $\mathbb{C}^{n}$ to an automorphism
$\Phi\in\mathrm{Aut}(\mathbb{C}^{n})$. Indeed, since

$|\phi_{j}(z)-z|<\epsilon_{j}<1<\mathrm{dist}(B_{j},\mathbb{C}^{n}\setminus
B_{j+1})\ \ \text{for every $j=1,2,\ldots$}$

and $\sum_{j=1}^{\infty}\epsilon_{j}<\infty$ (see (iij)–(ivj)), the sequence
$\Phi_{j}$ converges uniformly on compacts in the domain
$\Omega=\bigcup_{j=1}^{\infty}\Phi_{j}^{-1}(B_{j})\subset\mathbb{C}^{n}$ to a
biholomorphic map $\Phi:\Omega\to\mathbb{C}^{n}$ onto $\mathbb{C}^{n}$ (see
[18, Corollary 4.4.2]). Furthermore, condition (ivj) ensures that
$j\overline{\mathbb{B}}\subset\Omega$ for every $j$, so
$\Omega=\mathbb{C}^{n}$ and hence $\Phi\in\mathrm{Aut}(\mathbb{C}^{n})$.
Hence, the sequence of foliations
$\mathcal{F}_{j+1}=(\Phi_{j})_{*}\mathcal{F}$ converges uniformly on compacts
in $\mathbb{C}^{n}$ to a limit foliation $\mathcal{G}=\Phi_{*}(\mathcal{F})$.

It remains to show that the foliation $\mathcal{G}$ is
$\mathfrak{g}$-complete. We must show that for any divergent path
$\gamma:[0,1)\to\mathbb{C}^{n}$ contained in a leaf of $\mathcal{F}$, the path
$\tilde{\gamma}=\Phi\circ\gamma:[0,1)\to\mathbb{C}^{n}$ has infinite
$\mathfrak{g}$-length. (Since the foliation $\mathcal{F}$ is assumed to have
closed leaves of positive dimension, every divergent path in a leaf of
$\mathcal{F}$ is also divergent in $\mathbb{C}^{n}$. Note that
$\tilde{\gamma}$ is a divergent path in $\mathbb{C}^{n}$ contained in a leaf
of $\mathcal{G}$.) For every $k\in\mathbb{N}$ let
$\gamma_{k}=\Phi_{k}\circ\gamma:[0,1)\to\mathbb{C}^{n}$; this is a divergent
path contained in a leaf of the foliation
$\mathcal{F}_{k+1}=(\Phi_{k})_{*}\mathcal{F}$. Pick $j\in\mathbb{N}$ such that
$\gamma(0)\in j\mathbb{B}$. By (ivj) we have that
$\gamma_{j}(0)=\Phi_{j}(\gamma(0))\in(r_{j+1}-1)\mathbb{B}\subset B_{j+1}$,
and hence $\gamma_{j}([0,1))\subset E_{j+1}$ (see (3.5)). Conditions
(ij)–(iiij) imply that for every $k>j$ we have that $\gamma_{k}(0)\in
B_{j+1}=r_{j+1}\overline{\mathbb{B}}$ and the trace of $\gamma_{k}$ avoids the
labyrinths $\Gamma_{i}$ for $i=j+1,\ldots,k$. Since $\gamma_{k}$ is a
divergent path in $\mathbb{C}^{n}$, its $\mathfrak{g}$-length is at least
$\sum_{i=j+1}^{k}M_{i}$ by (ii). As $k\to\infty$, it follows that $\gamma_{k}$
converges to a divergent path
$\tilde{\gamma}=\Phi\circ\gamma:[0,1)\to\mathbb{C}^{n}$ which has infinite
$\mathfrak{g}$-length in view of (3.3). ∎

###### Proof of Theorem 1.7.

The proof will follow that of Theorem 1.5 if we find labyrinths $\Gamma$ in
the given Stein manifold $X$ which increase the length of divergent paths
avoiding $\Gamma$ by a given amount and whose pieces are holomorphically
contractible in $X$. (Note that the labyrinths used in the proof of Theorem
1.1, which are obtained by intersecting labyrinths in $\mathbb{C}^{N}$ whose
pieces are balls in affine real hyperplanes with the embedded submanifold
$X\subset\mathbb{C}^{N}$, need not satisfy this property.) To this end, we
shall prove the following lemma whose first part generalizes Lemma 2.1, as
well as [13, Lemma 2.4] by Charpentier and Kosiński, while the second part is
an analogue of Lemma 3.1 adjusted to this situation. As before, $\mathbb{B}$
denotes the unit ball of $\mathbb{C}^{N}$.

######  Lemma 3.2.

Let $X$ be a closed complex submanifold of $\mathbb{C}^{N}$, and let
$\mathfrak{g}$ be a Riemannian metric on $X$. Given numbers $0<r<s$ and $M>0$,
there is a compact set $\Gamma\subset X\cap(s\mathbb{B}\setminus
r\overline{\mathbb{B}})$ satisfying the following conditions.

  1. (a)

The compact set $\Gamma\cup(X\cap(r\overline{\mathbb{B}}))$ is
$\mathscr{O}(X)$-convex.

  2. (b)

$\Gamma=\bigcup_{i=1}^{m}\Gamma_{i}$ is the union of finitely many pairwise
disjoint compact sets $\Gamma_{i}$ such that every $\Gamma_{i}$ is convex in a
certain local holomorphic chart on $X$.

  3. (c)

Every piecewise smooth path $\gamma:[0,1]\to X\setminus\Gamma$ with
$\gamma(0)\in X\cap r\overline{\mathbb{B}}$ and $\gamma(1)\in X\setminus
s\mathbb{B}$ has $\mathfrak{g}$-length at least $M$.

Given such $\Gamma$ and assuming in addition that $X$ has the density
property, then for every closed subset $E\subsetneq X$ whose complement
$X\setminus E$ is not relatively compact in $X$ and for any $\epsilon>0$ there
exists an automorphism $\Theta\in\mathrm{Aut}(X)$ such that

  1. (A)

$\Theta(E)\cap\Gamma=\varnothing$, and

  2. (B)

$|\Theta(z)-z|<\epsilon$ for all $z\in X\cap\,r\overline{\mathbb{B}}$.

###### Proof.

We construct the labyrinth $\Gamma$ in two stages.

In the first stage, we apply Lemma 2.1 to find a labyrinth
$\Gamma^{0}=\bigcup_{j=1}^{k}\Gamma^{0}_{j}$ in the spherical shell
$S_{r,s}=s\mathbb{B}\setminus r\overline{\mathbb{B}}\subset\mathbb{C}^{N}$
which increases the $\mathfrak{g}$-length in $X$ by the given amount $M>0$ and
the set $\Gamma^{0}\cup r\overline{\mathbb{B}}$ is polynomially convex. Set
$B_{0}=X\cap\,r\overline{\mathbb{B}}$. It follows that each of the compact
sets $\Gamma^{X}_{j}=\Gamma^{0}_{j}\cap X$ ($j=1,\ldots,k$),
$\Gamma^{X}=\bigcup_{j=1}^{k}\Gamma^{X}_{j}$, and $\Gamma^{X}\cup B_{0}$ are
$\mathscr{O}(X)$-convex. The construction of such labyrinths in [9] shown that
the connected components $\Gamma^{0}_{j}$ of $\Gamma^{0}$ (which are closed
balls in affine real hyperplanes in $\mathbb{C}^{N}$) may be chosen with
arbitrarily small diameter (this is an immediate consequence of Pythagoras
theorem, see [1, Lemma 2.3]); in particular we can choose them small enough
such that $\Gamma^{X}_{j}=\Gamma^{0}_{j}\cap X$ is contained in a holomorphic
coordinate chart $U_{j}\subset X$ which is Runge in $X$ for every
$j=1,\ldots,k$. (Most of the sets $\Gamma^{X}_{j}$ are empty, and we discard
them from the above collection and adjust the indexes accordingly.) More
precisely, for any $\Gamma^{X}_{j}\neq\varnothing$ we pick a point
$p_{j}\in\Gamma^{X}_{j}$ and let $\Sigma_{j}=T_{p_{j}}X\cong\mathbb{C}^{n}$
with $n=\dim X$ be the tangent plane of $X$ at $p_{j}$, with the orthogonal
$\mathbb{C}$-linear projection $\pi_{j}:\mathbb{C}^{N}\to\Sigma_{j}$. Then, we
may assume that there is a Runge neighbourhood $U_{j}\subset X$ of
$\Gamma^{X}_{j}$ such that the restricted projection

$\pi_{j}|_{U_{j}}:U_{j}\to\pi_{j}(U_{j})=V_{j}\subset\Sigma_{j}\cong\mathbb{C}^{n}$

is a biholomorphic map onto a ball $V_{j}\subset\Sigma_{j}$ around the point
$p_{j}$. It follows that a compact set $K\subset U_{j}$ is
$\mathscr{O}(X)$-convex if and only if $\pi_{j}(K)$ is polynomially convex in
$\Sigma_{j}\cong\mathbb{C}^{n}$.

In the second stage, we choose for every $j=1,\ldots,k$ a compact
$\mathscr{O}(X)$-convex neighbourhood $B_{j}\subset U_{j}$ of $\Gamma^{X}_{j}$
such that the sets $B_{0}=X\cap\,r\overline{\mathbb{B}},B_{1},\ldots,B_{k}$
are pairwise disjoint and $\bigcup_{j=0}^{k}B_{j}$ is $\mathscr{O}(X)$-convex.
(Note that every compact $\mathscr{O}(X)$-set has a basis of compact
$\mathscr{O}(X)$-convex neighbourhoods.) We now apply the construction of
labyrinths with holomorphically contractible pieces in pseudoconvex domains in
[13, Theorem 1.1] by Charpentier and Kosiński, to find for each $j=1,\ldots,k$
a labyrinth $\Gamma_{j}\subset\mathring{B}_{j}\setminus\Gamma^{X}_{j}$ which
is $\mathscr{O}(B_{j})$-convex (and hence $\mathscr{O}(X)$-convex), its
connected components are holomorphically contractible sets in $X$ (in fact,
under the projection $\pi_{j}:U_{j}\to V_{j}\subset\Sigma_{j}$ they correspond
to closed balls in affine real hyperplanes of
$\Sigma_{j}\cong\mathbb{C}^{n}$), and the $\mathfrak{g}$-distance in
$X\setminus\Gamma_{j}$ from $\Gamma^{X}_{j}$ to $bB_{j}$ is at least $M$. In
other words, any path in $U_{j}$ from $\Gamma^{X}_{j}$ to $bB_{j}$ which
avoids $\Gamma_{j}$ has $\mathfrak{g}$-length at least $M$.

We claim that the labyrinth $\Gamma=\bigcup_{j=1}^{k}\Gamma_{j}\subset X\cap
S_{r,s}$ satisfies conditions (a)–(c) in the lemma. The set $\Gamma_{j}$ is
$\mathscr{O}(B_{j})$ convex for every $j=1,\ldots,k$. Since
$\bigcup_{j=0}^{k}B_{j}$ is $\mathscr{O}(X)$-convex, it follows that
$\Gamma\cup B_{0}=\Gamma\cup(X\cap r\overline{\mathbb{B}})$ is
$\mathscr{O}(X)$-convex, so (a) holds. Condition (b) holds by the
construction. Let $\gamma$ be a path as in part (c) avoiding $\Gamma$. If
$\gamma$ avoids the initial labyrinth $\Gamma^{X}=\Gamma^{0}\cap X$ then its
$\mathfrak{g}$-length is at least $M$ by the choice of $\Gamma^{0}$. If on the
other hand $\gamma$ intersects $\Gamma^{X}_{j}$ for some
$j\in\\{1,\ldots,k\\}$, then $\gamma$ connects a point in $bB_{j}$ to
$\Gamma^{X}_{j}$ avoiding the labyrinth $\Gamma_{j}$, hence its
$\mathfrak{g}$-length is at least $M$ by the choice of $\Gamma_{j}$. This
proves (c).

The second part of the lemma is obtained by following the proof of Lemma 3.1.
The only point deserving an explanation is the construction of the
automorphisms $\Psi_{j}$ $(j=1,\ldots,k)$ and $\Psi$ in the proof of Lemma
3.1. Recall that the labyrinth $\Gamma_{j}=\bigcup_{i=1}^{k_{j}}\Gamma_{j,i}$
has holomorpically contractible $\mathscr{O}(X)$-convex connected components
$\Gamma_{j,i}$. More precisely, there is a 1-parameter family of biholomorphic
contractions on a pseudoconvex Runge neighbourhood of $\Gamma_{j,i}$ in $X$
which shrinks this set within itself almost to a point. We can then move the
images of these small sets to $X\setminus(B_{0}\cup E)$ by an isotopy of
biholomorphic maps through pseudoconvex Runge domains in $X$, ensuring that
the traces of these isotopies for $j=1,\ldots,k$ are pairwise disjoint,
contained in $X\setminus B_{0}$, and their unions (together with $B_{0})$ are
Runge in $X$ for every value of the parameter. (The set $B_{0}$ remains fixed
during this process.) Assuming that the Stein manifold $X$ has the density
property, we can apply the approximation theorem for such isotopies of
injective holomorphic maps (see [18, Theorem 4.10.5]) to obtain an
automorphism $\Psi\in\mathrm{Aut}(X)$ as in the proof of Lemma 3.1. The
remainder of the proof is exactly as in the case $X=\mathbb{C}^{n}$. ∎

With Lemma 3.2 in hand, we obtain Theorem 1.7 by following the proof of
Theorem 1.5, and we leave the obvious details to the reader. ∎

###### Proof of Theorem 1.8.

Let $\mathcal{F}_{0}$ be a proper holomorphic foliation on a Stein manifold
$X$ with the density property, and let $\Omega$ be a pseudoconvex Runge domain
in $X$. By using Lemma 3.2 inductively, we find a normal exhaustion
$\Omega_{1}\Subset\Omega_{2}\Subset\cdots\subset\bigcup_{i=1}^{\infty}\Omega_{i}=\Omega$
by open relatively compact pseudoconvex Runge domains and for each
$i=1,2,\ldots$ a labyrinth
$\Gamma_{i}\subset\Omega_{i+1}\setminus\overline{\Omega}_{i}$ having the
properties (a)–(c) in Lemma 3.2 for a given constant $M_{i}>0$ chosen such
that $\sum_{i}M_{i}=+\infty$. Furthermore, we can ensure that the compact set
$\Gamma_{i+1}\cup\overline{\Omega}_{i}$ is $\mathscr{O}(\Omega)$-convex (and
hence also $\mathscr{O}(X)$-convex) for every $i\in\mathbb{N}$. Since the
components of the labyrinth $\Gamma=\bigcup_{i=1}^{\infty}\Gamma_{i}$ are
holomorphically contractible, we can apply the proof of Theorem 1.5 to
inductively twist the foliation $\mathcal{F}_{0}$ by a sequence of
automorphisms $\Phi_{i}\in\mathrm{Aut}(X)$ $(i=1,2,\ldots)$, chosen so that
they converge uniformly on compacts in $\Omega$, and the leaves of the
foliations $\mathcal{F}_{i}=(\Phi_{i})_{*}\mathcal{F}_{i-1}$ avoid more and
more components of $\Gamma$ as $i\to\infty$. The limit holomorphic foliation
$\mathcal{F}=\lim_{i\to\infty}\mathcal{F}_{i}$ on $\Omega$ is such that every
leaf avoids all but finitely many labyrinths $\Gamma_{i}$, and hence it is
$\mathfrak{g}$-complete. Furthermore, the construction ensures that every leaf
is a pseudoconvex Runge domain in a leaf of the initial foliation
$\mathcal{F}_{0}$ on $X$. The details can be found (for the case
$X=\mathbb{C}^{n}$) in [9] and [5]. ∎

In Theorems 1.5, 1.7, and 1.8 we focused on constructing complete nonsingular
holomorphic foliations. However, the proofs of these results also apply to
individual closed complex submanifolds and yield the following result.

######  Theorem 3.3.

Let $X$ be a Stein manifold with the density property, and let $Y$ be a closed
complex submanifold of $X$. The following assertions hold.

  1. (i)

Given a Riemannian metric $\mathfrak{g}$ on $X$ there is a holomorphic
automorphism $\Phi\in\mathrm{Aut}(X)$ such that the submanifold $\Phi(Y)$ is
$\mathfrak{g}$-complete.

  2. (ii)

Let $\Omega$ be a pseudoconvex Runge domain in $X$ such that $\Omega\cap
Y\neq\varnothing$. Given a Riemannian metric $\mathfrak{g}$ on $\Omega$ and a
connected compact subset $K\subset\Omega\cap Y$, there is a
$\mathfrak{g}$-complete closed complex submanifold of $\Omega$ which is
biholomorphic to a pseudoconvex Runge domain in $Y$ containing $K$.

By combining assertion (i) in Theorem 3.3 with the embedding theorems for
Stein manifolds (see [15, 27, 12]) we obtain the following corollary.

######  Corollary 3.4.

Every Stein manifold $Y$ of dimension $n\geq 1$ admits a proper
$\mathfrak{g}$-complete holomorphic embedding in
$(\mathbb{C}^{N},\mathfrak{g})$ for any
$N\geq\max\left\\{3,\left[\frac{3n}{2}\right]+1\right\\}$, and a proper
$\mathfrak{g}$-complete holomorphic embedding in $(X,\mathfrak{g})$ for any
Riemannian Stein manifold $X$ with the density property of dimension $\dim
X\geq 2n+1$.

Acknowledgements. Alarcón is partially supported by the State Research Agency
(AEI) via the grant no. PID2020-117868GB-I00 and the “Maria de Maeztu”
Excellence Unit IMAG, reference CEX2020-001105-M, funded by
MCIN/AEI/10.13039/501100011033/, Spain.

Forstnerič is supported by the European Union (ERC Advanced grant HPDR,
101053085) and grants P1-0291, J1-3005, and N1-0237 from ARRS, Republic of
Slovenia.

We thank an anonymous referee for useful remarks, and the editor for the
suggestion to make the introduction more accessible to a wider audience.

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# Neural-Symbolic Integration for Interactive Learning and Conceptual
Grounding

Benedikt Wagner
Department of Computer Science
City, University of London
London, EC1 0HB, UK
<EMAIL_ADDRESS>&Artur d’Avila Garcez
Department of Computer Science
City, University of London
London, EC1 0HB, UK
<EMAIL_ADDRESS>
###### Abstract

We propose neural-symbolic integration for abstract concept explanation and
interactive learning. Neural-symbolic integration and explanation allow users
and domain-experts to learn about the data-driven decision making process of
large neural models. The models are queried using a symbolic logic language.
Interaction with the user then confirms or rejects a revision of the neural
model using logic-based constraints that can be distilled into the model
architecture. The approach is illustrated using the Logic Tensor Network
framework alongside Concept Activation Vectors and applied to a Convolutional
Neural Network.

##  1 Introduction

The widespread application of Machine Learning (ML) has caused a surge in
interest and research around explainability. Although differing in motive,
there is increasing demand for comprehending the processes used by large
trained models to arrive at certain decisions, and for evaluating whether such
processes conform with human understanding. Explanations are offered in a
variety of media with varying levels of fidelity and intuition [Confalonieri
et al., 2021, Guidotti et al., 2018] with a substantial part achieved through
post-hoc explanation algorithms. In most cases, the ML system produces
statistical explanations which help determine how the output is derived. Most
current approaches focus on decoding the behaviour of a system by examining
input or output characteristics. However, using background knowledge may
provide explanations which are more in line with human conceptualisation, and
thus more useful in applications.
In the process of providing explanations, one of the limitations of
statistical methods is that in general they do not consider domain or
background knowledge. In light of the above limitations, it may be beneficial
to combine statistical techniques with symbolic approaches. We argue that the
ease of interpretation of symbolic representations through their interaction
with powerful neural network learning algorithms can offer a promising
direction for explainability and comprehensibility.
Futia and Vetrò [2020] underline the importance of neural-symbolic integration
for explainability by suggesting that traditional explainable AI (XAI) methods
lack the ability to provide explanations for the variety of target audiences.
While most of the explainability methods may be valuable at providing insight
to ML experts, domain experts in applications such as finance or healthcare
may struggle to interpret the explanations given. It is proposed that
_interactive integration with semantically-rich representations is key to
refining explanations targeted at different stakeholders_ Futia and Vetrò
[2020]. Indeed, having flexibility in the abstract representation of the
information that forms an explanation is key to leveraging domain expertise
through interaction and revision of the decision making process.
More specifically, neural-symbolic integration would allow us to overcome the
static nature of the current ML paradigm. Explainability methods of today do
not provide an ability to act on extracted information. Upon finding an
undesired property, the only way to influence the model is to retrain it until
a good model is found. However, retraining as a process may be unguided and
can only be influenced indirectly by the explanation through the collection of
additional data. The result is that many explanation methods become limited in
their usefulness, with retraining commonly resulting in catastrophic
forgetting of previously acquired information. The inclusion of symbolic
knowledge during training of deep networks in the form of first-order logic
constraints on the loss function has been shown to improve fairness while
maintaining the performance of the ML system [Wagner and Garcez, 2021] by
direct comparison with existing fairness-specific methods using various
metrics.
We propose to define XAI as the alignment of model behaviour with human values
achieved through model comprehensibility and revision. If one wishes to obtain
intuitive, human-like explanations then the alignment must take place at a
high level of abstraction with an ability to drill down to deeper explanations
as the need arises, as in the case of a child’s sequence of _why_ questions.
While the predominant low-level and statistical explanations are effective for
debugging, logical and reasoning-based explanations require a more abstract
knowledge representation using higher-level concepts.

Figure 1: Illustration of the neural-symbolic cycle: knowledge extraction will
be carried out by querying a deep network interactively and learning
continually, thereby repeatedly applying the neural-symbolic cycle until
values are deemed to have been aligned. The neural-symbolic cycle enables a
human-in-the-loop approach by offering a common ground for communication and
system interaction. Symbolic knowledge representations extracted from the
learning system at an adequate level of abstraction for communication with the
user will allow for knowledge consolidation and targeted revision of the
model.

##  2 Interactive Learning and Conceptual Grounding

Our aim is to query the neural network for symbolic knowledge so that a direct
interpretation of abstract representations and operations on those
representations become feasible. The approach seeks to explain and possibly
revise a decision making process post-hoc and to be model-agnostic.
Furthermore, to ensure that the model can adapt to the complexity of tasks in
the usual way as popularized by current ML, we aim to retain the advantages of
gradient-based, end-to-end learning. As a means of ensuring that the common
_communication layer_ is not hindered by irreconcilable disparities between
the symbolic (discrete) and neural (continuous) representations, we will need
to ensure that the model can be queried with human-interpretable operations at
an adequate level of abstraction.
We use neural-symbolic integration as the bridge at this communication layer:
logic provides the semantic precision required of the questions to be put to
the model. Although the use of logic may seem to be a barrier at first sight,
it is required to formalise knowledge and interaction with a precise
semantics. The gap that may exist can in principle be filled by building
wrappers to formulate logical queries e.g. using natural language [Singh et
al., 2020]. The core building block will be the usual logical operators. The
goal is to provide an intuition into the operations that the ML model has
inferred based on the observed data at a given task. The logical operators
_connect_ the symbol representations also in the usual way. Symbols are
tangible references which will be used to denote abstract concepts that arise
through learning of model-specific, data-driven representations. These
abstract concepts will be derived from within a trained model, giving rise to
explanations that are grounded on the model’s inherent representation and
operations.
The neurosymbolic framework adopted in this paper is that of Logic Tensor
Networks (LTN) as described in Serafini and Garcez [2016] and Badreddine et
al. [2020]. However, instead of treating the learning of the parameters from
data and knowledge as a single process, we emphasise the dynamic and flexible
nature of training from data followed by querying the trained model for
knowledge, followed by consolidating that knowledge in the form of constraints
for further training, as part of a cycle with stopping criteria defined by the
user. We make LTN iterative by saving the parametrization learned at each
cycle in our implementation while not requiring the use of any specific model
architecture, thus making the approach fully model-agnostic.
LTN implements a many-valued first-order logic (FOL) language $\mathcal{L}$
into deep networks. The syntax of LTN is that of FOL, with formulas consisting
of predicate symbols, here denoting concepts, and the connectives: negation
($\neg$), conjunction ($\land$), disjunction ($\lor$), implication
($\rightarrow$), as well as universal ($\forall$) and existential ($\exists$)
quantification.
To emphasize that symbols are interpreted according to their grounding onto
real numbers, LTN uses the term _grounding_ , denoted by $\mathcal{G}$, in
place of interpretation. Here, we are specifically interested in the grounding
of predicates which make up the symbols that refer to the abstract concepts
which will form the basis of our model explanation. Predicates are grounded as
mappings onto the interval $[0,1]$ representing the predicate’s degree of
truth given the input. In order to allow for flexible gradient-based learning
with logical constraints, the logical formulas are made differentiable. This
is done by defining the connectives according to fuzzy logic connectives
approximated by so-called t-norms, t-conorms and fuzzy implication and
negation. These are mathematical operations that satisfy a set of logical
properties. In the same manner, quantification such as $\forall$ (for all) and
aggregation of logical formulas into a set (i.e. a knowledge-base) are defined
using generalised mean. For an extensive explanation of LTN, we refer the
reader to [Badreddine et al., 2020].
Our adaptation of the LTN framework can be deployed after training as will be
demonstrated. Here, specific outputs and inner representations of any neural
network are mapped onto a predicate $P$ to obtain an explanation for $P$
w.r.t. other predicates (other outputs and inner representations) with the use
of the logical connectives. Once a relation among the predicates has been
established as a logical rule, one can impose additional constraints onto the
network with this rule to seek to understand the relationship between this and
other possible rules. If we consider all groundings of mappings of a network
to be learnable, they will all depend on a set of parameters $\bm{\theta}$.
The background knowledge consists of a (possibly empty) set of logical rules,
referred to as $\phi$, which entails all predicate mappings. Since the
grounding of a rule $\mathcal{G}_{\theta}(\phi)$ denotes the degree of truth
of $\phi$, one natural training signal is the degree of truth of all the
rules, including mappings in the knowledge-base $\mathcal{K}$. The objective
function is therefore to maximise the satisfiability of all the rules in
$\mathcal{K}$,
$\bm{\theta}^{\ast}=\operatorname*{arg\,max}_{\bm{\theta}\in\bm{\Theta}}\
\mathrm{Sat}_{A}(\mathcal{G}_{\theta}(\mathcal{K}))$, which is subject to an
aggregation $A$ of the rules in $\mathcal{K}$.

Figure 2: We produce local explanations (for individual inputs/images) and
global explanations (universally-quantified formulas) for the deep learning
model by querying specific neurons. We then reason about the generality of the
explanations given the truth queries extracted from the trained network. The
figure shows some of these queries associated with groundings in the neural
network and their satisfiability (sat) levels. Using linear probes to ground
the activation patterns of intermediate representations into the language of
LTN, we are able to utilise abstract concepts as explanation symbols in the
logic. Following querying, the neural model can be constrained based on a user
selection of logical formulas $\mathcal{K}_{new}$ for further learning. This
iterative process seeks to align the model with user values in the context of
background knowledge $\mathcal{K}$. In the figure, the Quagga is classified as
a zebra. A user’s desire to change such classification should trigger the
addition of knowledge into $\mathcal{K}_{new}$ informed by the queries to be
satisfied by the final trained model. Notice that training from data may begin
without any background knowledge which can be revised by querying user-defined
concepts and constraints deemed as necessary for the network to learn.

##  3 Demonstration of Grounding for Concept Interaction Explanation

To obtain conceptual explanations that provide comprehensible descriptions of
what has been learned, we must ground low-level information into reusable
concepts that are present at hidden representations within the network. Let us
illustrate the idea with an example which we have implemented in LTN to
explain a Convolutional Neural Network (CNN). We draw inspiration from the
TCAV approach [Kim et al., 2018] but modify it substantially for the
implementation in LTN. Consider any neural network that takes as input
$\bm{x}\in\mathbb{R}^{n}$, which projects onto any layer $l$ within the
network consisting of $m$ neurons, according to a function:
$f_{l}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$. In an iterative explanation
process, we seek to connect representations inside the network. There is no
restriction on which layer $l$ to use, but in a CNN this is generally the
layer immediately before the fully-connected layer (i.e. the classifier)
[Odense and d’Avila Garcez, 2020]. We adapt TCAV Kim et al. [2018] to enable
users to specify the concepts to be checked (queried) at any adequate level of
abstraction. Graziani et al. [2018] have shown in an application to medical
imaging that domain-related concepts can be valuable for gaining insight into
the decision making. The approach proposed in this paper extends this idea to
allow for complex concept interactions (as defined by the logic) and model
retraining using such logical rules as constraints. Using random examples
alongside a user-defined set of examples (images) that capture a concept, we
form a linear probe at layer $l$ which evaluates the activation values
produced by the examples and already known concepts from which further data
can be selected for use. In Kim et al. [2018], the linear probe at layer $l$
serves as a building block for the Concept Activation Vector used to calculate
conceptual sensitivities of inputs and classes. In this paper, we integrate
the concept mapping directly into the interactive framework, allowing concepts
to be combined into the logic, evaluated using fuzzy logic, and chosen for
further training of the neural network model. This linear probe works as a
classifier for our conceptual grounding, which is then integrated as a logical
predicate into the LTN framework. The top part of Figure 2 illustrates the
process. At each time that a user wishes to distil a model inference into
specific concepts $C$, they simply need to select a set $P_{C}$ of positive
examples and a set $N$ of negative examples. The linear probe then serves to
distinguish the activation values of the neurons in layer $l$ between
$\left\\{f_{l}(\bm{x}):\bm{x}\in P_{C}\right\\}$ and
$\left\\{f_{l}(\bm{y}):\bm{y}\in N\right\\}$. This has the advantage of not
being bound by pre-existing data or features. Independent of the original
task, examples (images) may be collected and any number of user-defined
concepts checked (queried) against the network.
As an example, we query a GoogLeNet model [Szegedy et al., 2015] trained on
ImageNet to explain the output class of zebras with respect to user-defined
concepts, as illustrated in Kim et al. [2018] and the bottom part of Figure 2.
We extract four different concept descriptions using images from the Broden
dataset [Bau et al., 2017] to dissect the _zebra_ classification into the
concepts of _stripes_ , _dots_ , _zigzags_ and an abstract representation of
the horse-family concept _equidae_ (horses, donkeys, zebras and others). We
learn the groundings of the activation patterns for the specified concepts
from 150 images of each concept and an equal number of negative examples for
each class. Subsequently, the truth-value of each query is calculated through
fuzzy logic inference using LTN. These queries can be specific to an image
(local) or aggregated across the entire set of examples (global).
The quantifier $\forall$ is used to aggregate across a set of data points by
replacing $x$ with every image available from the dataset thus evaluating the
model’s behavior across all available data. The following implication:
$\forall x:zebra(x)\rightarrow stripe(x)$, with the symbol $stripe(x)$ being
replaced by the corresponding concept grounding in the network, provides an
insight onto how important the concept $stripe(x)$ is for the CNN’s
classification output $zebra(x)$ given the set of images $x$.
Furthermore, we can combine several concepts using the logic: $\forall
x:equid(x)\land stripe(x)\rightarrow zebra(x)$ returns a truth value of $0.98$
across a set of 3000 examples from ImageNet, indicating that the CNN assigns
any horse-like object with stripes to the class of zebras. When applying a
universal quantifier, the user is able to evaluate the decision making process
of the model in general, by examining the concepts on all available data, even
if it has not been used for training, thereby producing a global explanation.
One example previously unknown to the model is the extinct quagga, an animal
characterised by a brown striped coat instead of the black and white pattern
of zebras which has been selected to illustrate the potential of local
explanations. The model identifies this animal correctly as a member of the
Equidae family, recognises the stripes on the animal and consequently
classifies the image as that of a zebra, as shown in Figure 2. By utilising
the trained linear probes of the activation vectors to ground individual
images, we generate local explanations that provide insight into why a
particular image might be classified in a certain way according to the model.

Upon identifying potential undesired behaviour, for example by querying known
exceptions, a user can add new rules into the knowledge-base (by adding
logical formulas into $\mathcal{K}_{new}$) for further training of the
network. In case the specification of quagga and zebra is to be changed, an
alternative inference process can be imposed on the CNN model. Recall that
quagga are currently considered by the CNN to be a subspecies of zebra.
Assuming that the user decides to change this, as an example, let us consider
introducing concept probes $bw(x)$ for _black and white_ objects and $col(x)$
for _colourful objects_ , and let us assume that these concepts are to be
regarded as mutually exclusive. Adding the following rule to
$\mathcal{K}_{new}$ as a new constraint to be satisfied by learning should
force the neural model to only classify black and white objects as zebras:
$\phi_{5}=\forall x:equid(x)\land stripe(x)\land\neg bw(x)\rightarrow\neg
zebra(x)$.111Notice that the satisfiability of this rule should be the same as
that of $\forall x:equid(x)\land stripe(x)\land col(x)\rightarrow\neg
zebra(x)$, as we apply a $softmax$ function to mutually exclusive concept
probes; in this case $\forall x:col(x)\leftrightarrow\neg bw(x)$.
Before further training, $\phi_{5}$ exhibits a low $sat$-level of
$sat_{\phi_{5}}=0.09$, as the model classifies all objects associated with the
$equid(x)$ and the $stripe(x)$ concepts to the $zebra$ class regardless of
their color. By retraining for only five iterations, the $sat$-level increases
to $sat_{\phi_{5}}=0.94$, which indicates that only black and white objects
(in conjunction with the _stripe_ and _equidae_ concepts) are now considered
to be zebras. Therefore, the example image of the quagga is no longer inferred
to be in the zebra class with $\mathcal{G}(zebra(x_{qua}))=0.08$, where
$x_{qua}$ denotes the image of a quagga. The neural model nevertheless
identifies correctly the equidae and stripe concepts in the quagga, with
$\mathcal{G}(equid(x_{qua})\land stripe(x_{qua}))=0.97$.

##  4 Conclusion

We proposed a method combining TCAV and LTN that allows for complex conceptual
explanation and interactive learning. It enables domain experts to learn about
the data-inferred decision making process of large ML models by querying the
model and defining constraints for further learning as part of an iterative
process. We have implemented the idea in LTN and used GoogLeNet model to
illustrate its use.
Future work will explore alternative model architectures to improve the
effectiveness of concept groundings. In particular, vision transformers are
capable of providing rich representations that may prove useful. Furthermore,
we shall investigate an alternative to the TCAV approach of providing examples
of concepts. Despite the fact that this may be useful in that it allows for
customizable concepts, it may be impractical at times to collect data. It
should be possible to accomplish the same goal by integrating multimodal
(vision and language) models that permit natural language description from
zero-shot classification.

## References

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  * Bau et al. [2017] D. Bau, B. Zhou, A. Khosla, A. Oliva, and A. Torralba.  Network dissection: Quantifying interpretability of deep visual representations.  In _Proceedings - 30th IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2017_ , 2017.  ISBN 9781538604571.  doi: 10.1109/CVPR.2017.354.
  * Confalonieri et al. [2021] R. Confalonieri, L. Coba, B. Wagner, and T. R. Besold.  A historical perspective of explainable Artificial Intelligence.  _Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery_ , 2021.  ISSN 19424795.  doi: 10.1002/widm.1391.
  * Futia and Vetrò [2020] G. Futia and A. Vetrò.  On the integration of knowledge graphs into deep learning models for a more comprehensible AI-Three challenges for future research, 2020.  ISSN 20782489.
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  * Szegedy et al. [2015] C. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, and A. Rabinovich.  Going deeper with convolutions.  In _Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition_ , 2015.  ISBN 9781467369640.  doi: 10.1109/CVPR.2015.7298594.
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# Notes on ergodic theorems in non-commutative symmetric spaces

Genady Ya. Grabarnik  St Johns University, Queens, NY, USA
<EMAIL_ADDRESS>
(Date: March 1, 2016)

###### Abstract.

In this paper we establish individual ergodic theorem for positive kernels (or
so called Danford Shwartz (DS+) operators acting on non commutative symmetric
spaces.

###### Key words and phrases:

Ergodic theory, Operator algebras, re-arrangement invariant spaces

###### 1991 Mathematics Subject Classification:

Primary 05C38, 15A15; Secondary 05A15, 15A18

##  1\. Introduction

The goal of the paper is to see that one of the results by Veksler [20] or
Muratov, Pashkova and Rubshtein [16] remains valid for the non commutative
case. 111 It become known to author that Litvinov and Chilin also saw this
result at the same time and wrote it at the same time. I suggested to them to
combine results and names on paper. Waiting for the answer.  Let $M,\tau$ be a
semifinite von Neumann algebra with semifinite normal faithful trace $\tau$.

In addition we assume that $M,\tau$ satisfy homogeneity property, it may be
presented in the form of the resonant property on trace, see for example [1].

The space of all measurable operators affiliated with $M,\tau$ in the Sigal
[19] sense is denoted by $L_{0}$, see for details [19, 5, 6].

Notions of $L_{1}(M,\tau)$ and $L_{\infty}(M,\tau)$ was naturally introduced
in the same paper.

Since we fix algebra $M$ and trace $\tau$, we omit them from the notations
from now on.

Let $F\in M$ be a set of finite linear combinations of orthogonal projections
with finite trace.

Space $R_{0}=(L_{1}+L_{\infty})_{0}$ is the closure of $F$ in the norm
$\|x\|=inf\\{\|x_{1}\|_{1}+\|x_{2}\|_{\infty},|x=x_{1}+x_{2},\
x_{1}\in\L_{1},x_{2}\in\L_{\infty}\\}$.

######  Remark 1.

The space $R_{0}$ is not necessary separable, it is sufficient to consider
$M=B(H)$ of all bounded operators in the Hilbert space $H$ with not separable
$H$ with natural trace $\tau$.

######  Definition 1.

Non-commutative re-arrangement invariant (or symmetric) space $L$ for the
fully symmetric case were introduced by Yeadon in [24]. For the definition of
the symmetric spaces we refer to the recent book of Lord, Sukochev, Zanin on
singular traces, with original proofs due to Kalton and Sukochev.

######  Definition 2.

Re-arrangement invariant space $L$ over $(M,\tau)$ is called minimal if $F$ is
dense in $L$ by norm of $L$.

##  2\. Embedding theorem

The following refinement of the embedding theorem [12] take place.

######  Theorem 1.

Let $L$ be a re-arrangement invariant space over $(M,\tau)$. Then

  1. (1)

If $L$ is minimal, then

$L_{1}\cap L_{\infty}\subseteq L\subseteq R_{0}$

  2. (2)

If $L$ is not minimal $L$, then

$L_{\infty}\subseteq L\subseteq L_{1}+L_{\infty}$

###### Proof.

Proof is given in the forthcoming paper of author and some co-authors. ∎

##  3\. Individual ergodic theorem for minimal symmetric spaces

###  3.1. The Largest Minimal Symmetric Space

The largest minimal symmetric space is $R_{0}=(L_{1}+L_{\infty})_{0}$ is a set
of all measurable integrable with trace operators plus bounded with not
increasing re-arrangement functions decreasing to 0.

The space $R_{0}$ is minimal symmetric space. Any minimal symmetric space is a
subset of $R_{0}$.

###  3.2. Positive double contraction on $R_{0}$

Any positive kernel ($T\in DS^{+}$) leaves $R_{0}$ invariant.

###  3.3. Mean Ergodic Theorem

Von Neumann Mean ergodic Theorem on $L_{2}$ follows from the general von
Neumann ergodic theorem for the contractions on the general Hilbert spaces.

Mean convergence on $L_{1}\cap L_{\infty}$ follows from the the fact that both
$L_{1}$ and $L_{\infty}$ are invariant under positive kernels. The space
$L_{1}\cap L_{\infty}$ itself is invariant under action of positive kernel,
and positive kernel is a contraction of the space $L_{1}\cap L_{\infty}$. The
von Neumann Ergodic theorem and the closedness of $L_{1}\cap L_{\infty}$ in
its norm implies mean ergodic theorem on $L_{1}\cap L_{\infty}$.

###  3.4. Mean convergence on $R_{0}$

######  Proposition 1.

The Cesaro averages $S_{n}(T)x$ converge to some $\tilde{x}$ in norm of
$R_{0}$.

###### Proof.

Follows from von Neumann ergodic theorem and the fact that $L_{2}$ is dense
$R_{0}$. Details. We show that $s_{l}(T)x,l=1,2,...$ is fundamental sequence
in $R_{0}$. Indeed, each $x\in R_{0}$ may be presented as $x=x_{1,n}+x_{2,n}$
with $x_{1,n}\in L_{1}$ and $x_{2,n}\in L_{\infty}$ with
$\|x_{2,n}\|_{\infty}<2^{-n}$. Then we can apply von Neumann or Yeadon’s Mean
Ergodic theorem 4.2 [24] for $L_{1}$ to $x_{1,n}$ and find $l(n)$ such that
for $l,m\geq l(n)$ holds

$\|s_{l}(T)x_{1,n}-s_{m}(T)x_{1,n}\|_{L_{1}}<2^{-n}.$

Then for $l,m\geq l(n)$

$\|s_{l}(T)x-s_{m}(T)x\|_{R_{0}}\leq\|s_{l}(T)x_{1,n}-s_{m}(T)x_{1,n}\|_{L_{1}}+$
$+\|s_{l}(T)x_{2,n}-s_{m}(T)x_{2,n}\|_{L_{\infty}}\leq 4*2^{-n},$

and, hence, the sequence $s_{l}(T)x$ is fundamental. Completeness of $R_{0}$
implies existence of $\tilde{x}\lim_{l\to\infty}s_{l}(T)x$. ∎

######  Remark 2.

Mean ergodic theorem for fully symmetric spaces is due to Yeadon [24], Theorem
4.2. Note that we do not require the space $L$ to be fully symmetric here.
Condition ii) of the theorem 4.2 [24] means that the space $L$ is minimal. The
space $R_{0}$ does not satisfy condition iii) in theorem 4.2 [24].

###  3.5. Individual Ergodic Theorem in $L_{1}$

Individual Ergodic theorem for $L_{1}$ was established by Yeadon [22], among
other authors.

###  3.6. Individual ergodic theorem for $R_{0}$

The goal of the section is to show double side almost everywhere convergence
for the operators from the $R_{0}$.

######  Definition 3.

The sequence $x_{n}$ from $L_{0}$ is called converging double side almost
everywhere to $x_{0}\in L_{0}$ if for every $\epsilon>0$ there exist
orthogonal projection $E\in M$ such that $\tau(1-E)<\epsilon$,
$E(x_{n}-x_{0})E\in M$ and $E(x_{n}-x_{0})E\to 0$.

######  Theorem 2.

Let $M,\tau,R_{0}$ are as above and $T$ is positive kernel on $M$. For $x\in
R_{0}$, Cesaro averages $S_{n}(T)x$ converge d.s.a.e. in $R_{0}$.

###### Proof.

The proof follows the line of the proof of Proposition 1 and uses Yeadon’s
individual ergodic theorem for $L_{1}$ [22], see also Chilin, Litvinov [cl1],
[cl2]

We show that $s_{l}(T)x,l=1,2,...$ is fundamental d.s.a.e. sequence in
$R_{0}$. Indeed, each $x\in R_{0}$ may be presented as $x=x_{1,n}+x_{2,n}$
with $x_{1,n}\in L_{1}$ and $x_{2,n}\in L_{\infty}$ with
$\|x_{2,n}\|_{\infty}<2^{-4*n}$. In turn, $x_{1,n}=x_{1,1,n}+x_{1,2,n}$, with
$x_{1,1,n}\in L_{\infty}$, $x_{1,2,n}\in L_{1}$ and
$\|x_{1,2,n}\|_{L_{1}}<2^{-8*n}$, $n=1,2,...$.

Then we can apply Yeadon’s Individual Ergodic theorem 1, [22] for $L_{1}$ to
$x_{1,1,n}$ and find $l(n)$ and projector $E(n)\in M$ such that
$\tau(1-E(n))<2^{-4*n}$ and for $l,m\geq l(n)$ holds

$\|E(n)(s_{l}(T)x_{1,1,n}-s_{m}(T)x_{1,1,n})E(n)\|_{L_{\infty}}\\}\to 0.$

We can represent $x_{1,2,n}=\sum_{k=1}^{\infty}x_{2,n,k}$, with $x_{2,n,k}\in
L_{\infty}$ and $\|x_{2,n,k}\|_{L_{1}}<2^{-8*(n+k)}$.

Then, we can find $E(1,n)=\wedge_{k}E(1,n,k)$, with $\tau(1-E(1,n))\leq
2^{-4*n}$ and $E(1,n)s_{l}(x_{1,2,n})E(1,n)\in L_{\infty}$ and
$\|E(1,n)s_{l}(x_{2,n})E(1,n)\|_{L_{\infty}}<2^{-4*n}$, where projections
$E(1,n,k)$ are obtain by Theorem 1 from [22] applied to $x_{2,n,k}$ and
$\epsilon=2^{-4(n+k)}$.

By choosing
$E(2,n)=\wedge_{k=1}^{\infty}E(1,n+k)\wedge\wedge_{k=1}^{\infty}E(n+k)$, we
have $\tau(1-E(2,n))<2^{-4*n}$.

Moreover, for $l,m\geq l(n)$

$\|E(2,n)(s_{l}(T)x-s_{m}(T)x)E(2,n)\|_{L_{\infty}}$
$\leq\|E(2,n)(s_{l}(T)x_{1,n}-s_{m}(T)x_{1,n})E(2,n)\|_{L_{\infty}}+\|E(2,n)(s_{l}(T)x_{2,n}-s_{m}(T)x_{2,n})E(2,n)\|_{L_{\infty}}$
$\leq\|E(2,n)(s_{l}(T)x_{1,1,n}-s_{m}(T)x_{1,1,n})E(2,n)\|_{L_{\infty}}+\|E(2,n)(s_{l}(T)x_{1,2,n}E(2,n)\|_{L_{\infty}}$
$+\|E(2,n)s_{m}(T)x_{1,2,n}E(2,n)\|_{L_{\infty}}+2*2^{-4*n}\leq 8*2^{-n},$

and, hence, the sequence $s_{l}(T)x$ is fundamental d.s.a.e. The sequence
$s_{l}(T)x$ also converges in norm in $R_{0}$ to $\tilde{x}$, hence it
converges in measure. This implies convergence of $s_{l}(T)x$ d.s.a.e. to
$\tilde{x}$. ∎

######  Remark 3.

In the case when space $L$ is not minimal, it contains $M$. Then it is
possible to show [16], that even in the commutative case, there exists ergodic
automorphism of the space with measure such that ergodic averages do not
converge almost everywhere.

######  Corollary 1.

Let $L$ be a minimal non-commutative symmetric space. Let $T$ be a positive
kernel such that $T$ leaves $L$ invariant and $T$ acts as contraction on $L$.
Then Cesaro averages $s_{n}(T)x$ converge in norm and d.s.a.e. for any $x\in
L$.

###### Proof.

Since the space $L$ is minimal, the set of $L_{1}\cap L_{\infty}$ is dense in
$L$. Since $L_{1}\cap L_{\infty}\subseteq L$, hence

$\|x\|_{L_{1}\cap L_{\infty}}\geq C*\|x\|_{L}$

for any $x\in L_{1}\cap L_{\infty}$, which in turn implies convergence of
Cesaro averages of $s_{n}(T)x$ in norm of $L$ for $x\in L_{1}\cap L_{\infty}$.
Fix real $\epsilon>0$. Find $x_{k}\in L_{1}\cap L_{\infty}$ with
$\|x-x_{k}\|_{L}<\epsilon/2$. Then Cesaro averages are $s_{n}(T)x$ are within
$\epsilon$ of the $\tilde{x_{k}}$, where
$\tilde{x_{k}}=\lim_{n\to\infty}s_{n}(T)x_{k}$. This implies norm convergence
of $s_{n}(T)x$.

The d.s.a.e. convergence follows from the embedding theorem 1, since a minimal
re-arrangement invariant non-commutative function space $L$ is a subspace of
$R_{0}$. ∎

######  Corollary 2.

Let $L$ be a minimal fully symmetric space. Let $T$ be a positive kernel on
$(M,\tau)$. Then $T$ leaves $L$ invariant and act on $L$ as contraction.
Moreover, the Cesaro averages $S_{n}(T)x$ converge d.a.e. for any $x\in L$.

######  Corollary 3.

(see Chilin Litvinov [3]). Let $L_{\Psi}$ be a non-commutative Orlicz space
with function $\Psi$ satisfying conditions $\delta_{2}$ and $\Delta_{2}$. Let
$T$ be a positive kernel. Then the Cesaro averages $S_{n}(T)x$ converge
d.s.a.e. for any $x\in L$.

###### Proof.

The Orlicz space $L_{\Psi}$ with function $\Psi$ satisfying conditions
$\delta_{2}$ and $\Delta_{2}$ is minimal [1, 12, 15]. Since the space
$L_{\Psi}$ is fully symmetric, it is interpolation space [1, 12] and hence $T$
leaves $L_{\Psi}$ invariant and acts on $L_{\Psi}$ as a contraction. Then we
are in the assumptions of the Corollary 2, and hence $S_{n}(T)x$ converges
d.a.e. . ∎

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