# Quasiregular curves and cohomology

Susanna Heikkilä  Department of Mathematics and Statistics, P.O. Box 68
(Pietari Kalmin katu 5), FI-00014 University of Helsinki, Finland
<EMAIL_ADDRESS>
###### Abstract.

Let $N$ be a closed, connected, and oriented Riemannian manifold, which admits
a quasiregular $\omega$-curve $\mathbb{R}^{n}\to N$ with infinite energy. We
prove that, if the de Rham class of $\omega$ is non-zero and belongs to a so-
called Künneth ideal, then there exists a non-trivial graded algebra
homomorphism $H_{\mathrm{dR}}^{*}(N)\to\bigwedge^{*}\mathbb{R}^{n}$ from the
de Rham algebra $H_{\mathrm{dR}}^{*}(N)$ of $N$ to the exterior algebra
$\bigwedge^{*}\mathbb{R}^{n}$. As an application, we give examples of pairs
$(N,\omega)$, where $N$ is a closed manifold and $\omega$ is a closed $n$-form
for $n<\dim N$, for which every quasiregular $\omega$-curve $\mathbb{R}^{n}\to
N$ is constant.

###### 2010 Mathematics Subject Classification:

Primary 30C65; Secondary 30L10, 32A30, 53C15, 57M12

This work was supported in part by the Academy of Finland project #332671.

##  1\. Introduction

A continuous map $f\colon M\to N$ between oriented Riemannian $n$-manifolds,
$n\geqslant 2$, is _$K$ -quasiregular for $K\geqslant 1$_ if $f\in
W_{\operatorname{loc}}^{1,n}(M,N)$ and

$\lVert Df\rVert^{n}\leqslant K(\star f^{*}\mathrm{vol}_{N})\text{ a.e.~{}in
}M;$

here $\lVert Df\rVert$ is the operator norm of the weak differential $Df$ and
$\star f^{*}\mathrm{vol}_{N}$ is the Hodge star dual of the $n$-form
$f^{*}\mathrm{vol}_{N}$. An oriented and connected $n$-manifold $N$ is
_quasiregularly elliptic_ , if there exists a non-constant quasiregular map
$\mathbb{R}^{n}\to N$.

In 1981, Gromov [3, p. 200] posed a question, whether all closed simply
connected manifolds are quasiregularly elliptic. In 2019, Prywes [12,
Corollary 1.2] answered Gromov’s question by giving examples of closed simply
connected $4$-manifolds which are not quasiregularly elliptic. The examples
were obtained as an application of Prywes’ cohomology dimension bound for
closed quasiregularly elliptic manifolds [12, Theorem 1.1]: _If $N$ is a
closed quasiregularly elliptic manifold, then $\dim
H_{\mathrm{dR}}^{k}(N)\leqslant\dim\bigwedge^{k}\mathbb{R}^{n}=\binom{n}{k}$
for each $k$._ In [6], Pankka and the author showed that, in fact,
$H_{\mathrm{dR}}^{*}(N)$ is Euclidean in the following sense: _If $N$ is a
closed quasiregularly elliptic manifold, then there exists an embedding of
graded algebras $H_{\mathrm{dR}}^{*}(N)\to\bigwedge^{*}\mathbb{R}^{n}$._ This
embedding yields a topological classification of closed simply connected
quasiregularly elliptic $4$-manifolds [6, Corollary 1.2].

In this paper, we prove a generalization of this cohomology embedding theorem
for closed manifolds admitting quasiregular curves.

For the definition of a quasiregular curve, let $M$ be an oriented Riemannian
$n$-manifold with $n\geqslant 2$. Let $N$ be an oriented Riemannian manifold
and let $\omega\in\Omega^{n}(N)$ be a closed non-vanishing form, i.e.,
$d\omega=0$ and $\omega(x)\neq 0$ for every $x\in N$. A continuous map
$F\colon M\to N$ is a _$K$ -quasiregular $\omega$-curve for $K\geqslant 1$_ if
$F\in W_{\operatorname{loc}}^{1,n}(M,N)$ and

$(\lVert\omega\rVert\circ F)\lVert DF\rVert^{n}\leqslant K(\star
F^{*}\omega)\text{ a.e.~{}in }M;$

here the _pointwise comass norm of $\omega$_ is given by

$\lVert\omega(x)\rVert=\max\\{\omega_{x}(v_{1},\ldots,v_{n})\colon
v_{1},\ldots,v_{n}\text{ unit vectors in }T_{x}N\\},$

$\lVert DF\rVert$ is the operator norm of the weak differential $DF$, and
$\star F^{*}\omega$ is the Hodge star dual of the $n$-form $F^{*}\omega$.

Quasiregular curves are introduced in [11] and include, for example,
quasiregular maps, holomorphic curves, and pseudoholomorphic curves.

Quasiregular curves lead to the following two notions of quasiregular
ellipticity: Let $N$ be an oriented and connected Riemannian manifold and let
$\omega\in\Omega^{n}(N)$ be a closed non-vanishing form with $n\geqslant 2$.
We say that the pair $(N,\omega)$ is _quasiregularly elliptic_ if there exists
a non-constant quasiregular $\omega$-curve $F\colon\mathbb{R}^{n}\to N$. We
also say that the pair $(N,\omega)$ is _infinite energy quasiregularly
elliptic_ if there exists a quasiregular $\omega$-curve
$F\colon\mathbb{R}^{n}\to N$ satisfying
$\int_{\mathbb{R}^{n}}F^{*}\omega=\infty$.

######  Remark 1.1.

We note that a pair $(N,\mathrm{vol}_{N})$, where $N$ is a closed $n$-manifold
satisfying $H_{\mathrm{dR}}^{*}(N)\neq H_{\mathrm{dR}}^{*}(\mathbb{S}^{n})$,
is infinite energy quasiregularly elliptic if and only if $N$ is
quasiregularly elliptic. The equivalence follows from the Bonk–Heinonen growth
result for quasiregular mappings [1, Theorem 1.11]: _Let $N$ be a closed,
connected, and oriented Riemannian $n$-manifold with $n\geqslant 2$ and
$H_{\mathrm{dR}}^{*}(N)\neq H_{\mathrm{dR}}^{*}(\mathbb{S}^{n})$. Then every
non-constant $K$-quasiregular map $f\colon\mathbb{R}^{n}\to N$ satisfies_

$\liminf_{r\to\infty}\frac{\int_{B^{n}(0,r)}f^{*}\mathrm{vol}_{N}}{r^{\varepsilon}}>0,$

_where $\varepsilon=\varepsilon(n,K)>0$._ To our knowledge, a similar growth
result is not known for quasiregular curves in full generality; for so-called
_signed quasiregular curves_ , see [5]. In particular, the growth theorem in
[5] guarantees infinite energy in all the examples presented in this paper;
for details see Theorem 2.3.

To state our main result, we introduce some terminology stemming from the
Künneth theorem: _Let $N=M\times P$ be a product of closed smooth manifolds.
Then the map_

$\bigoplus_{\ell=0}^{k}H_{\mathrm{dR}}^{\ell}(M)\otimes
H_{\mathrm{dR}}^{k-\ell}(P)\to H_{\mathrm{dR}}^{k}(N),\;c\otimes
c^{\prime}\mapsto\pi_{M}^{*}c\wedge\pi_{P}^{*}c^{\prime},$

_where $\pi_{M}\colon N\to M$ and $\pi_{P}\colon N\to P$ are the projections,
is an isomorphism for each $k$._

For a smooth manifold $N$, we say that the _Künneth ideal of $N$_ is the ideal
$K^{*}(N)=\oplus_{k=2}^{\dim N}K^{k}(N)$ of $H_{\mathrm{dR}}^{*}(N)$, where
the $k$th layer $K^{k}(N)$ is the vector space generated by products $c\wedge
c^{\prime}$ of de Rham classes $c\in H_{\mathrm{dR}}^{\ell}(N)$ and
$c^{\prime}\in H_{\mathrm{dR}}^{k-\ell}(N)$ with $1\leqslant\ell\leqslant
k-1$, i.e.,

$K^{k}(N)=\mathrm{span}_{\mathbb{R}}\left(\bigcup_{\ell=1}^{k-1}H_{\mathrm{dR}}^{\ell}(N)\wedge
H_{\mathrm{dR}}^{k-\ell}(N)\right).$

We are now ready to present our main result.

######  Theorem 1.2.

Let $(N,\omega)$ be infinite energy quasiregularly elliptic, where $N$ is
closed and $0\neq[\omega]_{\mathrm{dR}}\in K^{n}(N)$. Then there exists a
graded algebra homomorphism $\Phi\colon
H_{\mathrm{dR}}^{*}(N)\to\bigwedge^{*}\mathbb{R}^{n}$ for which
$\Phi[\omega]_{\mathrm{dR}}\neq 0$.

By the Poincaré duality and the Bonk–Heinonen growth theorem, Theorem 1.2
yields as a corollary the aforementioned embedding theorem in [6].

Before discussing further consequences of Theorem 1.2, we briefly mention
examples of non-trivial Künneth ideals, which are used later in this paper.
The first example is given by product manifolds, where the factors are
equidimensional.

######  Example 1.3.

Let $n\geqslant 2$. Let $N=N_{1}\times\cdots\times N_{m}$ and
$\omega=\sum_{i=1}^{m}\pi_{i}^{*}\mathrm{vol}_{N_{i}}\in\Omega^{n}(N)$, where
each $N_{i}$ is a closed, connected, and oriented Riemannian $n$-manifold with
$H_{\mathrm{dR}}^{*}(N_{i})\neq H_{\mathrm{dR}}^{*}(\mathbb{S}^{n})$ and each
$\pi_{i}\colon N\to N_{i}$ is the projection. Then, by the Künneth theorem and
Poincaré duality, $0\neq[\omega]_{\mathrm{dR}}\in K^{n}(N)$. In particular, if
$m=1$, then $0\neq[\mathrm{vol}_{N}]\in K^{n}(N)$.

Kähler manifolds also give examples of non-trivial Künneth ideals.

######  Example 1.4.

Let $(N,\omega_{\mathrm{sym}})$ be a Kähler manifold. Let $m$ be an integer
satisfying $4\leqslant 2m\leqslant\dim N$ and let
$\omega=\omega_{\mathrm{sym}}^{\wedge
m}=\omega_{\mathrm{sym}}\wedge\cdots\wedge\omega_{\mathrm{sym}}\in\Omega^{2m}(N)$.
Then $0\neq[\omega]_{\mathrm{dR}}\in K^{2m}(N)$.

### Examples of non-ellipticity

We obtain from Theorem 1.2 two families of examples of closed manifolds $N$
and closed non-vanishing forms $\omega$ for which the pair $(N,\omega)$ is not
quasiregularly elliptic. Our first family of examples consists of pairs
$(N,\omega)$, where $N$ is a $6$-manifold and $\omega$ is a $4$-form.

######  Corollary 1.5.

Let $g\geqslant 1$ and $T_{g}=\\#^{g}(\mathbb{S}^{1}\times\mathbb{S}^{1})$.
Let $\omega_{\mathrm{sym}}\in\Omega^{2}(\mathbb{C}P^{2})$ be the standard
symplectic form. Then
$(T_{g}\times\mathbb{C}P^{2},\pi_{1}^{*}\mathrm{vol}_{T_{g}}\wedge\pi_{2}^{*}\omega_{\mathrm{sym}})$
is quasiregularly elliptic if and only if $g=1$.

Corollary 1.5 also shows that the quasiregular ellipticity of a pair
$(N,\omega)$ depends on both $N$ and $\omega$. Indeed,
$(T_{g}\times\mathbb{C}P^{2},\pi_{2}^{*}\omega_{\mathrm{sym}})$ is
quasiregularly elliptic for every $g\geqslant 1$.

Our second family of not quasiregularly elliptic examples are pairs
$(N,\omega)$, where $N$ is a simply connected $8$-manifold and $\omega$ is a
$6$-form.

######  Corollary 1.6.

Let $\nu\geqslant 8$ and
$S_{\nu}=\\#^{\nu}(\mathbb{S}^{2}\times\mathbb{S}^{2})$. Let
$\omega_{\mathrm{sym}}\in\Omega^{2}(\mathbb{C}P^{2})$ be the standard
symplectic form. Then
$(S_{\nu}\times\mathbb{C}P^{2},\pi_{1}^{*}\mathrm{vol}_{S_{\nu}}\wedge\pi_{2}^{*}\omega_{\mathrm{sym}})$
is not quasiregularly elliptic.

It is not clear to us, whether the lower bound $\nu\geqslant 8$ is sharp in
Corollary 1.6.

### Prywes’ theorem for quasiregular curves

As a byproduct of the proof of Theorem 1.2, we also obtain the following
result, which can be interpreted as a relative of Prywes’ theorem for
quasiregular curves.

######  Theorem 1.7.

Let $(N,\omega)$ be infinite energy quasiregularly elliptic, where $N$ is a
closed manifold,
$\\{0\\}\neq\mathrm{span}_{\mathbb{R}}([\omega]_{\mathrm{dR}})=K^{n}(N)$, and
$\lVert\omega\rVert>0$. Let $M\subset N$ be a closed, connected, and oriented
$n$-submanifold satisfying
$\iota^{*}(H_{\mathrm{dR}}^{*}(N))=H_{\mathrm{dR}}^{*}(M)$ and
$\iota^{*}[\omega]_{\mathrm{dR}}\neq 0$, where $\iota\colon M\to N$ is the
inclusion. Then $\dim H_{\mathrm{dR}}^{k}(M)\leqslant\binom{n}{k}$ for
$k=0,\ldots,n$.

Theorem 1.7 has a topological analogue: _Let $F\colon\Sigma\to N$ be a
continuous map between smooth manifolds for which $F^{*}(K^{n}(N))\neq\\{0\\}$
for some $n\geqslant 1$. Let $M\subset N$ be a closed, connected, and oriented
$n$-submanifold satisfying
$\iota^{*}(H_{\mathrm{dR}}^{*}(N))=H_{\mathrm{dR}}^{*}(M)$ and
$\ker\iota^{*}\cap\,K^{n}(N)\subset\ker F^{*}$, where $\iota\colon M\to N$ is
the inclusion. Then there exists an embedding $H_{\mathrm{dR}}^{k}(M)\to
H_{\mathrm{dR}}^{k}(\Sigma)$ for each $k$._ For a formulation of Theorem 1.7
more similar to this topological analogue, see Theorem 6.4.

### Quasiregular curves with respect to closed non-exact forms

Instead of directly proving Theorem 1.2, we prove a stronger statement for
pairs $(N,\omega)$, where $\omega$ is a closed non-exact form, possibly
vanishing at some points on $N$.

######  Theorem 1.8.

Let $N$ be a closed, connected, and oriented Riemannian manifold. Let
$2\leqslant n\leqslant\dim N$ and let $\omega\in\Omega^{n}(N)$ be a closed
form with $0\neq[\omega]_{\mathrm{dR}}\in K^{n}(N)$. Let
$F\colon\mathbb{R}^{n}\to N$ be a quasiregular $\omega$-curve satisfying
$\int_{\mathbb{R}^{n}}F^{*}\omega=\infty$ and
$\inf_{\mathbb{R}^{n}}(\lVert\omega\rVert\circ F)>0$. Then there exists a
graded algebra homomorphism $\Phi\colon
H_{\mathrm{dR}}^{*}(N)\to\bigwedge^{*}\mathbb{R}^{n}$ for which
$\Phi[\omega]_{\mathrm{dR}}\neq 0$.

######  Remark 1.9.

Formally, $F$ is a quasiregular $\omega_{+}$-curve $\mathbb{R}^{n}\to N_{+}$,
where

$N_{+}=\\{x\in N\colon\omega(x)\neq 0\\}\quad\text{ and
}\quad\omega_{+}=\omega|_{N_{+}}.$

In Theorem 1.8, we assume that the form is not exact since, if $N$ is a
closed, connected, and oriented Riemannian manifold and
$\omega\in\Omega^{n}(N)$, $2\leqslant n\leqslant\dim N$, is an exact form,
then every quasiregular $\omega$-curve $F\colon\mathbb{R}^{n}\to N$ satisfying
$\inf_{\mathbb{R}^{n}}(\lVert\omega\rVert\circ F)>0$ is constant; see [11,
Theorem 1.7] and [11, Remark 3.2].

### Idea of the proof

The proof of Theorem 1.8 follows the proof of the cohomology embedding theorem
for closed quasiregularly elliptic manifolds in [6].

The proof consists of two key ideas. We reduce the global problem of Theorem
1.8 to a local setting. For the statements, let $N$ be an oriented Riemannian
manifold and let $\omega\in\Omega^{n}(N)$ be a closed non-exact form, where
$2\leqslant n\leqslant\dim N$. For constants $K,D\geqslant 1$, let
$\mathcal{F}_{K,D}(N,\omega)$ be the family of non-constant $K$-quasiregular
$\omega$-curves $F\colon B_{2}^{n}\to N$ satisfying the additional conditions

$\int_{B_{2}^{n}}F^{*}\omega\leqslant D\int_{B^{n}}F^{*}\omega$

and $\inf_{B_{2}^{n}}(\lVert\omega\rVert\circ F)\geqslant D^{-1}$. Here and in
what follows, $B^{n}$ and $B_{2}^{n}$ denote the Euclidean balls of radius $1$
and $2$ centered at the origin, respectively. In what follows, we also denote

$A_{\omega}(F)=\int_{B^{n}}F^{*}\omega>0$

for $F\in\mathcal{F}_{K,D}(N,\omega)$.

We are now ready to state a localized reformulation of Theorem 1.8; see also
[6, Theorem 1.3].

######  Theorem 1.10.

Let $N$ be a closed, connected, and oriented Riemannian manifold. Let
$2\leqslant n\leqslant\dim N$ and let $\omega\in\Omega^{n}(N)$ be a closed
form with $0\neq[\omega]_{\mathrm{dR}}\in K^{n}(N)$. Assume also that there
exists $K,D\geqslant 1$ satisfying

$\sup_{F\in\mathcal{F}_{K,D}(N,\omega)}A_{\omega}(F)=\infty.$

Then there exists a graded algebra homomorphism $\Phi\colon
H_{\mathrm{dR}}^{*}(N)\to\bigwedge^{*}\mathbb{R}^{n}$ for which
$\Phi[\omega]_{\mathrm{dR}}\neq 0$.

The reduction of Theorem 1.8 to Theorem 1.10 follows from Rickman’s Hunting
Lemma. After the reduction, we construct the homomorphism in Theorem 1.10 as
the evaluation of a weak limit of normalized pull-backs.

Let $N$ be a closed, connected, and oriented Riemannian manifold and let

$h\colon H_{\mathrm{dR}}^{*}(N)\to\Omega^{*}(N),\quad c\mapsto h_{c},$

be the map associating to each de Rham class $c\in H_{\mathrm{dR}}^{*}(N)$ its
unique harmonic representative $h_{c}\in c$; to see that the map $h$ is well-
defined see e.g. [16, Theorem 6.11]. Given $\omega$, $K$ and $D$, we define,
for each $F\in\mathcal{F}_{K,D}(N,\omega)$, a _normalized pull-back_
$F^{\\#}\colon H_{\mathrm{dR}}^{*}(N)\to\mathscr{W}^{*}(B_{2}^{n})$ by formula

$F^{\\#}c=A_{\omega}(F)^{-\frac{k}{n}}F^{*}(h(c))$

for $c\in H_{\mathrm{dR}}^{k}(N)$ and $k=0,\ldots,n$. Here and in what
follows, $\mathscr{W}^{*}(B_{2}^{n})$ is the graded algebra

$\mathscr{W}^{k}(B_{2}^{n})=\begin{cases}W^{d,\frac{n}{k}}(B_{2}^{n};{\bigwedge}^{k}\mathbb{R}^{n}),&\,\text{
for }1\leqslant k\leqslant n,\\\
\ker(d\colon\Omega^{0}(B_{2}^{n})\to\Omega^{1}(B_{2}^{n})),&\,\text{ for
}k=0,\end{cases}$

of Sobolev forms on $B_{2}^{n}$; for more details see Section 3.

The normalized pull-backs have the following weak limit property: if
$[\omega]_{\mathrm{dR}}\in K^{n}(N)$ and $(F_{j})$ is a sequence in
$\mathcal{F}_{K,D}(N,\omega)$ with the properties that
$A_{\omega}(F_{j})\to\infty$ and that the sequence

$\left(F_{j}^{\\#}|_{H_{\mathrm{dR}}^{k}(N)}\colon
H_{\mathrm{dR}}^{k}(N)\to\mathscr{W}^{k}(B_{2}^{n})\right)$

has a weak limit for each $k=1,\ldots,n-1$, then the limit operator extends to
a graded algebra homomorphism
$H_{\mathrm{dR}}^{*}(N)\to\mathscr{W}^{*}(B_{2}^{n})$ which maps
$[\omega]_{\mathrm{dR}}$ non-trivially. More precisely, we prove the following
theorem; see also [6, Theorem 1.7].

######  Theorem 1.11.

Let $N$ be a closed, connected, and oriented Riemannian manifold. Let
$2\leqslant n\leqslant\dim N$ and let $\omega\in\Omega^{n}(N)$ be a closed
form with $0\neq[\omega]_{\mathrm{dR}}\in K^{n}(N)$. Let $K,D\geqslant 1$. Let
$(F_{j})$ be a sequence in $\mathcal{F}_{K,D}(N,\omega)$ for which
$A_{\omega}(F_{j})\to\infty$ and $F_{j}^{\\#}\rightharpoonup L$, where

$F_{j}^{\\#},L\colon\bigoplus_{k=1}^{n-1}H_{\mathrm{dR}}^{k}(N)\to\bigoplus_{k=1}^{n-1}L^{\frac{n}{k}}(B_{2}^{n};{\bigwedge}^{k}\mathbb{R}^{n}).$

Then $L$ extends to a graded algebra homomorphism $L\colon
H_{\mathrm{dR}}^{*}(N)\to\mathscr{W}^{*}(B_{2}^{n})$ for which
$\int_{B^{n}}L[\omega]_{\mathrm{dR}}=1$.

In general, the choice of harmonic representative $h\colon
H_{\mathrm{dR}}^{*}(N)\to\Omega^{*}(N)$ is not an algebra homomorphism and
hence neither are the normalized pull-backs $F_{j}^{\\#}$. However, Theorem
1.11 shows that the limiting operator $L$ commutes with the exterior product.

### Organization of the article

In Section 2, we discuss how Theorem 1.8 reduces to Theorem 1.10. In Section
3, we recall the necessary basics of Sobolev forms and prove estimates for
normalized pull-backs. In Section 4, we prove Theorem 1.11. In Section 5, we
give the proof of Theorem 1.10. In Section 6, we prove Theorem 1.7. Lastly, in
Section 7, we prove Corollaries 1.5 and 1.6.

### Acknowledgements

The author thanks her thesis advisor Pekka Pankka for all his help improving
the manuscript. We also thank Eero Hakavuori and Toni Ikonen for comments and
discussions on these topics.

##  2\. Reduction of Theorem 1.8 to Theorem 1.10

In this section, we reduce Theorem 1.8 to Theorem 1.10 with the following
proposition; see also [6, Proposition 2.1] and [12, Section 4].

######  Proposition 2.1.

Let $N$ be a closed, connected, and oriented Riemannian manifold. Let
$\omega\in\Omega^{n}(N)$ be a closed form with $2\leqslant n\leqslant\dim N$
and $0\neq[\omega]_{\mathrm{dR}}\in K^{n}(N)$. Let $K\geqslant 1$ and let
$F\colon\mathbb{R}^{n}\to N$ be a $K$-quasiregular $\omega$-curve satisfying
$\int_{\mathbb{R}^{n}}F^{*}\omega=\infty$ and
$\inf_{\mathbb{R}^{n}}(\lVert\omega\rVert\circ F)>0$. Then there exist a
constant $D=D(n,\omega,F)>1$ and a sequence of pairs
$(a_{j},r_{j})\in\mathbb{R}^{n}\times(0,\infty)$ for which

$F\circ T_{j}\in\mathcal{F}_{K,D}(N,\omega)\quad\text{ and
}\quad\lim_{j\to\infty}A_{\omega}(F\circ T_{j})=\infty,$

where $T_{j}\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ is the affine map $x\mapsto
r_{j}x+a_{j}$. In particular,

$\sup_{F^{\prime}\in\mathcal{F}_{K,D}(N,\omega)}A_{\omega}(F^{\prime})=\infty.$

The proof of Proposition 2.1 is based on Rickman’s Hunting Lemma [14, Lemma
5.1]. The following formulation is due to Bonk and Poggi-Corradini [2, Lemma
2.1].

######  Lemma 2.2 (Rickman’s Hunting Lemma).

Let $\mu$ be an atomless Borel measure on $\mathbb{R}^{n}$ satisfying
$\mu(\mathbb{R}^{n})=\infty$ and $\mu(B)<\infty$ for every ball
$B\subset\mathbb{R}^{n}$. Then there exists a constant $D=D(n)>1$ with the
property that, for every $j\in\mathbb{N}$, there exists a ball
$B\subset\mathbb{R}^{n}$ for which

$j\leqslant\mu(2B)\leqslant D\mu(B).$

Proposition 2.1 is an immediate consequence.

###### Proof of Proposition 2.1.

Since $\int_{\mathbb{R}^{n}}F^{*}\omega=\infty$, the Borel measure $\mu$ on
$\mathbb{R}^{n}$ defined by $\mu(E)=\int_{E}F^{*}\omega$ satisfies the
assumptions of Lemma 2.2. Hence, there exists a constant $D=D(n)>1$ and a
sequence of balls $(B^{n}(a_{j},r_{j}))$ for which

$j\leqslant\int_{B^{n}(a_{j},2r_{j})}F^{*}\omega\leqslant
D\int_{B^{n}(a_{j},r_{j})}F^{*}\omega.$

Let $D^{\prime}=D^{\prime}(n,\omega,F)\geqslant D$ be a constant for which
$\inf_{\mathbb{R}^{n}}(\lVert\omega\rVert\circ F)\geqslant(D^{\prime})^{-1}$.
We obtain that the maps $F_{j}\colon B_{2}^{n}\to N$, $x\mapsto
F(r_{j}x+a_{j})$, form a sequence in $\mathcal{F}_{K,D^{\prime}}(N,\omega)$
satisfying $A_{\omega}(F_{j})\to\infty$. The claim follows. ∎

The following theorem gives a sufficient condition for a quasiregular curve to
have infinite energy. The theorem is a special case of the Bonk–Heinonen type
growth result for signed quasiregular curves; see [5, Theorem 1.1].

######  Theorem 2.3.

_([5, Theorem 1.1])_ Let $N$ be a connected and oriented Riemannian manifold.
Let $\omega\in\Omega^{n}(N)$ be a closed non-vanishing form with $2\leqslant
n\leqslant\dim N$ and

$\omega\in\mathrm{span}_{\mathbb{R}}\left(\bigcup_{k=1}^{n-1}\mathcal{Z}^{k}(N)\wedge\mathcal{Z}^{n-k}(N)\right),$

where $\mathcal{Z}^{k}(N)$ denotes the space of all smooth, bounded, and
closed $k$-forms on $N$. Then every non-constant $K$-quasiregular
$\omega$-curve $F\colon\mathbb{R}^{n}\to N$ satisfies

$\liminf_{r\to\infty}\frac{\int_{B^{n}(0,r)}F^{*}\omega}{r^{\varepsilon}}>0,$

where $\varepsilon=\varepsilon(n,K,\omega)>0$.

Theorem 2.3 implies trivially that non-constant quasiregular
$\omega_{\mathrm{sym}}^{\wedge m}$-curves $\mathbb{R}^{2m}\to N$, where
$\omega_{\mathrm{sym}}^{\wedge m}$ is as in Example 1.4, have infinite energy.
It also follows straightforwardly that non-constant quasiregular
$\left(\sum_{i=1}^{m}\pi_{i}^{*}\mathrm{vol}_{N_{i}}\right)$-curves
$\mathbb{R}^{n}\to N_{1}\times\cdots\times N_{m}$, where
$\sum_{i=1}^{m}\pi_{i}^{*}\mathrm{vol}_{N_{i}}$ is as in Example 1.3, have
infinite energy.

######  Corollary 2.4.

Let $n\geqslant 2$ and let $N_{1},\ldots,N_{m}$ be closed, connected, and
oriented Riemannian $n$-manifolds with $H_{\mathrm{dR}}^{*}(N_{i})\neq
H_{\mathrm{dR}}^{*}(\mathbb{S}^{n})$. Let $N=N_{1}\times\cdots\times N_{m}$
and $\omega=\sum_{i=1}^{m}\pi_{i}^{*}\mathrm{vol}_{N_{i}}\in\Omega^{n}(N)$.
Then every non-constant quasiregular $\omega$-curve $F\colon\mathbb{R}^{n}\to
N$ satisfies $\int_{\mathbb{R}^{n}}F^{*}\omega=\infty$.

###### Proof.

By [4, Proposition 2.13], it holds that

$\mathrm{vol}_{N_{i}}\in\mathrm{span}_{\mathbb{R}}\left(\bigcup_{k=1}^{n-1}\mathcal{Z}^{k}(N_{i})\wedge\mathcal{Z}^{n-k}(N_{i})\right)$

for each $i$. Thus,

$\omega\in\mathrm{span}_{\mathbb{R}}\left(\bigcup_{k=1}^{n-1}\mathcal{Z}^{k}(N)\wedge\mathcal{Z}^{n-k}(N)\right)$

and the claim follows by Theorem 2.3. ∎

##  3\. Preliminaries

### Sobolev forms

We begin by reviewing the basics of Sobolev spaces of differential forms. A
$(k+1)$-form $d\tau\in
L_{\operatorname{loc}}^{1}(B_{2}^{n};\bigwedge^{k+1}\mathbb{R}^{n})$ is the
_weak exterior derivative_ of a $k$-form $\tau\in
L_{\operatorname{loc}}^{1}(B_{2}^{n};\bigwedge^{k}\mathbb{R}^{n})$ if

$\int_{B_{2}^{n}}\tau\wedge
d\varphi=(-1)^{k+1}\int_{B_{2}^{n}}d\tau\wedge\varphi$

for every $\varphi\in\Omega_{0}^{n-k-1}(B_{2}^{n})$. For $k=0,\ldots,n-1$,
$W^{d,\frac{n}{k}}(B_{2}^{n};\bigwedge^{k}\mathbb{R}^{n})$ denotes the Sobolev
space of forms $\tau\in
L^{\frac{n}{k}}(B_{2}^{n};\bigwedge^{k}\mathbb{R}^{n})$ with a weak exterior
derivative $d\tau\in
L^{\frac{n}{k}}(B_{2}^{n};\bigwedge^{k+1}\mathbb{R}^{n})$. For $k=n$,
$W^{d,\frac{n}{k}}(B_{2}^{n};\bigwedge^{k}\mathbb{R}^{n})$ denotes
$L^{1}(B_{2}^{n};\bigwedge^{n}\mathbb{R}^{n})$.

Next, we recall the compatibility of the weak exterior derivative and the
pull-back of a continuous Sobolev map.

######  Lemma 3.1.

_([13, II.4.7])_ Let $N$ be a smooth manifold and let $F\in
W_{\operatorname{loc}}^{1,n}(B_{2}^{n},N)$ be a continuous map. Then
$dF^{*}\alpha=F^{*}d\alpha$ for every $\alpha\in\Omega^{k}(N)$ and
$k=0,\ldots,n-1$.

We also have an elementary estimate for the the pull-back of a continuous
Sobolev map. Namely, if $F\in W_{\operatorname{loc}}^{1,n}(B_{2}^{n},N)$ is a
continuous map into a smooth manifold $N$, then

(3.1) $\lVert F^{*}\alpha\rVert\leqslant(\lVert\alpha\rVert\circ F)\lVert
DF\rVert^{k}\text{ a.e.~{}in }B_{2}^{n}$

for $\alpha\in\Omega^{k}(N)$ and $k=0,\ldots,\dim N$.

Lemma 3.1 and the pointwise estimate (3.1) yield as an immediate consequence
the following.

######  Corollary 3.2.

Let $N$ be a closed smooth manifold and let $F\in W^{1,n}(B_{2}^{n},N)$ be a
continuous map. Then $F^{*}$ defines a graded linear map
$F^{*}\colon\Omega^{*}(N)\to\oplus_{k=0}^{n}L^{\frac{n}{k}}(B_{2}^{n};{\bigwedge}^{k}\mathbb{R}^{n})$.
In particular, $F^{*}|_{\mathcal{Z}^{k}(N)}\colon\mathcal{Z}^{k}(N)\to
W^{d,\frac{n}{k}}(B_{2}^{n};\bigwedge^{k}\mathbb{R}^{n})$ is well-defined for
$k=0,\ldots,n$.

### Estimates for normalized pull-backs

Next, we define the normalized pull-back of a differential form under a
quasiregular curve.

Let $N$ be a closed, connected, and oriented Riemannian manifold and let
$\omega\in\Omega^{n}(N)$ be a closed non-exact form with $2\leqslant
n\leqslant\dim N$. Let $K,D\geqslant 1$. For each
$F\in\mathcal{F}_{K,D}(N,\omega)$, let
$F^{!}\colon\Omega^{*}(N)\to\bigoplus_{k=0}^{n}L^{\frac{n}{k}}(B_{2}^{n};{\bigwedge}^{k}\mathbb{R}^{n})$
be the linear map

$F^{!}\alpha=A_{\omega}(F)^{-\frac{k}{n}}F^{*}\alpha\text{ for
}\alpha\in\Omega^{k}(N)\text{ and }k=0,\ldots,n;$

recall that

$A_{\omega}(F)=\int_{B^{n}}F^{*}\omega.$

We note that $F^{!}$ commutes with the exterior product and that $F^{!}\circ
h=F^{\\#}$.

The following elementary result gives a bound for the operator norm of
$F^{!}$. The proof is similar to the proof of [6, Lemma 3.1], so we omit the
details.

######  Lemma 3.3.

Let $N$ be a closed, connected, and oriented Riemannian manifold and let
$\omega\in\Omega^{n}(N)$ be a closed non-exact form with $2\leqslant
n\leqslant\dim N$. Let $K,D\geqslant 1$. Let
$F\in\mathcal{F}_{K,D}(N,\omega)$. Then

$\lVert F^{!}\alpha\rVert_{\frac{n}{k},B_{2}^{n}}\leqslant
KD^{2}\lVert\alpha\rVert_{\infty}$

for every $\alpha\in\Omega^{k}(N)$ and $k=0,\ldots,n$.

Lemma 3.3 implies that exact forms vanish in a weak sense under limits of
operators $F_{j}^{!}$ for sequences $(F_{j})$ in $\mathcal{F}_{K,D}(N,\omega)$
satisfying $A_{\omega}(F_{j})\to\infty$. Since the proof is similar to the
proof of [6, Lemma 3.2], we omit the details.

######  Lemma 3.4.

Let $N$ be a closed, connected, and oriented Riemannian manifold and let
$\omega\in\Omega^{n}(N)$ be a closed non-exact form with $2\leqslant
n\leqslant\dim N$. Let $K,D\geqslant 1$. Let $(F_{j})$ be a sequence in
$\mathcal{F}_{K,D}(N,\omega)$ satisfying $A_{\omega}(F_{j})\to\infty$. Then

$\lim_{j\to\infty}\int_{B_{2}^{n}}\varphi\wedge F_{j}^{!}d\alpha=0$

for $\alpha\in\Omega^{k}(N)$, $\varphi\in\Omega_{0}^{n-k-1}(B_{2}^{n})$, and
$k=0,\ldots,n-1$.

##  4\. Sobolev–Poincaré limits and their properties

In this section, our end goal is to prove Theorem 1.11. The proof is based on
Sobolev–Poincaré limits.

### Motivation and definition

We begin by recalling the Poincaré operator of Iwaniec and Lutoborski [8]:
_There exists a graded compact linear operator_

$T\colon\bigoplus_{k=1}^{n-1}L^{\frac{n}{k}}(B_{2}^{n};{\bigwedge}^{k}\mathbb{R}^{n})\to\bigoplus_{k=1}^{n-1}L^{\frac{n}{k}}(B_{2}^{n};{\bigwedge}^{k-1}\mathbb{R}^{n})$

_satisfying ${\operatorname{id}}=dT+Td$. In particular, $dT\omega=\omega$ for
every weakly closed form
$\omega\in\oplus_{k=1}^{n-1}L^{\frac{n}{k}}(B_{2}^{n};\bigwedge^{k}\mathbb{R}^{n})$.
_

Next, we show that, for a sequence $(F_{j})$ in $\mathcal{F}_{K,D}(N,\omega)$,
a weak limit of $(F_{j}^{\\#})$ is the weak exterior derivative of a limit of
$(TF_{j_{i}}^{\\#})$ for some subsequence. Before stating a precise
formulation, we recall for the reader’s convenience that, for
$F\in\mathcal{F}_{K,D}(N,\omega)$,

$F^{\\#}\colon
H_{\mathrm{dR}}^{*}(N)\to\mathscr{W}^{*}(B_{2}^{n}),\;H_{\mathrm{dR}}^{k}(N)\ni
c\mapsto A_{\omega}(F)^{-\frac{k}{n}}F^{*}(h(c))=F^{!}(h(c)),$

where $h\colon H_{\mathrm{dR}}^{*}(N)\to\Omega^{*}(N)$ is the choice of
harmonic representative. The proof is similar to the proof of [6, Lemma 4.1],
so we omit the details.

######  Lemma 4.1.

Let $N$ be a closed, connected, and oriented Riemannian manifold and let
$\omega\in\Omega^{n}(N)$ be a closed non-exact form with $2\leqslant
n\leqslant\dim N$. Let $K,D\geqslant 1$. Let $(F_{j})$ be a sequence in
$\mathcal{F}_{K,D}(N,\omega)$ for which $F_{j}^{\\#}\rightharpoonup L$, where

$F_{j}^{\\#},L\colon\bigoplus_{k=1}^{n-1}H_{\mathrm{dR}}^{k}(N)\to\bigoplus_{k=1}^{n-1}L^{\frac{n}{k}}(B_{2}^{n};{\bigwedge}^{k}\mathbb{R}^{n}).$

Then there exists a subsequence $(F_{j_{i}})$ and a graded linear operator

$\widehat{L}\colon\bigoplus_{k=1}^{n-1}H_{\mathrm{dR}}^{k}(N)\to\bigoplus_{k=1}^{n-1}W^{d,\frac{n}{k}}(B_{2}^{n};{\bigwedge}^{k-1}\mathbb{R}^{n})$

satisfying $TF_{j_{i}}^{\\#}\to\widehat{L}$. In particular, $d\widehat{L}=L$.

We are now ready to give the definition of a Sobolev–Poincaré limit. Let $N$
be a closed, connected, and oriented Riemannian manifold and let
$\omega\in\Omega^{n}(N)$ be a closed non-exact form with $2\leqslant
n\leqslant\dim N$. Let $K,D\geqslant 1$. Let $(F_{j})$ be a sequence in
$\mathcal{F}_{K,D}(N,\omega)$. We say that a graded linear operator

$\widehat{L}\colon\bigoplus_{k=1}^{n-1}H_{\mathrm{dR}}^{k}(N)\to\bigoplus_{k=1}^{n-1}W^{d,\frac{n}{k}}(B_{2}^{n};{\bigwedge}^{k-1}\mathbb{R}^{n})$

is a _Sobolev–Poincaré limit of $(F_{j}^{\\#})$_ if $d\widehat{L}$ is the weak
limit of the operators $F_{j}^{\\#}$ and $\widehat{L}$ is the limit of the
operators $TF_{j}^{\\#}$.

### Vague limits of products

The weak exterior derivative of a Sobolev–Poincaré limit for a sequence whose
normalizations grow without bound commutes with the exterior product. We begin
by showing the commutativity in a weak sense without assumptions on the growth
of the normalizations; see also [6, Proposition 4.2] and [12, Lemma 4.2]. For
a precise statement, we recall that a sequence $(\mu_{j})$ of Radon measures
on $B_{2}^{n}$ _converges vaguely to a Radon measure $\mu$ on $B_{2}^{n}$_ if

$\lim_{j\to\infty}\int_{B_{2}^{n}}\varphi\,\mathrm{d}\mu_{j}=\int_{B_{2}^{n}}\varphi\,\mathrm{d}\mu$

for every $\varphi\in C_{0}(B_{2}^{n})$.

######  Proposition 4.2.

Let $N$ be a closed, connected, and oriented Riemannian manifold and let
$\omega\in\Omega^{n}(N)$ be a closed non-exact form with $2\leqslant
n\leqslant\dim N$. Let $K,D\geqslant 1$. Let $(F_{j})$ be a sequence in
$\mathcal{F}_{K,D}(N,\omega)$ and suppose that the sequence $(F_{j}^{\\#})$
has a Sobolev–Poincaré limit
$\widehat{L}\colon\oplus_{k=1}^{n-1}H_{\mathrm{dR}}^{k}(N)\to\oplus_{k=1}^{n-1}W^{d,\frac{n}{k}}(B_{2}^{n};{\bigwedge}^{k-1}\mathbb{R}^{n})$.
Let $k_{1},k_{2}\in\\{1,\ldots,n-1\\}$ for which $k:=k_{1}+k_{2}\leqslant n$.
Then, for every $c_{i}\in H_{\mathrm{dR}}^{k_{i}}(N)$, we have

$F_{j}^{\\#}c_{1}\wedge F_{j}^{\\#}c_{2}\rightharpoonup
d\widehat{L}(c_{1})\wedge d\widehat{L}(c_{2}).$

Here, the weak convergence is the usual weak convergence in
$L^{\frac{n}{k}}(B_{2}^{n};\bigwedge^{k}\mathbb{R}^{n})$ if $k<n$ and the
vague convergence of measures if $k=n$.

###### Proof.

Let $c_{i}\in H_{\mathrm{dR}}^{k_{i}}(N)$. By Lemma 3.3, the sequence
$(F_{j}^{\\#}c_{1}\wedge F_{j}^{\\#}c_{2})=(F_{j}^{!}(h(c_{1})\wedge
h(c_{2})))$ is bounded in
$L^{\frac{n}{k}}(B_{2}^{n};\bigwedge^{k}\mathbb{R}^{n})$. Moreover,
$\Omega_{0}^{n-k}(B_{2}^{n})$ is dense in
$L^{\frac{n}{k}}(B_{2}^{n};\bigwedge^{k}\mathbb{R}^{n})$ with respect to the
$L^{\frac{n}{k}}$-norm if $k<n$ and $C_{0}^{\infty}(B_{2}^{n})$ is dense in
$C_{0}(B_{2}^{n})$ with respect to the $L^{\infty}$-norm. Thus, it suffices to
prove that

$\lim_{j\to\infty}\int_{B_{2}^{n}}\varphi\wedge F_{j}^{\\#}c_{1}\wedge
F_{j}^{\\#}c_{2}=\int_{B_{2}^{n}}\varphi\wedge d\widehat{L}c_{1}\wedge
d\widehat{L}c_{2}$

for every $\varphi\in\Omega_{0}^{n-k}(B_{2}^{n})$.

Let $\varphi\in\Omega_{0}^{n-k}(B_{2}^{n})$. By the triangle inequality, it
suffices to estimate

$\left\lvert\int_{B_{2}^{n}}\varphi\wedge(d\widehat{L}c_{1}-F_{j}^{\\#}c_{1})\wedge
d\widehat{L}c_{2}\right\rvert\text{ and
}\left\lvert\int_{B_{2}^{n}}\varphi\wedge
F_{j}^{\\#}c_{1}\wedge(d\widehat{L}c_{2}-F_{j}^{\\#}c_{2})\right\rvert.$

Since

$\varphi\wedge d\widehat{L}c_{2}\in
L^{\frac{n}{k_{2}}}(B_{2}^{n};{\bigwedge}^{n-k_{1}}\mathbb{R}^{n})\subset
L^{\frac{n}{n-k_{1}}}(B_{2}^{n};{\bigwedge}^{n-k_{1}}\mathbb{R}^{n})$

and $F_{j}^{\\#}c_{1}\rightharpoonup d\widehat{L}c_{1}$ in
$L^{\frac{n}{k_{1}}}(B_{2}^{n};\bigwedge^{k_{1}}\mathbb{R}^{n})$, the first
term converges to zero.

For the second term, observe that $dTF_{j}^{\\#}c_{2}=F_{j}^{\\#}c_{2}$. Thus,
we have the estimate

$\displaystyle\left\lvert\int_{B_{2}^{n}}\varphi\wedge
F_{j}^{\\#}c_{1}\wedge(d\widehat{L}c_{2}-F_{j}^{\\#}c_{2})\right\rvert$
$\displaystyle=\left\lvert\int_{B_{2}^{n}}\varphi\wedge F_{j}^{\\#}c_{1}\wedge
d(\widehat{L}c_{2}-TF_{j}^{\\#}c_{2})\right\rvert$
$\displaystyle=\left\lvert\int_{B_{2}^{n}}d\varphi\wedge
F_{j}^{\\#}c_{1}\wedge(\widehat{L}c_{2}-TF_{j}^{\\#}c_{2})\right\rvert$
$\displaystyle\leqslant C(n)\lVert
d\varphi\rVert_{\infty}\int_{B_{2}^{n}}\lVert
F_{j}^{\\#}c_{1}\rVert\lVert\widehat{L}c_{2}-TF_{j}^{\\#}c_{2}\rVert.$

By Hölder’s inequality and Lemma 3.3, we may estimate further by

$\displaystyle\int_{B_{2}^{n}}\lVert
F_{j}^{\\#}c_{1}\rVert\lVert\widehat{L}c_{2}-TF_{j}^{\\#}c_{2}\rVert$
$\displaystyle\quad\leqslant\lVert
F_{j}^{\\#}c_{1}\rVert_{\frac{n}{n-k_{2}},B_{2}^{n}}\lVert\widehat{L}c_{2}-TF_{j}^{\\#}c_{2}\rVert_{\frac{n}{k_{2}},B_{2}^{n}}$
$\displaystyle\quad\leqslant C(n,k_{1},k_{2})\lVert
F_{j}^{\\#}c_{1}\rVert_{\frac{n}{k_{1}},B_{2}^{n}}\lVert\widehat{L}c_{2}-TF_{j}^{\\#}c_{2}\rVert_{\frac{n}{k_{2}},B_{2}^{n}}$
$\displaystyle\quad\leqslant C(n,k_{1},k_{2})KD^{2}\lVert
h(c_{1})\rVert_{\infty}\lVert\widehat{L}c_{2}-TF_{j}^{\\#}c_{2}\rVert_{\frac{n}{k_{2}},B_{2}^{n}}.$

Since $TF_{j}^{\\#}c_{2}\to\widehat{L}c_{2}$ in
$L^{\frac{n}{k_{2}}}(B_{2}^{n};\bigwedge^{k_{2}}\mathbb{R}^{n})$, the claim
follows. ∎

In the next result, we show that the weak commutativity of the weak exterior
derivative of a Sobolev–Poincaré limit with the exterior product improves into
commutativity, provided that the normalizations grow without bound. The
argument in the proof is analogous to the one in the proof of [6, Lemma 4.3]
and we omit the details.

######  Lemma 4.3.

Let $N$ be a closed, connected, and oriented Riemannian manifold and let
$\omega\in\Omega^{n}(N)$ be a closed non-exact form with $2\leqslant
n\leqslant\dim N$. Let $K,D\geqslant 1$. Let $(F_{j})$ be a sequence in
$\mathcal{F}_{K,D}(N,\omega)$ for which $A_{\omega}(F_{j})\to\infty$ and
suppose that the sequence $(F_{j}^{\\#})$ has a Sobolev–Poincaré limit
$\widehat{L}$. Then

$d\widehat{L}(c_{1}\wedge c_{2})=d\widehat{L}c_{1}\wedge d\widehat{L}c_{2}$

for every $c_{1}\in H_{\mathrm{dR}}^{k_{1}}(N)$ and $c_{2}\in
H_{\mathrm{dR}}^{k_{2}}(N)$ with $k_{1},k_{2}\in\\{1,\ldots,n-1\\}$ satisfying
$k_{1}+k_{2}<n$.

### Extension to $K^{n}(N)$

In the setting of quasiregular mappings in [6], weak exterior derivatives of
Sobolev–Poincare limits are extended to $H_{\mathrm{dR}}^{\dim
N}(N)=\mathrm{span}_{\mathbb{R}}([\mathrm{vol}_{N}]_{\mathrm{dR}})$. Next, we
show that, in the setting of this paper, the weak exterior derivative of a
Sobolev–Poincare limit has a well-defined extension to $K^{n}(N)$ arising from
the algebraic structure of $K^{n}(N)$.

######  Lemma 4.4.

Let $N$ be a closed, connected, and oriented Riemannian manifold and let
$\omega\in\Omega^{n}(N)$ be a closed non-exact form with $2\leqslant
n\leqslant\dim N$. Let $K,D\geqslant 1$. Let $(F_{j})$ be a sequence in
$\mathcal{F}_{K,D}(N,\omega)$ for which $A_{\omega}(F_{j})\to\infty$ and
suppose that the sequence $(F_{j}^{\\#})$ has a Sobolev–Poincaré limit
$\widehat{L}$. Then the mapping

$d\widehat{L}_{n}\colon K^{n}(N)\to
L^{1}(B_{2}^{n};{\bigwedge}^{n}\mathbb{R}^{n}),\quad\sum_{i=1}^{m}c_{i}\wedge
c_{i}^{\prime}\mapsto\sum_{i=1}^{m}d\widehat{L}c_{i}\wedge
d\widehat{L}c_{i}^{\prime},$

is a well-defined linear map.

###### Proof.

Let $k_{1},\ldots,k_{m},\ell_{1},\ldots,\ell_{\nu}\in\\{1,\ldots,n-1\\}$. Let
$c_{i}\in H_{\mathrm{dR}}^{k_{i}}(N)$, $c_{i}^{\prime}\in
H_{\mathrm{dR}}^{n-k_{i}}(N)$, $e_{i}\in H_{\mathrm{dR}}^{\ell_{i}}(N)$, and
$e_{i}^{\prime}\in H_{\mathrm{dR}}^{n-\ell_{i}}(N)$ be de Rham classes
satisfying $\sum_{i=1}^{m}c_{i}\wedge
c_{i}^{\prime}=\sum_{i=1}^{\nu}e_{i}\wedge e_{i}^{\prime}$.

Let $\alpha\in\Omega^{n-1}(N)$ be a form satisfying

$\sum_{i=1}^{m}h(c_{i})\wedge h(c_{i}^{\prime})=\sum_{i=1}^{\nu}h(e_{i})\wedge
h(e_{i}^{\prime})+d\alpha$

and let $\varphi\in C_{0}^{\infty}(B_{2}^{n})$. By Proposition 4.2, we obtain

$\displaystyle\int_{B_{2}^{n}}\varphi\left(\sum_{i=1}^{m}d\widehat{L}c_{i}\wedge
d\widehat{L}c_{i}^{\prime}\right)$
$\displaystyle=\sum_{i=1}^{m}\lim_{j\to\infty}\int_{B_{2}^{n}}\varphi
F_{j}^{\\#}c_{i}\wedge F_{j}^{\\#}c_{i}^{\prime}$
$\displaystyle=\lim_{j\to\infty}\sum_{i=1}^{m}\int_{B_{2}^{n}}\varphi
F_{j}^{\\#}c_{i}\wedge F_{j}^{\\#}c_{i}^{\prime}$
$\displaystyle=\lim_{j\to\infty}\int_{B_{2}^{n}}\varphi\sum_{i=1}^{m}F_{j}^{!}(h(c_{i})\wedge
h(c_{i}^{\prime}))$ $\displaystyle=\lim_{j\to\infty}\int_{B_{2}^{n}}\varphi
F_{j}^{!}\left(\sum_{i=1}^{m}h(c_{i})\wedge h(c_{i}^{\prime})\right).$

A similar computation yields that

$\int_{B_{2}^{n}}\varphi\left(\sum_{i=1}^{\nu}d\widehat{L}e_{i}\wedge
d\widehat{L}e_{i}^{\prime}\right)=\lim_{j\to\infty}\int_{B_{2}^{n}}\varphi
F_{j}^{!}\left(\sum_{i=1}^{\nu}h(e_{i})\wedge h(e_{i}^{\prime})\right).$

Since, by Lemma 3.4, we have

$\displaystyle\left\lvert\int_{B_{2}^{n}}\varphi
F_{j}^{!}\left(\sum_{i=1}^{m}h(c_{i})\wedge
h(c_{i}^{\prime})\right)-\int_{B_{2}^{n}}\varphi
F_{j}^{!}\left(\sum_{i=1}^{\nu}h(e_{i})\wedge
h(e_{i}^{\prime})\right)\right\rvert$
$\displaystyle\quad=\left\lvert\int_{B_{2}^{n}}\varphi
F_{j}^{!}d\alpha\right\rvert\to 0,$

it follows that

$\sum_{i=1}^{m}d\widehat{L}c_{i}\wedge
d\widehat{L}c_{i}^{\prime}=\sum_{i=1}^{\nu}d\widehat{L}e_{i}\wedge
d\widehat{L}e_{i}^{\prime}.$

Hence, $d\widehat{L}_{n}$ is well-defined. ∎

The following proposition states that extending $d\widehat{L}$ with
$d\widehat{L}_{n}\colon K^{n}(N)\to
L^{1}(B_{2}^{n};\bigwedge^{n}\mathbb{R}^{n})$ is compatible with vague
convergence of measures on $B_{2}^{n}$; see also [6, Theorem 4.4].

######  Proposition 4.5.

Let $N$ be a closed, connected, and oriented Riemannian manifold and let
$\omega\in\Omega^{n}(N)$ be a closed non-exact form with $2\leqslant
n\leqslant\dim N$. Let $K,D\geqslant 1$. Let $(F_{j})$ be a sequence in
$\mathcal{F}_{K,D}(N,\omega)$ for which $A_{\omega}(F_{j})\to\infty$ and
suppose that the sequence $(F_{j}^{\\#})$ has a Sobolev–Poincaré limit
$\widehat{L}$. Let $\alpha\in\Omega^{n}(N)$ be a closed form satisfying
$[\alpha]_{\mathrm{dR}}\in K^{n}(N)$. Then

$\int_{B_{2}^{n}}\varphi
d\widehat{L}_{n}[\alpha]_{\mathrm{dR}}=\lim_{j\to\infty}\int_{B_{2}^{n}}\varphi
F_{j}^{!}\alpha$

for every $\varphi\in C_{0}(B_{2}^{n})$. In particular,
$F_{j}^{!}\alpha\rightharpoonup d\widehat{L}_{n}[\alpha]_{\mathrm{dR}}$
vaguely as measures and, if each $F_{j}^{!}\alpha$ is non-negative, then
$d\widehat{L}_{n}[\alpha]_{\mathrm{dR}}$ is non-negative.

###### Proof.

Let $[\alpha]_{\mathrm{dR}}=\sum_{i=1}^{m}c_{i}\wedge c_{i}^{\prime}$ be a
representation of $[\alpha]_{\mathrm{dR}}$ in $K^{n}(N)$ and let
$\beta\in\Omega^{n-1}(N)$ be a form satisfying

$\sum_{i=1}^{m}h(c_{i})\wedge h(c_{i}^{\prime})=\alpha+d\beta.$

Since $\lVert F_{j}^{!}\alpha\rVert_{1,B_{2}^{n}}\leqslant
KD^{2}\lVert\alpha\rVert_{\infty}$ by Lemma 3.3 and
$C_{0}^{\infty}(B_{2}^{n})$ is dense in $C_{0}(B_{2}^{n})$ with respect to the
$L^{\infty}$-norm, it suffices to prove that

$\int_{B_{2}^{n}}\varphi
d\widehat{L}_{n}[\alpha]_{\mathrm{dR}}=\lim_{j\to\infty}\int_{B_{2}^{n}}\varphi
F_{j}^{!}\alpha$

for every $\varphi\in C_{0}^{\infty}(B_{2}^{n})$.

By Proposition 4.2 and Lemma 3.4, we have

$\displaystyle\int_{B_{2}^{n}}\varphi d\widehat{L}_{n}[\alpha]_{\mathrm{dR}}$
$\displaystyle\quad=\int_{B_{2}^{n}}\varphi\left(\sum_{i=1}^{m}d\widehat{L}c_{i}\wedge
d\widehat{L}c_{i}^{\prime}\right)=\sum_{i=1}^{m}\lim_{j\to\infty}\int_{B_{2}^{n}}\varphi
F_{j}^{\\#}c_{i}\wedge F_{j}^{\\#}c_{i}^{\prime}$
$\displaystyle\quad=\lim_{j\to\infty}\int_{B_{2}^{n}}\varphi\sum_{i=1}^{m}F_{j}^{!}(h(c_{i})\wedge
h(c_{i}^{\prime}))=\lim_{j\to\infty}\int_{B_{2}^{n}}\varphi
F_{j}^{!}(\alpha+d\beta)$
$\displaystyle\quad=\lim_{j\to\infty}\int_{B_{2}^{n}}\varphi F_{j}^{!}\alpha$

for every $\varphi\in C_{0}^{\infty}(B_{2}^{n})$. This concludes the proof. ∎

For $\alpha=\omega$, we get the following improved version of Proposition 4.5;
see also [6, Corollary 4.5].

######  Corollary 4.6.

Let $N$ be a closed, connected, and oriented Riemannian manifold. Let
$2\leqslant n\leqslant\dim N$ and let $\omega\in\Omega^{n}(N)$ be a closed
form with $0\neq[\omega]_{\mathrm{dR}}\in K^{n}(N)$. Let $K,D\geqslant 1$. Let
$(F_{j})$ be a sequence in $\mathcal{F}_{K,D}(N,\omega)$ for which
$A_{\omega}(F_{j})\to\infty$ and suppose that the sequence $(F_{j}^{\\#})$ has
a Sobolev–Poincaré limit $\widehat{L}$. Then there exists a non-negative
measure $\mu\in L^{1}(B_{2}^{n};\bigwedge^{n}\mathbb{R}^{n})$ having following
properties:

  1. (1)

$\mu=d\widehat{L}_{n}[\omega]_{\mathrm{dR}}$, where $d\widehat{L}_{n}$ is the
extension of $d\widehat{L}$ to $K^{n}(N)$ as defined in Lemma 4.4,

  2. (2)

$F_{j}^{!}\omega\rightharpoonup\mu$ vaguely as measures on $B_{2}^{n}$,

  3. (3)

$F_{j}^{!}\omega\llcorner\bar{B}^{n}\rightharpoonup\mu\llcorner\bar{B}^{n}$
weakly as measures on $\bar{B}^{n}$, and

  4. (4)

$\mu(B^{n})=1$.

######  Remark 4.7.

The assumption $[\omega]_{\mathrm{dR}}\in K^{n}(N)$ is necessary; see e.g. the
example in [6, Section 8].

###### Proof of Corollary 4.6.

By Proposition 4.5, there exists a non-negative measure $\mu\in
L^{1}(B_{2}^{n};\bigwedge^{n}\mathbb{R}^{n})$ satisfying (1) and (2). Thus, it
suffices to show that $\mu$ satisfies also (3) and (4).

Let $\varphi\in C(\bar{B}^{n})$ be a non-negative function. Let
$\tilde{\varphi}\in C_{b}(B_{2}^{n})$ be a non-negative bounded function
satisfying $\tilde{\varphi}|_{\bar{B}^{n}}=\varphi$. Then
$\tilde{\varphi}F_{j}^{!}\omega\rightharpoonup\tilde{\varphi}\mu$ vaguely as
measures on $B_{2}^{n}$.

Let $0<r<1<R<2$. Let $\psi_{r}\in C_{0}(B^{n})$ and $\psi_{R}\in
C_{0}(B^{n}(0,R))$ satisfy $\chi_{B^{n}(0,r)}\leqslant\psi_{r}\leqslant 1$ and
$\chi_{B^{n}}\leqslant\psi_{R}\leqslant 1$. Since $\tilde{\varphi}\mu$ is non-
negative, we obtain

$\int_{\bar{B}^{n}(0,r)}\tilde{\varphi}\mu\leqslant\int_{B_{2}^{n}}\psi_{r}\tilde{\varphi}\mu=\lim_{j\to\infty}\int_{B_{2}^{n}}\psi_{r}\tilde{\varphi}F_{j}^{!}\omega\leqslant\liminf_{j\to\infty}\int_{B^{n}}\tilde{\varphi}F_{j}^{!}\omega$

and

$\int_{B^{n}(0,R)}\tilde{\varphi}\mu\geqslant\int_{B_{2}^{n}}\psi_{R}\tilde{\varphi}\mu=\lim_{j\to\infty}\int_{B_{2}^{n}}\psi_{R}\tilde{\varphi}F_{j}^{!}\omega\geqslant\limsup_{j\to\infty}\int_{B^{n}}\tilde{\varphi}F_{j}^{!}\omega.$

Since $\mu\in L^{1}(B_{2}^{n};\bigwedge^{n}\mathbb{R}^{n})$,

$\liminf_{j\to\infty}\int_{B^{n}}\tilde{\varphi}F_{j}^{!}\omega\geqslant\tilde{\varphi}\mu(B^{n})=\tilde{\varphi}\mu(\bar{B}^{n})\geqslant\limsup_{j\to\infty}\int_{B^{n}}\tilde{\varphi}F_{j}^{!}\omega.$

Thus

(4.1) $\lim_{j\to\infty}\int_{\bar{B}^{n}}\varphi
F_{j}^{!}\omega=\int_{\bar{B}^{n}}\varphi\mu.$

Equality (4.1) is then obtained for an arbitrary function $\varphi\in
C(\bar{B}^{n})$ by writing $\varphi=\max(\varphi,0)-\max(-\varphi,0)$.

Especially, since $\mu\in L^{1}(B_{2}^{n};\bigwedge^{n}\mathbb{R}^{n})$, we
have, for $\varphi=1$, that

$\displaystyle\mu(B^{n})$
$\displaystyle=\int_{\bar{B}^{n}}\mu=\lim_{j\to\infty}\int_{\bar{B}^{n}}F_{j}^{!}\omega=\lim_{j\to\infty}\int_{B^{n}}F_{j}^{!}\omega$
$\displaystyle=\lim_{j\to\infty}A_{\omega}(F_{j})^{-1}\int_{B^{n}}F_{j}^{*}\omega=1.$

This concludes the proof. ∎

### Homomorphism $H_{\mathrm{dR}}^{*}(N)\to\mathscr{W}^{*}(B_{2}^{n})$

Next, we show that, if the normalizations grow without bound, then the weak
exterior derivative of a Sobolev–Poincare limit extends to a homomorphism
$H_{\mathrm{dR}}^{*}(N)\to\mathscr{W}^{*}(B_{2}^{n})$; see also [6,
Proposition 5.1].

For a precise statement, we introduce the following notation. Let $N$ be a
closed, connected, and oriented Riemannian manifold and let
$\omega\in\Omega^{n}(N)$ be a closed non-exact form with $2\leqslant
n\leqslant\dim N$. Let $\pi\colon H_{\mathrm{dR}}^{n}(N)\to
H_{\mathrm{dR}}^{n}(N)$ be a projection onto $K^{n}(N)$, i.e., a linear map
satisfying $\pi\circ\pi=\pi$ and $\mathrm{im}\,\pi=K^{n}(N)$.

Let $K,D\geqslant 1$. Let $(F_{j})$ be a sequence in
$\mathcal{F}_{K,D}(N,\omega)$ for which $A_{\omega}(F_{j})\to\infty$ and let
$\widehat{L}$ be the Sobolev–Poincaré limit of $(F_{j}^{\\#})$. Let

$\Psi_{d\widehat{L}}^{\pi}\colon
H_{\mathrm{dR}}^{*}(N)\to\mathscr{W}^{*}(B_{2}^{n})$

be the graded linear map defined by

$H_{\mathrm{dR}}^{k}(N)\ni c\mapsto\begin{cases}\text{constant function
}h(c),&\,\text{if }k=0,\\\ d\widehat{L}c,&\,\text{if }1\leqslant k\leqslant
n-1,\\\ d\widehat{L}_{n}\pi c,&\,\text{if }k=n,\\\
0,&\,\text{otherwise.}\end{cases}$

For the reader’s convenience, we recall that $d\widehat{L}_{n}\colon
K^{n}(N)\to L^{1}(B_{2}^{n};\bigwedge^{n}\mathbb{R}^{n})$ is the linear map

$\sum_{i=1}^{m}c_{i}\wedge
c_{i}^{\prime}\mapsto\sum_{i=1}^{m}d\widehat{L}c_{i}\wedge
d\widehat{L}c_{i}^{\prime}.$

######  Proposition 4.8.

Let $N$ be a closed, connected, and oriented Riemannian manifold and let
$\omega\in\Omega^{n}(N)$ be a closed non-exact form with $2\leqslant
n\leqslant\dim N$. Let $\pi\colon H_{\mathrm{dR}}^{n}(N)\to
H_{\mathrm{dR}}^{n}(N)$ be a projection onto $K^{n}(N)$. Let $K,D\geqslant 1$.
Let $(F_{j})$ be a sequence in $\mathcal{F}_{K,D}(N,\omega)$ for which
$A_{\omega}(F_{j})\to\infty$ and suppose that the sequence $(F_{j}^{\\#})$ has
a Sobolev–Poincaré limit $\widehat{L}$. Then $\Psi_{d\widehat{L}}^{\pi}\colon
H_{\mathrm{dR}}^{*}(N)\to\mathscr{W}^{*}(B_{2}^{n})$ is a homomorphism.

###### Proof.

Let $k_{1},k_{2}\in\\{0,\ldots,\dim N\\}$ and let $c_{i}\in
H_{\mathrm{dR}}^{k_{i}}(N)$. Without loss of generality, we may assume that
$k_{1},k_{2}>0$ and that $k_{1}+k_{2}\leqslant n$.

If $k_{1}+k_{2}<n$, then, by Lemma 4.3, we have

$\Psi_{d\widehat{L}}^{\pi}(c_{1}\wedge c_{2})=d\widehat{L}(c_{1}\wedge
c_{2})=d\widehat{L}c_{1}\wedge
d\widehat{L}c_{2}=\Psi_{d\widehat{L}}^{\pi}c_{1}\wedge\Psi_{d\widehat{L}}^{\pi}c_{2}.$

Suppose $k_{1}+k_{2}=n$. Since $\pi|_{K^{n}(N)}={\operatorname{id}}$, we
obtain

$\Psi_{d\widehat{L}}^{\pi}(c_{1}\wedge c_{2})=d\widehat{L}_{n}\pi(c_{1}\wedge
c_{2})=d\widehat{L}_{n}(c_{1}\wedge c_{2})=d\widehat{L}c_{1}\wedge
d\widehat{L}c_{2}=\Psi_{d\widehat{L}}^{\pi}c_{1}\wedge\Psi_{d\widehat{L}}^{\pi}c_{2}.$

The claim follows by linearity. ∎

We are now ready to prove the following generalization of Theorem 1.11.

######  Theorem 4.9.

Let $N$ be a closed, connected, and oriented Riemannian manifold and let
$\omega\in\Omega^{n}(N)$ be a closed non-exact form with $2\leqslant
n\leqslant\dim N$. Let $K,D\geqslant 1$. Let $(F_{j})$ be a sequence in
$\mathcal{F}_{K,D}(N,\omega)$ for which $A_{\omega}(F_{j})\to\infty$ and
$F_{j}^{\\#}\rightharpoonup L$, where

$F_{j}^{\\#},L\colon\bigoplus_{k=1}^{n-1}H_{\mathrm{dR}}^{k}(N)\to\bigoplus_{k=1}^{n-1}L^{\frac{n}{k}}(B_{2}^{n};{\bigwedge}^{k}\mathbb{R}^{n}).$

Then $L$ extends to a graded algebra homomorphism $L\colon
H_{\mathrm{dR}}^{*}(N)\to\mathscr{W}^{*}(B_{2}^{n})$ for which
$F_{j}^{\\#}|_{K^{n}(N)}\rightharpoonup L|_{K^{n}(N)}$ vaguely as measures.
Moreover, if $[\omega]_{\mathrm{dR}}\in K^{n}(N)$, then

$\int_{B^{n}}L[\omega]_{\mathrm{dR}}=1\;\text{ and }\;Lc\neq 0\text{ if
}c\text{ is a factor of }[\omega]_{\mathrm{dR}},$

i.e., $c\in H_{\mathrm{dR}}^{*}(N)$ and there exists $c^{\prime}\in
H_{\mathrm{dR}}^{n-*}(N)$ for which $c\wedge
c^{\prime}=[\omega]_{\mathrm{dR}}$.

###### Proof.

By Lemma 4.1, there exists a subsequence $(F_{j_{i}})$ for which the operator
$L$ is the weak exterior derivative $d\widehat{L}$ of a Sobolev–Poincaré limit
$\widehat{L}$ of $(F_{j_{i}}^{\\#})$. Let $\pi\colon H_{\mathrm{dR}}^{n}(N)\to
H_{\mathrm{dR}}^{n}(N)$ be a projection onto $K^{n}(N)$. Then, by Proposition
4.8, the map $\Psi_{d\widehat{L}}^{\pi}\colon
H_{\mathrm{dR}}^{*}(N)\to\mathscr{W}^{*}(B_{2}^{n})$ is a graded algebra
homomorphism and $\Psi_{d\widehat{L}}^{\pi}$ extends $L$ by definition.
Furthermore, for each $c\in K^{n}(N)$, we have

$F_{j}^{\\#}c=F_{j}^{!}(h(c))\rightharpoonup
d\widehat{L}_{n}c=d\widehat{L}_{n}\pi c=\Psi_{d\widehat{L}}^{\pi}c$

vaguely as measures by Proposition 4.5.

Suppose that $[\omega]_{\mathrm{dR}}\in K^{n}(N)$. By Corollary 4.6, we obtain

$\int_{B^{n}}\Psi_{d\widehat{L}}^{\pi}[\omega]_{\mathrm{dR}}=\int_{B^{n}}d\widehat{L}_{n}\pi[\omega]_{\mathrm{dR}}=\int_{B^{n}}d\widehat{L}_{n}[\omega]_{\mathrm{dR}}=1.$

Hence, it remains to prove that $\Psi_{d\widehat{L}}^{\pi}c\neq 0$ if $c$ is a
factor of $[\omega]_{\mathrm{dR}}$.

Let $c\in H_{\mathrm{dR}}^{k}(N)$ and $c^{\prime}\in H_{\mathrm{dR}}^{n-k}(N)$
satisfy $c\wedge c^{\prime}=[\omega]_{\mathrm{dR}}$, where
$k\in\\{0,\ldots,n\\}$. Then, we have

$\Psi_{d\widehat{L}}^{\pi}c\wedge\Psi_{d\widehat{L}}^{\pi}c^{\prime}=\Psi_{d\widehat{L}}^{\pi}(c\wedge
c^{\prime})=\Psi_{d\widehat{L}}^{\pi}[\omega]_{\mathrm{dR}}\neq 0$

implying $\Psi_{d\widehat{L}}^{\pi}c\neq 0$. ∎

##  5\. Proof of Theorem 1.10

In this section, we recall the statement of Theorem 1.10 and give its proof.

######  Theorem 1.10.

Let $N$ be a closed, connected, and oriented Riemannian manifold. Let
$2\leqslant n\leqslant\dim N$ and let $\omega\in\Omega^{n}(N)$ be a closed
form with $0\neq[\omega]_{\mathrm{dR}}\in K^{n}(N)$. Assume also that there
exists $K,D\geqslant 1$ for which

$\sup_{F\in\mathcal{F}_{K,D}(N,\omega)}A_{\omega}(F)=\infty.$

Then there exists a graded algebra homomorphism $\Phi\colon
H_{\mathrm{dR}}^{*}(N)\to\bigwedge^{*}\mathbb{R}^{n}$ for which
$\Phi[\omega]_{\mathrm{dR}}\neq 0$.

Before giving the proof of Theorem 1.10, we state the following elementary
lemma.

######  Lemma 5.1.

Let $N$ be a closed, connected, and oriented Riemannian manifold and let
$\omega\in\Omega^{n}(N)$ be a closed non-exact form with $2\leqslant
n\leqslant\dim N$. Let $K,D\geqslant 1$. Let $(F_{j})$ be a sequence in
$\mathcal{F}_{K,D}(N,\omega)$. Then there exist a subsequence $(F_{j_{i}})$
and an operator

$L\colon\bigoplus_{k=1}^{n-1}H_{\mathrm{dR}}^{k}(N)\to\bigoplus_{k=1}^{n-1}L^{\frac{n}{k}}(B_{2}^{n},{\bigwedge}^{k}\mathbb{R}^{n})$

satisfying $F_{j_{i}}^{\\#}|_{H_{\mathrm{dR}}^{k}(N)}\rightharpoonup L$ for
$k=1,\ldots,n-1$.

###### Proof.

Since each $H_{\mathrm{dR}}^{k}(N)$ is finite-dimensional, the claim follows
immediately by Lemma 3.3 and the Banach–Alaoglu theorem. ∎

We are now ready to prove Theorem 1.10.

###### Proof of Theorem 1.10.

There exist a sequence $(F_{j})$ in $\mathcal{F}_{K,D}(N,\omega)$ and an
operator

$L\colon\bigoplus_{k=1}^{n-1}H_{\mathrm{dR}}^{k}(N)\to\bigoplus_{k=1}^{n-1}L^{\frac{n}{k}}(B_{2}^{n},{\bigwedge}^{k}\mathbb{R}^{n})$

satisfying $A_{\omega}(F_{j})\to\infty$ and
$F_{j}^{\\#}|_{\oplus_{k=1}^{n-1}H_{\mathrm{dR}}^{k}(N)}\rightharpoonup L$ by
Lemma 5.1. By Theorem 4.9, the operator $L$ extends to a graded algebra
homomorphism $L\colon H_{\mathrm{dR}}^{*}(N)\to\mathscr{W}^{*}(B_{2}^{n})$
satisfying $\int_{B^{n}}L[\omega]_{\mathrm{dR}}=1$.

Let $c_{1},\ldots,c_{m},c_{m+1},\ldots,c_{\nu}$ be a basis of
$H_{\mathrm{dR}}^{*}(N)$, where $c_{1},\ldots,c_{m}$ is a basis of $K^{n}(N)$.
Since $[\omega]_{\mathrm{dR}}\in K^{n}(N)$, there exist
$\lambda_{1},\ldots,\lambda_{m}\in\mathbb{R}$ satisfying
$[\omega]_{\mathrm{dR}}=\sum_{i=1}^{m}\lambda_{i}c_{i}$.

We fix point-wise Borel representatives for $Lc_{i}$ and $L(c_{i}\wedge
c_{\ell})$, where $i,\ell=1,\ldots,\nu$. Let

$E=\left\\{x\in B^{n}\colon\left(\sum_{i=1}^{m}\lambda_{i}Lc_{i}\right)(x)\neq
0\right\\}$

and

$E_{i,\ell}=\\{x\in B^{n}\colon(Lc_{i})(x)\wedge(Lc_{\ell})(x)=(Lc_{i}\wedge
c_{\ell})(x)\\}$

for $i,\ell=1,\ldots,\nu$. Now $m_{n}(E_{i,\ell})=m_{n}(B^{n})$ for each pair
$(i,\ell)\in\\{1,\ldots,\nu\\}^{2}$. Furthermore, $m_{n}(E)>0$ since

$\int_{E}\sum_{i=1}^{m}\lambda_{i}Lc_{i}=\int_{B^{n}}\sum_{i=1}^{m}\lambda_{i}Lc_{i}=\int_{B^{n}}L[\omega]_{\mathrm{dR}}=1.$

Thus there exists $x_{0}\in\cap_{i,\ell=1}^{\nu}E\cap E_{i,\ell}$.

Let $\Phi\colon H_{\mathrm{dR}}^{*}(N)\to\bigwedge^{*}\mathbb{R}^{n}$ be the
linear map defined by $c_{i}\mapsto(Lc_{i})(x_{0})$ for $i=1,\ldots,\nu$. By
definition, the map $\Phi$ is graded. Moreover, we have

$\Phi(c_{i}\wedge c_{\ell})=(L(c_{i}\wedge
c_{\ell}))(x_{0})=(Lc_{i})(x_{0})\wedge(Lc_{\ell})(x_{0})=\Phi c_{i}\wedge\Phi
c_{\ell}$

for each pair $(i,\ell)\in\\{1,\ldots,\nu\\}^{2}$ since $x_{0}\in E_{i,\ell}$.
Hence, the map $\Phi$ is a homomorphism by linearity. Finally,

$\Phi[\omega]_{\mathrm{dR}}=\Phi\left(\sum_{i=1}^{m}\lambda_{i}c_{i}\right)=\sum_{i=1}^{m}\lambda_{i}\Phi
c_{i}=\left(\sum_{i=1}^{m}\lambda_{i}Lc_{i}\right)(x_{0})\neq 0$

since $x_{0}\in E$. ∎

##  6\. Submanifolds controlled by quasiregular curves

In this section, we present two results, which together yield a stronger
version of Theorem 1.7. The first result is, in fact, a reformulation of
Theorem 1.8.

######  Theorem 6.1.

Let $N$ be a closed, connected, and oriented Riemannian manifold. Let
$2\leqslant n\leqslant\dim N$ and let $\omega\in\Omega^{n}(N)$ be a closed
form satisfying $0\neq[\omega]_{\mathrm{dR}}\in K^{n}(N)$. Let $K\geqslant 1$
and let $F\colon\mathbb{R}^{n}\to N$ be a $K$-quasiregular $\omega$-curve for
which $\int_{\mathbb{R}^{n}}F^{*}\omega=\infty$ and
$\inf_{\mathbb{R}^{n}}(\lVert\omega\rVert\circ F)>0$. Then there exists a
graded algebra homomorphism $\Psi\colon
H_{\mathrm{dR}}^{*}(N)\to\mathscr{W}^{*}(B_{2}^{n})$ satisfying

$\int_{B^{n}}\Psi[\omega]_{\mathrm{dR}}=1\quad\text{ and
}\quad\mathrm{core}\,_{\omega}(F)\cap K^{n}(N)\subset\ker\Psi.$

Here and in what follows, $\mathrm{core}\,_{\omega}(F)\subset
H_{\mathrm{dR}}^{n}(N)$ denotes the subspace of de Rham classes $c$ satisfying

$\lim_{j\to\infty}\frac{\int_{B^{n}(a_{j},r_{j})}|F^{*}(h(c))|}{\int_{B^{n}(a_{j},r_{j})}F^{*}\omega}=0$

for any sequence $(B^{n}(a_{j},r_{j}))$ with
$\int_{B^{n}(a_{j},r_{j})}F^{*}\omega\to\infty$.

###### Proof.

By Proposition 2.1, there exist $D\geqslant 1$ and
$(a_{j},r_{j})\in\mathbb{R}^{n}\times(0,\infty)$ for which $F_{j}:=F\circ
T_{j}\in\mathcal{F}_{K,D}(N,\omega)$ and $A_{\omega}(F_{j})\to\infty$, where
$T_{j}\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ is the affine map $x\mapsto
r_{j}x+a_{j}$. By passing to a subsequence, Lemma 5.1 and Theorem 4.9 together
yield that there exists a graded algebra homomorphism $L\colon
H_{\mathrm{dR}}^{*}(N)\to\mathscr{W}^{*}(B_{2}^{n})$ for which
$F_{j}^{\\#}|_{K^{n}(N)}\rightharpoonup L|_{K^{n}(N)}$ vaguely as measures and

$\int_{B^{n}}L[\omega]_{\mathrm{dR}}=1.$

It remains to show that $\mathrm{core}\,_{\omega}(F)\cap K^{n}(N)\subset\ker
L$.

Let $c\in\mathrm{core}\,_{\omega}(F)\cap K^{n}(N)$ and let $\varphi\in
C_{0}(B_{2}^{n})$. Then

$\int_{B_{2}^{n}}\varphi Lc=\lim_{j\to\infty}\int_{B_{2}^{n}}\varphi
F_{j}^{\\#}c=\lim_{j\to\infty}\int_{B_{2}^{n}}\varphi F_{j}^{!}(h(c)).$

On the other hand, since $F_{j}\in\mathcal{F}_{K,D}(N,\omega)$ and
$c\in\mathrm{core}\,_{\omega}(F)$, we have

$\displaystyle\left\lvert\int_{B_{2}^{n}}\varphi F_{j}^{!}(h(c))\right\rvert$
$\displaystyle=A_{\omega}(F_{j})^{-1}\left\lvert\int_{B_{2}^{n}}\varphi
F_{j}^{*}(h(c))\right\rvert$ $\displaystyle\leqslant
A_{\omega}(F_{j})^{-1}\lVert\varphi\rVert_{\infty}\int_{B_{2}^{n}}\left\lvert
F_{j}^{*}(h(c))\right\rvert$
$\displaystyle=\lVert\varphi\rVert_{\infty}\frac{\int_{B^{n}(a_{j},2r_{j})}\left\lvert
F^{*}(h(c))\right\rvert}{\int_{B^{n}(a_{j},r_{j})}F^{*}\omega}$
$\displaystyle\leqslant
D\lVert\varphi\rVert_{\infty}\frac{\int_{B^{n}(a_{j},2r_{j})}\left\lvert
F^{*}(h(c))\right\rvert}{\int_{B^{n}(a_{j},2r_{j})}F^{*}\omega}\to 0.$

Hence

$\int_{B_{2}^{n}}\varphi Lc=0$

for an arbitrary function $\varphi\in C_{0}(B_{2}^{n})$, which yields that
$Lc=0$. This concludes the proof. ∎

The next result is an elementary algebraic result based on the Poincaré
duality on a submanifold.

######  Proposition 6.2.

Let $N$ be a closed, connected, and oriented Riemannian manifold and let
$M\subset N$ be a closed, connected, and oriented $n$-submanifold with
$n\geqslant 2$. Let $c\in K^{n}(N)$ and let $\Psi\colon
H_{\mathrm{dR}}^{*}(N)\to\mathscr{W}^{*}(B_{2}^{n})$ be a graded algebra
homomorphism for which

$\int_{B^{n}}\Psi c=1\quad\text{ and }\quad\ker\iota^{*}\cap
K^{n}(N)\subset\ker\Psi,$

where $\iota\colon M\to N$ is the inclusion. Let $1\leqslant k\leqslant n-1$
be an index for which $\iota^{*}\colon H_{\mathrm{dR}}^{n-k}(N)\to
H_{\mathrm{dR}}^{n-k}(M)$ is surjective. Then
$\dim\iota^{*}H_{\mathrm{dR}}^{k}(N)\leqslant\binom{n}{k}$.

As a preparatory step, we recall the following elementary dimension bound for
subspaces of $\mathscr{W}^{*}(B_{2}^{n})$; see e.g. [12, Corollary 4.4] for a
proof.

######  Lemma 6.3.

Let $n\geqslant 2$ and $1\leqslant k\leqslant n-1$. Let
$u_{1},\ldots,u_{m}\in\mathscr{W}^{k}(B_{2}^{n})$ and
$v_{1},\ldots,v_{m}\in\mathscr{W}^{n-k}(B_{2}^{n})$ satisfy $u_{i}\wedge
v_{\ell}=\delta_{i\ell}w$, where $\delta_{i\ell}$ is the Kronecker delta and
$0\neq w\in L^{1}(B_{2}^{n};\bigwedge\mathbb{R}^{n})$. Then
$m\leqslant\dim\bigwedge^{k}\mathbb{R}^{n}$.

###### Proof of Proposition 6.2.

Let $\iota^{*}c_{1},\ldots,\iota^{*}c_{m}$ be a basis of
$\iota^{*}H_{\mathrm{dR}}^{k}(N)$. By Poincaré duality, there exist de Rham
classes $\iota^{*}c_{1}^{\prime},\ldots,\iota^{*}c_{m}^{\prime}\in
H_{\mathrm{dR}}^{n-k}(M)$ satisfying
$\iota^{*}c_{i}\wedge\iota^{*}c_{\ell}^{\prime}=\delta_{i\ell}[\mathrm{vol}_{M}]_{\mathrm{dR}}$,
where $\delta_{i\ell}$ is the Kronecker delta.

Since $\iota^{*}c\neq 0$, we may fix $\lambda\neq 0$ for which
$\lambda\iota^{*}c=[\mathrm{vol}_{M}]_{\mathrm{dR}}$. Then $c_{i}\wedge
c_{\ell}^{\prime}-\delta_{i\ell}\lambda c\in\ker\iota^{*}\cap\,K^{n}(N)$. Thus

$\Psi c_{i}\wedge\Psi c_{\ell}^{\prime}=\Psi(c_{i}\wedge
c_{\ell}^{\prime})=\Psi(\delta_{i\ell}\lambda c)=\delta_{i\ell}\lambda\Psi c,$

which yields $m\leqslant\dim\bigwedge^{k}\mathbb{R}^{n}=\binom{n}{k}$ by Lemma
6.3. ∎

Combining Theorem 6.1 and Proposition 6.2 yields immediately the following
variation of Theorem 1.7.

######  Theorem 6.4.

Let $N$ be a closed, connected, and oriented Riemannian manifold. Let
$2\leqslant n\leqslant\dim N$ and let $\omega\in\Omega^{n}(N)$ be a closed
form for which $0\neq[\omega]_{\mathrm{dR}}\in K^{n}(N)$. Let
$F\colon\mathbb{R}^{n}\to N$ be a quasiregular $\omega$-curve for which
$\int_{\mathbb{R}^{n}}F^{*}\omega=\infty$ and
$\inf_{\mathbb{R}^{n}}(\lVert\omega\rVert\circ F)>0$. Let $M\subset N$ be a
closed, connected, and oriented Riemannian $n$-submanifold satisfying
$\iota^{*}(H_{\mathrm{dR}}^{*}(N))=H_{\mathrm{dR}}^{*}(M)$ and
$\ker\iota^{*}\cap\,K^{n}(N)\subset\mathrm{core}\,_{\omega}(F)$, where
$\iota\colon M\to N$ is the inclusion. Then $\dim
H_{\mathrm{dR}}^{k}(M)\leqslant\binom{n}{k}$ for $k=0,\ldots,n$.

The following example shows that the assumption
$\ker\iota^{*}\cap\,K^{n}(N)\subset\mathrm{core}\,_{\omega}(F)$ cannot be
replaced by a weaker assumption $\iota^{*}[\omega]_{\mathrm{dR}}\neq 0$ in
Theorem 6.4.

######  Example 6.5.

Let $n\geqslant 2$. Let $M$ and $P$ be closed, connected, and oriented
Riemannian manifolds with $\dim M=n$ and $\dim P\geqslant n$. Let
$\xi\in\Omega^{n}(P)$ be a closed non-exact form. Let
$f\colon\mathbb{R}^{n}\to M$ be a constant map and let
$g\colon\mathbb{R}^{n}\to P$ be a quasiregular $\xi$-curve satisfying
$\int_{\mathbb{R}^{n}}g^{*}\xi=\infty$.

Let $N=M\times P$, $\omega=\pi_{M}^{*}\mathrm{vol}_{M}+\pi_{P}^{*}\xi$, and
$F=(f,g)\colon\mathbb{R}^{n}\to N$. Then $F$ is a quasiregular $\omega$-curve
with $\int_{\mathbb{R}^{n}}F^{*}\omega=\infty$.

Let also $\iota\colon M\to N$ be an inclusion $m\mapsto(m,p)$ for some fixed
$p\in P$. Then $\iota^{*}\colon H_{\mathrm{dR}}^{*}(N)\to
H_{\mathrm{dR}}^{*}(M)$ is surjective and
$\iota^{*}[\omega]_{\mathrm{dR}}=[\mathrm{vol}_{M}]_{\mathrm{dR}}\neq 0$.

Compare now Example 6.5 with the choices
$M=\\#^{2}(\mathbb{S}^{1}\times\mathbb{S}^{1})$,
$P=\mathbb{S}^{1}\times\mathbb{S}^{1}$, $\xi=\mathrm{vol}_{P}$, and
$g(x,y)=(e^{2\pi ix},e^{2\pi iy})$ to Theorem 6.4.

Considering further the case, where the ambient manifold is a product manifold
and the studied submanifold is one of the factors, we obtain the following
result.

######  Corollary 6.6.

Let $n\geqslant 2$. Let $M$ and $P$ be closed, connected, and oriented
Riemannian manifolds with $\dim M=n$ and $\dim P\geqslant n$. Let
$\xi\in\Omega^{n}(P)$ be a closed form for which $[\xi]_{\mathrm{dR}}\in
K^{n}(P)$. Let $N=M\times P$ and let
$\omega=\pi_{M}^{*}\mathrm{vol}_{M}+\pi_{P}^{*}\xi$. Let
$F\colon\mathbb{R}^{n}\to N$ be a quasiregular $\omega$-curve for which
$\int_{\mathbb{R}^{n}}F^{*}\omega=\infty$ and
$[\pi_{P}^{*}\xi]_{\mathrm{dR}}\in\mathrm{core}\,_{\omega}(F)$. Then there
exists a graded algebra homomorphism $\Psi\colon
H_{\mathrm{dR}}^{*}(M)\to\mathscr{W}^{*}(B_{2}^{n})$ satisfying

$\int_{B^{n}}\Psi[\mathrm{vol}_{M}]_{\mathrm{dR}}=1.$

###### Proof.

We may assume that $H_{\mathrm{dR}}^{*}(M)\neq
H_{\mathrm{dR}}^{*}(\mathbb{S}^{n})$. Then, by Theorem 6.1, there exists a
graded algebra homomorphism $\Psi\colon
H_{\mathrm{dR}}^{*}(N)\to\mathscr{W}^{*}(B_{2}^{n})$ satisfying

$\int_{B^{n}}\Psi[\omega]_{\mathrm{dR}}=1\quad\text{ and
}\quad\mathrm{core}\,_{\omega}(F)\cap K^{n}(N)\subset\ker\Psi.$

Since $\Psi[\pi_{P}^{*}\xi]_{\mathrm{dR}}=0$, it follows that
$\Psi\circ\pi_{M}^{*}\colon
H_{\mathrm{dR}}^{*}(M)\to\mathscr{W}^{*}(B_{2}^{n})$ is a graded algebra
homomorphism satisfying

$\int_{B^{n}}(\Psi\circ\pi_{M}^{*})[\mathrm{vol}_{M}]_{\mathrm{dR}}=\int_{B^{n}}\Psi[\omega]_{\mathrm{dR}}=1.$

∎

##  7\. Examples of pairs $(N,\omega)$ which are not quasiregularly elliptic

In this section, we show that Theorem 1.8 yields obstructions for non-constant
quasiregular curves from the Euclidean space to product manifolds containing
$\mathbb{C}P^{2}$ with respect to product forms containing the standard
symplectic form on $\mathbb{C}P^{2}$. In particular, for pairs
$(S_{\nu}\times\mathbb{C}P^{2},\pi_{1}^{*}\mathrm{vol}_{S_{\nu}}\wedge\pi_{2}^{*}\omega_{\mathrm{sym}})$
we obtain an obstruction for $\nu\geqslant 8$.

######  Proposition 7.1.

Let $\nu\geqslant 8$ and let
$S_{\nu}=\\#^{\nu}(\mathbb{S}^{2}\times\mathbb{S}^{2})$. Let
$\omega_{\mathrm{sym}}\in\Omega^{2}(\mathbb{C}P^{2})$ be the standard
symplectic form. Let $F\colon\mathbb{R}^{6}\to S_{\nu}\times\mathbb{C}P^{2}$
be a quasiregular curve with respect to
$\omega_{S_{\nu}}=\pi_{1}^{*}\mathrm{vol}_{S_{\nu}}\wedge\pi_{2}^{*}\omega_{\mathrm{sym}}$.
Then $F$ is constant.

Proposition 7.1 yields Corollary 1.6 as an immediate consequence.

On the other hand, for pairs
$(T_{g}\times\mathbb{C}P^{2},\pi_{1}^{*}\mathrm{vol}_{T_{g}}\wedge\pi_{2}^{*}\omega_{\mathrm{sym}})$
we obtain an obstruction for $g\geqslant 2$.

######  Proposition 7.2.

Let $g\geqslant 2$ and let
$T_{g}=\\#^{g}(\mathbb{S}^{1}\times\mathbb{S}^{1})$. Let
$\omega_{\mathrm{sym}}\in\Omega^{2}(\mathbb{C}P^{2})$ be the standard
symplectic form. Let $F\colon\mathbb{R}^{4}\to T_{g}\times\mathbb{C}P^{2}$ be
a quasiregular curve with respect to
$\omega_{T_{g}}=\pi_{1}^{*}\mathrm{vol}_{T_{g}}\wedge\pi_{2}^{*}\omega_{\mathrm{sym}}$.
Then $F$ is constant.

The obstruction in Proposition 7.2 is sharp in the following sense.

######  Proposition 7.3.

Let $T_{1}=\mathbb{S}^{1}\times\mathbb{S}^{1}$ and let
$\omega_{\mathrm{sym}}\in\Omega^{2}(\mathbb{C}P^{2})$ be the standard
symplectic form. There exists a non-constant map $F\colon\mathbb{R}^{4}\to
T_{1}\times\mathbb{C}P^{2}$ which is a quasiregular curve with respect to
$\omega_{T_{1}}=\pi_{1}^{*}\mathrm{vol}_{T_{1}}\wedge\pi_{2}^{*}\omega_{\mathrm{sym}}$.

Propositions 7.2 and 7.3 together yield immediately Corollary 1.5.

Before discussing the proofs of Propositions 7.1, 7.2, and 7.3, we briefly
recall the definition of the standard symplectic form on a complex projective
space. For $m\geqslant 1$, _the standard symplectic form on $\mathbb{C}P^{m}$_
is the unique form $\omega_{\mathrm{sym}}\in\Omega^{2}(\mathbb{C}P^{m})$
satisfying

$U^{*}\omega_{\mathrm{sym}}=\omega_{\mathrm{sym}}$

for each unitary $(m+1)\times(m+1)$ matrix $U\in U(m+1)$ and

$\iota^{*}\omega_{\mathrm{sym}}=\mathrm{vol}_{\mathbb{C}P^{1}},$

where $\iota\colon\mathbb{C}P^{1}\to\mathbb{C}P^{m}$ is the inclusion
$[z_{0}:z_{1}]\mapsto[z_{0}:z_{1}:0:\cdots:0]$ in homogeneous coordinates; see
e.g. [15, Section 3.3.2] for an explicit formula of $\omega_{\mathrm{sym}}$.

### Construction of a quasiregular $\omega_{T_{1}}$-curve $\mathbb{R}^{4}\to
T_{1}\times\mathbb{C}P^{2}$

In preparation for the proof of Proposition 7.3, we show that the natural
inclusion $\mathbb{C}P^{1}\hookrightarrow\mathbb{C}P^{2}$ is a quasiregular
curve with respect to the standard symplectic form.

######  Lemma 7.4.

Let $1\leqslant i\leqslant\ell$ and let
$\iota\colon\mathbb{C}P^{i}\to\mathbb{C}P^{\ell}$ be the inclusion
$[z_{0}:\cdots:z_{i}]\mapsto[z_{0}:\cdots:z_{i}:0:\cdots:0]$. Let
$\omega_{\mathrm{sym}}\in\Omega^{2}(\mathbb{C}P^{\ell})$ be the standard
symplectic form on $\mathbb{C}P^{\ell}$. Then $\iota$ is a $1$-quasiregular
curve with respect to $\omega_{\mathrm{sym}}^{\wedge
i}=\omega_{\mathrm{sym}}\wedge\cdots\wedge\omega_{\mathrm{sym}}\in\Omega^{2i}(\mathbb{C}P^{\ell})$.

###### Proof.

The claim follows immediately since
$\iota^{*}\omega_{\mathrm{sym}}\in\Omega^{2}(\mathbb{C}P^{i})$ is the standard
symplectic form on $\mathbb{C}P^{i}$. ∎

We also show that mappings of bounded length distortion $\mathbb{R}^{n}\to
N_{i}$ induce quasiregular curves $(\mathbb{R}^{n})^{m}\to
N_{1}\times\cdots\times N_{m}$; we refer to [10] and [7] for more discussion
on BLD-maps.

######  Lemma 7.5.

Let $n\geqslant 2$ and let $N_{1},\ldots,N_{m}$ be connected and oriented
Riemannian manifolds with $\dim N_{i}\geqslant n$. Let
$\omega_{i}\in\Omega^{n}(N_{i})$ be closed forms satisfying
$\lVert\pi_{1}^{*}\omega_{1}\wedge\cdots\wedge\pi_{m}^{*}\omega_{m}\rVert\leqslant
1$. Let $L\geqslant m$ and let $\psi_{i}\in
W_{\operatorname{loc}}^{1,n}(\mathbb{R}^{n},N_{i})$ be maps satisfying $\lVert
D\psi_{i}\rVert\leqslant L$ a.e. in $\mathbb{R}^{n}$ and
$\star\psi_{i}^{*}\omega_{i}\geqslant L^{-n}$ a.e. in $\mathbb{R}^{n}$. Then
the map $\psi\colon(\mathbb{R}^{n})^{m}\to N_{1}\times\cdots\times N_{m}$,
$(x_{1},\ldots,x_{m})\mapsto(\psi_{1}(x_{1}),\ldots,\psi_{m}(x_{m}))$, is a
$(L^{4n})^{m}$-quasiregular curve with respect to
$\pi_{1}^{*}\omega_{1}\wedge\cdots\wedge\pi_{m}^{*}\omega_{m}$.

###### Proof.

For a.e. $x=(x_{1},\ldots,x_{m})\in(\mathbb{R}^{n})^{m}$, we have

$\displaystyle\lVert D\psi(x)\rVert^{nm}$
$\displaystyle\leqslant\left(\sum_{i=1}^{m}\lVert
D\psi_{i}(x_{i})\rVert\right)^{nm}\leqslant m^{nm}\sum_{i=1}^{m}\lVert
D\psi_{i}(x_{i})\rVert^{nm}$ $\displaystyle\leqslant m^{nm+1}L^{nm}\leqslant
L^{3nm}\leqslant L^{4nm}\prod_{i=1}^{m}(\star\psi_{i}^{*}\omega_{i})(x_{i})$
$\displaystyle=L^{4nm}(\star\psi^{*}(\pi_{1}^{*}\omega_{1}\wedge\cdots\wedge\pi_{m}^{*}\omega_{m}))(x).$

Since
$(\pi_{1}^{*}\omega_{1}\wedge\cdots\wedge\pi_{m}^{*}\omega_{m})(\psi(x))\neq
0$ for a.e. $x\in(\mathbb{R}^{n})^{m}$, the claim follows. ∎

We are now ready to prove Proposition 7.3.

###### Proof of Proposition 7.3.

Let $\exp\colon\mathbb{R}^{2}\to T_{1}$ be the covering map
$(x,y)\mapsto(e^{2\pi ix},e^{2\pi iy})$. Let
$A\colon\mathbb{R}^{2}\to\mathbb{C}P^{1}$ be an Alexander map (for the
construction see e.g. [9, Section 6.5.4]) and let $L\geqslant 1$ a constant
satisfying $\lVert DA\rVert\leqslant L$ a.e. in $\mathbb{R}^{2}$ and $\star
A^{*}\mathrm{vol}_{\mathbb{C}P^{1}}\geqslant L^{-2}$ a.e. in $\mathbb{R}^{2}$.
Let $\iota\colon\mathbb{C}P^{1}\to\mathbb{C}P^{2}$ be the inclusion
$[z_{0}:z_{1}]\mapsto[z_{0}:z_{1}:0]$. Then, by Lemma 7.4,

$\lVert D(\iota\circ A)\rVert\leqslant\lVert D\iota\circ A\rVert\lVert
DA\rVert\leqslant L$

a.e. in $\mathbb{R}^{2}$ and

$\star(\iota\circ A)^{*}\omega_{\mathrm{sym}}=\star
A^{*}\iota^{*}\omega_{\mathrm{sym}}=\star
A^{*}\mathrm{vol}_{\mathbb{C}P^{1}}\geqslant L^{-2}$

a.e. in $\mathbb{R}^{2}$. Thus, by Lemma 7.5, the map

$F\colon\mathbb{R}^{4}\to
T_{1}\times\mathbb{C}P^{2},\quad(x,y,z,w)\mapsto(\exp(x,y),\iota(A(z,w))),$

is a quasiregular curve with respect to
$\pi_{1}^{*}\mathrm{vol}_{T_{1}}\wedge\pi_{2}^{*}\omega_{\mathrm{sym}}$. Since
$F$ is non-constant, this concludes the proof. ∎

### Obstruction for quasiregular $\omega_{T_{g}}$-curves $\mathbb{R}^{4}\to
T_{g}\times\mathbb{C}P^{2}$

The quasiregular $\omega_{T_{1}}$-curve $\mathbb{R}^{4}\to
T_{1}\times\mathbb{C}P^{2}$ constructed in the proof of Proposition 7.3 arose
from a pair $(\exp,\iota\circ A)$, where $\exp\colon\mathbb{R}^{2}\to T_{1}$
and $\iota\circ A\colon\mathbb{R}^{2}\to\mathbb{C}P^{2}$ are quasiregular
maps. By the classical quasiconformal theory, there does not exist a
quasiregular map $\mathbb{R}^{2}\to T_{g}$ if $g\geqslant 2$, so a priori
Proposition 7.2, we know that a quasiregular $\omega_{T_{g}}$-curve
$\mathbb{R}^{4}\to T_{g}\times\mathbb{C}P^{2}$ cannot be constructed using
quasiregular coordinate maps. Proposition 7.2 shows that there does not exist
any method of constructing a non-constant quasiregular $\omega_{T_{g}}$-curve
$\mathbb{R}^{4}\to T_{g}\times\mathbb{C}P^{2}$.

The proof of Proposition 7.2 is based on combining the following elementary
algebraic lemma with Theorem 1.8.

######  Lemma 7.6.

Let $N$ be a smooth manifold and let $\Phi\colon
H_{\mathrm{dR}}^{*}(N)\to\bigwedge^{*}\mathbb{R}^{n}$ be a graded algebra
homomorphism. Let $k,k^{\prime}>0$. Suppose $c\in H_{\mathrm{dR}}^{k}(N)$ and
$c^{\prime}\in H_{\mathrm{dR}}^{k^{\prime}}(N)$ are de Rham classes satisfying
$\Phi c\wedge\Phi c^{\prime}\neq 0$. Suppose $c_{1},\ldots,c_{m}\in
H_{\mathrm{dR}}^{1}(N)$ are de Rham classes for which $c\wedge c_{i}=0$ for
$i=1,\ldots,m$. Suppose $c_{1}^{\prime},\ldots,c_{m}^{\prime}\in
H_{\mathrm{dR}}^{k-1}(N)$ are de Rham classes satisfying $c_{i}\wedge
c_{\ell}^{\prime}=\delta_{i\ell}c$, where $\delta_{i\ell}c$ is the Kronecker
delta. Then $m<n$.

###### Proof.

For every $(\lambda_{1},\ldots,\lambda_{m})\in\mathbb{R}^{m}\setminus\\{0\\}$,
we have

$\displaystyle\left(\sum_{i=1}^{m}\lambda_{i}\Phi
c_{i}\right)\wedge\left(\sum_{\ell=1}^{m}\lambda_{\ell}\Phi
c_{\ell}^{\prime}\right)$
$\displaystyle=\sum_{i,\ell=1}^{m}\lambda_{i}\lambda_{\ell}\Phi
c_{i}\wedge\Phi
c_{\ell}^{\prime}=\sum_{i,\ell=1}^{m}\lambda_{i}\lambda_{\ell}\Phi(c_{i}\wedge
c_{\ell}^{\prime})$ $\displaystyle=\sum_{i=1}^{m}\lambda_{i}^{2}\Phi
c=\left(\sum_{i=1}^{m}\lambda_{i}^{2}\right)\Phi c\neq 0,$

so $\sum_{i=1}^{m}\lambda_{i}\Phi c_{i}\neq 0$. We conclude that $\Phi
c_{1},\ldots,\Phi c_{m}\in\bigwedge^{1}\mathbb{R}^{n}$ are linearly
independent, which yields that $m\leqslant\dim\bigwedge^{1}\mathbb{R}^{n}=n$.

Suppose now that $m=n$. Then $\Phi c_{1},\ldots,\Phi c_{m}$ is a basis of
$\bigwedge^{1}\mathbb{R}^{n}$ and hence $\\{\Phi
c_{i_{1}}\wedge\cdots\wedge\Phi c_{i_{k^{\prime}}}\\}_{I}$ is a basis of
$\bigwedge^{k^{\prime}}\mathbb{R}^{n}$. Since $\Phi
c^{\prime}\in\bigwedge^{k^{\prime}}\mathbb{R}^{n}$, there exist coefficients
$\mu_{I}\in\mathbb{R}$ satisfying

$\Phi c^{\prime}=\sum_{I}\mu_{I}\Phi c_{i_{1}}\wedge\cdots\wedge\Phi
c_{i_{k^{\prime}}}.$

Hence, we arrive at a contradiction

$\displaystyle\Phi c\wedge\Phi c^{\prime}$ $\displaystyle=\Phi
c\wedge\left(\sum_{I}\mu_{I}\Phi c_{i_{1}}\wedge\cdots\wedge\Phi
c_{i_{k^{\prime}}}\right)=\sum_{I}\mu_{I}\Phi c\wedge\Phi
c_{i_{1}}\wedge\cdots\wedge\Phi c_{i_{k^{\prime}}}$
$\displaystyle=\sum_{I}\mu_{I}\Phi(c\wedge c_{i_{1}}\wedge\cdots\wedge
c_{i_{k^{\prime}}})=0.$

∎

We are now ready to prove Proposition 7.2.

###### Proof of Proposition 7.2.

Suppose that $F$ is non-constant. By Theorem 2.3, we have
$\int_{\mathbb{R}^{n}}F^{*}\omega_{T_{g}}=\infty$. Thus, by Theorem 1.8, there
exists a graded algebra homomorphism $\Phi\colon
H_{\mathrm{dR}}^{*}(T_{g}\times\mathbb{C}P^{2})\to\bigwedge^{*}\mathbb{R}^{4}$
for which $\Phi[\omega_{T_{g}}]_{\mathrm{dR}}\neq 0$.

Let $c_{1},\ldots,c_{2g}$ be basis of $H_{\mathrm{dR}}^{1}(T_{g})$ satisfying
$c_{i}\wedge c_{i+1}=[\mathrm{vol}_{T_{g}}]_{\mathrm{dR}}$ for
$i=1,3,\ldots,2g-1$ and $c_{i}\wedge c_{\ell}=0$ for every pair
$(i,\ell)\in\\{1,\ldots,2g\\}^{2}$ with $|2\ell-2i-1|>1$. Denote
$c_{i}^{\prime}=c_{i+1}$ for $i=1,3,\ldots,2g-1$ and $c_{i}^{\prime}=-c_{i-1}$
for $i=2,4,\ldots,2g$. Since

$\Phi\pi_{1}^{*}[\mathrm{vol}_{T_{g}}]_{\mathrm{dR}}\wedge\Phi\pi_{2}^{*}[\omega_{\mathrm{sym}}]_{\mathrm{dR}}=\Phi[\omega_{T_{g}}]_{\mathrm{dR}}\neq
0,$
$\pi_{1}^{*}[\mathrm{vol}_{T_{g}}]_{\mathrm{dR}}\wedge\pi_{1}^{*}c_{i}=\pi_{1}^{*}([\mathrm{vol}_{T_{g}}]_{\mathrm{dR}}\wedge
c_{i})=0,$

and

$\pi_{1}^{*}c_{i}\wedge\pi_{1}^{*}c_{\ell}^{\prime}=\pi_{1}^{*}(c_{i}\wedge
c_{\ell}^{\prime})=\pi_{1}^{*}(\delta_{i\ell}[\mathrm{vol}_{T_{g}}]_{\mathrm{dR}})=\delta_{i\ell}\pi_{1}^{*}[\mathrm{vol}_{T_{g}}]_{\mathrm{dR}},$

Lemma 7.6 yields that $2g<4$. This is a contradiction since $g\geqslant 2$. ∎

### Obstruction for quasiregular $\omega_{S_{\nu}}$-curves $\mathbb{R}^{4}\to
S_{\nu}\times\mathbb{C}P^{2}$

Proposition 7.1 follows from Theorem 1.8 and the following elementary
observation. Since the proof is similar to the proof of Lemma 7.6, we omit the
details.

######  Lemma 7.7.

Let $N$ be a smooth manifold and let $\Phi\colon
H_{\mathrm{dR}}^{*}(N)\to\bigwedge^{*}\mathbb{R}^{n}$ be a graded algebra
homomorphism. Let $k\geqslant 2$ and suppose $c\in H_{\mathrm{dR}}^{k}(N)$ is
a de Rham class satisfying $\Phi c\neq 0$. Let $1\leqslant k^{\prime}\leqslant
k-1$. Suppose $c_{1},\ldots,c_{m}\in H_{\mathrm{dR}}^{k^{\prime}}(N)$ and
$c_{1}^{\prime},\ldots,c_{m}^{\prime}\in H_{\mathrm{dR}}^{k-k^{\prime}}(N)$
are de Rham classes for which $c_{i}\wedge c_{\ell}^{\prime}=\delta_{i\ell}c$,
where $\delta_{i\ell}c$ is the Kronecker delta. Then
$m\leqslant\binom{n}{k^{\prime}}$.

###### Proof of Proposition 7.1.

Suppose that $F$ is non-constant. By Theorem 2.3, we have
$\int_{\mathbb{R}^{n}}F^{*}\omega_{S_{\nu}}=\infty$. Thus, by Theorem 1.8,
there exists a graded algebra homomorphism $\Phi\colon
H_{\mathrm{dR}}^{*}(S_{\nu}\times\mathbb{C}P^{2})\to\bigwedge^{*}\mathbb{R}^{6}$
for which $\Phi[\omega_{S_{\nu}}]_{\mathrm{dR}}\neq 0$.

Let $c_{1},\ldots,c_{2\nu}$ be basis of $H_{\mathrm{dR}}^{2}(S_{\nu})$
satisfying $c_{i}\wedge c_{i+1}=[\mathrm{vol}_{S_{\nu}}]_{\mathrm{dR}}$ for
$i=1,3,\ldots,2\nu-1$ and $c_{i}\wedge c_{\ell}=0$ for every pair
$(i,\ell)\in\\{1,\ldots,2\nu\\}^{2}$ with $|2\ell-2i-1|>1$. Denote
$c_{i}^{\prime}=c_{i+1}$ for $i=1,3,\ldots,2\nu-1$ and
$c_{i}^{\prime}=c_{i-1}$ for $i=2,4,\ldots,2\nu$. Since
$\Phi\pi_{1}^{*}[\mathrm{vol}_{S_{\nu}}]_{\mathrm{dR}}\neq 0$ and

$\pi_{1}^{*}c_{i}\wedge\pi_{1}^{*}c_{\ell}^{\prime}=\pi_{1}^{*}(c_{i}\wedge
c_{\ell}^{\prime})=\pi_{1}^{*}(\delta_{i\ell}[\mathrm{vol}_{S_{\nu}}]_{\mathrm{dR}})=\delta_{i\ell}\pi_{1}^{*}[\mathrm{vol}_{S_{\nu}}]_{\mathrm{dR}},$

Lemma 7.7 yields that $2\nu\leqslant\binom{6}{2}=15$. This is a contradiction
since $\nu\geqslant 8$. ∎

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