text:
\section{introduction}

The recent development of quantum information raised the problem
of finding a satisfactory quantum generalization of the classical
random walks. The relevance of this problem for quantum information
has been emphasized in the last past years, see e.g.
\cite{0}--\cite{2}, \cite{3}, \cite{4}, \cite{5}--\cite{9},
\cite{10} for different
solutions of this problem. However, these proposals introduce some
features which are not
quite satisfactory from the mathematical point of view.

First of all, these constructions are based on special models and
a general mathematical definition of quantum random walk seems
to be lacking.
Second, all these constructions are based on a quantum evolution,
unitary in some cases, irreversible in others. Now, random walks are
particular cases of Markov processes and it is well known that, while
in the classical case a Markov evolution uniquely determines the law
of the corresponding stochastic process, this is in general false in
the quantum case.
Finally, a desirable requirement for a quantum extension of a family
of classical processes, is that there should exist a standard procedure
to embed the original classical family into its quantum extension.
In a satisfactory quantum generalization of the classical random walks,
all these requirements should be precisely formulated and fulfilled.

We dealth with these problems in the previous paper
\cite{AFi3}, by defining a nontrivial quantum lifting of classical
Markov chains by using the Schur multiplication (known also as the
Hadamard multiplication, see e.g. \cite{B}). In the above mentioned
paper, we outlined this construction for
general classical Markov chains, concentrating our attention to
Markov chains with finite state--space.

In the present paper we study in detail the
construction of entangled Markov chains based on classical Markov
chains with infinite state--space. We refer the reader to \cite{AFi3}
and the above quoted papers for a discussion about the motivations,
the potential applications to quantum
information, and further details.

We obtain nontrivial examples of random walks
satisfying some natural requirements. Namely, they are quantum Markov
chains such that their restrictions to at least one maximal Abelian subalgebra,
are classical random walks. They are
uniquely determined, up to arbitrary
phases, by these classical restrictions. Finally, taking into account
possible applications to information theory, they are purely generated
(i.e. generated by isometries, see \cite{FNW1, FNW2}, and \cite{AFi3}
for a discussion about this point).

The entangled Markov chains so constructed, generate in a natural
way, states on the $C^{*}$--infinite tensor product
${\displaystyle\overline{\bigcup_{J\subset{\mathbb Z}}
\overline{\otimes}_{J}F}^{C^{*}}}$ (denoted in the sequel
${\displaystyle\bigotimes_{{\mathbb Z}}F}$ by an abuse of notation),
provided that the underlying classical
Markov chains admit invariant distributions. Here,
$F$ is any infinite dimensional type $\ty{I}$ factor, $J$ is a
finite interval of ${\mathbb Z}$, and the bar denotes the von Neumann tensor product
between von Neumann algebras.

The present paper is organized as follows. Section \ref{s2} is
devoted to prove that the Schur multiplication is well defined also
in infinite dimensional case. Namely, the Schur multiplication
generates a (commutative) multiplication also
for infinite dimensional type
$\ty{I}$ factors. In Section \ref{s3}, we extend to the infinite
dimensional case, the
definition of entangled
lifting of classical Markov chains by using the Schur
multiplication. The ergodic properties of the entangled Markov chains
with a stationary distribution are investigated in Section \ref{s3a}.
As for finite dimensional case, we see that the ergodic properties of the
entangled chain are determined by those of the underlying classical
chain. Contrary to finite dimensional case, the problem concerning the
pureness of strongly clustering infinite dimensional entangled Markov
chains is left open.
Yet, some very simple examples (see Section \ref{s4}) of pure Markov chains go
towards the conjecture that any strongly clustering entangled Markov
chain should generate a pure state on ${\displaystyle\bigotimes_{{\mathbb Z}}F}$.
Section \ref{s4} contains also the description
of entangled Markov processes arising from classical random walks
on discrete groups. Then, starting from random walks on discrete
Infinite Conjugacy Class (ICC for short) groups, we
provide examples of quantum random walks based on type $\ty{II_{1}}$
factors.

\section{the Schur multiplication in the infinite dimensional case}
\label{s2}

Let $I$ be an index set. Consider the space
$\stackrel{\circ}{{\mathbb M}}_{I}$ consisting of all the $I\times I$ matrices
with complex entries. The subset of all bounded $I\times I$ matrices is denoted
as ${\mathbb M}_{I}$. It gives the most general type $\ty{I}$ $W^{*}$--factor,
as $I$ varies among all cardinalities. Define the map
$\F:{\mathbb M}_{I}\mapsto\stackrel{\circ}{{\mathbb M}}_{I\times I}$ as
\begin{equation}
\label{1}
\F(A)_{(i,j)(k,l)}:=A_{ik}\delta}  \def\D{\Delta_{ij}\delta}  \def\D{\Delta_{kl}\,.
\end{equation}
Notice that the map $\F$ is identity preserving.

Let $A,B\in{\mathbb M}_{I}$. We can define the {\it Schur multiplication} as
\begin{equation*}
(A\diamond B)_{ij}:=A_{ij}B_{ij}\,.
\end{equation*}
Taking into account that
$$
(A\otimes B)_{(i,j)(k,l)}=A_{ik}B_{jl}\,,
$$
we can extend the Schur multiplication to a map
$m:{\mathbb M}_{I\times I}\mapsto \stackrel{\circ}{{\mathbb M}}_{I}$ by putting
\begin{equation}
\label{13a}
m(X)_{ij}:=X_{(i,i)(j,j)}\,.
\end{equation}
\begin{lem}
\label{3}
Let $A\in{\mathbb M}_{I}$. Then $\F(A)\in{\mathbb M}_{I\times I}$.
\end{lem}
\begin{proof}
We compute
\begin{align*}
|\langle\F(A)x,y\rangle|
\equiv&|\sum\overline{y_{ij}}A_{ik}\delta}  \def\D{\Delta_{ij}\delta}  \def\D{\Delta_{kl}x_{kl}|\\
=|\sum\overline{y_{ii}}A_{ik}x_{kk}|
\leq&\|A\|_{{\mathbb M}_{I}}\|x\|_{\ell^{2}(I\times I)}\|y\|_{\ell^{2}(I\times I)}
\end{align*}
which leads to $\|\F(A)\|_{{\mathbb M}_{I\times I}}\leq\|A\|_{{\mathbb M}_{I}}$.
\end{proof}

From now on, we consider the map $\F$ as a bounded map from
${\mathbb M}_{I}$ to ${\mathbb M}_{I\times I}$.
\begin{prop}
\label{4}
The map $\F:{\mathbb M}_{I}\mapsto{\mathbb M}_{I\times I}$ is an
identity preserving $*$--morphism.
\end{prop}
\begin{proof}
It is easily seen that $\F$ is $*$--preserving
and multiplicative.
\end{proof}

Let $\rho\in L^{1}({\mathbb M}_{I})_{+}$. It is immediate to show that
\begin{equation}
\label{12}
\mathop{\rm Tr}\otimes\mathop{\rm Tr}(\F(\rho))=\mathop{\rm Tr}(\rho)
\end{equation}
that is the positive element $\F(\rho)$ is trace--class. Furthermore, if
$\sigma} \def\S{\Sigma\in L^{1}({\mathbb M}_{I\times I})_{+}$, then
\begin{equation}
\label{121212}
0\leq\mathop{\rm Tr}(m(\sigma} \def\S{\Sigma))\leq\mathop{\rm Tr}\otimes\mathop{\rm Tr}(\sigma} \def\S{\Sigma)\,,
\end{equation}
that is the positive element $m(\sigma} \def\S{\Sigma)$ is trace--class as well.

For the sake of completeness we report the following
\begin{thm}
\label{11}
Let $\F:{\mathbb M}_{I}\mapsto{\mathbb M}_{I\times I}$ be the linear map given in \eqref{1}.

Then $\F$ is a normal faithful identity preserving $*$--morphism
of ${\mathbb M}_{I}$ into ${\mathbb M}_{I\times I}$.

Moreover if $p\geq1$, $\F$ restricts itself to
isometries of $L^{p}({\mathbb M}_{I})$ into $L^{p}({\mathbb M}_{I\times I})$.
\end{thm}
\begin{proof}
Let $\rho\in L^{1}({\mathbb M}_{I\times I})_{+}$, and $\{X_{\alpha}\}\subset{\mathbb M}_{I}$
be a net converging to $X\in{\mathbb M}_{I}$
in the $*$--weak topology. We have by \eqref{121212},
\begin{align*}
\lim_{\alpha}\mathop{\rm Tr}\otimes\mathop{\rm Tr}(\rho\F(X_{\alpha}))&=\lim_{\alpha}\mathop{\rm Tr}(m(\rho)X_{\alpha})\\
=\mathop{\rm Tr}(m(\rho)X)&=\mathop{\rm Tr}\otimes\mathop{\rm Tr}(\rho\F(X))\,,
\end{align*}
where $m$ is the
Schur multiplication. Namely, $\F$ is normal. The first part follows by Proposition
\ref{4}, taking
into account that a normal $*$--morphism between von Neumann factors
is automatically faithful.

Let now $p\geq1$, and $T\in L^{p}({\mathbb M}_{I})$.
We get by the previous results,
\begin{align*}
\|\F(T)\|^{p}_{p}\equiv&\mathop{\rm Tr}\otimes\mathop{\rm Tr}((\F(T)^{*}\F(T))^{p/2})\\
=&\mathop{\rm Tr}\otimes\mathop{\rm Tr}(\F((T^{*}T)^{p/2}))\\
=&\mathop{\rm Tr}((T^{*}T)^{p/2})\equiv\|T\|^{p}_{p}\,.
\end{align*}
\end{proof}

The main properties of the Schur multiplication \eqref{13a} are
summarized in the following
\begin{thm}
The Schur multiplication \eqref{13a} defines a completely positive
identity preserving normal map from ${\mathbb M}_{I\times I}$ into ${\mathbb M}_{I}$.

Moreover, $m=\big(\F\lceil_{L^{1}({\mathbb M}_{I})}\big)^{*}$ where $\F$ is the map given in
\eqref{1}.
\end{thm}
\begin{proof}
An easy computation gives $\big(\F\lceil_{L^{1}({\mathbb M}_{I})}\big)^{*}=m$.
Then, taking into account the
properties of $\F$, it remains to check the normality. Let
$\{X_{\alpha}\}\subset{\mathbb M}_{I\times I}$ be a net converging to $X\in{\mathbb M}_{I\times I}$
in the $*$--weak topology, and $\rho\in L^{1}({\mathbb M}_{I})_{+}$.
We get by \eqref{12},
\begin{align*}
\lim_{\alpha}\mathop{\rm Tr}(\rho m(X_{\alpha}))&=\lim_{\alpha}\mathop{\rm Tr}\otimes\mathop{\rm Tr}(\F(\rho)X_{\alpha})\\
=\mathop{\rm Tr}\otimes\mathop{\rm Tr}(\F(\rho)X)&=\mathop{\rm Tr}(\rho m(X))\,,
\end{align*}
that is $m$ is normal.
\end{proof}

Taking into account the above results, we have that
$m\lceil_{L^{1}({\mathbb M}_{I\times I})}$ is bounded and
$\F=\big(m\lceil_{L^{1}({\mathbb M}_{I\times I})}\big)^{*}$.

From the above considerations, the Schur multiplication $m$
can be defined as a completely positive
identity preserving normal map from $F\overline{\otimes}F$ to $F$ for arbitrary
type I factor $F$, provided that a complete system of matrix units
$e:=\{e_{ij}\}\subset F$ is kept fixed.

\section{infinite dimensional entangled Markov chains}
\label{s3}

Let $I$ be any index set which is kept fixed during the analysis.
Consider a copy ${\mathcal M}_{j}$ of the algebra
${\mathbb M}_{I}\equiv{\mathbb M}_{I}({\mathbb C})$ of all bounded
$I\times I$ matrices with complex entries, together with a copy
${\mathcal D}_{j}$ of the maximal Abelian subalgebra
$\ell^{\infty}(I)$ of ${\mathbb M}_{I}({\mathbb C})$. For each finite subset
$J\subset{\mathbb Z}$, we put
$$
{\mathfrak M}}\def\gpm{{\mathfrak m}_{J}:=\overline{\bigotimes}_{j\in J}{\mathcal M}_{j}\,,\quad
{\mathfrak D}_{J}:=\overline{\bigotimes}_{j\in J}{\mathcal D}_{j}\,,
$$
where $\overline{\otimes}$ is the usual von Neumann tensor product
between von Neumann algebras. If $F\subset G$, we consider the natural
embedding $a_F\mapsto a_F\otimes{1}\!\!{\rm I}_{G\backslash F}$.

The local algebra
$$
{\mathfrak M}}\def\gpm{{\mathfrak m}:=\overline{\big(\lim_{\stackrel{\longrightarrow}
{J\uparrow{\mathbb Z}}}{\mathfrak M}}\def\gpm{{\mathfrak m}_{J}\big)}^{\,C^{*}}
$$
is the $C^{*}$--inductive limit
associated to the directed system $\{{\mathfrak M}}\def\gpm{{\mathfrak m}_{J}\}_{J\subset{\mathbb Z}}$, $J$
finite subsets of ${\mathbb Z}$.
For our purpouse, we consider also the maximal Abelian subalgebra
$$
{\mathfrak D}:=\overline{\big(\lim_{\stackrel{\longrightarrow}
{J\uparrow{\mathbb Z}}}{\mathfrak D}_{J}\big)}^{\,C^{*}}
$$
of ${\mathfrak M}}\def\gpm{{\mathfrak m}$ made of the $C^{*}$--inductive limit
associated to the directed system $\{{\mathfrak D}_{J}\}_{J\subset{\mathbb Z}}$ as before.

For general $C^{*}$--algebras ${\mathfrak A}$, ${\mathfrak B}$, a
completely positive identity pre\-ser\-ving linear map
${\mathcal E}:{\mathfrak A}\otimes{\mathfrak B}\mapsto{\mathfrak B}$ will be called in the sequel, a {\it transition
expectation}.

Consider a completely positive normal map $P:{\mathbb M}_{I}\mapsto{\mathbb M}_{I}$.
Such a map is said to be {\it Schur identity preserving} if
${\mathcal E}:{\mathbb M}_{I}\overline{\otimes}{\mathbb M}_{I}\mapsto{\mathbb M}_{I}$ given by
\begin{equation}
\label{14}
{\mathcal E}=m\circ(\mathop{\rm id}\otimes P)
\end{equation}
is identity preserving
(where ``$\circ$'' stands for composition of maps). This condition means
$$
P({1}\!\!{\rm I})_{ii}=1\,,\quad i\in I\,.
$$

Following \cite{AFi3}, Definition 1,
any such a $P$ is said to be
an {\it entangled Markov operator} if $P({1}\!\!{\rm I})\neq{1}\!\!{\rm I}$.

We
consider also unbounded entangled Markov operators
$P:{\mathbb M}_{I}\mapsto\stackrel{\circ}{{\mathbb M}}_{I}$ such that the
associated maps
\eqref{14} are well defined (i.e. bounded) transition expectations, see below.

Let $P$ be an entangled Markov operator. Consider ${\mathcal E}$ as above.
Put ${\mathcal E}_{A}(B):={\mathcal E}(A\otimes B)$ and
consider a ``initial distribution'' $\rho$ which is a positive normalized
element in $L^{1}({\mathbb M}_{I})$ satisfying
$\rho=\rho\circ{\mathcal E}_{{1}\!\!{\rm I}}$.\footnote{In the infinite dimensional case, an
invariant distribution does not always exist in general.}
A state $\omega} \def\Om{\Omega\in{\mathcal S}({\mathfrak M}}\def\gpm{{\mathfrak m})$ is uniquely
determined by all the ``finite dimensional distributions''
$$
\omega} \def\Om{\Omega(A_{1}\otimes\cdots\otimes A_{n}):=
\rho({\mathcal E}_{A_{1}}\circ\cdots\circ{\mathcal E}_{A_{n}}({1}\!\!{\rm I}))\,.
$$

Such a state is a translation invariant quantum Markov chain on ${\mathfrak M}}\def\gpm{{\mathfrak m}$
generated by the triplet $({\mathbb M}_{I},{\mathcal E},\rho)$ following the terminology of
\cite{FNW2}.
It generalizes the construction given in \cite{AFi3} to the infinite
dimensional case.
For further details about the quantum Markov chains, we refer
to \cite{A0, BrKiJoWe00, FNW1, FNW2},
and the references cited therein.

Now we specialize the matter to the quantum Markov chains generated in a
canonical way by classical Markov chains with an infinite state--space,
that is to {\it infinite dimensional
entangled Markov chains}.

Let $\Pi\in\stackrel{\circ}{{\mathbb M}}_{I}$ be a stochastic matrix. Define,
for $A\in{\mathbb M}_{I}$,
\begin{equation}
\label{z1}
P(A)_{ij}:=\sum_{k,l\in I}\sqrt{\Pi_{ik}\Pi_{jl}}A_{kl}\,.
\end{equation}
By Holder inequality, we have
\begin{align*}
&\big|P(A)_{ij}\big|\equiv
\big|\sum_{k,l}\sqrt{\Pi_{ik}}A_{kl}\sqrt{\Pi_{jl}}\big|\\
&\leq\|A\|_{{\mathbb M}_{I}}\big(\sum_{k}\Pi_{ik}\big)^{1/2}
\big(\sum_{k}\Pi_{jk}\big)^{1/2}\equiv\|A\|_{{\mathbb M}_{I}}\,,
\end{align*}
that is $P(A)\in\stackrel{\circ}{{\mathbb M}}_{I}$.\footnote{The entangled operator
$P$ given in \eqref{z1} is not bounded
in general. However, we define the entangled Markov chain directly in terms of the
transition expectation given in \eqref{14}. This allows us to treat
entangled Markov chains arising from any stochastic matrix without
affecting our analysis, see Proposition \ref{p1}.}

We show that the transition expectation ${\mathcal E}$ is purely generated,
following
the terminology of \cite{FNW2}.
\begin{prop}
\label{p1}
We have for the map ${\mathcal E}=m\circ(\mathop{\rm id}\otimes P)$, with $P$ as
in \eqref{z1},
$$
{\mathcal E}(A)=V^{*}AV\,,\quad A\in{\mathbb M}_{I\times I}\,,
$$
where
the isometry $V:\ell^{2}(I)\mapsto\ell^{2}(I\times I)$ is given by
\begin{equation}
\label{rotto1}
Ve_{i}=\sum_{j\in I}\sqrt{\Pi_{ij}}e_{i}\otimes e_{j}\,,
\end{equation}
and $\{e_{i}\}_{i\in I}$ is the canonical basis of $\ell^{2}(I)$.

Hence, ${\mathcal E}$ extends to a completely positive, identity preserving normal map of
${\mathbb M}_{I}\overline{\otimes}{\mathbb M}_{I}\cong {\mathbb M}_{I\times I}$ into ${\mathbb M}_{I}$.
\end{prop}
\begin{proof}
It is immediate to show that Formula \eqref{rotto1} defines a bounded operator.
The proof follows as
$$
V^{*}e_{i}\otimes e_{j}=\sqrt{\Pi_{ij}}e_{i}\,.
$$
\end{proof}

Let now $\pi:=\{\pi_{j}\}_{j\in I}$ be an invariant measure for $\Pi$. Define
the matrix $Q(\pi)\in\stackrel{\circ}{{\mathbb M}}_{I}$ given by
\begin{equation}
\label{z2}
Q(\pi)_{ij}:=\sum_{k\in I}\pi_{k}\sqrt{\Pi_{ki}\Pi_{kj}}\,.
\end{equation}
By Holder inequality, we get
\begin{align*}
&Q(\pi)_{ij}\equiv\sum_{k}\sqrt{\pi_{k}\Pi_{ki}}\sqrt{\pi_{k}\Pi_{kj}}\\
\leq&\big(\sum_{k}\pi_{k}\Pi_{ki}\big)^{1/2}
\big(\sum_{k}\pi_{k}\Pi_{kj}\big)^{1/2}\equiv\sqrt{\pi_{i}\pi_{j}}\,,
\end{align*}
that is, $Q(\pi)$ is well defined. Moreover, it is
positive by construction. Furthermore, $Q(\pi)$ defines a
bounded positive form on ${\mathbb M}_{I}$ if and only if $\pi$ defines a
positive form on $\ell^{\infty}(I)$, with
$\|Q(\pi)\|_{L^{1}({\mathbb M}_{I})}=\|\pi\|_{\ell^{1}(I)}$.
\begin{prop}
\label{inv}
$Q$ given in \eqref{z2} maps the set of
invariant measures
for $\Pi$ into the set of normal semifinite weights on ${\mathbb M}_{I}$ invariant for
${\mathcal E}_{{1}\!\!{\rm I}}\equiv {1}\!\!{\rm I}\diamond P(\,\cdot\,)$,
with $P$ given in \eqref{z1}.

$Q$ restricts itself to
a one--to--one correspondence between the invariant probability
distributions
for $\Pi$ and the normal states invariant for
${\mathcal E}_{{1}\!\!{\rm I}}$.
\end{prop}
\begin{proof}
Taking into account the proof of Proposition 2.4 of \cite{AFi3}, it
is enough to prove semifiniteness and normality. Let $\pi$ be as
above. Let $\gpm_{Q(\pi)}$ be the definition--domain of the weight
$Q(\pi)$ (\cite{T2}, Definition VII.1.3). Then
$$
\gpm_{0}:=\bigcup\big\{{\mathbb M}_{F}\,\big|\,F\,\text{finite subset of}\,
I\big\}\subset\gpm_{Q(\pi)}
$$
and
$\gpm_{0}''={\mathbb M}_{I}$, that is $Q(\pi)$ is semifinite.

We write, with an abuse of notation,
$Q(\pi)(A)=\pi({\mathcal E}({1}\!\!{\rm I}\otimes A))$ as ${\mathcal E}({1}\!\!{\rm I}\otimes A)\in\ell^{\infty}(I)$.
The normality follows by Fatou Lemma (i.e. $\pi$ is normal on
$\ell^{\infty}(I)$), taking into account that ${\mathcal E}$ is a normal map.
\end{proof}

Let $(\Pi,\pi)$ consist of a stochastic matrix as above, and an invariant
probability measure for it, respectively. Put ${\mathbb M}:={\mathbb M}_{I}$.
The {\it entangled Markov chain} associated to
$(\Pi,\pi)$ is the translation invariant locally normal state on ${\mathfrak M}}\def\gpm{{\mathfrak m}$
generated by the triplet $({\mathbb M},{\mathcal E},Q(\pi))$. It is immediate to show
that the quantum chain reduces itself to the classical one, when
restricted to ${\mathfrak D}$.

An invariant distribution (i.e. an invariant
probability measure) does not always exist for a given infinite
stochastic matrix, see e.g. Theorem 6.2.1 of \cite{Ci}. However, by
Proposition \ref{inv}, one can
define on ${\mathfrak M}}\def\gpm{{\mathfrak m}$ a translation invariant locally normal weight
starting from the triplet $({\mathbb M},{\mathcal E},Q(\pi))$, $\pi$ being an
invariant measure for $\Pi$ if the last exists.\footnote{An invariant measure
for the stochastic matrix $\Pi$ always exists if the latter contains
recurrent indices (``states'' in the terminology of classical Markov
chains), see e.g. \cite{Ci}, Theorem 6.2.25 and Theorem 5.3.14.}

Notice that, by Proposition \ref{p1}, an entangled Markov chain is always
generated by an isometry (purely generated in the terminology of
\cite{FNW2}).

As it was shown in \cite{AFi3}, the entangled Markov chains are
defined up to arbitrary phases. Namely, let
$\stackrel{\circ}{{\mathbb M}}_I\!\!({\mathbb T})$ be the set consisting of all the
$I\times I$ matrices with entries in
the unit circle ${\mathbb T}$. If $\chi\in{\mathbb M}_I\!({\mathbb T})$,
then the map
$P_{\chi}$ defined as
\begin{equation}
\label{circle}
P_{\chi}(A)_{ij}:=\sum_{k,l\in I}\overline{\chi_{ik}}\chi_{jl}
\sqrt{\Pi_{ik}\Pi_{jl}}a_{kl}
\end{equation}
gives rise to an entangled Markov operator as well. The corresponding
quantum measures $Q_{\chi}(\pi)$ associated to the entangled Markov
operator in \eqref{circle} are given by
$$
Q_{\chi}(\pi)_{ij}=\sum_{k\in I}\pi_{k}
\chi_{ki}\overline{\chi_{kj}}\sqrt{\Pi_{ki}\Pi_{kj}}\,,
$$
provided that the classical chain $\Pi$ admits the invariant measure $\pi$.

\section{ergodic properties}
\label{s3a}

The present section is devoted to the investigation of
the ergodic properties of infinite dimensional entangled
Markov chains. The following analysis parallels the corresponding one
of Section 3 of
\cite{AFi3} relative to the finite dimensional case.

Taking into account Formula (3.15) of \cite{Ci}, we write
\begin{equation}
\label{111}
\Pi=p_0\Pi p_0+\sum_{\theta}  \def\Th{\Theta\in\Th}(p_0\Pi p_\theta}  \def\Th{\Theta+p_\theta}  \def\Th{\Theta\Pi
p_\theta}  \def\Th{\Theta)+\sum_{\lambda} \def\L{\Lambda\in\L}(p_0\Pi p_\lambda} \def\L{\Lambda+p_\lambda} \def\L{\Lambda\Pi p_\lambda} \def\L{\Lambda)\,.
\end {equation}

Here, $\Th$, $\L$ label the recurrent--null, and the
recurrent--positive classes of $\Pi$, respectively, and $p_{0}$ is
the selfadjoint projection associated to the indices relative to the
transient states of the classical chain under consideration.
Furthermore, for each $\Pi_{\lambda} \def\L{\Lambda}:=p_\lambda} \def\L{\Lambda\Pi p_\lambda} \def\L{\Lambda$,
\begin{equation*}
\Pi_{\lambda} \def\L{\Lambda}=\sum_{j_{\lambda} \def\L{\Lambda}=1}^{m_{\lambda} \def\L{\Lambda}}
p_{\lambda} \def\L{\Lambda,j_{\lambda} \def\L{\Lambda}}\Pi_{\lambda} \def\L{\Lambda}p_{\lambda} \def\L{\Lambda,j_{\lambda} \def\L{\Lambda}+1}
=:\sum_{j_{\lambda} \def\L{\Lambda}=1}^{m_{\lambda} \def\L{\Lambda}}
p_{\lambda} \def\L{\Lambda,j_{\lambda} \def\L{\Lambda}}\Pi_{j_{\lambda} \def\L{\Lambda},j_{\lambda} \def\L{\Lambda}+1}p_{\lambda} \def\L{\Lambda,j_{\lambda} \def\L{\Lambda}+1}
\end{equation*}
with the convention that $m_{\lambda} \def\L{\Lambda}+1=1$. Here, $m_{\lambda} \def\L{\Lambda}$ is the (finite) period
corresponding to indices (``states'' in the terminology of classical
Markov chains) of the ergodic class $\lambda} \def\L{\Lambda$, with the convention that
$m_{\lambda} \def\L{\Lambda}=1$ in the aperiodic case. Notice that the indices (i.e.
``states'')
corresponding to an ergodic class are at most denumerable.
We suppose that $\L$ is nonvoid.\footnote{This is always the case in
the finite dimensional case, where the class of recurrent--null
states is void by finiteness, see \cite{S} for further details.} In
this case, there exist stationary distributions for $\Pi$, and by
Proposition \ref{inv}, stationary distributions for ${\mathcal E}$ given by \eqref{14}.
Let $\pi$ be
any such a distribution. It has the form
\begin{equation*}
x=\sum_{\lambda} \def\L{\Lambda\in\L}\alpha_{\lambda} \def\L{\Lambda}x_{\lambda} \def\L{\Lambda}\,,\quad
\sum_{\lambda} \def\L{\Lambda\in\L}\alpha_{\lambda} \def\L{\Lambda}=1\,,\,\, \alpha_{\lambda} \def\L{\Lambda}\geq0\,,
\,\,\lambda} \def\L{\Lambda\in\L\,,
\end{equation*}
$x_{\lambda} \def\L{\Lambda}$ being the unique stationary distribution for the stochastic
matrix $\Pi_{\lambda} \def\L{\Lambda}$.
Let $\L_{\pi}\subset\L$ be the set of the ergodic classes $\lambda} \def\L{\Lambda$ of $\Pi$
such that
$\alpha_{\lambda} \def\L{\Lambda}>0$. Of
course, the cardinality of $\L_{\pi}$ is at most denumerable. If
\begin{equation}
\label{1113}
p:=\sum_{\lambda} \def\L{\Lambda\in\L}p_{\lambda} \def\L{\Lambda}
\end{equation}
is the support in $\ell^{\infty}(I)$ of $\pi$ considered in a natural
way as an element of ${\mathbb M}$,
define $\tilde{\mathcal E}:{\mathbb M}\otimes{\mathbb M}_{p}\mapsto{\mathbb M}_{p}$
given by
\begin{equation}
\label{113}
\tilde{\mathcal E}:=p{\mathcal E}\lceil_{{\mathbb M}_{p}}(\,\cdot\,)p\,.
\end{equation}

Let $\eta$ be the action of the completely reducible part $p\Pi p$ on
the set of projections
$\big\{\{p_{\lambda} \def\L{\Lambda,j_{1}},
\dots,p_{\lambda} \def\L{\Lambda,j_{m_{\lambda} \def\L{\Lambda}}}\}\big\}_{\lambda} \def\L{\Lambda\in\L}$, $p$ being given in
\eqref{1113}.
Such an action leaves each ergodic component
$\{p_{\lambda} \def\L{\Lambda,j_{\lambda} \def\L{\Lambda}}\}_{\lambda} \def\L{\Lambda=1}^{m_{\lambda} \def\L{\Lambda}}$ globally invariant, acting
cyclically on it. Choose any projection, say
$\bar p_{\lambda} \def\L{\Lambda}:=p_{\lambda} \def\L{\Lambda,\bar j}$, in each ergodic class $\lambda} \def\L{\Lambda\in\L_{\pi}$,
where $\L_{\pi}$ labels the ergodic components present in the stationary
distribution $\pi$ as described above. Define for $\{A_{1},\dots,A_{n}\}\subset{\mathbb M}$,
\begin{align*}
&\varphi} \def\F{\Phi_{\lambda} \def\L{\Lambda}(A_{1}\otimes\cdots\otimes A_{n}):=\pi(\bar p_{\lambda} \def\L{\Lambda})^{-1}
Q(\pi)({\mathcal E}_{A_{1}}\circ\cdots\circ{\mathcal E}_{A_{n}}(\eta^{-n}\bar p_{\lambda} \def\L{\Lambda}))\,,\\
&\omega} \def\Om{\Omega_{\lambda} \def\L{\Lambda}(A_{1}\otimes\cdots\otimes A_{n}):=\pi(p_{\lambda} \def\L{\Lambda})^{-1}
Q(\pi)({\mathcal E}_{A_{1}}\circ\cdots\circ{\mathcal E}_{A_{n}}(p_{\lambda} \def\L{\Lambda}))\,.
\end{align*}

Let $\omega} \def\Om{\Omega$ be the entangled Markov chain on ${\mathfrak M}}\def\gpm{{\mathfrak m}$ associated to the
triplet $({\mathbb M},{\mathcal E},Q(\pi))$.
It is straightforward to verify that
$$
\omega} \def\Om{\Omega_{\lambda} \def\L{\Lambda}=\frac{1}{m_{\lambda} \def\L{\Lambda}}\sum_{k=1}^{m_{\lambda} \def\L{\Lambda}}
\varphi} \def\F{\Phi_{\lambda} \def\L{\Lambda}\circ\tau^{k}\,,
$$
\begin{equation}
\label{cczz}
\omega} \def\Om{\Omega=\sum_{\lambda} \def\L{\Lambda\in\L_{\pi}}\frac{\pi(p_{\lambda} \def\L{\Lambda})}{m_{\lambda} \def\L{\Lambda}}\sum_{k=1}^{m_{\lambda} \def\L{\Lambda}}
\varphi} \def\F{\Phi_{\lambda} \def\L{\Lambda}\circ\tau^{k}\,,
\end{equation}
where $\tau$ is the one--step shift on the chain.

The states $\omega} \def\Om{\Omega_{\lambda} \def\L{\Lambda}$, $\varphi} \def\F{\Phi_{\lambda} \def\L{\Lambda}$ describe the decomposition of $\omega} \def\Om{\Omega$ into
ergodic and completely ergodic components,
respectively.\footnote{The state $\varphi} \def\F{\Phi_{\lambda} \def\L{\Lambda}$ is only
$m_{\lambda} \def\L{\Lambda}$--step translation invariant, $m_{\lambda} \def\L{\Lambda}$ being the period of
the component $\lambda} \def\L{\Lambda$, and keeps track of the localization (modulo a
period), see \cite{AFi3}, Section 5 for the precise way to define
$\varphi} \def\F{\Phi_{\lambda} \def\L{\Lambda}$ on ${\mathfrak M}}\def\gpm{{\mathfrak m}$.}
\begin{thm}
\label{main}
Let $(\Pi,\pi)$ consist of a stochastic matrix
and a stationary distribution for it, with $\L_{\pi}\neq\emptyset$.
Consider the entangled Markov chain
$\omega} \def\Om{\Omega$ on ${\mathfrak M}}\def\gpm{{\mathfrak m}$ generated by the triplet $({\mathbb M},{\mathcal E},Q(\pi))$.
The following assertions hold true.
\begin{itemize}
\item[(i)] The state $\omega} \def\Om{\Omega$ is ergodic w.r.t the spatial translations
if and only if the set $\L_{\pi}$ is a singleton.
\item[(ii)] The state $\omega} \def\Om{\Omega$ is strongly clustering w.r.t the spatial translations
if and only if the set $\L_{\pi}$ is a singleton, and in addition,
the corresponding block in
the decomposition \eqref{111} of $\Pi$ is aperiodic.
\end{itemize}
\end{thm}
\begin{proof}
It is immediate to verify that
$\omega} \def\Om{\Omega$ is given by \eqref{cczz}. Furthermore, the states appearing in the
r.h.s. of \eqref{cczz} give rise to different states when restricted to the
Abelian subalgebra ${\mathfrak D}$
of ${\mathfrak M}}\def\gpm{{\mathfrak m}$. So, they are mutually different. It is then enough to show
that the $\omega} \def\Om{\Omega_{\lambda} \def\L{\Lambda}$ are ergodic w.r.t. the one--step shift, and
the $\varphi} \def\F{\Phi_{\lambda} \def\L{\Lambda}$ are strongly clustering w.r.t. the $m_{\lambda} \def\L{\Lambda}$--step shift,
respectively.

Let $A=A_{1}\otimes\cdots\otimes A_{r}$, $B=B_{1}\otimes\cdots\otimes B_{s}$,
we compute by applying Lemma 2.1 of \cite{AFi3},
\begin{align*}
&\frac{1}{n}\sum_{k=1}^{n}\omega} \def\Om{\Omega_{\lambda} \def\L{\Lambda}(A\tau^{k}(B))
=\pi(p_{\lambda} \def\L{\Lambda})^{-1}\\
\times&Q(\pi)\bigg({\mathcal E}_{A_{1}}\circ\cdots\circ{\mathcal E}_{A_{r}}
\circ\bigg(\frac{1}{n}\sum_{k=0}^{n-1}{\mathcal E}_{{1}\!\!{\rm I}}^{k}\bigg)
\bigg({\mathcal E}_{{1}\!\!{\rm I}}\circ\tilde{\mathcal E}_{B_{1}}\circ\cdots\circ\tilde{\mathcal E}_{B_{s}}
(p_{\lambda} \def\L{\Lambda})\bigg)\bigg)\,,
\end{align*}
where $\tilde{\mathcal E}$ is given by \eqref{113}. Define, componentwise, the element
$v\in\ell^{1}(I)$ as
$$
v_{i}:=Q(\pi)\big({\mathcal E}_{A_{1}}\circ\cdots\circ{\mathcal E}_{A_{r}}(e_{ii})\big)\,.
$$
Define the element $D\in\ell^{\infty}(I)_{p_{\lambda} \def\L{\Lambda}}\subset{\mathbb M}_{I}$ as
$$
D:={\mathcal E}_{{1}\!\!{\rm I}}\circ\tilde{\mathcal E}_{B_{1}}\circ\cdots\circ\tilde{\mathcal E}_{B_{s}}(p_{\lambda} \def\L{\Lambda})\,.
$$
We get by Dominated
Convergence Theorem,
\begin{align*}
&\lim_{n}Q(\pi)\bigg({\mathcal E}_{A_{1}}\circ\cdots\circ{\mathcal E}_{A_{r}}
\circ\bigg(\frac{1}{n}\sum_{k=0}^{n-1}{\mathcal E}_{{1}\!\!{\rm I}}^{k}(D)\bigg)\bigg)\\
=&\lim_{n}Q(\pi)\bigg({\mathcal E}_{A_{1}}\circ\cdots\circ{\mathcal E}_{A_{r}}
\circ\bigg(\frac{1}{n}\sum_{k=0}^{n-1}\Pi_{\lambda} \def\L{\Lambda}^{k}D\bigg)\bigg)\\
=&\lim_{n}\bigg\langle v,\bigg(\frac{1}{n}
\sum_{k=0}^{n-1}\Pi_{\lambda} \def\L{\Lambda}^{k}\bigg)D\bigg\rangle\\
=&\pi(p_{\lambda} \def\L{\Lambda})^{-1}Q(\pi)({\mathcal E}_{B_{1}}\circ\cdots\circ{\mathcal E}_{B_{s}}
(p_{\lambda} \def\L{\Lambda}))
Q(\pi)({\mathcal E}_{A_{1}}\circ\cdots\circ{\mathcal E}_{A_{r}}(p_{\lambda} \def\L{\Lambda}))\,.
\end{align*}
Here, $\langle\,\cdot\,,\,\cdot\,\rangle$ is the natural pairing
between $\ell^{1}$ and $\ell^{\infty}$, and
the last equality follows by Theorem 6.1 of \cite{KSK}.

Collecting together, we have
$$
\lim_{n}\frac{1}{n}\sum_{k=1}^{n}\omega} \def\Om{\Omega_{\lambda} \def\L{\Lambda}(A\tau^{k}(B))=\omega} \def\Om{\Omega_{\lambda} \def\L{\Lambda}(A)\omega} \def\Om{\Omega_{\lambda} \def\L{\Lambda}(B)\,,
$$
that is, $\omega} \def\Om{\Omega_{\lambda} \def\L{\Lambda}$ is ergodic.
The mixing property w.r.t. the $m_{\lambda} \def\L{\Lambda}$--step shift for the $\varphi} \def\F{\Phi_{\lambda} \def\L{\Lambda}$
is proven in the same way, taking into account Theorem 6.38 of \cite{KSK}.

Now, the quantum chain is ergodic iff $\L_{\pi}$ is a singleton (i.e.
$\L_{\pi}=\{\lambda} \def\L{\Lambda_{0}\}$),  which
corresponds to the case when there is only one summand in the first
sum in \eqref{cczz}.
The quantum chain is strongly clustering when, in addition, also the second sum in
\eqref{cczz} consists of one element, that is when, for the period, $m_{\lambda} \def\L{\Lambda_{0}}=1$.
\end{proof}

\section{some applications}
\label{s4}

We are going to consider some interesting examples of
infinite dimensional entangled Markov chains.

\medskip

We begin with the classical chain $(\Pi,\pi)$, $\pi$ being an
invariant distribution for the stochastic matrix $\Pi$.
Consider the collections
$\{D^{(k)}\}_{k\in{\mathbb N}\backslash\{0\}}$ of
the density matrices relative to the local algebras
$$
{\mathfrak M}}\def\gpm{{\mathfrak m}_{[1,k]}:=\overline{\bigotimes}_{1\leq j\leq k}{\mathcal M}_{j}
$$
and their translates,
arising from the entangled Markov chain $({\mathbb M},{\mathcal E},Q(\pi))$ associated
to $(\Pi,\pi)$.
Here, ${\mathcal E}$ is given in \eqref{14}, with the
entangled operator $P$ given in \eqref{z1}.

\begin{prop}
\label{dms}
We have for the collections
$\{D^{(k)}\}_{k\in{\mathbb N}\backslash\{0\}}$ of
the density matrices,
\begin{equation}
\label{elm}
D^{(k)}_{(i_{1},\dots,i_{k})(j_{1},\dots,j_{k})}=
\mathop{\rm Tr}{}_{{\mathbb M}}\bigg({\mathcal E}_{D_{\pi}}\circ{\mathcal E}_{e_{i_{1}j_{1}}}
\circ\cdots\circ{\mathcal E}_{e_{i_{k}j_{k}}}({1}\!\!{\rm I})\bigg)\,,
\end{equation}
where $D_{\pi}:=\sum_{k\in I}\pi_{k}e_{kk}$ is the diagonal embedding
of $\pi$ in $L^{1}({\mathbb M})$, and the $e_{ij}$ are the canonical matrix units
of ${\mathbb M}$.
\end{prop}
\begin{proof}
A simple computation.
\end{proof}

We specialize the situation to the entangled Markov chain generated
by the the aperiodic stochastic projection $Q$ with
matrix elements $q_{ij}=\pi_{j}>0$, $i\in{\mathbb N}$ and
$\pi\in\ell^{1}$. We obtain for the sequences of density matrices given in
\eqref{elm},
$$
D^{(k)}=\underbrace{D^{(1)}\otimes\cdots\otimes D^{(1)}}_{k\text{--times}}\,,
$$
with $D^{(1)}_{ij}=\sqrt{\pi_{i}\pi_{j}}$. The state $\omega} \def\Om{\Omega$
on ${\mathfrak M}}\def\gpm{{\mathfrak m}$ associated to these examples of entangled Markov chains, is then an infinite
product vector state based on the vector
$\{\sqrt{\pi_{i}}\}_{i\in{\mathbb N}}\in\ell^{2}$ (\cite{vN}).
Namely, $\omega} \def\Om{\Omega$ is a pure state, see \cite{Sa}, Proposition 4.4.4.\footnote{This non generic
situation exhibits a low degree of entanglement. These examples are
then not
suitable for possible applications to quantum information theory.}

This very simple example, together with the result in Theorem 3.4 of
\cite{AFi3} relative to the pureness of finite dimensional entangled
Markov chains, allows us to conjecture that strongly
clustering infinite dimensional entangled Markov chains generate pure
states on ${\mathfrak M}}\def\gpm{{\mathfrak m}$. This might be proved by studying the algebraic properties of
the sequence of the ranges of the $D^{(k)}$ (see \cite{FNW1}),
taking into account
\eqref{elm}. For this aim, we report another useful formula for the density
matrices $D^{(k)}$.
Define the matrix $\G\in\stackrel{\circ}{{\mathbb M}}_{I\times I}$ as
$$
\G_{(i,j)(k,l)}:=\sqrt{\Pi_{ij}\Pi_{kl}}\,.
$$
We get
$$
D^{(k)}=\underbrace{m\otimes\cdots\otimes m}_{k\text{--times}}
\big(Q(\pi)\otimes\underbrace{\G\otimes\cdots\otimes
\G}_{(k-1)\text{--times}}\otimes P({1}\!\!{\rm I})\big)
$$
where $m$ is the Schur multiplication, $P$ is the entangled Markov
operator associated to the stochastic matrix $\Pi$ via \eqref{z1}, and $Q(\pi)$
is the trace--class matrix associated to the stationary distribution
$\pi$ via \eqref{z2}.

\medskip

We pass to the entangled Markov processes based on random walks on
discrete groups. We note that most of such Markov chains
generate merely locally normal weights on ${\mathfrak M}}\def\gpm{{\mathfrak m}$. For some basic facts
about random walks on groups, see e.g. \cite{R}.

Let $G$ be a discrete group and $\mu$ a probability measure on it
which is kept fixed during the analysis. The {\it right} and
{\it left} random walks on $G$ are given by the doubly stochastic
transition matrices
$\Pi^{r}$, $\Pi^{l}$ respectively, with
$$
\Pi^{r}_{gh}:=\mu(g^{-1}h)\,,\quad \Pi^{l}_{gh}:=\mu(gh^{-1})\,.
$$

Let $P^{r}$, $P^{l}$ be the corresponding entangled Markov operators
obtained by \eqref{z1}.\footnote{As $\Pi^{r}$, $\Pi^{l}$ are doubly
stochastic, the associated entangled operators $P^{r}$, $P^{l}$ are
bounded.} An easy computation gives rise for $g\in G$,
\begin{equation}
\label{tras}
\text{ad}(\lambda} \def\L{\Lambda(g))\circ P^{r}=P^{r}\circ\text{ad}(\lambda} \def\L{\Lambda(g))\,,\,
\text{ad}(\rho(g))\circ P^{l}=P^{l}\circ\text{ad}(\rho(g))\,,
\end{equation}
where $\lambda} \def\L{\Lambda$, $\rho$ are the left and right translations on $G$. Here,
\eqref{tras} follows by the corresponding equivariance properties of
$\Pi^{r}$, $\Pi^{l}$. Denote by $R(G)$, $L(G)$ the von Neumann
algebras generated by the right and left translations on $G$.
\begin{prop}
We have for the entangled Markov operators $P^{r}$, $P^{l}$, and
for the transition expectations ${\mathcal E}^{r}$, ${\mathcal E}^{l}$ constructed as
in \eqref{14},
\begin{align*}
&P^{r}(R(G))\subset R(G)\,,\quad P^{l}(L(G))\subset L(G)\,,\\
&{\mathcal E}^{r}(R(G)\overline{\otimes}R(G))\subset R(G)\,,\quad
{\mathcal E}^{l}(L(G)\overline{\otimes}L(G))\subset L(G)\,.
\end{align*}
\end{prop}
\begin{proof}
Taking into account that $\rho(g)_{xy}=\delta}  \def\D{\Delta_{x,yg^{-1}}$,
$\lambda} \def\L{\Lambda(g)_{xy}=\delta}  \def\D{\Delta_{x,gy}$, we easily obtain
$$
P^{r}(R(G))\subset L(G)'\equiv R(G)\,,
\quad P^{l}(L(G))\subset R(G)'\equiv L(G)\,.
$$
The proof follows as, for the Schur multiplication,
$$
m(R(G)\overline{\otimes}R(G))\subset R(G)\,,\quad
m(L(G)\overline{\otimes}L(G))\subset L(G)\,.
$$
\end{proof}

We end by noticing that, starting with a discrete ICC group, one
can construct entangled Markov processes on
${\displaystyle\overline{\bigcup_{J\subset{\mathbb Z}}
\overline{\otimes}_{J}R}^{C^{*}}}$ ($J$ runs on all finite subsets of ${\mathbb Z}$),
$R$ being the algebra $L(G)$
generated by left translations (equally well the algebra $R(G)$
generated by right translations). Then, we provide entangled Markov
processes based on type $\ty{II_{1}}$ factors.

We hope to return elsewhere on all the questions left open in this
section.

\section*{acknowledgements}

The author is grateful to L. Accardi for suggesting the problem and
for fruitful discussions.


text:
\section{Introduction}\label{sec:intro}

It is classical that a bounded convex domain with smooth boundary in $\R^n$ has a smooth convex
defining function; see Herbig and McNeal \cite{HerbigMcNeal2012} and the references therein
(in particular, Weinstock \cite[pp.\ 402--403]{Weinstock1975} and
Gilbarg and Trudinger \cite[pp.\ 354--357]{GilbargTrudinger1983}).
In this paper we prove the analogous result for $p$-convex domains for any
$2\le p<n$. This class of domains, and the associated $p$-plurisubharmonic
functions, are closely related to $p$-dimensional minimal submanifolds; in particular, to minimal
surfaces when $p=2$. They were studied by Harvey and Lawson \cite{HarveyLawson2013IUMJ};
see also \cite{HarveyLawson2011ALM,HarveyLawson2012AM}
and \cite[Sect.\ 8.1]{AlarconForstnericLopez2021}.

Let $\bx=(x_1,\ldots,x_n)$ denote the coordinates on $\R^n$.
Given a $\Cscr^2$ function $\rho:\Omega\to \R$ on a domain $\Omega\subset \R^n$,
we denote by $\nabla^2\rho(\bx)=\Hess_\rho(\bx)$ the Hessian of $\rho$ at $\bx\in \Omega$,
i.e., the quadratic form on $T_\bx\R^n=\R^n$
represented by the matrix $\big(\frac{\di^2\rho}{\di x_i\di x_j}(\bx)\big)$.
The trace of $\Hess_\rho(\bx)$ equals the Laplacian of $\rho$ at $\bx$: $\tr\, \Hess_\rho(\bx)=\Delta \rho(\bx)$.
For $2\le p\le n-1$ we denote by $G_p(\R^n)$ the Grassman manifold of $p$-planes in $\R^n$.
Given $\Lambda\in G_p(\R^n)$, we denote by $\tr_\Lambda \Hess_\rho(\bx)\in\R$
the trace of the restriction of $\Hess_\rho(\bx)$ to $\Lambda$.
Our main result is the following.

\begin{theorem}\label{th:main1}
Let $D$ be a bounded domain in $\R^n,\ n\ge 3,$ with boundary of class $\Cscr^{r,\alpha}$
for some $r\in \{2,3,\ldots\}$ and $0<\alpha\le 1$, and let $p\in \{2,\ldots,n-1\}$. If
the principal interior curvatures $\nu_1(\bx) \le \nu_2(\bx)\le \cdots\le \nu_{n-1}(\bx)$ of $bD$
at every point $\bx\in bD$ satisfy
\begin{equation}\label{eq:p-convexity}
	\nu_1(\bx) + \nu_2(\bx) +\cdots + \nu_p(\bx)\ge 0,
\end{equation}
then there exists a defining function $\rho$ of class $\Cscr^{r,\alpha}$ for $D$ such that
\begin{equation}\label{eq:p-psh}
	\tr_\Lambda \Hess_\rho(\bx)\ge 0\ \ \text{for all}\ \bx\in \overline D\ \text{and}\ \Lambda\in G_p(\R^n).
\end{equation}
If there are no points $\bx\in bD$ with $\nu_i(\bx)=0$ for $i=1,\ldots,p$
(such a point is called $p$-flat), then $\rho$ can be chosen such that strict inequality holds in
\eqref{eq:p-psh} for all $\bx\in D$ and $\Lambda\in G_p(\R^n)$.
\end{theorem}

We say that $bD$ is $p$-convex if condition \eqref{eq:p-convexity} holds for every
$\bx\in bD$, and is strongly $p$-convex at $\bx\in bD$ if strict inequality holds in \eqref{eq:p-convexity}.
Note that $1$-convexity is the usual convexity. In this case, Theorem \ref{th:main1} also holds
and it coincides with the main result of \cite{HerbigMcNeal2012}
(see also \cite[pp.\ 402--403]{Weinstock1975} and \cite[pp.\ 354--357]{GilbargTrudinger1983}).

A $\Cscr^2$ function $\rho$ satisfying \eqref{eq:p-psh} is said to be $p$-plurisubharmonic on $\overline D$;
it is strictly $p$-plurisubharmonic at $\bx$ if strict inequality holds at this point.
Recall that $\rho$ is $p$-plurisubharmonic
on a domain $\Omega\subset \R^n$ is and only if the restriction $\rho|_M$ to any
minimal $p$-dimensional submanifold $M\subset \Omega$ is a subharmonic function
on $M$ in the induced metric, $\Delta_M(\rho|_M)\ge 0$ (cf.\ \cite{HarveyLawson2013IUMJ}
and \cite[Sect.\ 8.1]{AlarconForstnericLopez2021}).
For affine subspaces this follows by observing that if $\bv_1,\ldots, \bv_p \in \R^n$ is an orthonormal
basis of $\Lambda\in G_p(\R^n)$ and we set
$
	\tilde\rho(u_1,\ldots, u_p)=\rho\big(\bx+\sum_{j=1}^p u_j\bv_j\big),
$
then
\[
	\tr_\Lambda \Hess_\rho(\bx) = \Delta \tilde \rho(0).
\]
The analogous formula holds on a minimal submanifold (cf.\ \cite[Equation (2.10)]{HarveyLawson2013IUMJ}).
Denoting by $\lambda_1\le\lambda_2\le\cdots\le\lambda_n$ the
eigenvalues of $\Hess_\rho(\bx)$, we have
\[
	\min_{\Lambda \in G_p(\R^n)} \tr_\Lambda \Hess_\rho(\bx)
	= \lambda_1+\lambda_2+\cdots + \lambda_p.
\]
(See \cite[Lemma 2.5 and Corollary 2.6]{HarveyLawson2013IUMJ} or Lemma \ref{lem:trace} below.)
A domain $D\subset\R^n$
is said to be $p$-convex if it admits a strongly $p$-plurisubharmonic exhaustion function
$\rho:D\to\R$, i.e., such that the set $\{\bx \in D:\rho(\bx)\le c\}$ is compact for every $c\in\R$.

Suppose now that $\rho$ is a $\Cscr^2$ defining function for $D$, i.e., $D=\{\rho<0\}$
and $d\rho\ne 0$ on $bD=\{\rho=0\}$. Condition \eqref{eq:p-convexity} is then equivalent to
the following restricted version of \eqref{eq:p-psh}:
\begin{equation}\label{eq:p-convexity2}
	\tr_\Lambda \Hess_\rho(\bx)\ge 0\ \ \text{for every point}\ \bx\in bD\
	\text{and $p$-plane}\ \Lambda \subset T_\bx bD.
\end{equation}
Indeed, choosing $\rho$ such that $|\nabla\rho|=1$ on $bD$, the Hessian $\Hess_\rho(\bx)$
at the point $\bx\in bD$ restricted to the tangent space $T_\bx bD$ is the second fundamental form
of $bD$ at $\bx$ with respect to the inner normal vector $-\nabla\rho(\bx)$
(see \cite[Remark 3.11]{HarveyLawson2013IUMJ}), and
\[
	\min_{\Lambda\subset T_\bx bD}
	\tr_\Lambda \Hess_\rho(\bx) = \nu_1(\bx)+\nu_2(\bx)+\cdots +\nu_p(\bx).
\]
(The minimum is over all $p$-planes $\Lambda\subset T_\bx bD$. See
\cite[Remark 3.12]{HarveyLawson2013IUMJ} and Lemma \ref{lem:trace}.)
For a bounded domain $D\subset \R^n$ with smooth boundary we have
(see \cite[Summary 3.16]{HarveyLawson2013IUMJ})
\[
	\text{$D$ is $p$-convex} \ \Longleftrightarrow\ \text{$bD$ is $p$-convex},
\]
and these are further equivalent to the condition that $-\log\mathrm{dist}(\cdotp,bD)$ is a
$p$-plurisubharmonic function in an interior collar around $bD$. However, the existence
of a $p$-plurisub\-harmonic defining function is a more convenient property, yielding applications
which are not possible by using unbounded $p$-plurisubharmonic exhaustion functions.

With this terminology, Theorem \ref{th:main1} can be stated as follows.

\begin{theorem}\label{th:main-simple}
Every bounded $p$-convex domain $D\subset \R^n$ $(2\le p<n)$ with smooth boundary
admits a smooth defining function $\rho$ which is $p$-plurisubharmonic on $\overline D$.
If in addition $bD$ has no $p$-flat points then $\rho$ can be chosen strongly
$p$-plurisubharmonic on $D$.
\end{theorem}

Following the terminology introduced in \cite{DrinovecForstneric2016TAMS} and used in
\cite{AlarconForstnericLopez2021}, $2$-convex domains are called {\em minimally convex}
and $2$-plurisubharmonic functions are called {\em minimal plurisubharmonic}
(as they pertain to minimal surfaces).
Thus, a bounded minimally convex domain $D\subset \R^n$,
$n\ge 3$, with smooth boundary admits a defining function which is minimal plurisubharmonic on
$\overline D$. This has the following corollary. Let $\D=\{z\in \C:|z|<1\}$ and $\D^*=\D\setminus \{0\}$.

\begin{corollary}\label{cor:extension}
Let $D$ be a bounded minimally convex domain in $\R^n,\ n\ge 3$, with $\Cscr^r$ boundary
for some $r>2$. If $M$ is a connected open conformal surface and $f:M\to \R^n$ is a conformal
harmonic map with $f(M)\subset \overline D$, then either $f(M)\subset D$ or $f(M)\subset bD$.
If $K\subset M$ is a removable set for bounded harmonic functions, then
every conformal harmonic map $M\setminus K\to D$ extends to a conformal harmonic map $M\to D$.
In particular, every conformal harmonic map $\D^*\to D$ from the punctured disc
extends to a conformal harmonic map $\D\to D$.
\end{corollary}

\begin{proof}
Let $\rho$ be a defining function for $D$ such that \eqref{eq:p-psh} holds for $p=2$.
The first claim follows from the maximum principle applied to the subharmonic function
$\rho\circ f:M\to (-\infty,0]$, and the second claim is an immediate consequence.
The last claim follows by observing that every bounded harmonic function on $\D^*$
extends to a harmonic function on $\D$.
\end{proof}

\begin{remark}
If $bD$ fails to be minimally convex at a point $\bp\in bD$, then there is an embedded minimal
disc $f:\D\to D\cup \{\bp\}$ with $f(0)=\bp$ and $f(\D\setminus \{0\})\subset D$
(see \cite[Lemma 3.13]{HarveyLawson2013IUMJ}). Hence, the conclusion of
Corollary \ref{cor:extension} fails in this case.
\end{remark}

\begin{remark}
Corollary \ref{cor:extension} generalizes the classical fact that two immersed
minimal surfaces in $\R^3$ which touch at a point but locally at that point lie on one side of one another
must coincide. In fact, assume that $D\subset\R^3$ is a bounded domain whose smooth
boundary contains a domain $\Sigma$ in an immersed minimal surface $\wt \Sigma\subset\R^3$
(hence, $\Sigma$ is embedded),
and $bD$ is strongly minimally convex at each point $\bp\in bD\setminus \overline \Sigma$.
Then, the domain $D$ is minimally convex. Let $\rho$ be a minimal plurisubharmonic defining function for $D$
provided by Theorem \ref{th:main1}. Given a connected open conformal surface $M$ and a
nonconstant conformal harmonic map $f:M\to \overline D$, the maximum principle for the
subharmonic function $\rho\circ f:M\to (-\infty,0]$ shows that either $f(M)\subset D$
or $f(M)\subset \Sigma$.
\end{remark}

We shall see that a suitable convexification of the signed distance function to $bD$,
\begin{equation}\label{eq:delta}
	\delta(\bx) = \mathrm{dist}(\bx, D) - \mathrm{dist}(\bx,\R^n\setminus D), \quad \bx\in\R^n
\end{equation}
satisfies the conclusion Theorem \ref{th:main1} in an interior collar around $bD$.
Assuming that $bD$ is of H\"older class $\Cscr^{r,\alpha}$ for some
$r\in \{2,3,\ldots\}$ and $0<\alpha\le 1$, $\delta$ is of the same class $\Cscr^{r,\alpha}$
in a neighbourhood $V\subset \R^n$ of $bD$ (see Li and Nirenberg \cite{LiNirenberg2005}).
This gives the following corollary.

\begin{corollary}
Let $D$ be a bounded $p$-convex domain with $\Cscr^r$ boundary in $\R^n$
for some $2\le p<n$ and real number $r>2$. Then every domain
$D_t = \{\bz\in D:\delta(\bz)<t\}$ for $t<0$ close to $0$ is $p$-convex,
and it is strongly $p$-convex if $bD$ does not contain any $p$-flat points.
\end{corollary}

On a domain $\Omega\subset\C^n$ and for a complex line
$\Lambda\subset \C^n$, $\tr_\Lambda \Hess_\rho(\bx)$ is the Levi form of $\rho$ at $\bx\in\Omega$
on any unit vector $\xi\in \Lambda$. It follows that every minimal plurisubharmonic
function on a domain in $\C^n$ is also plurisubharmonic in the usual sense of complex analysis,
and every 2-convex domain $D$ in $\C^n$ is pseudoconvex; the converse fails.
Assuming that $bD$ is of class $\Cscr^r$ for some $r>2$,
Levi pseudoconvexity of $bD$ is characterized by nonnegativity of the Hessian of $\delta$ applied
in complex tangent directions to $bD$. From the differential geometric viewpoint, Levi pseudoconvexity
says that the mean sectional curvature of $bD$ in complex tangent directions is nonnegative
(see \eqref{eq:HDelta} and \eqref{eq:Levi}).
This leads to an observation regarding strong pseudoconvexity of domains
$D_t = \{\bz\in D:\delta(\bz)<t\}$ for $t<0$ close to $0$; see Theorem \ref{th:main3}.

\section{Proof of Theorem \ref{th:main1}}\label{sec:main1}
Let $Q$ be a real symmetric $n\times n$ matrix for $n\ge 2$ and
$\Qcal$ be the associated quadratic form
$
	\Qcal(\bx)=Q\bx\,\cdotp \bx,\ \bx\in\R^n.
$
(The dot denotes the Euclidean inner product.)
Let $1\le p < n$. Given a $p$-plane $\Lambda\in G_p(\R^n)$, we denote by $\tr_\Lambda \Qcal$
the trace of the restricted quadratic form $\Qcal|_\Lambda$.  The following result
is \cite[Lemma 2.5 and Corollary 2.6]{HarveyLawson2013IUMJ};
we include a simple proof.

\begin{lemma}\label{lem:trace}
Let $\lambda_1\le \lambda_2\le\cdots\le \lambda_n$ be the eigenvalues of $Q$.
For every $\Lambda\in G_p(\R^n)$ we have that
$
	\tr_\Lambda \Qcal \ge \lambda_1+\lambda_2+\cdots+\lambda_p,
$
with equality if and only if $\Lambda$ admits an orthonormal basis $\bv_1,\ldots,\bv_p$
such that $Q\bv_i=\lambda_i\bv_i$ for $i=1,\ldots,p$.
\end{lemma}

\begin{proof}
This is obvious for $p=1$, and we proceed by induction on $p$.
We may assume that in the standard basis $\be_1,\ldots,\be_n$ of $\R^n$ the matrix
$Q$ is diagonal with $Q\be_i=\lambda_i \be_i$ for $i=1,\ldots,n$.
The subspace $\Lambda'=\Lambda\cap (\{0\}\times\R^{n-1})$ has dimension at least $p-1$.
Pick an orthonormal basis $\bv_1,\bv_2,\ldots,\bv_n$ of $\Lambda$ with
$\bv_2,\ldots,\bv_n \in \Lambda'$. Then,
$
	\tr_\Lambda \Qcal = Q\bv_1\,\cdotp \bv_1 + \sum_{j=2}^p Q\bv_j\,\cdotp \bv_j.
$
Clearly, $Q\bv_1\,\cdotp \bv_1 \ge \lambda_1$
with equality if and only if $Q\bv_1=\lambda_1\bv_1$. Since $\Lambda'':=\mathrm{span}\{\bv_2,\ldots,\bv_p\}$
is contained in the invariant subspace $\{0\}\times\R^{n-1}$ of $Q$ with eigenvalues
$\lambda_2\le \cdots \le \lambda_{n}$, the inductive hypothesis
gives $\sum_{j=2}^p Q\bv_j\,\cdotp \bv_j\ge  \lambda_2+\cdots +\lambda_p$,
with equality if and only if $\Lambda''$ is spanned by eigenvectors for eigenvalues
$\lambda_2,\ldots,\lambda_p$. This completes the induction step.
\end{proof}

\begin{proof}[Proof of Theorem \ref{th:main1}]
Let $D\subset \R^n$ be a bounded domain with $\Cscr^{r,\alpha}$ boundary
for some integer $r\ge 2$ and real number $0<\alpha\le 1$.
Denote by $\delta$ the signed distance function \eqref{eq:delta} from $bD$.
By Li and Nirenberg \cite{LiNirenberg2005}, $\delta$ is of the same class $\Cscr^{r,\alpha}$
in a collar neighbourhood $V\subset \R^n$ of $bD$.
We recall some further properties of $\delta$, referring to
Bellettini \cite[Theorem 1.18, p.\ 14]{Bellettini2013}
and Gilbarg and Trudinger \cite[Section 14.6]{GilbargTrudinger1983}.
Shrinking $V$ around $bD$ if necessary, there is a $\Cscr^r$ projection
$\pi:V\to bD$ such that for any $\bx\in V$, $\bp=\pi(\bx)\in bD$ is the unique nearest
point to $\bx$ on $bD$. The gradient $\nabla \delta$ has constant norm $|\nabla\delta|=1$ on $V$,
and it has constant value on the intersection of $V$ with the normal line
$N_\bp=\bp+\R\,\cdotp \nabla\delta(\bp)$ at $\bp\in bD$.
By a translation and an orthogonal rotation we may assume that $\bp=\zero,\ T_\zero bD=\{x_n=0\}$,
and the standard basis vectors $\be_1,\be_2,\ldots,\be_n=\nabla\delta(\zero)$ of $\R^n$
diagonalize the matrix $H$ of $\Hess_\delta(\zero)$:
\[
	H\be_i=\nu_i \be_i,\quad\ i=1,\ldots,n-1;\quad H\be_n=0.
\]
The restriction of $\Hess_\delta(\zero)$ to the tangent space $T_\zero bD=\R^{n-1}\times \{0\}$
is the second fundamental form of $bD$ at $\zero$ and $\nu_1,\ldots,\nu_{n-1}$
are the principal curvatures of $bD$ at $\zero$ from the side $-\be_n=-\nabla\delta(\zero)$.
By reordering we may assume that $\nu_1\le \nu_2\le \cdots\le \nu_{n-1}$.
For any point $\bx=(0,\ldots,0,t)\in N_\zero \cap V$ the basis $\be_1,\ldots,\be_n$
also diagonalizes $\Hess_\delta(\bx)$, with the eigenvalues
\begin{equation}\label{eq:nux}
	\nu_i(\bx)=\frac{\nu_i}{1 + t \nu_i},\quad\ i=1,\ldots,n.
\end{equation}
In particular, $\nu_n(\bx)=0$. Note that for $t<0$ (which corresponds to
$\bx\in N_\zero \cap V \cap D$) we have $\nu_i(\bx)\ge \nu_i$, while
for $t>0$ (i.e., $\bx\in N_\zero \cap V \setminus \overline D$) we have
$\nu_i(\bx)\le \nu_i$. In both cases equality holds if and only if $\nu_i=0$.
Furthermore, $\nu_1(\bx) \le \cdots \le \nu_{n-1}(\bx)$ for all $\bx \in N_\zero\cap V$.

Assume now that $bD$ is $p$-convex at $\zero$ in the sense that condition \eqref{eq:p-convexity}
holds, i.e.,
\[
	\nu_1+\nu_2+\cdots+\nu_p\ge 0.
\]
By Lemma \ref{lem:trace} applied with $\Qcal=\Hess_\delta(\zero)$
restricted to $T_\zero bD$, this is equivalent to
\[
	\tr_\Lambda \Hess_\delta(\zero) \ge 0\ \ \text{for every $p$-plane $\Lambda\subset T_\zero bD$}.
\]
Consider the family of domains $D_t=\{\delta<t\}$ for $t$ near $0$.
As $t$ increases, the domains $D_t$ strictly increase, and $D_0=D$. The tangent space
to $bD_t=\{\delta=t\}$ at $\bx=(0,\ldots,0,t)\in bD_t$ is $\R^{n-1}\times \{0\}$,
$\be_n=\nabla\delta(\bx)$ is the unit outer normal vector at $\bx$, and the numbers
$\nu_i(\bx)$ \eqref{eq:nux} for $i=1,\ldots,n-1$ are the principal normal curvatures of $bD_t$ at $\bx$.
From what has been said, we see that at $\bx=(0,\ldots,0,t)\in bD_t$ with $t<0$ we have that
\[
	\nu_1(\bx)+\nu_2(\bx)+\cdots+\nu_p(\bx)\ge \nu_1+\nu_2+\cdots+\nu_p\ge 0
\]
with equality if and only if $\nu_1=\nu_2=\cdots=\nu_p=0$, i.e., the point $\zero\in bD$
is $p$-flat. Since this analysis holds at any point $\bp\in bD$, we conclude that for $t<0$
close to $0$ the boundary $bD_t$ is $p$-convex, and it is strongly $p$-convex if and only if
$bD$ does not contain any $p$-flat points.

It remains to find a defining function for $D$, having the domains $D_t$ as sublevel
sets, which is $p$-plurisubharmonic on $\overline D$, and is strongly $p$-plurisubharmonic
on $D$ provided that $bD$ has no $p$-flat points.
Consider the function $h\circ \delta:V\to\R$, where $h$ is a smooth convex increasing
function on $\R$ with $h(0)=0$, $\dot h(0)=1$, and $\ddot h(0)=a>0$ (to be determined). Then,
\[
	\Hess_{h\circ \delta} =
	(\dot h\circ \delta) \Hess_\delta + (\ddot h\circ \delta) \nabla \delta\, \cdotp (\nabla \delta)^T.
\]
Assume that for $\bp\in bD$ the orthonormal vectors $\bv_1,\ldots,\bv_{n-1},\bv_n$
diagonalize $\Hess_\delta(\bp)$, where $T_\bp D=\mathrm{span}\{\bv_1,\ldots,\bv_{n-1}\}$ and
$\bv_n=\nabla\delta(\bp)$. Then the vectors $\bv_1,\ldots,\bv_{n-1}$ lie in the kernel of the matrix
$\nabla \delta(\bp)\, \cdotp \nabla \delta(\bp)^T$, while $\bv_n$ is an eigenvector with eigenvalue $1$.
Hence, the basis $\bv_1,\ldots,\bv_n$ diagonalizes
$\Hess_{h\circ \delta}(\bx)$ at every point $\bx=\bp+ \delta(\bx)\bv_n\in N_\bp\cap V$,
the eigenvalues corresponding to $\bv_1,\ldots,\bv_{n-1}$
get multiplied by the number $\dot h(d(\bx))$, which is close to $1$ if $\bx$ is close to $\bp$,
and the eigenvalue in the normal direction $\bv_n=\nabla \delta(\bx)$ is $\ddot h(\delta(\bx))\approx a$.
By choosing $a$ such that $a+\lambda_1(\bp)>0$ for all $\bp\in bD$
and shrinking the neighbourhood $V$ around $bD$,
the function $h\circ \delta$ is $p$-plurisubharmonic on $\overline D\cap V$,
and it is strongly $p$-plurisubharmonic on $D\cap V$ if $bD$ has no $p$-flat points.
Choose $c<0$ such that $\{c\le \delta\le 0\}\subset \overline D\cap V$.
Let $\phi:\R\to\R$ be a convex increasing function such that $\phi(t)=2c/3$ for $t\le c$ and
$\phi(t)=t$ for $t\ge c/2$. Then, $\rho=\phi\circ h\circ \delta$ is a well-defined function
on $\overline D\cup V$ which equals $h\circ \delta$ outside $D_{c/2}$ and is
$p$-plurisubharmonic on $\overline D$.

Assume now that $h\circ \delta$ is strongly $p$-plurisubharmonic on a collar
$\{c\le \delta<0\}$. Then, $\rho=\phi\circ h\circ \delta$ is strongly $p$-plurisubharmonic
on $\{c/2 \le \delta<0\}\subset D\cap V$. We take
\[
	\tilde \rho(\bx) = \rho(\bx) + \epsilon \chi(\bx)\cdotp |\bx|^2,
\]
where $\chi:\R^n\to [0,1]$ is a smooth function which equals $1$ on $\overline D_{c/3}$
and has compact support contained in $D$. The support of the differential of $\chi$ is then compact
and contained in the set $\{c/2< \delta<0\}$ where $\rho$ is strongly $p$-plurisubharmonic.
Note that $|\bx|^2=\sum_{i=1}^n x_i^2$ is strongly $p$-plurisubharmonic on $\R^n$.
Choosing $\epsilon>0$ small enough therefore ensures that $\tilde \rho$ is strongly
plurisubharmonic on $D$. This proves Theorem \ref{th:main1}.
\end{proof}

Let us look more closely at the case $p=2$ of Theorem \ref{th:main1}, which is of interest for the
theory of $2$-dimensional minimal surfaces. In particular, every $2$-convex domain in $\R^n$, $n\ge 3$,
admits plenty of proper minimal surfaces parameterized by conformal harmonic immersions from
any given bordered Riemann surface (see \cite[Theorems 8.3.1 and 8.3.11]{AlarconForstnericLopez2021}).
In this case, condition \eqref{eq:p-convexity2} (which is equivalent to condition \eqref{eq:p-convexity}
in Theorem \ref{th:main1}) can be expressed in more intrinsic differential geometric terms, thereby
exposing a connection to Levi pseudoconvexity studied in complex analysis; see the following section
for the latter.

Let $D$ be a smoothly bounded domain in $\R^n$, $n\ge 3$.
Given a point $\bp\in bD$ and a $2$-plane $\Lambda\subset T_\bp bD$,
let $S(\bp,\Lambda)$ be the surface obtained by intersecting $bD$ with the affine 3-plane
$\bp\in \Sigma_\bp\cong \R^3$ spanned by $\Lambda$ and the normal vector to $bD$ at $\bp$.
Let $\Qcal_\bp$ denote the second fundamental form of $bD$ at $\bp$.
The principal curvatures $\lambda_1(\bp,\Lambda),\lambda_2(\bp,\Lambda)$ of
$S(\bp,\Lambda)$ at $\bp$ are the eigenvalues of the restricted quadratic form
$\Qcal_\bp|_{\Lambda}$. Their sum
\begin{equation}\label{eq:Hp}
	H(\bp,\Lambda) =  \lambda_1(\bp,\Lambda)+\lambda_2(\bp,\Lambda)
\end{equation}
is the mean curvature of $S(\bp,\Lambda)$ at $\bp$, and their product
\begin{equation}\label{eq:Kp}
	K(\bp,\Lambda) = \lambda_1(\bp,\Lambda)\,\cdotp \lambda_2(\bp,\Lambda)
\end{equation}
is the Gaussian curvature of $S(\bp,\Lambda)$ at $\bp$. These numbers are,
respectively, the sectional mean curvature and the sectional Gaussian curvature of
$bD$ at $\bp$ on the 2-plane $\Lambda$. While $H$ reflects the way how $bD$ sits in
$\R^n$, $K$ is an intrinsic quantity depending only on the induced metric on $bD$.
If $\rho$ is a defining function for $D$ such that $|\nabla\rho(\bp)|=1$,
then $\Qcal_\bp$ equals the Hessian $\Hess_\rho(\bp)$ restricted to $T_\bp bD$,
and hence
\begin{equation}\label{eq:HDelta}
	H(\bp,\Lambda) = \tr_\Lambda \Hess_\rho(\bp) = \Delta(\rho|_{\bp+\Lambda})(\bp).
\end{equation}
If $H(\bp,\Lambda)=0$ then $K(\bp,\Lambda) = -\lambda_1(\bp,\Lambda)^2\le 0$,
and $K(\bp,\Lambda)=0$ if and only if $\Qcal_\bp|_\Lambda$ vanishes.
Theorem \ref{th:main1} for $p=2$ can now be stated as follows.

\begin{theorem}\label{th:main2}
Let $D$ be a bounded domain in $\R^n,\ n\ge 3$, with $\Cscr^{r,\alpha}$ boundary
for some $r\ge 2$ and $0<\alpha\le 1$. If
\begin{equation}\label{eq:MCboundary}
	H(\bp,\Lambda)\ge 0\ \
	\text{for every point $\bp\in bD$ and 2-plane $\Lambda\subset T_\bp bD$},
\end{equation}
then $D$ admits a $\Cscr^{r,\alpha}$ defining function $\rho$ satisfying \eqref{eq:p-psh}
with $p=2$ (i.e., $\rho$ is minimal plurisubharmonic on $\overline D$).
If in addition there is no $(\bp,\Lambda)$ as above such that
$H(\bp,\Lambda)= K(\bp,\Lambda)=0$, then $\rho$ can be chosen strongly
minimal plurisubharmonic on $D$.
\end{theorem}

\section{A remark on Levi pseudoconvex domains}\label{sec:Levi}
For a smoothly bounded domain $D$ in a complex Euclidean space $\C^n, n\ge 2$, the $2$-convexity
condition \eqref{eq:p-convexity2} applied only to complex lines $\Lambda\subset T_\bp bD$
is precisely the Levi pseudoconvexity condition.
Indeed, if $\bv\in \Lambda=\C \bv$ is a unit vector then
\begin{equation}\label{eq:Levi}
	(dd^c \rho)_\bp (\bv,J\bv)=\Delta(\rho|_{\bp+\Lambda})(\bp) =
	\tr_\Lambda \Hess_\rho(\bp) = H(\bp,\Lambda),
\end{equation}
where $J$ is the standard almost complex structure operator on $\C^n$.

It is well known that if $D$ is a bounded pseudoconvex domain in $\C^n$ then the function
$-\log \mathrm{dist}(\cdotp,bD)$ is plurisubharmonic on $D$, and hence the domains
$D_t=\{z\in D:\mathrm{dist}(z,bD)>t\}=\{\delta<-t\}$ for $t>0$ are pseudoconvex. This is a stronger result
than we can get for minimally convex domains, and it does not require any smoothness of $bD$.
However, the following observation related to Theorem \ref{th:main2} seems worthwhile recording.

\begin{theorem}\label{th:main3}
Let $D$ be a bounded pseudoconvex domain in $\C^n,\ n\ge 2,$
with $\Cscr^{r,\alpha}$ boundary for some $r\ge 2$ and $0<\alpha\le 1$, and
let $\delta$ be given by \eqref{eq:delta}.
If the sectional curvature $K(\bp,\Lambda)$ \eqref{eq:Kp} is nonzero
for every point $\bp\in bD$ and complex line $\Lambda\subset T_\bp bD$ on which the
Levi form \eqref{eq:Levi} vanishes, then the domains $D_t=\{\delta<t\}$ for $t<0$
close to $0$ are strongly pseudoconvex.
\end{theorem}

\begin{proof}
Let $\delta$ denote the signed distance function \eqref{eq:delta} to $bD$.
Fix a point $\bp\in bD$ and let $\bv=\nabla \delta(\bp)$.
Let $J$ denote the standard almost complex structure on
$\C^n$, which corresponds to multiplication by $\imath$ on complex tangent vectors.
The tangent space to $bD$ at $\bp$ is an orthogonal direct
sum $T_\bp bD= T_\bp^\C bD\oplus E$, where $\Sigma:=T_\bp^\C bD=T_\bp bD\cap J(T_\bp bD)$
is the maximal complex subspace of $T_\bp bD$ (a complex hyperplane)
and $E$ is the real line orthogonal to $T_\bp^\C bD$. Note that $J(E)=\R \bv$
is the normal line to $bD$ at $\bp$. The Levi pseudoconvexity condition is that
$H(\bp,\Lambda) = \tr_\Lambda \Hess_\delta(\bp) \ge 0$ for every complex line
$\Lambda\subset T_\bp^\C bD$ (see \eqref{eq:Levi}).

Let $D_t=\{\delta<t\}$ for $t<0$ close to $0$ and $\bx=\bp+t\bv \in bD_t$.
Then, $T_\bx bD_t$ is the orthogonal complement to $\bv$, and hence $T_\bx bD_t=\Sigma\oplus E$, i.e.,
the same orthogonal decomposition of $T_\bx bD_t$ holds along the line $N_\bp=\bp+\R\bv$.
To prove the theorem, it suffices to show that $\tr_\Lambda \Hess_\delta(\bx) \ge H(\bp,\Lambda)$,
with equality if and only if $H(\bp,\Lambda)=K(\bp,\Lambda)=0$.
This will imply that $bD_t$ for $t<0$ close to $0$ is Levi pseudoconvex, and it is strongly Levi pseudoconvex if and only if there are no complex lines $\Lambda\subset T_\bp bD$, $\bp\in bD$,
with $H(\bp,\Lambda)=K(\bp,\Lambda)=0$.

Let $\bv_1,\dots,\bv_{n-1}\in T_\bp bD$ be orthonormal eigenvectors of $\Hess_\delta(\bp)$
with the eigenvalues $\nu_1\le \nu_2\le \cdots\le \nu_{n-1}$.
Fix a complex line $\Lambda\subset T_\bp bD$. For any orthonormal basis
$\xi=\sum_{i=1}^{n-1} \xi_i \bv_i$ and $\eta=\sum_{i=1}^{n-1} \eta_i \bv_i$ for $\Lambda$
we have that
\begin{equation}\label{eq:decrease}
	\tr_\Lambda \Hess_\delta(\bx) = \sum_{i=1}^{n-1} \nu_i(\bx)(\xi_i^2+\eta_i^2)
	\ge \sum_{i=1}^{n-1} \nu_i(\xi_i^2+\eta_i^2)
	= \tr_\Lambda \Hess_\delta(\bp) = H(\bp,\Lambda),
\end{equation}
where we used $\nu_i(\bx)\ge \nu_i$ (see \eqref{eq:nux}).
Equality holds if and only if $\xi_i^2+\eta_i^2=0$ whenever
$\nu_i\ne 0$ (since for such $i$ we have $\nu_i(\bx)>\nu_i$).
Clearly this holds if and only if the vectors $\xi$ and $\eta$
are contained in the kernel of $\Hess_\delta(\bp)$. Since this argument holds for every
orthonormal basis of $\Lambda$, we infer that equality holds in \eqref{eq:decrease}
if and only if $H(\bp,\Lambda)=K(\bp,\Lambda)=0$.
\end{proof}

\begin{remark}\label{rem:pshdef}
(A) Regarding plurisubharmonic defining functions,
there is a major difference between the minimal case (cf.\ Theorem \ref{th:main1})
and the complex (Levi) case. It has been known since the 1970s
through examples of Diederich and Forn{\ae}ss \cite{DiederichFornaess1977MA}
and Forn{\ae}ss \cite[Example 5]{Fornaess1979} that there exist bounded
weakly pseudoconvex domains $D\subset\C^2$ with smooth real analytic boundaries
which do not admit a defining function that is plurisubharmonic on $\overline D$,
in the sense that its Levi form is nonnegative at every point of $\overline D$ in all
complex directions. Furthermore, on a pseudoconvex domain admitting a plurisubharmonic defining
function the $\dibar$--Neumann problem is exactly regular; see Boas and Straube
\cite{BoasStraube1991}. This is not true on all weakly pseudoconvex domains;
it fails in particular on certain worm domains according to
Barrett \cite{Barrett1992} and Christ \cite{Christ1996}.
Hence, Theorem \ref{th:main2} has no analogue in the complex case.
The reason is that Levi pseudoconvexity provides much less information
since it only pertains to complex lines, as opposed to all $2$-planes tangent to the boundary.
In particular, it does not give any condition in the direction
normal to the maximal complex subspace of $T_\bp bD$.

(B) On the other hand, it was shown by Diederich and Forn\ae ss in \cite{DiederichFornaess1978MM}
that locally near any boundary point of a smoothly bounded weakly pseudoconvex domain
in $\C^n$ one can make a local holomorphic change of coordinates so that the level
sets of the signed distance function are strongly pseudoconvex in the interior
of the new local domain. The point is that, while a biholomorphic map
preserves the sign of the Levi form on complex directions (and hence of the function $H(\bp,\Lambda)$
in \eqref{eq:Levi}), the sectional Gaussian curvature $K(\bp,\Lambda)$ \eqref{eq:Kp} may very well change
from zero to a nonzero value. If we ensure that this happens on all complex tangent directions
on which the Levi form vanishes, then Theorem \ref{th:main3} applies.
\end{remark}

\begin{comment}
Theorem 3.1 is unknown to me.
The closest I know is that locally near a boundary point one can make a
holomorphic change of coordinates so that the level sets of delta
are strongly pseudoconvex.
For example, if you start with the flat domain in $\C^2_{(z,w)}$
given by $\Re w<0$, then we can change it into $\Re w+\Re z^2<0$
and then the level sets of delta in the new coordinates are strongly pseudoconvex
\end{comment}

\smallskip
\noindent {\bf Acknowledgement.}
I wish to thank John Erik Forn\ae ss for the information discussed in Remark \ref{rem:pshdef} (B)
and for the relevant reference \cite{DiederichFornaess1978MM}. I also thank the referee for
remarks which helped me to improve the presentation.

redpajama_set_name: RedPajamaArXiv