\label{subsection:boundary-of-Reinhardt-domain}

Assume that $n \geq 2$.
Let $M$ be the boundary of the bounded Reinhardt domain
\begin{equation}
	\Omega
	\coloneqq \Set{ w = (w^{1}, \dots , w^{n + 1}

\begin{theorem}
\label{thm:infinite-geometric-moduli-complex-hyperbolic}
	The complement $S^{2 n + 1} \setminus S_{\bbC}^{2 k + 1}$ satisfies
	$\# \GModuli(S^{2 n + 1} \setminus S_{\bbC}^{2 k + 1}) = \infty$ if $0 \leq k < (n - 2) / 2$.
\end{theorem}

We add some remarks on these theorems.
Let $V$ be a real linear subspace in $\bbC^{n + 1}$
and set $\Lambda \coloneqq V \cap S^{2 n + 1} \subset S^{2 n + 1}$.
If $V$ is totally real or complex and $\dim_{H} \Lambda < n$,
then we proved that $\# \GModuli(S^{2 n + 1} \setminus \Lambda) = \infty$.
The key point of the proof is that
there exists a torsion-free convex cocompact discrete subgroup $\Gamma$ of $PU(n + 1, 1)$
whose limit set coincides with $\Lambda$.
Then the quotient of $S^{2 n + 1} \setminus \Lambda$ by $\Gamma$ satisfies \Cref{keyassumption}.
On the other hand,
when $V$ is neither totally real nor complex,
there exist no discrete subgroups in $PU(n + 1, 1)$
such that its limit set coincides with $\Lambda$,
and hence our argument does not apply.
As far as we know,
there are no existence results for complete contact forms
with constant Tanaka--Webster scalar curvature on $S^{2 n + 1} \setminus \Lambda$

We will also apply \cref{thm:infinite-geometric-moduli-general}
to circle bundles over compact \Kahler manifolds (\cref{thm:infinite-geometric-moduli-circle-bundle})
and the boundary of a Reinhardt domain (\cref{subsection:boundary-of-Reinhardt-domain}),
which is neither spherical nor Sasakian.

This paper is organized as follows.
In \cref{section:CR-geometry},
we review basic materials in CR geometry.
In \cref{section:CR-Yamabe-problem},
we recall the CR Yamabe problem and summarize known existence results that motivate the present work.
In \cref{section:complex-hyperbolic-geometry},
we collect several facts from complex hyperbolic geometry
that are needed to analyze complements of $\bbR$- and $\bbC$-spheres in the CR sphere.
In \cref{section:infinite-tower},
we study an existence problem for infinite towers of finite connected coverings;
this constitutes the technical core of the paper.
In \cref{section:proof-of-main-result},
we use this framework to prove \cref{thm:infinite-geometric-moduli-general}.
In \cref{section:complements-of-spheres},
we apply this theorem to complements of $\bbR$- and $\bbC$-spheres in the CR sphere.
Finally, in \cref{section:other-applications},
we discuss further applications to circle bundles over compact \Kahler manifolds
and to the boundary of a Reinhardt domain.

\section{CR geometry}
\label{section:CR-geometry}

\subsection{CR manifolds}
\label{subsection:CR-manifolds}

Let $M$ be a smooth $(2 n + 1)$-dimensional manifold.
A \emph{CR structure} is a rank $n$ complex subbundle $T^{1, 0} M$
of the complexified tangent bundle $T M \otimes \bbC$ such that
\begin{equation}
	T^{1, 0} M \cap T^{0, 1} M = 0, \qquad
	\comm{\Gamma(T^{1, 0} M)}{\Gamma(T^{1, 0} M)} \subset \Gamma(T^{1, 0} M),
\end{equation}
where $T^{0, 1} M$ is the complex conjugate of $T^{1, 0} M$ in $T M \otimes \bbC$.
Set $H M = \Re T^{1, 0} M$
and let $J \colon H M \to H M$ be the unique complex structure on $H M$
such that
\begin{equation}
	T^{1, 0} M = \Ker(J - \sqrt{- 1} \colon H M \otimes \bbC \to H M \otimes \bbC).
\end{equation}
A typical example of a CR manifold is a real hypersurface $M$ in an $(n + 1)$-dimensional complex manifold $X$;
this $M$ has the induced CR structure
\begin{equation}
	T^{1, 0} M
	\coloneqq T^{1, 0} X |_{M} \cap (T M \otimes \bbC).
\end{equation}
In particular,
the unit sphere
\begin{equation}
	S^{2 n + 1}
	\coloneqq \Set{z \in \bbC^{n + 1} | \abs{z}^{2} = 1}
\end{equation}
has the \emph{canonical} CR structure $T^{1, 0} S^{2 n + 1}$.

Let $(M, T^{1, 0} M)$ and $(M^{\prime}, T^{1, 0} M^{\prime})$ be CR manifolds.
A smooth map $f \colon M \to M^{\prime}$ is called a \emph{CR map}
if $f_{\ast} (T^{1, 0} M) \subset T^{1, 0} M^{\prime}$.
If $f$ is a diffeomorphism,
then we call $f$ a \emph{CR diffeomorphism}.
A CR manifold $(M, T^{1, 0} M)$ is said to be \emph{spherical}
if it is locally isomorphic to $(S^{2 n + 1}, T^{1, 0} S^{2 n + 1})$.

A CR structure $T^{1, 0} M$ is said to be \emph{strictly pseudoconvex}
if there exists a smooth real-valued one-form $\theta$ annihilating exactly $H M$
such that the \emph{Levi form}
\begin{equation}
	L_{\theta}(V, W)
	\coloneqq d \theta(V, J W)
	\qquad (V, W \in H M)
\end{equation}
is positive definite.
We call such a one-form a \emph{contact form}.
The triple $(M, T^{1, 0} M, \theta)$ is called a \emph{pseudo-Hermitian manifold}.
Denote by $T$ the \emph{Reeb vector field} with respect to $\theta$;
that is,
the unique vector field satisfying
\begin{equation}
	\theta(T) = 1, \qquad T \contr d \theta = 0.
\end{equation}
Let $(Z_{\alpha})$ be a local frame of $T^{1, 0} M$,
and set $Z_{\ovxa} = \overline{Z_{\alpha}}$.
Then
$(T, Z_{\alpha}, Z_{\ovxa})$ gives a local frame of $T M \otimes \bbC$,
called an \emph{admissible frame}.
Its dual frame $(\theta, \theta^{\alpha}, \theta^{\ovxa})$
is called an \emph{admissible coframe}.
The two-form $d \theta$ is written as
\begin{equation}
	d \theta
	= \sqrt{- 1} h_{\alpha \ovxb} \theta^{\alpha} \wedge \theta^{\ovxb},
\end{equation}
where $(h_{\alpha \ovxb})$ is a positive definite Hermitian matrix.
We use $h_{\alpha \ovxb}$ and its inverse $h^{\alpha \ovxb}$
to raise and lower indices of tensors.

The flat model of pseudo-Hermitian manifolds is the \emph{Heisenberg group} $\bbH^{2 n + 1}$,
that is,
the Lie group with the underlying manifold $\bbR \times \bbC^{n}$ and the multiplication
\begin{equation}
	(t, z) \cdot (t^{\prime}, z^{\prime})
	\coloneqq (t + t^{\prime} + 2 \Im (z \cdot \ovz^{\prime}), z + z^{\prime}).
\end{equation}
For $\alpha = 1, \dots , n$,
we introduce a left-invariant complex vector field $Z_{\alpha}$ by
\begin{equation}
	Z_{\alpha}
	\coloneqq \pdv{}{z^{\alpha}} + \sqrt{- 1} \ovz^{\alpha} \pdv{}{t}.
\end{equation}
The canonical CR structure $T^{1, 0} \bbH^{2 n + 1}$
is spanned by $Z_{1}, \dots , Z_{n}$.
Define a left-invariant one-form $\theta$ on $\bbH^{2 n + 1}$ by
\begin{equation}
	\theta
	\coloneqq d t + \sqrt{-1} \sum_{\alpha = 1}^{n}
		(z^{\alpha} d \ovz^{\alpha} - \ovz^{\alpha} d z^{\alpha}).
\end{equation}
Then $\theta$ annihilates $T^{1, 0} \bbH^{2 n + 1}$
and the Levi form $L_{\theta}$ satisfies
$L_{\theta}(Z_{\alpha}, Z_{\ovxb}) = 2 \delta_{\alpha \beta}$;
in particular,
$(\bbH^{2 n + 1}, T^{1, 0} \bbH^{2 n + 1})$ is a strictly pseudoconvex CR manifold
and $\theta$ is a contact form on $\bbH^{2 n + 1}$.

A CR manifold $(M, T^{1, 0} M)$ is said to be \emph{embeddable}
if there exists a smooth embedding $F \colon M \to \bbC^{N}$
such that $F_{\ast}(T^{1, 0} M) \subset T^{1, 0} \bbC^{N}$.
It is known that a compact strictly pseudoconvex CR manifold $(M, T^{1, 0} M)$
of dimension at least five must be embeddable~\cite{Boutet_de_Monvel1975}.
On the other hand,
there are many non-embeddable compact strictly pseudoconvex CR manifolds of dimension three;
see for example \cite{Burns-Epstein1990-Embed}.
We say that $(M, T^{1, 0} M)$ is \emph{universally embeddable}
if every finite cover of $(M, T^{1, 0} M)$ is embeddable.
This condition is strictly stronger than the embeddability condition~\cite{Case-Yang2025}.

Assume that $M$ is connected and fix a contact form $\theta$ on $M$.
We can also endow $M$ with the \emph{Carnot--Carath\'{e}odory distance} $d_{\CC}$ on $M$ as follows.
For any $p, q \in M$,
we can find a smooth path $c \colon \clcl{0}{1} \to M$
such that $c(0) = p$, $c(1) = q$, and $\theta(c^{\prime}(t)) = 0$.
Then $d_{\CC}(p, q)$ is the infimum of the length of such curves.
Denote by $\dim_{H} A$ the Hausdorff dimension of $A \subset M$ with respect to $d_{\CC}$.
The Carnot--Carath\'{e}odory distance $d_{\CC}$ induces the standard topology of $M$,
but the Hausdorff dimension $\dim_{H} M$ of $M$ is equal to $2 n + 2$,
which does not coincide with its topological dimension.
See \cite{Gromov1996} for more details.

\subsection{Tanaka--Webster connection}
\label{subsection:TW-connection}

A contact form $\theta$ induces a canonical connection $\nabla$ on $T M$,
called the \emph{Tanaka--Webster connection} with respect to $\theta$.
It is defined by
\begin{equation}
	\nabla T
	= 0,
	\quad
	\nabla Z_{\alpha}
	= \omega_{\alpha} {}^{\beta} Z_{\beta},
	\quad
	\nabla Z_{\ovxa}
	= \omega_{\ovxa} {}^{\ovxb} Z_{\ovxb},
\end{equation}
where $\omega_{\ovxa} {}^{\ovxb} = \overline{\omega_{\alpha} {}^{\beta}}$,
with the following structure equations:
\begin{gather}
\label{eq:str-eq-of-TW-conn1}
	d \theta^{\beta}
	= \theta^{\alpha} \wedge \omega_{\alpha} {}^{\beta}
	+ A^{\beta} {}_{\ovxa} \theta \wedge \theta^{\ovxa}, \\
\label{eq:str-eq-of-TW-conn2}
	d h_{\alpha \ovxb}
	= \omega_{\alpha} {}^{\gamma} h_{\gamma \ovxb}
		+ h_{\alpha \ovxg} \omega_{\ovxb} {}^{\ovxg}.
\end{gather}
The tensor $A_{\alpha \beta} = \overline{A_{\ovxa \ovxb}}$
is symmetric and is called the \emph{Tanaka--Webster torsion}.
The \emph{sub-Laplacian} $\Delta_{b}$ is defined by
\begin{equation}
	\Delta_{b} u
	\coloneqq - \nabla^{\alpha} \nabla_{\alpha} u - \nabla^{\ovxb} \nabla_{\ovxb} u.
\end{equation}

The curvature form
$\Omega_{\alpha} {}^{\beta} \coloneqq d \omega_{\alpha} {}^{\beta}
- \omega_{\alpha} {}^{\gamma} \wedge \omega_{\gamma} {}^{\beta}$
of the Tanaka--Webster connection satisfies
\begin{equation}
\label{eq:curvature-form-of-TW-connection}
	\Omega_{\alpha} {}^{\beta}
	= R_{\alpha} {}^{\beta} {}_{\rho \ovxs} \theta^{\rho} \wedge \theta^{\ovxs}
		\qquad \text{modulo $\theta, \theta^{\rho} \wedge \theta^{\gamma},
		\theta^{\ovxg} \wedge \theta^{\ovxs}$}.
\end{equation}
We call the tensor $R_{\alpha} {}^{\beta} {}_{\rho \ovxs}$
the \emph{Tanaka--Webster curvature}.
This tensor has the symmetry
\begin{equation}
	R_{\alpha \ovxb \rho \ovxs}
	= R_{\rho \ovxb \alpha \ovxs}
	= R_{\alpha \ovxs \rho \ovxb}.
\end{equation}
Contraction of indices gives the \emph{Tanaka--Webster Ricci curvature}
$\Ric_{\rho \ovxs} = R_{\alpha} {}^{\alpha} {}_{\rho \ovxs}$
and the \emph{Tanaka--Webster scalar curvature}
$R_{\theta} = \Ric_{\rho} {}^{\rho}$.
When $n \geq 2$,
the \emph{Chern tensor} $S_{\alpha \ovxb \rho \ovxs}$ is the completely trace-free part
of $R_{\alpha \ovxb \rho \ovxs}$;
this tensor is the CR analogue of the Weyl tensor in conformal geometry.
It is known that the Chern tensor vanishes identically
if and only if $(M, T^{1, 0} M)$ is spherical~\cite{Chern-Moser1974}.

Fix a contact form $\theta$ on $M$.
Then any contact form $\whxth$ is of the form $u^{2 / n} \theta$,
where
\begin{equation}
	u \in C^{\infty}_{+}(M) \coloneqq \Set{u \in C^{\infty}(M) | u > 0}.
\end{equation}
Under this conformal change,
the Tanaka--Webster scalar curvature $R_{\wtxth}$ with respect to $\whxth$ satisfies
\begin{equation}
	R_{\wtxth}
	= u^{- 1 - 2 / n} \sbra*{(2 + 2 / n) \Delta_{b} u + R_{\theta} u};
\end{equation}
see \cite{Jerison-Lee1987}*{p.\ 174} for example.

\subsection{Sasakian manifolds}
\label{subsection:Sasakian-manifolds}

Sasakian manifolds constitute an important class of pseudo-Hermitian manifolds.
See~\cite{Boyer-Galicki2008} for a comprehensive introduction.

A \emph{Sasakian manifold} is a pseudo-Hermitian manifold $(S, T^{1, 0}S, \theta)$
with vanishing Tanaka--Webster torsion.
This condition is equivalent to the requirement that the Reeb vector field $T$ of $\theta$
preserves the CR structure $T^{1, 0} S$.

A typical example of a Sasakian manifold
is the circle bundle associated with a negative Hermitian holomorphic line bundle.
Let $Y$ be an $n$-dimensional complex manifold
and $(L, h)$ a Hermitian holomorphic line bundle over $Y$
such that
\begin{equation}
	\omega = - \sqrt{- 1} \Theta_{h} = d d^{c} \log h
\end{equation}
is a \Kahler form on $Y$,
where $d^{c} \coloneqq (\sqrt{- 1} / 2)(\delb - \partial)$.
Consider the circle bundle
\begin{equation}
	S = \Set{ v \in L | h(v, v) = 1}
\end{equation}
over $Y$,
which is a real hypersurface in the total space of $L$
and has the canonical CR structure $T^{1, 0} S$.
The one-form $\theta = d^{c} \log h |_{S}$ is a contact form on $S$
and its Tanaka--Webster scalar curvature $R_{\theta}$ coincides with $p^{\ast} R_{\omega}$,
where $p \colon S \to Y$ is the projection
and $R_{\omega}$ is the scalar curvature of $\omega$.
Moreover,
the CR manifold $(S, T^{1, 0} S)$ is spherical if and only if
$(Y, \omega)$ is Bochner-flat.
See \cite{Takeuchi2022-Chern}*{Section 2.3} for more details.

\subsection{Schoen's Theorem for CR automorphism group}

The CR sphere and the Heisenberg group play a distinguished role
in CR geometry from the viewpoint of symmetry.
To explain this,
we begin with some definitions.

\begin{definition}
	Let $(M, T^{1, 0} M)$ be a strictly pseudoconvex CR manifold of dimension $2 n + 1$.
	The \emph{CR automorphism group} $\Aut_{\CR}(M, T^{1, 0} M)$
	consists of all CR diffeomorphisms $\varphi \in \Diff(M)$
	equipped with the compact-open topology.
	We say that $\Aut_{\CR}(M, T^{1, 0} M)$ \emph{acts properly} on $M$
	if $\Set{\varphi \in \Aut_{\CR}(M, T^{1, 0} M) | \varphi(K) \cap K \neq \emptyset}$
	is relatively compact in $\Aut_{\CR}(M, T^{1, 0} M)$ for any compact set $K \subset M$.
\end{definition}

Schoen's theorem for CR automorphism group states that
$\Aut_{\CR}(M, T^{1, 0} M)$ acts properly except for
$(S^{2 n + 1}, T^{1, 0} S^{2 n + 1})$ or $(\bbH^{2 n + 1}, T^{1, 0} \bbH^{2 n + 1})$.

\begin{theorem}[\cite{Schoen1995}*{Theorems 3.3' and 3.4'}]
\label{thm:Schoen's-theorem}
	Let $(M, T^{1, 0} M)$ be a strictly pseudoconvex CR manifold of dimension $2 n + 1$.
	If the CR automorphism group $\Aut_{\CR}(M, T^{1, 0} M)$ of $(M, T^{1, 0} M)$ does not act properly on $M$,
	then $(M, T^{1, 0} M)$ is CR diffeomorphic to
	$(S^{2 n + 1}, T^{1, 0} S^{2 n + 1})$ or $(\bbH^{2 n + 1}, T^{1, 0} \bbH^{2 n + 1})$.
\end{theorem}

\begin{theorem}
\label{thm:infinite-geometric-moduli-circle-bundle}
	If the scalar curvature $R_{\omega}$ of $\omega$ is positive
	and $\pi_{1}(Y)$ has infinite profinite completion,
	then $\# \GModuli(\wtS, T^{1, 0} \wtS) = \infty$.
\end{theorem}

\label{sec:Introduction}

The existence of complete metrics with constant scalar curvature
is one of the most fundamental problems in Riemannian and conformal geometry.
In the compact case,
this problem, known as the Yamabe problem, was solved affirmatively;
see \cite{Lee-Parker1987} and the references therein.
A typical non-compact setting arises
when one considers the complement of a closed subset in a compact manifold.
In this situation,
the problem is referred to as the singular Yamabe problem.
If the ambient manifold is the round sphere,
this problem has been studied by several authors~\cites{Schoen1989,Mazzeo-Smale1991,Bettiol-Piccione-Santaro2016,Bettiol-Piccione2018}.
Related to our result,
in joint work with Andrade, Piccione, and Wei~\cite{Andrade-Case-Piccione-Wei2025-preprint},
the first author proved that
$S^{n} \setminus S^{k}$ admits infinitely many non-homothetic periodic conformally round metrics
with positive constant scalar curvature provided that $n \geq 2 k + 3$.

Based on the analogy between conformal and CR geometry,
we consider the existence problem for complete contact forms with constant Tanaka--Webster scalar curvature.
We first recall the status of this problem
on compact, connected, strictly pseudoconvex CR manifolds $(M, T^{1, 0} M)$ of dimension $2 n + 1$.
The \emph{CR Yamabe functional} $\frakF$ is defined by
\begin{equation}
\label{eq:CR-Yamabe-functional}
	\frakF(\theta)
	\coloneqq \Vol_{\theta}(M)^{- n / (n + 1)} \int_{M} R_{\theta} \, \vol_{\theta},
\end{equation}
where $\theta$ is a contact form on $M$,
$R_{\theta}$ is the Tanaka--Webster scalar curvature associated with $\theta$,
$\vol_{\theta} \coloneqq \theta \wedge (d \theta)^{n}$,
and $\Vol_{\theta}(M) \coloneqq \int_{M} \vol_{\theta}$.
The infimum of this functional is called the \emph{CR Yamabe constant} $Y(M, T^{1, 0} M)$.
A critical point of this functional corresponds to a contact form with constant Tanaka--Webster scalar curvature.
Jerison and Lee~\cites{Jerison-Lee1987,Jerison-Lee1988,Jerison-Lee1989} proved that
this functional admits a minimizer if $n \geq 2$ and $(M, T^{1, 0} M)$ is not spherical,
or if $(M, T^{1, 0} M)$ is CR diffeomorphic to the CR sphere.
Gamara and Yacoub~\cites{Gamara-Yacoub2001,Gamara2001} established the existence of a critical point
in the remaining case via the method of critical points at infinity.
The contact form constructed by their method is not necessarily a CR Yamabe minimizer.
In fact,
Cheng, Malchiodi, and Yang~\cite{Cheng-Malchiodi-Yang2023} showed that the Rossi sphere,
which is an example of a non-embeddable CR three-manifold,
admits no CR Yamabe minimizers.
We also remark that $(M, T^{1, 0} M)$ admits a CR Yamabe minimizer
if $n = 1$ and $(M, T^{1, 0} M)$ is embeddable~\cites{Cheng-Malchiodi-Yang2017,Takeuchi2020-Paneitz}.

In this paper,
we consider the non-compact case.
Let $(M, T^{1, 0} M)$ be a (possibly non-compact) connected strictly pseudoconvex CR manifold.
Denote by $\CSC(M, T^{1, 0} M)$ the set of complete contact forms on $M$
with constant Tanaka--Webster scalar curvature.
We say that two contact forms $\theta_{1}$ and $\theta_{2}$ on $M$ are \emph{homothetic}
if there exist a CR diffeomorphism $\varphi$ and a constant $c > 0$
such that $\varphi^{\ast} \theta_{1} = c \theta_{2}$;
otherwise they are said to be \emph{non-homothetic}.
The \emph{geometric moduli space} of complete contact forms
with constant Tanaka--Webster scalar curvature on $(M, T^{1, 0} M)$ is the quotient
\begin{equation}
	\GModuli(M, T^{1, 0} M)
	\coloneqq \CSC(M, T^{1, 0} M) / \sim,
\end{equation}
where $\theta_{1} \sim \theta_{2}$ if $\theta_{1}$ and $\theta_{2}$ are homothetic.
Jerison and Lee proved that $\# \GModuli(M, T^{1, 0} M) = 1$
if $M$ is compact and $Y(M, T^{1, 0} M) \leq 0$~\cite{Jerison-Lee1987}*{Theorem 7.1}
or $(M, T^{1, 0} M) = (S^{2 n + 1}, T^{1, 0} S^{2 n + 1})$~\cite{Jerison-Lee1988}*{Theorem A}.
It is also known that $\# \GModuli(M, T^{1, 0} M) = 1$
if $M$ is compact and admits an Einstein contact form~\cite{Wang2015}*{Theorem 4}.

A typical non-compact situation arises when one considers the complement of a closed subset
in a compact strictly pseudoconvex CR manifold.
Let $(M, T^{1, 0} M)$ be a compact, connected, strictly pseudoconvex CR manifold of dimension $2 n + 1$,
let $\theta$ be a contact form on $M$,
and let $\Lambda$ be a closed subset of $M$.
One seeks complete contact forms with constant Tanaka--Webster scalar curvature on $M \setminus \Lambda$;
we abbreviate the CR structure $T^{1, 0} (M \setminus \Lambda)$ for simplicity.
This leads to the \emph{singular CR Yamabe equation}
\begin{equation}
\label{eq:singular-CR-Yamabe-problem}
	\begin{cases}
		(2 + 2 / n) \Delta_{b} u + R_{\theta} u = \lambda u^{1 + 2 / n} \text{\ on $M \setminus \Lambda$}, \\
		\lim_{x \to \Lambda} u(x) = \infty,
	\end{cases}
\end{equation}
for some constant $\lambda \in \bbR$ and $0 < u \in C^{\infty}(M \setminus \Lambda)$.
The condition $\lim_{x \to \Lambda} u(x) = \infty$ is necessary for the contact form
$\whxth = u^{2 / n} \theta$ to be complete on $M \setminus \Lambda$.
Chanillo and Yang~\cite{Chanillo-Yang2025-preprint}*{Theorem 2.1} proved that $\dim_{H} \Lambda \leq n$
if there exists a solution of \cref{eq:singular-CR-Yamabe-problem} with $\lambda \geq 0$,
where $\dim_{H}$ is the Hausdorff dimension with respect to the Carnot--Carath\'{e}odory distance;
see \cref{subsection:CR-manifolds} for the definition.
Regarding existence results,
Nayatani~\cite{Nayatani1999}*{p.\ 221} constructed a complete contact form $\theta^{\prime}$
on $S^{2 n + 1} \setminus S_{\bbC}^{2 k + 1}$ with $R_{\theta'} = (n + 1) (n - 2 k - 2)$,
where $S_{\bbC}^{2 k + 1}$ is the $\bbC$-sphere
\begin{equation}
	S_{\bbC}^{2 k + 1}
	\coloneqq \Set{(z^{1}, \dots , z^{k + 1}, 0, \dots , 0) \in S^{2 n + 1} | z^{i} \in \bbC}.
\end{equation}
Guidi, Maalaoui, and Martino~\cite{Guidi-Maalaoui-Martino2024}*{Theorem 1.1}
constructed the same contact form by a different method.
They also constructed an infinite family of complete contact forms
with constant Tanaka--Webster scalar curvature on $S^{2 n + 1} \setminus S_{\bbC}^{1}$
by using a bifurcation method~\cite{Guidi-Maalaoui-Martino2024}*{Theorem 1.3}.
Furthermore,
they proved that $S^{2 n + 1} \setminus \tau(S_{\bbC}^{1})$ admits a complete contact form
with constant Tanaka--Webster scalar curvature
if $\tau \in \Diff(S^{2 n + 1})$ satisfies certain conditions \cite{Guidi-Maalaoui-Martino2024}*{Theorem 1.2}.
Afeltra~\cite{Afeltra2020}*{Theorem 1.1} constructed infinitely many $\bbZ$-periodic
contact forms on $S^{2 n + 1} \setminus \{p, - p\}$ via the Lyapunov--Schmidt method.

In this paper,
we establish the existence of infinitely many pairwise
non-homothetic complete contact forms with constant Tanaka--Webster scalar curvature
on certain non-compact strictly pseudoconvex CR manifolds.
In particular,
we prove that the geometric moduli space is infinite
on complements of $\bbR$- and $\bbC$-spheres in the CR sphere.
From this perspective,
our results can be viewed as CR analogues of the existence of
infinitely many non-homothetic complete constant scalar curvature metrics
in the non-compact setting of conformal geometry.

To explain our result in the general setting,
we start with a compact, connected, strictly pseudoconvex CR manifold $(M, T^{1, 0} M)$ of dimension $2 n + 1$
satisfying the following assumption:

\begin{keyassumption}{$(\blacklozenge)$}
\label{keyassumption}
	Assume that the CR Yamabe constant of $(M, T^{1, 0} M)$ is positive.
	If $\dim M = 3$,
	then we further assume that $(M, T^{1, 0} M)$ is universally embeddable;
	see \cref{subsection:CR-manifolds} for definition.
	If $\dim M = 5$ and $(M, T^{1, 0} M)$ is spherical,
	then we additionally assume that the developing map $\Phi \colon \wtM \to S^{5}$ is injective.
\end{keyassumption}

These additional assumptions in low dimensions are imposed
to guarantee the existence of CR Yamabe minimizers on finite covers of $M$.
If the fundamental group $\pi_{1}(M)$ of $M$ has infinitely many normal subgroups of finite index,
then there exist infinitely many pairwise non-homothetic periodic contact forms on $(\wtM, T^{1, 0} \wtM)$.

In this paper,
we consider the non-compact case.
Let $(M, T^{1, 0} M)$ be a (possibly non-compact) connected strictly pseudoconvex CR manifold.
Denote by $\CSC(M, T^{1, 0} M)$ the set of complete contact forms on $M$
with constant Tanaka--Webster scalar curvature.
We say that two contact forms $\theta_{1}$ and $\theta_{2}$ on $M$ are \emph{homothetic}
if there exist a CR diffeomorphism $\varphi$ and a constant $c > 0$
such that $\varphi^{\ast} \theta_{1} = c \theta_{2}$;
otherwise they are said to be \emph{non-homothetic}.
The \emph{geometric moduli space} of complete contact forms
with constant Tanaka--Webster scalar curvature on $(M, T^{1, 0} M)$ is the quotient
\begin{equation}
 \GModuli(M, T^{1, 0} M)
 \coloneqq \CSC(M, T^{1, 0} M) / \sim,
\end{equation}
where $\theta_{1} \sim \theta_{2}$ if $\theta_{1}$ and $\theta_{2}$ are homothetic.
Jerison and Lee proved that $\# \GModuli(M, T^{1, 0} M) = 1$
if $M$ is compact and $Y(M, T^{1, 0} M) \leq 0$~\cite{Jerison-Lee1987}*{Theorem 7.1}
or $(M, T^{1, 0} M) = (S^{2 n + 1}, T^{1, 0} S^{2 n + 1})$~\cite{Jerison-Lee1988}*{Theorem A}.
It is also known that $\# \GModuli(M, T^{1, 0} M) = 1$
if $M$ is compact and admits an Einstein contact form~\cite{Wang2015}*{Theorem 4}.

A typical non-compact situation arises when one considers the complement of a closed subset
in a compact strictly pseudoconvex CR manifold.
Let $(M, T^{1, 0} M)$ be a compact, connected, strictly pseudoconvex CR manifold of dimension $2 n + 1$,
let $\theta$ be a contact form on $M$,
and let $\Lambda$ be a closed subset of $M$.
One seeks complete contact forms with constant Tanaka--Webster scalar curvature on $M \setminus \Lambda$;
we abbreviate the CR structure $T^{1, 0} (M \setminus \Lambda)$ for simplicity.
This leads to the \emph{singular CR Yamabe equation}
\begin{equation}
\label{eq:singular-CR-Yamabe-problem}
 \begin{cases}
 (2 + 2 / n) \Delta_{b} u + R_{\theta} u = \lambda u^{1 + 2 / n} \text{\ on $M \setminus \Lambda$}, \\
 \lim_{x \to \Lambda} u(x) = \infty,
 \end{cases}
\end{equation}
for some constant $\lambda \in \bbR$ and $0 < u \in C^{\infty}(M \setminus \Lambda)$.
The condition $\lim_{x \to \Lambda} u(x) = \infty$ is necessary for the contact form
$\whxth = u^{2 / n} \theta$ to be complete on $M \setminus \Lambda$.
Chanillo and Yang~\cite{Chanillo-Yang2025-preprint}*{Theorem 2.1} proved that $\dim_{H} \Lambda \leq n$
if there exists a solution of \cref{eq:singular-CR-Yamabe-problem} with $\lambda \geq 0$,
where $\dim_{H}$ is the Hausdorff dimension with respect to the Carnot--Carath\'{e}odory distance;
see \cref{subsection:CR-manifolds} for the definition.
Regarding existence results,
Nayatani~\cite{Nayatani1999}*{p.\ 221} constructed a complete contact form $\theta^{\prime}$
on $S^{2 n + 1} \setminus S_{\bbC}^{2 k + 1}$ with $R_{\theta'} = (n + 1) (n - 2 k - 2)$,
where $S_{\bbC}^{2 k + 1}$ is the $\bbC$-sphere
\begin{equation}
 S_{\bbC}^{2 k + 1}
 \coloneqq \Set{(z^{1}, \dots , z^{k + 1}, 0, \dots , 0) \in S^{2 n + 1} | z^{i} \in \bbC}.
\end{equation}
Guidi, Maalaoui, and Martino~\cite{Guidi-Maalaoui-Martino2024}*{Theorem 1.1}
constructed the same contact form by a different method.
They also constructed an infinite family of complete contact forms
with constant Tanaka--Webster scalar curvature on $S^{2 n + 1} \setminus S_{\bbC}^{1}$
by using a bifurcation method~\cite{Guidi-Maalaoui-Martino2024}*{Theorem 1.3}.
Furthermore,
they proved that $S^{2 n + 1} \setminus \tau(S_{\bbC}^{1})$ admits a complete contact form
with constant Tanaka--Webster scalar curvature
if $\tau \in \Diff(S^{2 n + 1})$ satisfies certain conditions \cite{Guidi-Maalaoui-Martino2024}*{Theorem 1.2}.
Afeltra~\cite{Afeltra2020}*{Theorem 1.1} constructed infinitely many $\bbZ$-periodic
contact forms on $S^{2 n + 1} \setminus \{p, - p\}$ via the Lyapunov--Schmidt method.

To explain our result in the general setting,
we start with a compact, connected, strictly pseudoconvex CR manifold $(M, T^{1, 0} M)$ of dimension $2 n + 1$
satisfying the following assumption:

\begin{keyassumption}{$(\blacklozenge)$}
\label{keyassumption}
 Assume that the CR Yamabe constant of $(M, T^{1, 0} M)$ is positive.
 If $\dim M = 3$,
 then we further assume that $(M, T^{1, 0} M)$ is universally embeddable;
 see \cref{subsection:CR-manifolds} for definition.
 If $\dim M = 5$ and $(M, T^{1, 0} M)$ is spherical,
 then we additionally assume that the developing map $\Phi \colon \wtM \to S^{5}$ is injective.
\end{keyassumption}

These additional assumptions in low dimensions are imposed
to guarantee the existence of CR Yamabe minimizers on finite covers of $M$.
If the fundamental group $\pi_{1}(M)$ of $M$ has infinitely many normal subgroups of finite index,
then there exist infinitely many pairwise non-homothetic periodic contact forms on $(\wtM, T^{1, 0} \wtM)$.

These additional assumptions in low dimensions are imposed
to guarantee the existence of CR Yamabe minimizers on finite covers of $M$.
If the fundamental group $\pi_{1}(M)$ of $M$ has infinitely many normal subgroups of finite index,
then there exist infinitely many pairwise non-homothetic periodic contact forms on $(\wtM, T^{1, 0} \wtM)$.

The novelty of our work lies in the fact that
our results do not follow from existing constructions in the literature
and rely on a different geometric mechanism.
In particular,
our approach applies to examples that are not accessible by bifurcation or ODE-based methods,
which have played a central role in previous studies.

Our proof is similar to that of \cite{Andrade-Case-Piccione-Wei2025-preprint}*{Theorem 3.3}.
Since $\pi_{1}(M)$ has infinite profinite completion,
we can construct a tower of finite connected coverings over $M$.
\Cref{keyassumption} implies that each such cover has a positive CR Yamabe minimizer.
Its lift to the universal cover defines a periodic contact form
with positive constant Tanaka--Webster scalar curvature.
Suppose to the contrary that $\# \GModuli(\wtM, T^{1, 0} \wtM) < \infty$.
Then the CR diffeomorphism group $\Aut_{\CR}(\wtM, T^{1, 0} \wtM)$ does not act properly.
By a result of Schoen~\cite{Schoen1995},
it follows that $(\wtM, T^{1, 0} \wtM)$ is CR diffeomorphic to the Heisenberg group.
We conclude from the characterization of CR automorphisms of the Heisenberg group that
$Y(M, T^{1, 0} M) = 0$, a contradiction.

\begin{proposition}
\label{prop:existence-of-tower-implies-Heisenberg}
 Let $(M, T^{1, 0} M)$ be a compact, connected, strictly pseudoconvex CR manifold of dimension $2 n + 1$.
 Suppose there are an infinite tower $(\pi_{j} \colon M_{j} \to M_{j - 1})_{j = 1}^{\infty}$
 of finite connected coverings of degree $(m_{j})_{j = 1}^{\infty}$ over $M$
 and a sequence $(\theta_{j})_{j = 0}^{\infty}$
 of unit volume CR Yamabe minimizers on $(M_{j}, T^{1, 0} M_{j})$
 such that for each $j \in \bbZ_{> 0}$,
 there exist $\Phi_{j} \in \Aut_{\CR}(\wtM, T^{1, 0} \wtM)$ and $c_{j} \in \bbR_{> 0}$
 satisfying $\Phi_{j}^{\ast} \wtxp^{\ast} \theta_{0} = c_{j} \wtxp_{j}^{\ast} \theta_{j}$.
 Then $Y(M, T^{1, 0} M) \leq 0$.
\end{proposition}

\begin{proposition}
\label{prop:existence-of-non-homothetic}
 Let $(M, T^{1, 0} M)$ be a compact, connected, strictly pseudoconvex CR manifold of dimension $2 n + 1$
 such that \Cref{keyassumption} holds and $\pi_{1}(M)$ has infinite profinite completion.
 Then there exist a regular infinite tower $(\pi_{j} \colon M_{j} \to M_{j - 1})_{j = 1}^{\infty}$
 of finite connected coverings over $M$
 and a sequence $(\theta_{j})_{j = 0}^{\infty}$
 of unit volume CR Yamabe minimizers on $(M_{j}, T^{1, 0} M_{j})$
 such that for any $j \in \bbZ_{> 0}$,
 the contact forms $((\Pi_{j}^{l})^{\ast} \theta_{l})_{l = 0}^{j}$ on $M_{j}$
 are pairwise non-homothetic
 and have positive constant Tanaka--Webster scalar curvature.
\end{proposition}

\begin{proof}[Proof of \cref{thm:infinite-geometric-moduli-real-hyperbolic}]
 Since $S^{2 n + 1} \setminus S_{\bbR}^{k}$ is simply connected,
 $\pi_{1}(M_{\Gamma}) \cong \Gamma$
 and $\wtM_{\Gamma}$ is CR diffeomorphic to $S^{2 n + 1} \setminus S_{\bbR}^{k}$.
 In particular,
 the developing map of $M_{\Gamma}$ is injective.
 We need to check that \Cref{keyassumption} holds.
 It follows from $\delta_{\Gamma} < n$
 that $Y(M_{\Gamma}, T^{1, 0} M_{\Gamma}) > 0$.
 As we noted,
 the developing map of $M_{\Gamma}$ is injective.
 Moreover,
 $M_{\Gamma}$ is universally embeddable
 since any finite connected cover of $M_{\Gamma}$ is of the form $M_{\Gamma^{\prime}}$
 for a subgroup $\Gamma^{\prime} \subset \Gamma$ of finite index.
 Furthermore,
 $\pi_{1}(M_{\Gamma})$ has infinite profinite completion
 since it is infinite and residually finite according to \cref{prop:residually-finite-PU}.
 \cref{thm:infinite-geometric-moduli-general} implies that
 $\# \GModuli(S^{2 n + 1} \setminus S_{\bbR}^{k}) = \infty$.
\end{proof}

\begin{proof}[Proof of \cref{thm:infinite-geometric-moduli-complex-hyperbolic}]
 Since $S^{2 n + 1} \setminus S_{\bbC}^{2 k + 1}$ is simply connected,
 $\pi_{1}(M_{\Gamma}) \cong \Gamma$
 and $\wtM_{\Gamma}$ is CR diffeomorphic to $S^{2 n + 1} \setminus S_{\bbC}^{2 k + 1}$.
 It follows from $\delta_{\Gamma} < n$
 that $Y(M_{\Gamma}, T^{1, 0} M_{\Gamma})$ is positive,
 which guarantees \Cref{keyassumption}.
 Moreover,
 $\pi_{1}(M_{\Gamma})$ has infinite profinite completion
 since it is infinite and residually finite according to \cref{prop:residually-finite-PU}.
 \cref{thm:infinite-geometric-moduli-general} implies that
 $\# \GModuli(S^{2 n + 1} \setminus S_{\bbC}^{2 k + 1}) = \infty$.
\end{proof}

Assume that $n \geq 2$.
Let $M$ be the boundary of the bounded Reinhardt domain
\begin{equation}
 \Omega
 \coloneqq \Set{ w = (w^{1}, \dots , w^{n + 1}) \in \bbC^{n + 1} | \rho(w) < 0 },
\end{equation}
where
\begin{equation}
 \rho(w) \coloneqq \sum_{j = 1}^{n + 1} (\log \abs{w^{j}})^{2} - 1.
\end{equation}
This $M$ has the contact form $\theta \coloneqq d^{c} \rho |_{M}$.
Consider the holomorphic map
\begin{equation}
 \Psi \colon \bbC^{n + 1} \to \bbC^{n + 1}; \qquad
 (z^{1}, \dots , z^{n + 1}) \mapsto (\exp z^{1}, \dots , \exp z^{n + 1}).
\end{equation}
The pull-back $\Psi^{*} \rho$ coincides with
\begin{equation}
 \wtxr(z)
 \coloneqq \sum_{j = 1}^{n + 1} (\Re z^{j})^{2} - 1,
\end{equation}
and the pre-image of $\Omega$ by $\Psi$ is the tube domain
\begin{equation}
 \wtxco
 \coloneqq \Set{ z = (z^{1}, \dots , z^{n + 1}) \in \bbC^{n + 1} |
 \sum_{j = 1}^{n + 1} (\Re z^{j})^{2} < 1 }.
\end{equation}
The holomorphic map $\Psi$ induces a CR map $\psi \colon \wtM \coloneqq \bdry \wtxco \to M$.
Note that $\psi \colon \wtM \to M$ is the universal covering map.
The authors~\cite{Case-Takeuchi2023}*{Section 8} proved that
$R_{\theta}$ is a positive constant and the Chern tensor is nowhere vanishing.
In particular,
$Y(M, T^{1, 0} M) > 0$ and $(M, T^{1, 0} M)$ is not spherical,
which implies that $(M, T^{1, 0} M)$ satisfies \Cref{keyassumption}.
The fundamental group $\pi_{1}(M)$ of $M$ is isomorphic to $\bbZ^{n + 1}$,
which has infinite profinite completion.
Therefore \cref{thm:infinite-geometric-moduli-general} implies $\# \GModuli(\wtM, T^{1, 0} \wtM) = \infty$.

\begin{keyassumption}{$(\blacklozenge)$}
\label{keyassumption}
	Assume that the CR Yamabe constant of $(M, T^{1, 0} M)$ is positive.
	If $\dim M = 3$,
	then we further assume that $(M, T^{1, 0} M)$ is universally embeddable;
	see \cref{subsection:CR-manifolds} for definition.
	If $\dim M = 5$ and $(M, T^{1, 0} M)$ is spherical,
	then we additionally assume that the developing map $\Phi \colon \wtM \to S^{5}$ is injective.
\end{keyassumption}

\begin{theorem}
\label{thm:infinite-geometric-moduli-general}
	Let $(M, T^{1, 0} M)$ be a compact, connected, strictly pseudoconvex CR manifold of dimension $2 n + 1$
	such that \Cref{keyassumption} holds
	and $\pi_{1}(M)$ has infinite profinite completion.
	Then $\# \GModuli(\wtM, T^{1, 0} \wtM) = \infty$.
\end{theorem}

The novelty of our work lies in the fact that
our results do not follow from existing constructions in the literature
and rely on a different geometric mechanism.
In particular,
our approach applies to examples that are not accessible by bifurcation or ODE-based methods,
which have played a central role in previous studies.

Our proof is similar to that of \cite{Andrade-Case-Piccione-Wei2025-preprint}*{Theorem 3.3}.
Since $\pi_{1}(M)$ has infinite profinite completion,
we can construct a tower of finite connected coverings over $M$.
\Cref{keyassumption} implies that each such cover has a positive CR Yamabe minimizer.
Its lift to the universal cover defines a periodic contact form
with positive constant Tanaka--Webster scalar curvature.
Suppose to the contrary that $\# \GModuli(\wtM, T^{1, 0} \wtM) < \infty$.
Then the CR diffeomorphism group $\Aut_{\CR}(\wtM, T^{1, 0} \wtM)$ does not act properly.
By a result of Schoen~\cite{Schoen1995},
it follows that $(\wtM, T^{1, 0} \wtM)$ is CR diffeomorphic to the Heisenberg group.
We conclude from the characterization of CR automorphisms of the Heisenberg group that
$Y(M, T^{1, 0} M) = 0$, a contradiction.

As an application,
consider the complements of an $\bbR$-sphere or a $\bbC$-sphere in $S^{2 n + 1}$,
where the $\bbR$-sphere
\begin{equation}
	S_{\bbR}^{k}
	\coloneqq \Set{(x^{1}, \dots , x^{k + 1}, 0, \dots , 0) \in S^{2 n + 1} | x^{i} \in \bbR}
\end{equation}
is the intersection of the totally real linear subspace $\bbR^{k + 1} \times \{0\}$ with $S^{2 n + 1}$.
There exists a torsion-free discrete subgroup $\Gamma$ of $\Aut_{\CR}(S^{2 n + 1} \setminus S_{\bbR}^{k})$
such that the quotient of $S^{2 n + 1} \setminus S_{\bbR}^{k}$ by $\Gamma$
satisfies the assumptions of \cref{thm:infinite-geometric-moduli-general}.

\begin{theorem}
\label{thm:infinite-geometric-moduli-real-hyperbolic}
	The complement $S^{2 n + 1} \setminus S_{\bbR}^{k}$ satisfies
	$\# \GModuli(S^{2 n + 1} \setminus S_{\bbR}^{k}) = \infty$ if $0 \leq k \leq n - 1$.
\end{theorem}

Similarly,
we consider the $\bbC$-sphere $S_{\bbC}^{2 k + 1}$,
which is the intersection of the complex linear subspace $\bbC^{k + 1} \times \{0\}$ with $S^{2 n + 1}$.
A similar argument to that for the $\bbR$-sphere implies the following result.

\begin{theorem}
\label{thm:infinite-geometric-moduli-complex-hyperbolic}
	The complement $S^{2 n + 1} \setminus S_{\bbC}^{2 k + 1}$ satisfies
	$\# \GModuli(S^{2 n + 1} \setminus S_{\bbC}^{2 k + 1}) = \infty$ if $0 \leq k < (n - 2) / 2$.
\end{theorem}

We add some remarks on these theorems.
Let $V$ be a real linear subspace in $\bbC^{n + 1}$
and set $\Lambda \coloneqq V \cap S^{2 n + 1} \subset S^{2 n + 1}$.
If $V$ is totally real or complex and $\dim_{H} \Lambda < n$,
then we proved that $\# \GModuli(S^{2 n + 1} \setminus \Lambda) = \infty$.
The key point of the proof is that
there exists a torsion-free convex cocompact discrete subgroup $\Gamma$ of $PU(n + 1, 1)$
whose limit set coincides with $\Lambda$.
Then the quotient of $S^{2 n + 1} \setminus \Lambda$ by $\Gamma$ satisfies \Cref{keyassumption}.
On the other hand,
when $V$ is neither totally real nor complex,
there exist no discrete subgroups in $PU(n + 1, 1)$
such that its limit set coincides with $\Lambda$,
and hence our argument does not apply.
As far as we know,
there are no existence results for complete contact forms
with constant Tanaka--Webster scalar curvature on $S^{2 n + 1} \setminus \Lambda$

We will also apply \cref{thm:infinite-geometric-moduli-general}
to circle bundles over compact \Kahler manifolds (\cref{thm:infinite-geometric-moduli-circle-bundle})
and the boundary of a Reinhardt domain (\cref{subsection:boundary-of-Reinhardt-domain}),
which is neither spherical nor Sasakian.

This paper is organized as follows.
In \cref{section:CR-geometry},
we review basic materials in CR geometry.
In \cref{section:CR-Yamabe-problem},
we recall the CR Yamabe problem and summarize known existence results that motivate the present work.
In \cref{section:complex-hyperbolic-geometry},
we collect several facts from complex hyperbolic geometry
that are needed to analyze complements of $\bbR$- and $\bbC$-spheres in the CR sphere.
In \cref{section:infinite-tower},
we study an existence problem for infinite towers of finite connected coverings;
this constitutes the technical core of the paper.
In \cref{section:proof-of-main-result},
we use this framework to prove \cref{thm:infinite-geometric-moduli-general}.
In \cref{section:complements-of-spheres},
we apply this theorem to complements of $\bbR$- and $\bbC$-spheres in the CR sphere.
Finally, in \cref{section:other-applications},
we discuss further applications to circle bundles over compact \Kahler manifolds
and to the boundary of a Reinhardt domain.

The proof of Theorem~\ref{thm:infinite-geometric-moduli-general} is described as being "similar to that of \cite{Andrade-Case-Piccione-Wei2025-preprint}*{Theorem 3.3}." Since $\pi_{1}(M)$ has infinite profinite completion, "we can construct a tower of finite connected coverings over $M$." The assumption \Cref{keyassumption} "implies that each such cover has a positive CR Yamabe minimizer," and "[i]ts lift to the universal cover defines a periodic contact form with positive constant Tanaka--Webster scalar curvature."

Assuming for contradiction that "$\# \GModuli(\wtM, T^{1, 0} \wtM) < \infty$," it follows that "the CR diffeomorphism group $\Aut_{\CR}(\wtM, T^{1, 0} \wtM)$ does not act properly." By Schoen's result \cite{Schoen1995}, "it follows that $(\wtM, T^{1, 0} \wtM)$ is CR diffeomorphic to the Heisenberg group." Using "the characterization of CR automorphisms of the Heisenberg group," they conclude that "$Y(M, T^{1, 0} M) = 0$, a contradiction."

In establishing the main theorem, the proof is described as being “similar to that of Andrade–Case–Piccione–Wei, *Andrade-Case-Piccione-Wei2025-preprint* (2025), Theorem 3.3.” Since $\pi_{1}(M)$ has infinite profinite completion, “we can construct a tower of finite connected coverings over $M$.”

We use the following assumption.
\begin{keyassumption}{$(\blacklozenge)$}
\label{keyassumption}
	Assume that the CR Yamabe constant of $(M, T^{1, 0} M)$ is positive.
	If $\dim M = 3$,
	then we further assume that $(M, T^{1, 0} M)$ is universally embeddable;
	see \cref{subsection:CR-manifolds} for definition.
	If $\dim M = 5$ and $(M, T^{1, 0} M)$ is spherical,
	then we additionally assume that the developing map $\Phi \colon \wtM \to S^{5}$ is injective.
\end{keyassumption}
This assumption “implies that each such cover has a positive CR Yamabe minimizer,” and “[i]ts lift to the universal cover defines a periodic contact form with positive constant Tanaka–Webster scalar curvature.”

Assuming for contradiction that “$\# \GModuli(\wtM, T^{1, 0} \wtM) < \infty$,” it follows that “the CR diffeomorphism group $\Aut_{\CR}(\wtM, T^{1, 0} \wtM)$ does not act properly.” By Schoen’s result (\cite{Schoen1995}), “it follows that $(\wtM, T^{1, 0} \wtM)$ is CR diffeomorphic to the Heisenberg group.” Using “the characterization of CR automorphisms of the Heisenberg group,” they conclude that “$Y(M, T^{1, 0} M) = 0$, a contradiction,” and this completes the proof of the main theorem.

\label{thm:infinite-geometric-moduli-general}
	Let $(M, T^{1, 0} M)$ be a compact, connected, strictly pseudoconvex CR manifold of dimension $2 n + 1$
	such that the following key assumption holds.

\begin{keyassumption}{$(\blacklozenge)$}
\label{keyassumption}
	Assume that the CR Yamabe constant of $(M, T^{1, 0} M)$ is positive.
	If $\dim M = 3$,
	then we further assume that $(M, T^{1, 0} M)$ is universally embeddable;
	see \cref{subsection:CR-manifolds} for definition.
	If $\dim M = 5$ and $(M, T^{1, 0} M)$ is spherical,
	then we additionally assume that the developing map $\Phi \colon \wtM \to S^{5}$ is injective.
\end{keyassumption}

	and $\pi_{1}(M)$ has infinite profinite completion.
	Then $\# \GModuli(\wtM, T^{1, 0} \wtM) = \infty$.,

Question:
Let $(M,T^{1,0}M)$ be a compact, connected, strictly pseudoconvex CR manifold of dimension $2n+1$, and let $\widetilde M$ denote its universal cover. Assume that the CR Yamabe constant of $(M,T^{1,0}M)$ is positive. In addition, if $\dim M=3$, assume that $(M,T^{1,0}M)$ is universally embeddable; and if $\dim M=5$ and $(M,T^{1,0}M)$ is spherical, assume that the developing map $\Phi:\widetilde M\to S^5$ is injective. Also assume that the fundamental group $\pi_1(M)$ has infinite profinite completion. For a CR manifold $X$, let $\CSC(X,T^{1,0}X)$ be the set of complete contact forms on $X$ with constant Tanaka--Webster scalar curvature, and define the geometric moduli space
\[
\GModuli(X,T^{1,0}X):=\CSC(X,T^{1,0}X)/\sim,
\]
where $\theta_1\sim\theta_2$ means that there exist a CR diffeomorphism $\varphi$ and a constant $c>0$ such that $\varphi^*\theta_1=c\theta_2$. Which conclusion about $\GModuli(\widetilde M,T^{1,0}\widetilde M)$ is valid under these hypotheses?

The geometric moduli space on the universal cover is infinite: \[\#\GModuli(\widetilde M,T^{1,0}\widetilde M)=\infty.\] Equivalently, $\widetilde M$ carries infinitely many pairwise non-homothetic complete contact forms with constant Tanaka--Webster scalar curvature.

Choices:
B. The geometric moduli space on the universal cover is finite and nonempty: \[1\le \#\GModuli(\widetilde M,T^{1,0}\widetilde M)<\infty.\] In particular, $\widetilde M$ admits only finitely many homothety classes of complete contact forms with constant Tanaka--Webster scalar curvature.
C. The geometric moduli space on the universal cover is nonempty: \[\#\GModuli(\widetilde M,T^{1,0}\widetilde M)\ge 1.\] Equivalently, $\widetilde M$ carries at least one complete contact form with constant Tanaka--Webster scalar curvature.
D. Every complete contact form on $\widetilde M$ with constant Tanaka--Webster scalar curvature is homothetic to every other one; equivalently, \[\#\GModuli(\widetilde M,T^{1,0}\widetilde M)=1.\]
E. The same infinitude conclusion already holds on the base manifold itself: \[\#\GModuli(M,T^{1,0}M)=\infty.\] Equivalently, $M$ carries infinitely many pairwise non-homothetic complete contact forms with constant Tanaka--Webster scalar curvature.

dropped infinitude of distinct homothety classes

universal-cover target of periodic lifts


\begin{thm}\label{open and discrete thm}
    Let $\Omega\subset\mathbb{R}^n$ be a domain, let $y_0\in\mathbb{R}^n$, and let $p, q \in [1, \infty]$. Suppose that
    $f\in W^{1,n}_{\rm loc}(\Omega,\mathbb{R}^n)$ has a value of finite distortion at $y_0\in\mathbb{R}^n$ with data $(K,\Sigma)$, where $K:\Omega\to[1,\infty)$ and $\Sigma:\Omega\to[0,\infty)$ are measurable functions such that
    \begin{equation}\label{K,Sigma condition}
        K\in L^p_{\rm loc}(\Omega),
        \quad{\rm and}\quad
        \frac{\Sigma}{K}\in L^{q}_{\rm loc}(\Omega).
    \end{equation}
    If $p > n-1$ and $p^{-1} + q^{-1} < 1$, then either $f\equiv y_0$ a.e.\ in $\Omega$, or the continuous representative of $f$ satisfies the following conditions:
    \begin{itemize}[noitemsep]
        \item $f^{-1}\{y_0\}$ is a discrete subset of $\Omega$,
        \item at every $x_0\in f^{-1}\{y_0\}$ the local index $i(x_0,f)$ is positive, and
        \item for every neighborhood $U$ of a point $x_0\in f^{-1}\{y_0\}$, we have $y_0\in  {\rm int}\ f(U)$.
    \end{itemize}
\end{thm}

\begin{ex}\label{ex:planar_counterexample}
    We recall that in \cite[Theorem 1.5]{dolevzalova2024mappings}, for some value of $r > 0$, a mapping $h \colon \B^2(0,r)\setminus\{0\} \to \C$ was constructed with the following properties: $h \in W^{1,2}(\B^2(0, r), \C)$, $\abs{Dh}^2 \le K J_h + \Sigma$ with
    \[
        K \in L^1(\B^2(0,r), [1, \infty)) \qquad \text{and} \qquad \Sigma \in L^\infty(\B^2(0, r)),
    \]
    and $h$ is of the form
    \[
        h(z) = \abs{z} + i \varphi(z)
    \]
    where $\varphi \colon \B^2(0, r) \setminus \{0\} \to [0, \infty)$ is continuous with $\lim_{z \to 0} \varphi(z) = \infty$. We may assume that $r < \pi$ by restricting $h$ if necessary.

    We consider the mapping $f \colon \B^2(0,r) \to \C$ defined by $f(0) = 0$ and
    \[
        f(z) = e^{ih(z)}, \qquad \text{ for } z \in \B^2(0, r) \setminus \{0\}.
    \]
    Since the complex exponential is $1$-Lipschitz on $(-\infty, 0] \times \R$, which contains the image of $ih$, it follows that $f \in W^{1,2}(\B^2(0, r), \C)$. Moreover, by the conformality of the complex exponential, we have $\abs{Df(z)} = \abs{f(z)} \abs{Dh(z)}$ and $J_{f}(z) = \abs{f(z)}^2 J_h(z)$; thus, we have
    \[
        \abs{Df(z)}^2 \le K(z) J_f(z) + \abs{f(z)}^2 \Sigma(z)
    \]
    for a.e.\ $z \in \B^2(0, r)$, where $K \in L^1(\B^2(0, r))$ and $\Sigma \in L^\infty(\B^2(0,r))$. However, since $0 \le \Im(ih) < \pi$ for all $z \in \B^2(0, r)$, the image of $f$ is contained in the closed upper half-plane; since $f(0) = 0$, the map $f$ hence cannot be sense-preserving at $0$, and does not map $\B^2(0, r)$ into a neighborhood of $0$.
\end{ex}

Let $\Omega\subset\mathbb{R}^n$, $n \ge 2$,  be a domain. A  mapping $f \colon \Omega\to\mathbb{R}^n$ belonging to the local Sobolev space  $W^{1,n}_{\mathrm{loc}}(\Omega,\mathbb{R}^n)$ is said to have  \emph{finite distortion} if there exists a measurable function $K \colon \Omega \to [1,\infty)$ such that
\begin{equation}\label{eq: finite distortion}
    |Df(x)|^n\le K(x)J_f(x)
\end{equation}
for almost every $x\in\Omega$. Here, $\left| Df(x) \right|$ denotes the operator norm of the weak differential of  $f$ at $x$, and $J_f(x)$ its Jacobian determinant.

The special case in which $K$ is bounded gives rise to the class of {\em mappings of bounded distortion}, also known as {\em quasiregular mappings}.  A cornerstone of the theory of quasiregular maps is Reshetnyak’s theorem \cite{reshetni_ak1989space,reshetnyak1968condition}, which asserts that such mappings are continuous, and either constant, or open, discrete, and sense preserving. This result places quasiregular mappings among the natural nonlinear analogues of holomorphic functions in higher dimensions.

In the case without a uniform bound on $K$, the continuity of $W^{1,n}_{\mathrm{loc}}$-regular mappings of finite distortion was established by Gol'dstein and Vodop'yanov \cite{vodop1976quasiconformal}, and later extended to larger Orlicz–Sobolev classes by Iwaniec, Koskela, and Onninen \cite{iwaniec2001mappings}.  In addition to initiating a systematic study of mappings of finite distortion within geometric function theory, the paper \cite{iwaniec2001mappings} also introduced a naming convention for a sequence of papers, which is also reflected in the title of the present paper.

The questions of openness and discreteness for mappings of finite distortion are considerably more subtle.  In the planar case, Iwaniec and \v{S}ver\'ak \cite{iwaniec1993mappings} showed that every non-constant mapping  $f\in W^{1,2}_{\mathrm{loc}}(\Omega,\mathbb{R}^2)$ of finite distortion with $K\in L^1_{\mathrm{loc}}(\Omega)$ is discrete and open, relying on techniques from the theory of Beltrami-type equations. Their work led to the conjecture that, in higher dimensions, non-constant mappings  $f\in W^{1,n}_{\mathrm{loc}}(\Omega,\mathbb{R}^2)$  with $K \in L_{\mathrm{loc}}^{n-1} (\Omega )$ should enjoy the same properties. The exponent $n-1$ is critical, as shown by Ball’s example \cite{ball1981global}, which produces, for every $p<n-1$, a Lipschitz mapping of finite distortion with $K\in L^p(\Omega)$ that is neither open nor discrete.

In dimensions $n \ge 3$, Heinonen and Koskela \cite{heinonen1993sobolev} proved that quasi-light mappings of finite distortion with $K\in L^p_{\mathrm{loc}} (\Omega)$,  $p>n-1$,  are open and discrete. The quasi-lightness assumption was subsequently removed by Manfredi and Villamor \cite{manfredi1995mappings}. Bj\"orn \cite{bjorn2002mappings} further extended these results to distortion functions in suitable Orlicz spaces strictly containing $\bigcup_{p>n-1}L^p_{\mathrm{loc}}(\Omega)$.  In the conjectured borderline case  $K \in L_{\mathrm{loc}}^{n-1}(\Omega)$ Hencl and Mal\'y \cite{hencl2002mappings} established openness and discreteness under the additional assumption of quasi-lightness, while Hencl and Rajala~\cite{henclrajala2013} later constructed counterexamples showing that this assumption cannot, in general, be removed when $n \ge 3$, two decades after the corresponding positive planar result of Iwaniec and \v{S}ver\'ak.

\subsection{Quasiregular values and values of finite distortion}

In recent years, a theory of \emph{quasiregular values} has emerged, providing single-value analogues of many fundamental results in the theory of quasiregular mappings. The origins of the theory trace back to a definition by Astala, Iwaniec, and Martin \cite[Section 8.5]{iwaniec2001geometric}, where they formulate an $n$-dimensional counterpart to a class of mappings used in the solution to the planar Calder\'on problem by Astala and P\"aiv\"arinta \cite{astala2006calderon}. The term ``quasiregular value'' was later coined in a series of papers by Kangasniemi and Onninen \cite{kangasniemi2022heterogeneous, kangasniemi2024correction, kangasniemi2025single, kangasniemi2024quasiregular, kangasniemi2024linear}, where the basic theory of these mappings was developed; see also the recent result \cite{heikkila2025quasiregular} by Heikkil\"a and Kangasniemi.

In this article, we consider an analogous generalization of mappings of finite distortion. This condition first appeared in the work of Dole\v{z}alov\'a, Kangasniemi and Onninen \cite{dolevzalova2024mappings}, though our choice of terminology corresponds to the more recent work \cite{kangasniemi2025continuity}.
\begin{defn}
    Let $\Omega\subset\mathbb{R}^n$ be a domain, let $y_0\in\mathbb{R}^n$, let $K \colon \Omega \to [1, \infty)$ be measurable, and let $\Sigma\in L^{1}_\mathrm{loc}(\Omega)$. Suppose that $f\in W^{1,n}_{\rm loc}(\Omega,\mathbb{R}^n)$. We say that $f$ has a \emph{value of finite distortion} at $y_0$ with data $(K, \Sigma)$ if
    \begin{align}\label{K,Sigma-quasiregular}
        |Df(x)|^n\le K(x)J_f(x)+\Sigma(x)|f-y_0|^n
    \end{align}
    for a.e.\ $x\in\Omega$. Alternatively, we say that $f$ satisfies the \emph{$(K, \Sigma)$-distortion inequality with defect} if
    \begin{align}\label{K,Sigma-distortion}
        |Df(x)|^n\le K(x)J_f(x)+\Sigma(x)
    \end{align}
    for a.e.\ $x \in \Omega$.
\end{defn}

If $K \in L^p_\mathrm{loc}(\Omega)$ and $\Sigma/K \in L^q_\mathrm{loc}(\Omega)$ with $p, q \in [1, \infty]$, then the solutions $f \in W^{1,n}_{\mathrm{loc}}(\Omega,\mathbb{R}^n)$ of \eqref{K,Sigma-quasiregular} or \eqref{K,Sigma-distortion} are continuous precisely in the cases where $p^{-1} + q^{-1} < 1$. Counterexamples in the complementary range  $p^{-1} + q^{-1} \ge 1$ were constructed in \cite{dolevzalova2024mappings}, while the sharp positive result for $p^{-1} + q^{-1} < 1$  was established recently by Kangasniemi and Onninen~ \cite{kangasniemi2025continuity}.

The main result of the present article is a Reshetnyak-type theorem for mappings possessing a value of finite distortion. The corresponding result for quasiregular values was proved in \cite[Theorem 1.2]{kangasniemi2025single}.

Let $\Omega\subset\mathbb{R}^n$, $n \ge 2$, be a domain. A mapping $f \colon \Omega\to\mathbb{R}^n$ belonging to the local Sobolev space $W^{1,n}_{\mathrm{loc}}(\Omega,\mathbb{R}^n)$ is said to have \emph{finite distortion} if there exists a measurable function $K \colon \Omega \to [1,\infty)$ such that
\begin{equation}\label{eq: finite distortion}
 |Df(x)|^n\le K(x)J_f(x)
\end{equation}
for almost every $x\in\Omega$. Here, $\left| Df(x) \right|$ denotes the operator norm of the weak differential of $f$ at $x$, and $J_f(x)$ its Jacobian determinant.

The questions of openness and discreteness for mappings of finite distortion are considerably more subtle. In the planar case, Iwaniec and \v{S}ver\'ak \cite{iwaniec1993mappings} showed that every non-constant mapping $f\in W^{1,2}_{\mathrm{loc}}(\Omega,\mathbb{R}^2)$ of finite distortion with $K\in L^1_{\mathrm{loc}}(\Omega)$ is discrete and open, relying on techniques from the theory of Beltrami-type equations. Their work led to the conjecture that, in higher dimensions, non-constant mappings $f\in W^{1,n}_{\mathrm{loc}}(\Omega,\mathbb{R}^2)$ with $K \in L_{\mathrm{loc}}^{n-1} (\Omega )$ should enjoy the same properties. The exponent $n-1$ is critical, as shown by Ball’s example \cite{ball1981global}, which produces, for every $p<n-1$, a Lipschitz mapping of finite distortion with $K\in L^p(\Omega)$ that is neither open nor discrete.

In dimensions $n \ge 3$, Heinonen and Koskela \cite{heinonen1993sobolev} proved that quasi-light mappings of finite distortion with $K\in L^p_{\mathrm{loc}} (\Omega)$, $p>n-1$, are open and discrete. The quasi-lightness assumption was subsequently removed by Manfredi and Villamor \cite{manfredi1995mappings}. Bj\"orn \cite{bjorn2002mappings} further extended these results to distortion functions in suitable Orlicz spaces strictly containing $\bigcup_{p>n-1}L^p_{\mathrm{loc}}(\Omega)$. In the conjectured borderline case $K \in L_{\mathrm{loc}}^{n-1}(\Omega)$ Hencl and Mal\'y \cite{hencl2002mappings} established openness and discreteness under the additional assumption of quasi-lightness, while Hencl and Rajala~\cite{henclrajala2013} later constructed counterexamples showing that this assumption cannot, in general, be removed when $n \ge 3$, two decades after the corresponding positive planar result of Iwaniec and \v{S}ver\'ak.

In this article, we consider an analogous generalization of mappings of finite distortion. This condition first appeared in the work of Dole\v{z}alov\'a, Kangasniemi and Onninen \cite{dolevzalova2024mappings}, though our choice of terminology corresponds to the more recent work \cite{kangasniemi2025continuity}.
\begin{defn}
 Let $\Omega\subset\mathbb{R}^n$ be a domain, let $y_0\in\mathbb{R}^n$, let $K \colon \Omega \to [1, \infty)$ be measurable, and let $\Sigma\in L^{1}_\mathrm{loc}(\Omega)$. Suppose that $f\in W^{1,n}_{\rm loc}(\Omega,\mathbb{R}^n)$. We say that $f$ has a \emph{value of finite distortion} at $y_0$ with data $(K, \Sigma)$ if
 \begin{align}\label{K,Sigma-quasiregular}
 |Df(x)|^n\le K(x)J_f(x)+\Sigma(x)|f-y_0|^n
 \end{align}
 for a.e.\ $x\in\Omega$. Alternatively, we say that $f$ satisfies the \emph{$(K, \Sigma)$-distortion inequality with defect} if
 \begin{align}\label{K,Sigma-distortion}
 |Df(x)|^n\le K(x)J_f(x)+\Sigma(x)
 \end{align}
 for a.e.\ $x \in \Omega$.
\end{defn}

If $K \in L^p_\mathrm{loc}(\Omega)$ and $\Sigma/K \in L^q_\mathrm{loc}(\Omega)$ with $p, q \in [1, \infty]$, then the solutions $f \in W^{1,n}_{\mathrm{loc}}(\Omega,\mathbb{R}^n)$ of \eqref{K,Sigma-quasiregular} or \eqref{K,Sigma-distortion} are continuous precisely in the cases where $p^{-1} + q^{-1} < 1$. Counterexamples in the complementary range $p^{-1} + q^{-1} \ge 1$ were constructed in \cite{dolevzalova2024mappings}, while the sharp positive result for $p^{-1} + q^{-1} < 1$ was established recently by Kangasniemi and Onninen~ \cite{kangasniemi2025continuity}.

The main result of the present article is a Reshetnyak-type theorem for mappings possessing a value of finite distortion. The corresponding result for quasiregular values was proved in \cite[Theorem 1.2]{kangasniemi2025single}.

\begin{align}\label{K,Sigma-distortion}
        |Df(x)|^n\le K(x)J_f(x)+\Sigma(x)
    \end{align}

\begin{align}\label{K,Sigma-quasiregular}
        |Df(x)|^n\le K(x)J_f(x)+\Sigma(x)|f-y_0|^n
    \end{align}

The main result of the present article is a Reshetnyak-type theorem for mappings possessing a value of finite distortion. The corresponding result for quasiregular values was proved in \cite[Theorem 1.2]{kangasniemi2025single}.

The assumption $p^{-1} + q^{-1} < 1$ in Theorem \ref{open and discrete thm} is essential. For one, solutions of \eqref{K,Sigma-quasiregular} need not even admit a continuous representative otherwise. Moreover, even a continuous solution of \eqref{K,Sigma-quasiregular} with $p^{-1} + q^{-1} \ge 1$ may fail to be open and sense-preserving in $f^{-1}\{y_0\}$; see Example \ref{ex:planar_counterexample}. We also note that when $p^{-1} + q^{-1} < 1$, it follows from \eqref{K,Sigma condition} that $\Sigma \in L^r_\loc(\Omega)$ with $r = (p^{-1} + q^{-1})^{-1} > 1$. Such higher integrability of $\Sigma$ is necessary, as demonstrated by the example given in \cite[Example 8.1]{kangasniemi2025single}.

\begin{thm}\label{thm:lusin_N}
 Let $\Omega\subset\mathbb{R}^n$ be a domain, and let $p,q\in[1,\infty]$.
 Suppose that
 $f\in W^{1,n}_{\loc}(\Omega,\mathbb{R}^n)$ satisfies the $(K,\Sigma)$-distortion inequality with defect or has a value of finite distortion at a point $y_0 \in \R^n$ with data $(K, \Sigma)$, where $K:\Omega\to[0,\infty)$ and $\Sigma:\Omega\to[0,\infty)$ are measurable functions such that
 \begin{equation*}
 K\in L^p_{\rm loc}(\Omega) \quad{\rm and}\quad
 \frac{\Sigma}{K}\in L^{q}_{\rm loc}(\Omega).
 \end{equation*}
 If $p^{-1}+q^{-1}<1$, then $f$ satisfies the Lusin (N) -property.
\end{thm}

\begin{lem}\label{Duk bounded lemma}
 Let $\Omega\subset\mathbb{R}^n$ be a domain, let $y_0\in\mathbb{R}^n$, and let $p>n-1,q>1$ with $p^{-1}+q^{-1}<1$.
 Suppose that
 $f\in W^{1,n}_{\rm loc}(\Omega,\mathbb{R}^n)$ has a value of finite distortion at $y_0\in\mathbb{R}^n$ with data $(K, \Sigma)$, where $K:\Omega\to[1,\infty)$ and $\Sigma:\Omega\to[0,\infty)$ are measurable functions such that
 \begin{equation}
 K\in L^p_{\rm loc}(\Omega) \quad{\rm and}\quad
 \frac{\Sigma}{K}\in L^{q}_{\rm loc}(\Omega).
 \end{equation}
 Let $u_k=\min(\log\log|f-y_0|^{-1},k)$, then there exists a neighborhood $V$ of $y_0$ such that
 \begin{equation}\label{Duk uniformly bounded}
 \int_D|\nabla u_k|^\beta\le C_D<\infty,
 \end{equation}
 for $\beta=\frac{pn}{p+1}<n$, all $k\in\mathbb{Z}_{>0}$ and all domains $D$ compactly contained in $U = f^{-1}(V)$, where $C_D=C_D(U,D,n,p,K,\Sigma)$ is independent of $k$.
\end{lem}
\begin{proof}
 We select $V = \B^n(y_0, 1/e)$ and let $U=f^{-1} V$, which is open since $f$ is continuous by \cite[Corollary 1.2]{kangasniemi2025continuity}. Let $D$ be a domain that is compactly contained in $U$, let $\varepsilon > 0$, and select $\eta\in C^\infty_0(U,[0,1])$ such that $\eta\equiv 1$ on $D$.

\begin{lem}\label{totally disconnected}
 Let $\Omega\subset\mathbb{R}^n$ be a domain, let $y_0\in\mathbb{R}^n$, and let $p>n-1,q>1$ with $p^{-1}+q^{-1}<1$.
 Suppose that $f\in W^{1,n}_{\rm loc}(\Omega,\mathbb{R}^n)$ has a value of finite distortion at $y_0\in\mathbb{R}^n$ with data $(K, \Sigma)$, where $K:\Omega\to[1,\infty)$ and $\Sigma:\Omega\to[0,\infty)$ are measurable functions such that
 \begin{equation}
 K\in L^p_{\rm loc}(\Omega) \quad{\rm and}\quad
 \frac{\Sigma}{K}\in L^{q}_{\rm loc}(\Omega).
 \end{equation}
 Then either $f\equiv y_0$ or $\mathcal{H}^{1}(f^{-1}\{y_0\})=0$. In particular, in the later case $f^{-1}\{y_0\}$ is totally disconnected.
\end{lem}

\begin{lem}\label{lem: Integral of J_g on annulus}
 Let $\Omega\subset\mathbb{R}^n$ be a domain, let $y_0\in\mathbb{R}^n$, and let $p>n-1,q>1$ be with $p^{-1}+q^{-1}<1$.
 Suppose that
 $f\in W^{1,n}_{\rm loc}(\Omega,\mathbb{R}^n)$ has a value of finite distortion at $y_0\in\mathbb{R}^n$ with data $(K,\Sigma)$, where $K:\Omega\to[1,\infty)$ and $\Sigma:\Omega\to[0,\infty)$ are measurable functions such that
 \begin{equation*}
 K\in L^p(\Omega) \quad{\rm and}\quad
 \frac{\Sigma}{K}\in L^{q}(\Omega).
 \end{equation*}
 Suppose that $U$ is a non-empty component of $f^{-1}\mathbb{B}^n(y_0,\varepsilon)$, where $\varepsilon>0$ is small enough that $\overline{U}\subset\Omega$ and $m_n(U)<c$ with $c=c(n,q,K,\Sigma,\Omega)>0$ given by Lemma \ref{non negative lemma}. If ${\rm deg}(f,U)=0$ and $g$ is defined as in \eqref{spherical-log- formula}. Then we have $g\in W^{1,\beta}(U,\mathbb{R}\times\mathbb{S}^{n-1})$ for some $p<n$, the integral of $J_g$ over $U$ vanishes, and for every $0<t<\varepsilon$, we have
 \begin{equation}\label{Jacobian over annulus}
 \int_{\{x\in U:\ |f-y_0|<t\}}J_g=0.
 \end{equation}
\end{lem}

\begin{cor}\label{cor: positive local index}
 Let $\Omega\subset\mathbb{R}^n$ be a domain, let $y_0\in\mathbb{R}^n$, and let $p>n-1,q>1$ with $p^{-1}+q^{-1}<1$. Suppose that
 $f\in W^{1,n}_{\rm loc}(\Omega,\mathbb{R}^n)$ has a value of finite distortion at $y_0\in\mathbb{R}^n$ with data $(K,\Sigma)$, where $K:\Omega\to[1,\infty)$ and $\Sigma:\Omega\to[0,\infty)$ are measurable functions such that
 \begin{equation*}
 K\in L^p_{\rm loc}(\Omega) \quad{\rm and}\quad
 \frac{\Sigma}{K}\in L^{q}_{\rm loc}(\Omega).
 \end{equation*}
 Then for every $x_0\in f^{-1}\{y_0\}$, the local index $i(x_0,f)$ is well-defined and we have $i(x_0,f)>0$.
\end{cor}

The final property to be deduced is the local openness property.
\begin{lem}\label{lem:open}
 Let $\Omega\subset\mathbb{R}^n$ be a domain, let $y_0\in\mathbb{R}^n$, and let $p>n-1,q>1$ with $p^{-1}+q^{-1}<1$. Suppose that $f\in W^{1,n}_{\rm loc}(\Omega,\mathbb{R}^n)$ has a value of finite distortion at $y_0\in\mathbb{R}^n$ with data $(K,\Sigma)$, where $K:\Omega\to[1,\infty)$ and $\Sigma:\Omega\to[0,\infty)$ are measurable functions such that
 \begin{equation*}
 K\in L^p_{\rm loc}(\Omega) \quad{\rm and}\quad
 \frac{\Sigma}{K}\in L^{q}_{\rm loc}(\Omega).
 \end{equation*}
 Then for every $x_0 \in f^{-1}\{y_0\}$ and every neighborhood $V$ of $x_0$, we have $y_0\in{\rm int}\ f(V)$.
\end{lem}

The assumption $p^{-1} + q^{-1} < 1$ in Theorem \ref{open and discrete thm} is essential. For one, solutions of \eqref{K,Sigma-quasiregular} need not even admit a continuous representative otherwise. Moreover, even a continuous solution of \eqref{K,Sigma-quasiregular} with $p^{-1} + q^{-1} \ge 1$ may fail to be open and sense-preserving in $f^{-1}\{y_0\}$; see Example \ref{ex:planar_counterexample}. We also note that when $p^{-1} + q^{-1} < 1$, it follows from \eqref{K,Sigma condition} that  $\Sigma \in L^r_\mathrm{loc}(\Omega)$ with $r = (p^{-1} + q^{-1})^{-1} > 1$. Such higher integrability of $\Sigma$ is necessary, as demonstrated by the example given in \cite[Example 8.1]{kangasniemi2025single}.

The assumption that $K\in L_{\mathrm{loc}}^p (\Omega)$ with $p>n-1$ is also necessary. For $n \ge 3$, this is already required in the special case  $\Sigma \equiv 0$; see~\cite{henclrajala2013}. For $n = 2$, although Reshetnyak's theorem remains valid for mappings of finite distortion when $p = 1$, the fundamental difference in our setting is that if $p = 1$, then there exists no $q \in [1, \infty]$ with $p^{-1} + q^{-1} < 1$; for a counterexample with $p = 1$ and $q = \infty$, see again Example \ref{ex:planar_counterexample}.

Finally, one cannot expect Reshetnyak-type conclusions at the preimages of values  $y\neq y_0$, nor for solutions of the defect inequality \eqref{K,Sigma-distortion}. For instance, the mapping  $f(x)=(x_1,0,...,0)$, satisfies the $(K,1)$-distortion inequality with defect for any constant $K\ge 1$, yet $f$ is neither
constant nor discrete or open.  Further counterexamples are given in  \cite[Example 8.4]{kangasniemi2025single}, where a mapping $f \colon \mathbb{R}^n \to \mathbb{R}^n$ is constructed that has a $(1, \Sigma)$-quasiregular value at $y_0 = 0$ with a constant $\Sigma$, has no values of finite distortion with a locally integrable $\Sigma$ at other $y \in \mathbb{R}^n \setminus \{0\}$, and fails to be sense preserving at the preimages of any $y \ne 0$.

\subsection{Methods and other notable results}

The proof of Theorem \ref{open and discrete thm} follows a similar outline as that of \cite[Theorem 1.2]{kangasniemi2025single}, which in turn inherits many of its steps from the original proofs for quasiregular maps and mappings of finite distortion. In particular, if $f$ is not identically $y_0$, we first show that $f^{-1}\{y_0\}$ is totally disconnected, by showing that the Hausdorff dimension of $f^{-1}\{y_0\}$ is less than 1. We then show a local positivity result for the topological degree of $f$, which in turn is used to derive all three parts of Theorem \ref{open and discrete thm}.

However, there are two parts in the proof where our approach differs significantly from that of \cite{kangasniemi2025single}. The first such difference is centered around the Lusin (N) -property; recall that if $\Omega\subset\mathbb{R}^n$ is a domain and $f:\Omega\to\mathbb{R}^n$ is continuous, then $f$ satisfies the {\em Lusin (N) -property} if, for every $m_n$-nullset $E\subset\Omega$, the image $f E$ is an $m_n$-nullset. The Lusin (N) -property holds for quasiregular mappings by a result of Reshetnyak, see e.g.\ \cite{reshetni_ak1989space}, and also for $W^{1,n}_\mathrm{loc}$-regular mappings of finite distortion by a result of Kauhanen, Koskela and Mal\'y \cite{kauhanen2001mappings}.

Notably, the proof of Theorem \ref{open and discrete thm} relies on $f$ satisfying the Lusin (N) -property. When $K$ is constant, this follows from a higher integrability and/or H\"older regularity result for $f$. However, in the case with variable $K$, there are no available higher integrability results, and the modulus of continuity shown in \cite{kangasniemi2025continuity} does not directly imply the Lusin (N) -property for $W^{1,n}$-maps; see the counterexamples of Koskela, Mal\'y, and Z\"urcher \cite{koskela2015luzin}, as well as the related positive result of Zapadinskaya \cite{zapadinskaya2014holder}.

For this reason, the Lusin (N) -property for solutions of \eqref{K,Sigma-quasiregular} and \eqref{K,Sigma-distortion} ends up being a result of independent interest. We record this result here.

\begin{thm}\label{thm:lusin_N}
    Let $\Omega\subset\mathbb{R}^n$ be a domain, and let $p,q\in[1,\infty]$.
    Suppose that
    $f\in W^{1,n}_{\mathrm{loc}}(\Omega,\mathbb{R}^n)$ satisfies the $(K,\Sigma)$-distortion inequality with defect or has a value of finite distortion at a point $y_0 \in \mathbb{R}^n$ with data $(K, \Sigma)$, where $K:\Omega\to[0,\infty)$ and $\Sigma:\Omega\to[0,\infty)$ are measurable functions such that
    \begin{equation*}
        K\in L^p_{\rm loc}(\Omega) \quad{\rm and}\quad
        \frac{\Sigma}{K}\in L^{q}_{\rm loc}(\Omega).
    \end{equation*}
    If $p^{-1}+q^{-1}<1$, then $f$ satisfies the Lusin (N) -property.
\end{thm}

The proof of Theorem \ref{thm:lusin_N} uses the key estimate \cite[Theorem 1.6]{kangasniemi2025continuity} behind the continuity of $W^{1,n}$-solutions to \eqref{K,Sigma-distortion}, along with ideas from the proof of the Lusin (N) -property for continuous pseudomonotone $W^{1,n}$-maps by Mal\'y and Martio \cite[Theorem A]{maly1995lusin}. We highlight that Theorem \ref{thm:lusin_N} in fact uses \cite[Theorem 1.6]{kangasniemi2025continuity} in a sharper manner than the main continuity result of its original paper. In particular, the continuity of $W^{1,n}$-solutions to \eqref{K,Sigma-distortion} arises from them being almost weakly monotone, see \cite[Definition 1.4]{kangasniemi2025continuity}, but the proof of Theorem \ref{thm:lusin_N} does not work for arbitrary almost weakly monotone $W^{1,n}$-maps.

The other major difference between the proofs of Theorem \ref{open and discrete thm} and \cite[Theorem 1.2]{kangasniemi2025single} occurs when we deduce that the topological degree of $f$ cannot vanish locally. This follows by showing that if it does vanish, then one obtains a continuity result for $\log \lvert f - y_0 \rvert$ which leads to a contradiction. The original version \cite[Lemma 5.5]{kangasniemi2025single} of this step uses technical estimates from \cite[Section 6]{kangasniemi2022heterogeneous}, but this approach is critically reliant on $K$ being constant.

Instead, we obtain this continuity result by again using the recent new techniques from \cite{kangasniemi2025continuity}. Notably, we cannot directly use the main result of \cite{kangasniemi2025continuity}, since when $K$ is non-constant, it is not clear to us whether $\log \lvert f - y_0 \rvert$ is in a $W^{1,n}$-space in this case. However, with some adjustments, the underlying ideas are still sufficient to obtain the desired result.

The proof of Theorem \ref{open and discrete thm} is said to follow “a similar outline” as \cite[Theorem 1.2]{kangasniemi2025single} (and ultimately the classical arguments for quasiregular maps and mappings of finite distortion). If $f$ is not identically $y_0$, the argument proceeds as follows:

1. First show that $f^{-1}\{y_0\}$ is totally disconnected “by showing that the Hausdorff dimension of $f^{-1}\{y_0\}$ is less than 1.”

2. Then prove “a local positivity result for the topological degree of $f$,” and use this positivity “to derive all three parts of Theorem \ref{open and discrete thm}.”

Two proof steps are highlighted as substantially different from \cite{kangasniemi2025single}:

- A key input is that “the proof of Theorem \ref{open and discrete thm} relies on $f$ satisfying the Lusin (N)-property.” Since for variable $K$ there are “no available higher integrability results” and the known modulus of continuity “does not directly imply the Lusin (N)-property,” the paper proves Lusin (N) separately (Theorem \ref{thm:lusin_N}).

- To show “that the topological degree of $f$ cannot vanish locally,” they argue that “if it does vanish, then one obtains a continuity result for $\log \lvert f-y_0\rvert$ which leads to a contradiction.” Because the earlier approach depended critically on $K$ being constant, they instead “obtain this continuity result by again using the recent new techniques from \cite{kangasniemi2025continuity},” with adjustments since for non-constant $K$ it is unclear whether “$\log\lvert f-y_0\rvert$ is in a $W^{1,n}$-space.”

The proof of Theorem \ref{open and discrete thm} is said to follow “a similar outline” as \cite[Theorem 1.2]{kangasniemi2025single} (and ultimately the classical arguments for quasiregular maps and mappings of finite distortion). If $f$ is not identically $y_0$, the argument proceeds as follows:

1. First show that $f^{-1}\{y_0\}$ is totally disconnected “by showing that the Hausdorff dimension of $f^{-1}\{y_0\}$ is less than 1.”

2. Then prove “a local positivity result for the topological degree of $f$,” and use this positivity to establish the main theorem.

Two proof steps are highlighted as substantially different from \cite{kangasniemi2025single}:

- A key input is that in establishing the main theorem one needs that $f$ satisfies the Lusin (N)-property. Since for variable $K$ there are “no available higher integrability results” and the known modulus of continuity “does not directly imply the Lusin (N)-property,” the paper proves Lusin (N) separately. We first prove the following theorem.

\begin{thm}\label{thm:lusin_N}
    Let $\Omega\subset\mathbb{R}^n$ be a domain, and let $p,q\in[1,\infty]$.
    Suppose that
    $f\in W^{1,n}_{\loc}(\Omega,\mathbb{R}^n)$ satisfies the $(K,\Sigma)$-distortion inequality with defect or has a value of finite distortion at a point $y_0 \in \R^n$ with data $(K, \Sigma)$, where $K:\Omega\to[0,\infty)$ and $\Sigma:\Omega\to[0,\infty)$ are measurable functions such that
    \begin{equation*}
        K\in L^p_{\rm loc}(\Omega) \quad{\rm and}\quad
        \frac{\Sigma}{K}\in L^{q}_{\rm loc}(\Omega).
    \end{equation*}
    If $p^{-1}+q^{-1}<1$, then $f$ satisfies the Lusin (N) -property.
\end{thm}

- To show “that the topological degree of $f$ cannot vanish locally,” they argue that “if it does vanish, then one obtains a continuity result for $\log \lvert f-y_0\rvert$ which leads to a contradiction.” Because the earlier approach depended critically on $K$ being constant, they instead “obtain this continuity result by again using the recent new techniques from \cite{kangasniemi2025continuity},” with adjustments since for non-constant $K$ it is unclear whether “$\log\lvert f-y_0\rvert$ is in a $W^{1,n}$-space.”

Question:
Let \(\Omega\subset \mathbb{R}^n\) be a domain, let \(y_0\in\mathbb{R}^n\), and let \(p,q\in[1,\infty]\). Suppose \(f\in W^{1,n}_{\mathrm{loc}}(\Omega,\mathbb{R}^n)\) has a value of finite distortion at \(y_0\) with data \((K,\Sigma)\), meaning that for measurable functions \(K:\Omega\to[1,\infty)\) and \(\Sigma:\Omega\to[0,\infty)\),
\[
|Df(x)|^n\le K(x)J_f(x)+\Sigma(x)|f(x)-y_0|^n
\]
for almost every \(x\in\Omega\), where \(|Df(x)|\) is the operator norm of the weak differential and \(J_f(x)\) is the Jacobian determinant. Assume also that
\[
K\in L^p_{\mathrm{loc}}(\Omega),\qquad \frac{\Sigma}{K}\in L^q_{\mathrm{loc}}(\Omega),
\]
and that \(p>n-1\) and \(p^{-1}+q^{-1}<1\). Which statement holds for \(f\)?

Either \(f\equiv y_0\) almost everywhere in \(\Omega\), or \(f\) has a continuous representative such that \(f^{-1}\{y_0\}\) is a discrete subset of \(\Omega\), the local index \(i(x_0,f)\) is positive for every \(x_0\in f^{-1}\{y_0\}\), and for every neighborhood \(U\) of any point \(x_0\in f^{-1}\{y_0\}\), one has \(y_0\in \operatorname{int} f(U)\).

Choices:
B. Either \(f\equiv y_0\) almost everywhere in \(\Omega\), or \(f\) has a continuous representative such that \(f^{-1}\{y_0\}\) is a discrete subset of \(\Omega\), the local index \(i(x_0,f)\) is nonnegative for every \(x_0\in f^{-1}\{y_0\}\), and for every neighborhood \(U\) of any point \(x_0\in f^{-1}\{y_0\}\), one has \(y_0\in \partial f(U)\cup \operatorname{int} f(U)\).
C. Either \(f\equiv y_0\) almost everywhere in \(\Omega\), or \(f\) has a continuous representative such that \(f^{-1}\{y_0\}\) is a discrete subset of \(\Omega\).
D. Either \(f\equiv y_0\) almost everywhere in \(\Omega\), or \(f\) has a continuous representative such that \(f^{-1}\{y_0\}\) is a discrete subset of \(\Omega\), the local index \(i(x_0,f)\) is positive for every \(x_0\in f^{-1}\{y_0\}\), and there exists a neighborhood \(U_0\) of each point \(x_0\in f^{-1}\{y_0\}\) for which \(y_0\in \operatorname{int} f(U_0)\).
E. Either \(f\equiv y_0\) almost everywhere in \(\Omega\), or \(f\) has a continuous representative such that \(f^{-1}\{y_0\}\) has Hausdorff \(1\)-measure zero in \(\Omega\), the local index \(i(x_0,f)\) is positive for every \(x_0\in f^{-1}\{y_0\}\), and for every neighborhood \(U\) of any point \(x_0\in f^{-1}\{y_0\}\), one has \(y_0\in \operatorname{int} f(U)\).

strict positivity of local degree/opening conclusion

dropped positivity of local index and local openness conclusion

quantifier on local openness at preimages of y_0

replacing discreteness by merely \(\mathcal{H}^1\)-null preimage