# Homologically area-minimizing surfaces mod $v$ have at worst codimension 2
singular sets asymptotically

Zhenhua Liu  Dedicated to Xunjing Wei

###### Abstract.

In [9], De Lellis and coauthors have proved a sharp regularity theorem for
area-minimizing currents in finite coefficient homology. They prove that area-
minimizing mod $v$ currents are smooth outside of a singular set of
codimension at least $1.$ Classical examples like triple junctions demonstrate
that their result is sharp.

Surprisingly, even though their regularity theorem cannot be improved for any
fixed $v$, if one instead fixes the homology class, then $v$ asymptotically
one can always achieve more regularity. For any integral homology class
$[\Sigma]$ on any Riemannian manifold, we show that for $v$ large, any area-
minimizing mod $v$ current in $[\Sigma\mod v]$ must be an integral current,
thus having a singular set of codimension at least $2$ in general and of
codimension at least $7$ in the hypersurface case. Similar results are
obtained for Plateau problems in Euclidean space. Our work is inspired by
Morgan’s work [44] and based on De Lellis’ and coauthors’ work [9].

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries and Lemmas
  3. 3 Proof of Theorem 1.2,1.4,1.3 and Corollary 1
  4. 4 Proof of Theorem 1.5 and Theorem 1.6

##  1\. Introduction

In this manuscript, area-minimizing surfaces refer to area-minimizing integral
currents and mod $v$ currents for integers $v\geq 2$, which roughly speaking
are submanifolds counted with multiplicity in $\mathbb{Z}$ or
$\mathbb{Z}/v\mathbb{Z}$ coefficient, respectively, minimizing the area
functional with homologous competitors.

For $\mathbb{Z}$-coefficient homology, Almgren’s Big Regularity Theorem ([3])
and De Lellis-Spadaro-Minter-Skorobogatova ([13][14][15][12]) show that
$n$-dimensional area minimizing integral currents are smooth manifolds outside
of a rectifiable singular set of dimension at most $n-2.$ (In the codimension
1 case, the dimension of the singular set can be reduced to $n-7$ by
[23][46].)

For $\mathbb{Z}/v\mathbb{Z}$-coefficient homology, De Lellis-Hirsch-Marchese-
Stuvard ([9]) show that $n$-dimensional area minimizing integral currents are
smooth manifolds outside of a singular set of dimension at most $n-1$. (In the
hypersurface case, the singular set consists of transversely intersecting
hypersurfaces along a common boundary outside of a set of codimension at least
$2$ [10][11][40]. This refined regularity theory however will not be needed in
this manuscript. In the hypersurface case [53] suffices for our purposes.)

Concrete examples like holomorphic subvarieties and triple junctions show that
both the $\mathbb{Z}$ and the $\mathbb{Z}/v\mathbb{Z}$ regularity theorems
above are sharp, respectively. Thus, it seems impossible that the dimension
difference in regularity theorems between the $\mathbb{Z}$ and
$\mathbb{Z}/v\mathbb{Z}$ coefficient can be bridged in general.

However, pioneering work by Frank Morgan ([44]) has shown that for boundaries
with at most $v/2$ components on convex sets, mod $v$ hypersurface solutions
to Plateau problems must be integral, thus having a singular set of
codimension at least $7$. To the author’s knowledge, Morgan’s theorem ([44])
remains the only known instance in the literature about such unexpected
improvement of regularity and is largely regarded as a special case by
experts. To quote verbatim Minter-Wickramasekera ([40]),
”although in general such singularities in T obviously do arise, they can in
certain special circumstances be ruled out; for instance, by a theorem of F.
Morgan [44]”

However, inspired by [44] and in sharp contrast to the literature, we show
that asymptotically we can always annihilate the difference in regularity
between $\mathbb{Z}$ and $\mathbb{Z}/v\mathbb{Z}$ coefficient area-minimizers,
in the setting of both homological area-minimizers and Plateau problems.

######  Theorem 1.1.

Let $[\Sigma]$ be a $d$-dimensional integral homology class on a
$d+c$-dimensional smooth compact closed Riemannian manifold $M^{d+c}$, with
$d,c\geq 1.$ Then for $v$ large enough, any area-minimizing mod $v$ current in
$[\Sigma\mod v]$ is smooth outside of a singular set of codimension at least
$2$ for $c\geq 2$ and codimension at least $7$ for $c\geq 1.$

The above theorem is a direct corollary of the following much stronger result.

######  Theorem 1.2.

Assume

  * •

$d,c\geq 1,v\geq 2$ are integers,

  * •

$M^{d+c}$ is a compact closed (not necessarily orientable) smooth manifold,

  * •

$S$ is a finite collection of $d$-dimensional integral homology classes on
$M$,

  * •

$K$ is a compact subset of the space of Riemannian metrics on $M.$

Then there exists an open set $\Omega_{K,S}$ in the space of Riemannian
metrics with $K\subset\Omega_{K,S}$, and an integer $\nu_{K,S},$ such that,

  * •

for any metric $g\in\Omega_{K,S}$, any $v$ with $v\geq\nu_{K,S}$, any mod $v$
area-minimizing current $T$ in any homology class $[\Sigma\mod v]$ with
$[\Sigma]\in S$ is an integral current, thus smooth outside of a singular set
of dimension at most $d-2$ for $c\geq 2$ and at most $d-7$ for $c=1$,

######  Remark 1.

Compact subsets in the space of Riemannian metrics come easily by taking
finite-dimensional smooth families of Riemannian metrics.

######  Remark 2.

Note that the assumption of an integral $[\Sigma]$ before taking mod $v$
cannot be dropped. It is equivalent to saying that $[\Sigma\mod v]$ has
integral representatives. By the universal coefficient theorem, there are
indeed homology classes that can never be represented by integral currents if
$H_{d-1}(M,\mathbb{Z})$ has suitable nontrivial torsion, e.g.,
$M=\mathbb{RP}^{d+c}$.

######  Remark 3.

We do not claim that $T\in[\Sigma]$. We only know that $T\in[\Sigma]+v[\Pi]$
for some integral homology class $[\Pi].$ When $H_{d}(M,\mathbb{Z})$ is
torsion-free, then it is indeed true that for $v$ large enough $T\in[\Sigma],$
with a proof similar to that of Lemma 3.1. However, for $H_{d}(M,\mathbb{Z})$
with torsion subgroups, concrete examples show that one can have
$T\in[\Sigma]+v[\Pi]$ for $v\to\infty$ and possibly different $[\Pi].$

A much stronger conclusion is true if we consider the set of all integral
homology classes mod $v.$

######  Theorem 1.3.

With the same assumption as in Theorem 1.2, in any smooth metric on $M$, we
have

$\displaystyle\liminf_{v\to\infty}\frac{\\#\\{H_{d}(M,\mathbb{Z}/v\mathbb{Z})\ni[\Pi]\textnormal{
having only integral area-
minimizers}\\}}{\\#\\{H_{d}(M,\mathbb{Z}/v\mathbb{Z})\ni[\Pi]\textnormal{
admitting integral representatives}\\}}>0.$

Here $\\#$ means the number of elements in the set.

######  Remark 4.

By the universal coefficient theorem, the denominator above equals

$\displaystyle\\#H_{d}(M,\mathbb{Z})\otimes\mathbb{Z}/v\mathbb{Z}.$

Consequently $\\#H_{d}(M,\mathbb{Z})\otimes\mathbb{Z}/v\mathbb{Z}\geq
v^{b_{d}}\to\infty$ as $v\to\infty$, where $b_{d}$ is the $d$-th Betti number
of $M$.

For Plateau problems with boundaries, similar conclusions hold in Euclidean
space.

######  Theorem 1.4.

Suppose

  * •

$d\geq 1,c\geq 1$ are integers,

  * •

$\Gamma=\sum_{j}t_{j}\Gamma_{j}$ is a smooth integral current, with
$t_{j}\in\mathbb{Z},$ $\Gamma_{j}\subset\mathbb{R}^{d+c}$ a finite collection
of pairwise disjoint connected orientable $d$-dimensional compact submanifolds
of $\mathbb{R}^{d+c}.$

Then there is $\nu_{\Gamma}>0,$ such that for $v\geq\nu_{\Gamma}$, any area-
minimizing mod $v$ current $T$ with

$\partial T=\Gamma\mod v$

is an integral current with boundary $\Gamma$, thus smooth in the interior
outside of a singular set of dimension at most $d-2$ for $c\geq 2$ and at most
$d-7$ for $c=1$.

######  Remark 5.

Theorem 1.3 and Theorem 1.4 can be strengthened in the style of Theorem 1.2
about smooth families of boundaries and smooth families of metrics.

The above phenomenon is deeply connected to the Lavrentiev gap of the minimal
area between integral and mod $v$ coefficient homology. To discuss these, we
need some definitions.

######  Definition 1.1.

With the same assumption as in Theorem 1.2, let $[\Sigma]$ be a
$d$-dimensional integral homology class, define

$\displaystyle\inf_{\mathbb{Z}}\mathbf{M}_{g}([\Sigma])$
$\displaystyle=\inf_{Z\in[\Sigma]}\mathbf{M}_{g}(Z),$
$\displaystyle\inf_{\mathbb{Z}}\mathbf{M}_{g}([\Sigma\mod v])$
$\displaystyle=\inf_{[\Pi]\in
H_{d}(M,\mathbb{Z})}\inf_{\mathbb{Z}}\mathbf{M}_{g}([\Sigma]+v[\Pi]),$
$\displaystyle\inf_{\mathbb{Z}/v\mathbb{Z}}\mathbf{M}_{g}([\Sigma\mod v])$
$\displaystyle=\inf_{Z\in[\Sigma\mod v]\in
H_{d}(M,\mathbb{Z}/v\mathbb{Z})}\mathbf{M}_{g}^{v}(Z),$
$\displaystyle\left\lVert[\Sigma]\right\rVert_{g}$
$\displaystyle=\inf_{Z\in[\Sigma]\in H_{d}(M,\mathbb{R})}\mathbf{M}_{g}(Z).$

Here the subscript $g$ is the Riemannian metric and the superscript $v$
indicates mass mod $v$. The inclusion into
$\mathbb{Z},\mathbb{Z}/v\mathbb{Z},\mathbb{R}$ coefficient homology means the
natural map by universal coefficient theorem and the objects inside the
homology classes are integral currents, mod $v$ currents, and real flat
chains, respectively.

To simplify notations, we will write $\left\lVert\Sigma\right\rVert_{g}$
instead of $\left\lVert[\Sigma]\right\rVert_{g}.$ Also, when there is no
confusion, we will omit the subscript $g.$

######  Remark 6.

The reader might wonder why we use $\left\lVert\cdot\right\rVert_{g}$ instead
of $\inf_{\mathbb{R}}\mathbf{M}_{g}([\Sigma]).$ Federer ([24]) proved that
$\left\lVert\cdot\right\rVert_{g}$ is a well-defined continuous norm on the
real homology groups of $M$ and this is a crucial fact we use many times.
Details can be found in Section 2.8.

First of all, by definition we have

(1.1) $\displaystyle\inf_{\mathbb{Z}}\mathbf{M}_{g}([\Sigma\mod
v])\geq\inf_{\mathbb{Z}/v\mathbb{Z}}\mathbf{M}_{g}([\Sigma\mod v]).$

######  Definition 1.2.

Define

$\displaystyle K_{[\Sigma\mod v]}$

to be the set of metrics $g$ on $M$, so that
$\inf_{\mathbb{Z}}\mathbf{M}_{g}([\Sigma\mod
v])=\inf_{\mathbb{Z}/v\mathbb{Z}}\mathbf{M}_{g}([\Sigma\mod v]).$

######  Remark 7.

When $d=1,v=2,$ by regularity of $1$-d minimizing currents ([23]), equality
always holds in (1.1).

We have

######  Corollary 1.

  1. (1)

$K_{[\Sigma\mod v]}$ is always a (possibly empty) closed set, and equality in
(1.1) holds if and only if some area-minimizing current mod $v$ in
$[\Sigma\mod v]$ is integral.

  2. (2)

There exists $\nu>0,$ so that

$\displaystyle\bigcap_{v\geq\nu}K_{[\Sigma\mod v]}$

is a closed set with a non-empty interior.

  3. (3)

$\displaystyle\bigcup_{v}K_{[\Sigma\mod v]}$

equals the set of all smooth Riemannian metrics on $M.$

We show that Theorem 1.2 is sharp in the sense that $K_{[\Sigma\mod
v]}^{\complement}$ is non-empty in many cases as follows.

######  Theorem 1.5.

With the same assumption as in Theorem 1.2, suppose

  * •

$c\geq d\geq 2,$ or $c=1,d\geq 2,$ $v$ is an integer with $v\equiv 2(\mod 4)$,
and $R$ is a positive real number,

  * •

$[\Sigma]$ is an integral homology class so that $2[\Sigma\mod v]=0.$

Then

  * •

there exists a smooth Riemannian metric $g$ on $M$ so that

$\displaystyle\frac{\inf_{\mathbb{Z}}\mathbf{M}_{g}([\Sigma\mod
v])}{\inf_{\mathbb{Z}/v\mathbb{Z}}\mathbf{M}_{g}([\Sigma\mod v])}\geq R$

In other words, the minimum area among mod $v$ currents is arbitrarily small
compared to integral currents, and $K_{[\Sigma\mod v]}^{\complement}$ is a
non-empty open set.

######  Remark 8.

Note that if the $d$-th Betti number of $M$ is non-zero, then such $[\Sigma]$
always exists.

######  Remark 9.

Frank Morgan has communicated to the author the hypersurface case of the above
theorem, and the credit goes to him for the proof of $c=1,d\geq 2$.
Furthermore, Morgan has pointed out that on compact manifolds with
$\mathbb{Z}/v\mathbb{Z}$ symmetry, one can construct mod $v$ area-minimizers
that have mod $v$ junction-like singularities, i.e., non-empty codimension 1
strata. Thus, Theorem 1.2 is indeed sharp.

Similar result holds for Theorem 1.4.

######  Theorem 1.6.

Let $d\geq 2,c\geq 1$ be integers. Then for any $v\geq 2,$ $R>0,$ there exists
a smooth $d-1$-dimensional connected embedded compact oriented submanifold
$\Gamma$ of $\mathbb{R}^{d+c}$, so that

$\displaystyle\frac{\inf\\{\mathbf{M}(T)|\partial
T=\Gamma\\}}{\inf\\{\mathbf{M}^{v}(T)|\partial T=\Gamma\mod v\\}}\geq R.$

The above results can be compared with Federer’s work [24] and the author’s
work [36] on the Lavrentiev gap between $\mathbb{Z}$ coefficient and
$\mathbb{R}$ coefficient minimal mass, which is more prevalent. Also, compared
to Theorem 1.2, it is not expected that real coefficient mass minimizers in
general lie in integral homology.

###  1.1. Sketch of proof

Consider any mod $v$ area-minimizer in $[\Sigma].$ Take any of its
representing lifts to rectifiable currents, denoted by $T$. The key
observation is that the monotonicity formula gives a priori density upper
bounds on mod $v$ minimizers, forcing every point to have a density smaller
than $v/2$ for $v$ large. This strict density bound has three consequences:

  1. (1)

the boundary of $T$ is supported in the singular set,

  2. (2)

the codimension $1$ strata of the singular set is empty,

  3. (3)

as per Theorem 1.6 of [20], the top strata singular points have codimension at
least two.

These observations, combined with Almgren’s stratification of singular sets,
imply that the boundary of the rectifiable lift, $T$, has at least codimension
$2$. By classical properties of integral flat chains, this means $T$ must be
an integral cycle.

A more quantitative approach to these arguments proves Theorem 1.2 and 1.3.
Corollary 1 follows directly from these results. For the Plateau problems in
Euclidean space (Theorem 1.4), the method is analogous. Theorem 1.5 leverages
Frank Morgan’s calibration modulo $v$ ([44]) and Yongsheng Zhang’s
constructions ([57][58]), while Theorem 1.6 is based on a simple perturbation
of $v$ multiples of a $d-1$ dimensional unit sphere.

## Acknowledgements

I cannot thank my advisor Professor Camillo De Lellis enough for his
unwavering support while I have been recovering from illness. I feel so lucky
that I have Camillo as my advisor. Many thanks go to him for countless helpful
suggestions regarding this manuscript and others. The author also wants to
thank Professor Frank Morgan for his constant support and pioneering work that
has inspired many constructions in the author’s works. Many thanks go to him
for countless helpful conversations. A special thank goes to Professor Bruno
Martelli, whose Mathoverflow answer [38] is essential to the $c=1$ case of
Theorem 1.5.

##  2\. Preliminaries and Lemmas

In this section, we will fix our notations and prove some basic lemmas.

###  2.1. Manifolds and neighborhoods

We will reserve $M$ to denote an ambient smooth compact closed Riemannian
manifold. Submanifolds will be denoted by $N,L,Q$, etc. We will use the
following sets of definitions and notations.

######  Definition 2.1.

Let $N$ be a smooth submanifold of a Riemannian manifold $(M,g).$

  * •

$B_{r}^{(M,g)}(N)$ denotes a tubular neighborhood of $N$ inside $M$ in the
intrinsic metric on $M$ of radius $r$. When there is no a priori choice of
metric, then use an arbitrary metric.

  * •

$\textnormal{Inj}_{g}(M)$ denotes the injectivity radius of $M$ in metric $g.$

  * •

$\mathbf{T}_{p}M$ denotes the tangent space to $M$ at $p.$ We will often
regard $\mathbf{T}_{p}N$ as a subspace of $\mathbf{T}_{p}M.$

  * •

$\exp^{\perp}_{N\subset M}$ denotes the normal bundle exponential map of the
inclusion $N\subset M.$

  * •

$\textnormal{FocalRad}_{N}^{M}$ is the focal radius of $N$ in $M,$ i.e., the
radius below which the normal bundle exponential map remains injective.

  * •

$\pi_{N}^{M}$ denotes the nearest distance/normal bundle exponential map
projection from $M$ to $N$ in the metric intrinsic of $M$ in $B_{r}^{N}(M)$
with $r$ less than the focal radius of $N$ inside $M$.

######  Remark 10.

We will often drop the sup/subscripts $M,N$ when there is no confusion.

###  2.2. Algebraic topology

The main reference is [32]. For a compact manifold $M,$ $H_{d}(M,\mathbb{Z})$
is a finitely generated abelian group. We say a class is torsion if some
multiple of it is zero, and call a class non-torsion otherwise.

We fix a decomposition of the homology group as follows:

  * •

${[\Sigma_{1}],\cdots,[\Sigma_{b}],[\tau_{1}],\cdots,[\tau_{a}]}$ is a
$\mathbb{Z}$-linear independent generating set of $H_{d}(M,\mathbb{Z})$ over
$\mathbb{Z},$

  * •

all $[\Sigma_{j}]$ are non-torsion and all $[\tau_{j}]$ are torsion,

  * •

$N=\max_{j}\deg[\tau_{j}],$ where the degree is the least positive integer
which multiplies the torsion class to zero.

The reader should be familiar with the universal coefficient theorem, which
will remain fundamental to all of our discussions. The natural map from
integral to mod $v$ homology will be denoted by $[\Sigma]\mapsto[\Sigma\mod
v].$

###  2.3. Integral currents and flat chains

In this manuscript, we need several notions of currents. For a comprehensive
introduction, the standard references are [49] and [22]. For our purposes, the
reader suffices to know the following from 4.1.24 in [22]

  * •

$d$-dimensional rectifiable currents are finite mass limits of the Lipschitz
image of oriented polyhedron chains under flat topology.

  * •

$d$-dimensional integral currents are $d$-dimensional rectifiable currents
with boundary being $d-1$-dimensional rectifiable currents.

  * •

$d$-dimensional integral flat chains decompose as sum of $d$-dimensional
rectifiable currents and boundary of $d+1$-dimensional rectifiable currents.

By 4.2.16 of [22], integral currents are rectifiable currents with finite
boundary mass. Rectifiable currents are integral flat chains with finite mass.
We will use the following definition of the irreducibility of currents.

######  Definition 2.2.

A closed integral current $T$ is irreducible in $U,$ if we cannot write
$T=S+W,$ with $\partial S=\partial W=0,$ and $S,W$ nonzero, and
$\mathbf{M}(T)=\mathbf{M}(S)+\mathbf{M}(W).$

######  Remark 11.

It is easy to see that the definition of irreducibility above is equivalent to
1.c.i of Section 2.7 of [35], if we assume the other assumptions in that
Section.

Often we will abuse the notations by using $T$ to denote $\textnormal{spt}T$.
Also, we will use the differential geometry convention of closedness. An
integral current $T$ is closed if $\partial T=0$ and $T$ has compact support.

###  2.4. Mod $v$ currents

By mod $v$ currents, we mean flat chains modulo $v$. The main reference is
Section 4.2.26 of [23], [56] and [9]. Roughly speaking, they are the closure
of polyhedron chains mod $v$ under mod $v$ flat topology. For our purposes, it
suffices to know that they form a complete set under mod $v$ flat topology
(page 424 of [23]), and the chain complex formed by them induces precisely the
homology with mod $v$ coefficient (4.4.5 of [23]).

There is a natural map from integral flat chains to flat chains mod $v$, i.e.,
taking its mod $v$ equivalence class ([56], 4.2.26 of [22]). We will use
$T\mapsto T\mod v$ to denote this map.

Conversely, every mod $v$ current $T$ of finite mass has at least one
representative modulo $v$ (page 430 of [22]), i.e., a rectifiable current $S$
with the same induced mod $v$ mass measure and in the same mod $v$ equivalence
class as $T.$

Throughout the manuscript, we will speak of mod $v$ currents that are integral
currents. The definition is as follows.

######  Definition 2.3.

Suppose

  * •

$T$ is a mod $v$ current, $\Gamma$ is an integral current,

  * •

$\partial T=\Gamma\mod v.$

We say $T$ is an integral current if

  * •

at least one representative modulo $v$, say $S$, of $T$ is an integral
current,

  * •

$\textnormal{spt}\partial S\subset\textnormal{spt}\Gamma.$

###  2.5. Mass and comass

The comass of a $d$-dimensional differential form $\phi$ in a metric $g$ is
defined as

$\displaystyle\operatorname{comass}_{g}\phi=\sup_{x}\sup_{P\subset\mathbf{T}_{x}M}\phi(\frac{P}{|P|_{g}}),$

where $P$ ranges over $d$-dimensional oriented planes in the tangent space to
$M.$

The mass of an integral current is defined as

$\displaystyle\mathbf{M}_{g}(T)=\sup_{\operatorname{comass}_{g}(\phi)\leq
1}T(\phi).$

The definition of mass mod $v$ is more complicated, but roughly

$\mathbf{M}^{v}(T)$

equals the mass of the least mass of rectifiable currents in the same modulo
class (page 430 of [22]).

###  2.6. Representing homology classes by submanifolds

In this section, we will collect several facts about representing homology
classes using submanifolds.

######  Lemma 2.1.

For $d\geq 2,c\geq 2,$ there exists a connected embedded $d$-dimensional non-
orientable submanifold $L^{d}$ in $\mathbb{R}^{d+c}.$

###### Proof.

It is well known that $\mathbb{RP}^{2}$ embeds into the unit ball in
$\mathbb{R}^{4},$ e.g., [33]. Consider the following map
$u:\mathbb{R}^{4}\times\mathbb{R}^{d-1}\to\mathbb{R}^{d+2}$ defined by

$\displaystyle
u(x_{1},\cdots,x_{4},y_{1},\cdots,y_{d-1})=((4+x_{1})y_{1},\cdots,(4+x_{1})y_{d-1},x_{2},x_{3},x_{4}).$

Straightforward calculation shows that $u$ is an embedding restricted to
$B_{2}^{4}(0)\times S^{d-2}.$ Thus, $u(\mathbb{RP}^{2}\times S^{d-2})$ is an
embedding of $\mathbb{RP}^{2}\times S^{d-2}$ into $\mathbb{R}^{d+2}$. Note
that $\mathbb{RP}^{2}\times S^{d-2}$ is not orientable. (Suppose not. Equip
the product with a product metric. Interior product of the volume form on the
sphere factor with the volume form on the product gives a volume form on
$\mathbb{RP}^{2},$ a contradiction.) Now consider any standard embedding
$\mathbb{R}^{d+2}\subset\mathbb{R}^{d+c}$ and compose it with $u.$ We are
done. ∎

######  Lemma 2.2.

With the assumptions in Theorem 1.6, there exists a closed connected non-
orientable smooth submanifold $Q$ of $M,$ so that

$\displaystyle[\Sigma\mod v]=[\frac{v}{2}Q\mod v].$

###### Proof.

Let us deal with the case of $c\geq d\geq 2$ first. As mentioned in the second
paragraph of [5], [51] and [50] show that for any integral homology class
$[\Sigma]$ with $d,c\geq 1$, there is an odd integer $\lambda$ so that
$\lambda[\Sigma]$ can be represented by continuous maps from smooth manifolds.
See also [4] for sharp bounds on $\lambda$. By Whitney’s approximation theorem
(Theorem 6.26 of [34]) and the denseness of embeddings in the mapping space
(Theorem 4.7.7 of [52]) when $c>d,$ we can get a smoothly embedded submanifold
$N$ representing $\lambda[\Sigma].$ When $c=d,$ the above argument gives an
immersed representative of $\lambda[\Sigma]$ with transverse double points
only. However, one can always replace the double points with necks to get a
smoothly embedded representative of $\lambda[\Sigma]$ as well.

To sum it up, for some odd number $\lambda$, $\lambda[\Sigma]$ can be
represented by an embedded orientable smooth submanifold $N$. Since the
codimension is larger than $1$, using connected sums, one can assume that $N$
is connected (detailed argument in the proof of Lemma 2.1 in [36]). Since
$\lambda,\frac{v}{2}$ is odd and $[2\Sigma\mod v]=0$, we deduce that

$\displaystyle[\frac{v}{2}N\mod v]=[\frac{v}{2}\lambda\Sigma\mod
v]=[\Sigma\mod v].$

Pick a point $p\in N,$ and a small coordinate $(x_{1},\cdots,x_{d+c})$ ball
$B_{r}(p)$ around $p$ so that $N$ becomes the $x_{1},\cdots,x_{d}$-plane. By
Lemma 2.1, there exists a non-orientable manifold $L^{d}$ that can be embedded
into $B_{r}(p)\cap\\{x_{d+c}>0\\},$ since the latter is diffeomorphic to
$d+c$-dimensional standard ball. Pick a point $q$ on $L$ and take a curve
$\gamma$ from $p$ to $q$ that is disjoint from $N,L$ except at the endpoints.
Make a tubular neighborhood of $\gamma$ into a neck and make a connected sum
of $N\\#L$. Note that $N\\#L$ is not orientable, otherwise, $L$ minus a small
disk is orientable. Then adding the disk will keep orientability, a
contradiction.

Set $Q=N\\#L$. Then we claim that $[\frac{v}{2}Q\mod v]=[\frac{v}{2}N\mod
v]=[\Sigma\mod v]$.

To see this, let us show that $[\frac{v}{2}L\mod v]$ is a boundary. Note that
$\frac{v}{2}L$ is indeed a mod $v$ cycle. Consider $\mathbb{RP}^{2}$ as an
oriented chain with boundary $2\mathbb{RP}^{1}$ and take the chain structure
to $L=\mathbb{RP}^{2}\times S^{d-2}.$ Since the $d$-dimensional mod $v$
homology of $B_{r}(p)$ is trivial by homotopy invariance, we deduce that
$\frac{v}{2}L$ is indeed a mod $v$ boundary.

The necks we use to do connected sums are tubular neighborhoods of $\gamma.$
By taking a continuous family of ambient diffeomorphisms that shrink a
neighborhood around $\gamma$ to $\gamma$, we deduce that
$[\frac{v}{2}Q]=[\frac{v}{2}N]+[\frac{v}{2}L]$ by homotopy invariance ([32]).
We are done with the case $c\geq d\geq 2$.

For the case of $c=1,d\geq 2,$ by Corollary 3.28 of [32],
$H_{d}(M,\mathbb{Z})\equiv\mathbb{Z}^{b_{d}},$ with $b_{d}$ the $d$-th Betti
number of $M.$ It is straightforward to check that our assumptions imply that
$[\Sigma\mod v]=\frac{v}{2}[\Pi\mod v]$ for some primitive homology class
$\Pi\in H_{d}(M,\mathbb{Z}).$ By point 2 of Representation Theorem of [39],
$[\Pi]$ can be represented by a connected closed smooth hypersurface $N$ and
$M\setminus N$ has one path component with two ends. Instead of doing
connected sums with $L,$ which does not exist for $c=1,$ we do a self-
connected sum to destroy the orientation of $N,$ as pointed out by [38]. Since
$N\setminus M$ has only one path component, take a smooth curve in $M\setminus
N$ to connect its two ends. Do a connected sum of $N$ along $\gamma$ with
itself. Call the result $Q.$ It is straightforward to check that $Q$ is
unorientable and $Q$ is homologous to $N$ as mod $2$ chains. Now apply the
same argument as in the case of $c\geq d\geq 2,$ we deduce that $[\Sigma\mod
v]=[\frac{v}{2}Q\mod v].$ We are done. ∎

######  Lemma 2.3.

With the same assumptions as in Theorem 1.2, suppose

  * •

$[\Pi]\in H_{d}(M,\mathbb{Z}/v\mathbb{Z})$ is a mod $v$ homology class with
$v\geq 2$,

  * •

there is a smooth connected not necessarily orientable manifold $Q$ and an
integer $0<t\leq\frac{v}{2},$

  * •

$[\Pi]=[tQ].$

Then for any $R>1,$ there exists a smooth metric $g$, and a neighborhood $U$
around $Q,$ such that

  * •

$U$ deformation retracts onto $Q$,

  * •

$tQ$ is homologically mod $v$ area-minimizing in $(M,g)$,

  * •

any stationary varifold that is not compactly contained in $U$ must have mass
at least $Rt\mathbf{M}(Q)$.

###### Proof.

First, equip $M$ with an arbitrary smooth metric $h.$ Then take
$U=B_{\frac{r}{3}}(Q)$ with any chosen $r<\textnormal{FocalRad}_{Q}^{M}.$ By
construction, $U$ deformation retracts onto $Q.$

Even if $Q$ is not orientable, $\left\lVert(\pi_{Q}^{M})dvol_{Q}\right\rVert$
is a well-defined smooth function in $B_{r}(Q),$ where $dvol_{Q}$ is any
locally chosen volume form of $Q.$

By Remark 3.5 of [57] and Lemma 5.1 of [35], we deduce that $\pi_{Q}^{M}$ is
an area-non-increasing projection in the metric
$\left\lVert(\pi_{Q}^{M})dvol_{Q}\right\rVert^{\frac{2}{d}}h.$

For any $R>1,$ using the proof of Theorem 4.1 in [58], there exists a smooth
function $f$, so that $f$ equals $1$ near $Q,$ and in $fh,$ $\pi_{Q}^{(M,h)}$
is still area-non-increasing in $B_{r}^{h}(Q).$ Moreover, any stationary
varifold in $(M,fh)$ that is not compactly contained in
$B_{\frac{r}{3}}^{h}(Q)$ has area at least $R\mathbf{M}_{h}(tQ)$.

We claim that $tQ$ is mod $v$ area-minimizing in $(M,fh)$. To see this,
suppose $T$ is an area-minimizing current mod $v$ in $[\Sigma]=[tQ].$ Then by
construction, $T$ must be contained in $B_{\frac{r}{3}}^{h}(Q),$ otherwise
having more area than $tQ.$ However, if $T$ is contained in
$B_{\frac{r}{3}}(Q),$ then

$\displaystyle\mathbf{M}^{v}(\pi_{Q}^{(M,h)}(T))\leq\mathbf{M}^{v}(T),$

thus $\pi_{Q}^{(M,h)}(T)$ must also be area-minimizing mod $v.$ Since
$\pi_{N}^{M}$ is a deformation retract, we deduce that $\pi_{Q}^{M}(T)$ is
supported in $Q$ and quals to $tQ\mod v.$ Since $1\leq t\leq\frac{v}{2},$ we
must have $\mathbf{M}^{v}(\pi_{Q}^{M}(T))=\mathbf{M}(tQ).$ We are done. ∎

###  2.7. A priori density bounds

In this subsection, we will collect several a priori density bounds about mod
$v$ currents.

######  Lemma 2.4.

With the same assumptions as in Theorem 1.2, let $g$ be a smooth Riemannian
metric on $M.$ Then there exist an open set of smooth Riemannian metrics
$\Omega_{g}$ which contains $g$, and constants $c_{g},r_{g}>0$, such that,

  * •

for any $d$-dimensional stationary integral varifold $V$ on $M$ in metric
$g^{\prime}\in\Omega_{g}$, any point $p\in M$ and radii $r\in(0,r_{g}],$ the
function

$\displaystyle\exp(c_{g}r)V(B_{r}^{g^{\prime}}(p))r^{-d}$

is monotonically increasing in $r$, where $V(B_{r}^{g^{\prime}}(p))$ is the
measure $V$ of the radius $r$ ball centered at $p$ in $g^{\prime}.$

  * •

the density $\theta_{p}$ of $V$ stationary in any $g^{\prime}\in\Omega_{g},$
at any point $p\in M$ is well-defined and there is a constant $C>0$, so that

$\displaystyle\theta_{p}<CV(M).$

###### Proof.

This is folklore and we will only give a sketch of the proof. First consider
everything in the metric $g,$ for any point $p\in M$ and
$\rho=\frac{1}{2}\textnormal{Inj}_{g}(M),$ adopt a normal coordinate
$(x_{1},\cdots,x_{d+c})$ in $B_{\rho}(p)$. Let $r$ denote the distance to $p$
then by Chapter 2 in [28] we have

$\displaystyle r\nabla r=\sum_{j}x_{j}\partial_{j}.$

A straightforward calculation using Taylor expansion shows that

$\displaystyle|\langle\nabla e(r\nabla r),e\rangle-1|\leq c,$

where $e$ is any unit length tangent vector in $B_{\rho}(p).$ Here the
constant $c$ depends is controlled by the second derivatives of the metric.

Arguing as in the first two sections of [7], we deduce that
$\exp(cr)V(B_{r}^{g}(p))r^{-d}$ is monotonically increasing in $r$.

Now if one varies the point $p$, since $c$ is controlled by $\left\lVert
g\right\rVert_{C^{3}},$ by the compactness of $M,$ we deduce the existence of
a uniform $c$ that works for all $p.$ Note that all of the above is
independent of $V.$

By Section 8 Theorem in [21], $\textnormal{Inj}_{g}(M)$ depends continuously
on the metric. Thus, there exists an open neighborhood $\Omega_{g}$ in the
space of Riemannian metrics so that $g\in\Omega_{g}$ and
$r_{g}=\frac{1}{2}\inf_{g^{\prime}\in\Omega_{g}}\textnormal{Inj}_{g^{\prime}}(M)>0.$
However for each $g^{\prime}\in\Omega_{g}$, running the above argument gives a
monotonicity formula with a constant $c$ that is controlled by $C^{3}$ norm of
the metric. Thus, by shrinking $\Omega_{g}$ if necessary, we can get a uniform
upper bound $c_{r}$. Since the product of non-negative monotonically
increasing functions preserves the monotonicity, we are done with the first
two claims.

The last claim follows from the first two points and

(2.1) $\displaystyle\theta_{p}=\lim_{r\to
0}\frac{V(B_{r}^{g^{\prime}}(p))}{r^{d}}\leq\exp(c_{g}r_{g})V(B_{r_{g}}^{g^{\prime}}(p))\leq
CV(M).$

∎

######  Lemma 2.5.

With the same assumptions as in Theorem 1.2, there exists an open set
$\Omega_{K,S}$ in the space of Riemannian metrics with $K\subset\Omega_{K,S}$,
and a real number $\theta_{K,S}$ such that

  * •

for any metric $g\in\Omega_{K,S}$, any $v$ with $v\geq 2$, any area-minimizing
mod $v$ current $T$ in any homology class $[\Sigma\mod v]$ with $[\Sigma]\in
S$ has density at most $\theta_{K,S}$ at any point.

###### Proof.

First, it suffices to deal with the case $S=\\{[\Sigma]\\},$ $K=\\{g\\}$,
i.e., both having only one element. To see this, suppose we have already
proved the lemma for the one-element-only case. Then by finiteness of $S$,
setting $\Omega_{g,S}=\cap_{[\Sigma]\in S}\Omega_{g,[\Sigma]}$ and
$\theta_{g,S}=\max_{[\Sigma]\in S}\theta_{g,S}$ proves the case for $K=g$ and
$S$ arbitrary. Now, $\cup_{g\in K}\Omega_{g,S}$ forms an open cover of $K,$ so
we can choose a finite subcover $K\subset\cup_{j=1}^{k}\Omega_{g_{j},S}.$
Finally, setting $\Omega_{K,S}=\cup_{j=1}^{n}\Omega_{g_{j},S}$ and
$\theta_{K,S}=\max_{j=1}^{n}\theta_{g_{j},S}$ and we are done.

With the reduction in hand, it suffices to prove the one-element-only case.

Apply Lemma 2.4 to deduce that there exists an open set of Riemannian metrics
$\Omega_{g}\supset g,$ so that for any stationary varifold $V$ in any metric
$g^{\prime}\in\Omega_{g}$, for any point $p,$ we have

(2.2) $\displaystyle\theta_{p}<CV(M).$

Let $T$ be an area-minimizing mod $v$ current in $[\Sigma\mod v]$ in metric
$g^{\prime}\in\Omega_{g}$, by (2.2) we deduce that

$\theta_{p}<C\mathbf{M}^{v}(T).$

Let $S$ be an area-minimizing integral current in $[\Sigma].$ Then

$\displaystyle\mathbf{M}^{v}(T)\leq\mathbf{M}^{v}(S)\leq\mathbf{M}(S).$

Thus, we have

(2.3) $\displaystyle\theta_{p}<C\mathbf{M}(S),$

However, $\mathbf{M}(S)$, the minimal integral area in $[\Sigma]$, depends
continuously on the metric. Thus, by shrinking the $\Omega_{g}$ if necessary,
we can deduce a uniform upper bound. Since $C$ in Lemma 2.4 is independent of
the metric in $\Omega_{g}$, we are done. ∎

######  Lemma 2.6.

With the same assumptions as in Theorem 1.4, there exists $\theta>0$ so that
for any $v\geq 2,$ any mod $v$ area-minimizing current $T$ with $\partial
T=\Gamma\mod v$, the density of $T$ at any point is smaller than $\theta.$

###### Proof.

Let $S$ be an area-minimizing integral current with boundary $\Gamma.$ Then we
always have $\mathbf{M}^{v}(T)\leq\mathbf{M}^{v}(S)\leq\mathbf{M}(S).$

Since the measure $V_{T}$ of $T$ is a stationary integral varifold in
$\Gamma^{\complement},$ by compactness of $\Gamma$ and (2) Theorem of Section
3.4 in [2], there exists $s>0,c>0$ so that for any $0<r\leq s$, and any
$p\in\Gamma$,

(2.4) $\displaystyle\exp(cr)V_{T}(B_{r}(p))r^{-d}$

is monotonically increasing. This implies that

$\displaystyle\theta_{p}\leq\exp(cs)s^{-d}\mathbf{M}^{v}(T)\leq\exp(cs)s^{-d}\mathbf{M}(S),$

for any point $p\in\Gamma.$

On the other hand, for points $p$ at least $\frac{1}{2}s$ away from $\Gamma$,
by monotonicity formula in [1], we have

$\displaystyle\theta_{p}\leq(\frac{1}{2}s)^{-d}\mathbf{M}^{v}(T)=2^{d}s^{-d}\mathbf{M}(S).$

For $p$ of distance at most $\frac{1}{2}s$ from $\Gamma,$ let $q$ be a point
minimizing $|p-q|.$ Then

$\displaystyle\theta_{p}\leq$
$\displaystyle|p-q|^{-d}V_{T}(B_{|p-q|}(p))\leq|p-q|^{-d}V_{T}(B_{2|p-q|}(q))$
$\displaystyle\leq$ $\displaystyle 2^{d}\exp(c(s-2|p-q|))s^{-d}V(B_{s}(q))\leq
2^{d}\exp(cs)s^{-d}\mathbf{M}(S)$

Combining all the three cases, we are done. ∎

###  2.8. Norms on homology

In this subsection, we will collect several facts about the minimal mass
(Definition 1.1) in homology. First of all the reader needs to know that
$\left\lVert\cdot\right\rVert_{g}$ is non-zero on an integral homology class
if and only if the class is non-torsion ([24]). Thus, adding torsion classes
does not change the norm.

######  Lemma 2.7.

Let $\left\lVert\cdot\right\rVert$ be a continuous norm on $\mathbb{R}^{n}$
and $\\{v_{1},\cdots,v_{n}\\}$ be a basis. Then there exists a $C>0$ that
depends continuously on $\left\lVert\cdot\right\rVert,$ so that

$\displaystyle\sum_{j}|a_{j}|\left\lVert v_{j}\right\rVert\leq
C\left\lVert\sum_{j}a_{j}v_{j}\right\rVert,$

with $a_{j}\in\mathbb{R}.$

###### Proof.

For $v=\sum_{j}a_{j}v_{j}$ with $a_{j}\in\mathbb{R}$ define

$\displaystyle\left\lVert v\right\rVert^{\infty}=\sum_{j}|a_{j}|\left\lVert
v_{j}\right\rVert.$

It is straightforward to verify that $v$ is a continuous function on
$\mathbb{R}^{n}$ and $v$ is a norm on $\mathbb{R}^{n}.$ Thus, the lemma is
equivalent to $\left\lVert v\right\rVert^{\infty}/\left\lVert v\right\rVert$
is bounded. By homogeneity, it suffices to verify this for $\left\lVert
v\right\rVert=1.$ By continuity of $\left\lVert\cdot\right\rVert^{\infty},$
and the compactness of $\left\lVert v\right\rVert=1$, we are done. ∎

######  Lemma 2.8.

With the same assumptions as in Theorem 1.2, for any fixed metric, there is a
constant $D>0,$ so that for any non-torsion homology class $[\Sigma]\in
H_{d}(M,\mathbb{Z})$ we have

$\displaystyle\frac{\inf_{\mathbb{Z}}\mathbf{M}([\Sigma])}{\left\lVert\Sigma\right\rVert}\leq
D.$

######  Remark 12.

For torsion class $[\Sigma]$, $\left\lVert\Sigma\right\rVert=0$ while
$\inf_{\mathbb{Z}}\mathbf{M}([\Sigma])>0.$

###### Proof.

In the notation of Section 2.2, we can write

$\Sigma=\sum_{j}\alpha_{j}[\Sigma_{j}]+\beta_{j}[\tau_{j}],$

with $\alpha_{j}\in\mathbb{Z},\beta_{j}\in[-N,N]$, where we regard terms with
ill-defined subscript as zero. By Lemma 2.7 there is $C>0,$ so that

$\displaystyle\left\lVert\Sigma\right\rVert=\left\lVert\sum_{j}[\Sigma_{j}]\right\rVert\geq
C^{-1}\sum_{j}|\alpha_{j}|\left\lVert[\Sigma_{j}]\right\rVert.$

By the mediant inequality and non-torsion of $[\Sigma]$ (i.e.,
$\max_{j}|\alpha_{j}|\geq 1$), we have

$\displaystyle\frac{\inf_{\mathbb{Z}}\mathbf{M}([\Sigma])}{\left\lVert\Sigma\right\rVert}$
$\displaystyle\leq$ $\displaystyle
C\frac{\sum_{j}|\alpha_{j}|\inf_{\mathbb{Z}}\mathbf{M}([\Sigma_{j}])+|\beta_{j}|\inf_{\mathbb{Z}}\mathbf{M}([\tau_{j}])}{\sum_{j}|\alpha_{j}|\left\lVert\Sigma_{j}\right\rVert}$
$\displaystyle=$ $\displaystyle
C\frac{\sum_{j}|\alpha_{j}|\inf_{\mathbb{Z}}\mathbf{M}([\Sigma_{j}])}{\sum_{j}|\alpha_{j}|\left\lVert\Sigma_{j}\right\rVert}+C\frac{\sum_{j}|\beta_{j}|\inf_{\mathbb{Z}}\mathbf{M}([\tau_{j}])}{\sum_{j}|\alpha_{j}|\left\lVert\Sigma_{j}\right\rVert}$
$\displaystyle\leq$ $\displaystyle
C\max_{j}\frac{\inf_{\mathbb{Z}}\mathbf{M}([\Sigma_{j}])}{\left\lVert[\Sigma_{j}]\right\rVert}+CNa\frac{\max_{j}\inf_{\mathbb{Z}}\mathbf{M}([\tau_{j}])}{\min_{j}\left\lVert\Sigma_{j}\right\rVert},$

where $N,a$ are constants defined in the decomposition of
$H_{d}(M,\mathbb{Z})$ as in Section 2.2. The last line is independent of
$\alpha_{j},\beta_{j},\cdots$. We are done. ∎

###  2.9. Regularity theorems

In this subsection, we will prove some essential lemmas about the regularity
of mod $v$ currents.

######  Lemma 2.9.

Let $T$ be a mod $v$ area-minimizing current in a not necessarily complete
open Riemannian manifold $M$, with $v\geq 2.$ Let $S$ be a rectifiable current
that is representative modulo $v$ of $T$ (page 430 of [22]). Then

$\displaystyle\textnormal{spt}\partial
S\subset(\textnormal{Reg}(T)\cap\\{p|\theta_{p}<\frac{v}{2}\\})^{\complement},$

where $\textnormal{Reg}(T)$ denotes the regular part of $T.$

###### Proof.

Since $\theta\leq\frac{v}{2}$ $d$-dimensional a.e. for $S$ (page 430 of [22]),
it suffices to prove that if $p\in\textnormal{spt}\partial
S\cap\textnormal{Reg}(T),$ then $\theta_{p}=\frac{v}{2}.$ Since $p$ is in the
regular set of $T,$ there exists a neighborhood $B_{r}(p)$ in which $S$
restricted to $B_{r}(p)$ equals $\theta_{p}N$ for some smooth submanifold $N$
of $B_{r}(p)$, so that in some coordinate system $(x_{1},\cdots,x_{d+c})$, $N$
is the a smooth ball contained in the $x_{1}\cdots x_{d}$-plane. By Corollary
1.4 of [56], there exists a integral flat chain $R$ in $B_{r}(p)$ so that

$\displaystyle S\mathop{\hbox{\vrule
height=7.0pt,width=0.3pt,depth=0.0pt\vrule
height=0.3pt,width=5.0pt,depth=0.0pt}}\nolimits B_{r}(p)=vR+\theta_{p}N$

as a rectifiable current. Since $S,N$ are both of finite mass, we deduce that
$R$ is also of finite mass, thus a rectifiable current by 4.2.16 of [22].

By 4.1.28(5) of [22], there exist a $d$-dimensional rectifiable set $B$, so
that

$\displaystyle R(\phi)=\int_{B}\Theta\langle\eta,\phi\rangle
d\mathcal{H}^{d},$

with $\phi$ any smooth $d$-dimensional form, $\eta$ an $\mathcal{H}^{d}$
measurable simple unit vector field, $\Theta$ is $\mathcal{H}^{d}$ a.e. the
positive integer density of $S.$

By page 430 of [22], the support of $S$ is contained in $N,$ thus the support
of $R$ is also contained in $N,$ i.e., $B\subset N.$ This implies that
$\eta=\pm\mathbf{T}N$ $\mathcal{H}^{d}$-a.e.

Thus, we have

$\displaystyle S\mathop{\hbox{\vrule
height=7.0pt,width=0.3pt,depth=0.0pt\vrule
height=0.3pt,width=5.0pt,depth=0.0pt}}\nolimits
B_{r}(p)(\phi)=\int_{N\setminus B}\theta_{p}\langle\mathbf{T}_{x}N,\phi\rangle
d\mathcal{H}^{d}(x)+\int_{B}(\theta_{p}\pm
v\Theta)\langle\mathbf{T}_{x}N,\phi\rangle d\mathcal{H}^{d}(x).$

By 4.1.28 of [22], we deduce that

$\displaystyle\mathbf{M}(S\mathop{\hbox{\vrule
height=7.0pt,width=0.3pt,depth=0.0pt\vrule
height=0.3pt,width=5.0pt,depth=0.0pt}}\nolimits B_{r}(p))=\int_{N\setminus
B}\theta_{p}d\mathcal{H}^{d}+\int_{B}|\theta_{p}\pm v\Theta|d\mathcal{H}^{d}.$

However, since $\Theta$ is a positive integer on $B$ $\mathcal{H}^{d}$-a.e.,
and $\theta_{p}\leq\frac{v}{2}$, we have

$\displaystyle|\theta_{p}\pm v\Theta|\geq\frac{v}{2},$

where equality holds if and only if $\theta_{p}=\frac{v}{2},\Theta=1.$

Thus, we have

$\displaystyle\mathbf{M}(S\mathop{\hbox{\vrule
height=7.0pt,width=0.3pt,depth=0.0pt\vrule
height=0.3pt,width=5.0pt,depth=0.0pt}}\nolimits
B_{r}(p))\geq\theta_{p}\mathbf{M}(N)=\mathbf{M}^{v}(S\mathop{\hbox{\vrule
height=7.0pt,width=0.3pt,depth=0.0pt\vrule
height=0.3pt,width=5.0pt,depth=0.0pt}}\nolimits B_{r}(p)).$

However, by page 430 of [22], $\mathbf{M}(S\mathop{\hbox{\vrule
height=7.0pt,width=0.3pt,depth=0.0pt\vrule
height=0.3pt,width=5.0pt,depth=0.0pt}}\nolimits
B_{r}(p))=\mathbf{M}^{v}(S\mathop{\hbox{\vrule
height=7.0pt,width=0.3pt,depth=0.0pt\vrule
height=0.3pt,width=5.0pt,depth=0.0pt}}\nolimits B_{r}(p))$. Since $\Theta>0$
$\mathcal{H}^{d}$ a.e., then we must have $\theta_{p}=\frac{v}{2},$ $\Theta=1$
on $B$ $\mathcal{H}^{d}$ a.e. We are done. ∎

######  Lemma 2.10.

With the same assumption as in Lemma 2.5, assume

  * •

either $\partial T=0\mod v$ or $\partial T=\Gamma\mod v,$ where
$\Gamma=\sum_{j}t_{j}\Gamma_{j},$ with $\Gamma_{j}$ a finite collection of
disjoint compact connected oriented submanifolds of $M$ and
$\frac{v}{2}>t_{j}\geq 1$ integers,

  * •

the density of $T$ at any point $p\in M$ satisfies $\theta_{p}<\frac{v}{2}$.

Then $T$ is an integral current with either $\partial T=0$ or $\partial
T=\Gamma$, respectively.

###### Proof.

Let $S$ be a rectifiable current that is a representative modulo $v$ of $T.$
It suffices to prove that $\partial S=\Gamma.$

Let us first deal with the case of $\partial T=0\mod v$ using interior density
bound. Let us analyze strata by strata using Almgren stratification ([54]).

By Lemma 2.5, $\partial S$ intersect the regular set of $T$ only at those of
density $\theta\geq\frac{v}{2},$ which is empty by our assumptions. Thus,
$\partial S$ is supported in the singular set of $T.$

By Theorem 1.6 of [9] the flat tangent cone singular points of any mod $v$
area-minimizing current with density less than $\frac{v}{2}$ is of Hausdorff
codimension at least $2.$ In the hypersurface case, we can also use [53],
which gives the stronger conclusion that no such points exist.

Next, note that points with $d-1$-symmetric tangent cones are empty since
every such cone must be $\mathbb{R}^{d-1}$ times $v$ rays originating from the
origin, which has density $\frac{v}{2}.$

Since the $d-j$-strata of the singular set has Hausdorff dimension at most
$d-j$ ([54]), we deduce that the singular set of $T$ is of Hausdorff dimension
at most $d-2.$ Consequently, $\textnormal{spt}\partial S$ has Hausdorff
dimension at most $d-2.$ By 4.1.20 of [22], a $d-1$-dimensional flat chain
supported in a $d-2$-dimensional integral geometric measure zero set must
equal to zero. Since Hausdorff measure zero implies integral geometric measure
zero (2.10.6 of [22]), we deduce that $\partial S=0.$ Thus, $T$ is an integral
current.

Now let us deal with the case of $\partial T=\Gamma\mod v\not=0\mod v.$ Again,
arguing as above, we deduce that $\partial S$ must be supported in $\Gamma.$
By 4.1.31 of [22], we deduce that

$\displaystyle\partial S=\sum_{j}s_{j}\Gamma_{j},$

with $s_{j}\in\mathbb{Z}$.

Since $\partial S=\Gamma\mod v,$ we have

$s_{j}-t_{j}=l_{j}v$

with $l_{j}\in\mathbb{Z}.$ Now consider the Almgren stratification of $T$ at
the boundary points $p\in\Gamma$, then for $\mathcal{H}^{d-1}$ a.e., the
tangent cone must be $\mathbb{R}^{d-1}$ times $k$ (depending on $p$) number of
rays from origin, with $\frac{k}{2}<\frac{v}{2}$. Now if we orient the rays
using the orientation of $S,$ then there are $a$ rays pointing away from the
origin and $b$ rays pointing towards the origin, with

$\displaystyle a-b=s_{j}=t_{j}+l_{j}v,a+b=k.$

This implies that

$\displaystyle 0$ $\displaystyle\leq
a=\frac{1}{2}t_{j}+\frac{1}{2}l_{j}v+\frac{k}{4}<\frac{1}{4}v+\frac{1}{2}l_{j}v+\frac{v}{4},$
$\displaystyle 0$ $\displaystyle\leq
b=-\frac{1}{2}t_{j}-\frac{1}{2}l_{j}v+\frac{k}{4}<-\frac{1}{2}l_{j}v+\frac{v}{4}.$

This gives

$\displaystyle-1<l_{j}<\frac{1}{2}.$

Thus, we must have $l_{j}=0.$ We are done.∎

##  3\. Proof of Theorem 1.2,1.4,1.3 and Corollary 1

###  3.1. Proof of the theorems

For Theorem 1.2, apply Lemma 2.5 and Lemma 2.10.

For Theorem 1.4 apply Lemma 2.6 and Lemma 2.10.

For Theorem 1.3, by the universal coefficient theorem, a class in
$H_{d}(M,\mathbb{Z}/v\mathbb{Z})$ admits integral representatives if and only
if it lies in the image of tensoring with $\mathbb{Z}/v\mathbb{Z},$ i.e.,
taking a class modulo $v.$

Write

$H_{d}(M,\mathbb{Z})=\mathbb{Z}^{b_{d}}\oplus G,$

where $b_{d}$ is the $d$-th Betti number of $M$ and $G$ denotes the torsion
subgroup of $M.$ Then we have

$\displaystyle
H_{d}(M,\mathbb{Z})\otimes\mathbb{Z}/v\mathbb{Z}=(\mathbb{Z}/v\mathbb{Z})^{b_{d}}\oplus(G\otimes\mathbb{Z}/v\mathbb{Z}).$

Thus, the denominator in our theorem equals
$v^{b_{d}}\times\\#G\otimes\mathbb{Z}/v\mathbb{Z}.$

For $[\Sigma]\in G$, since $G$ is finite, by Theorem 1.2, there exists $\nu>0$
so that for any $v\geq\nu,$ all homology classes in
$G\otimes\mathbb{Z}/v\mathbb{Z}$ admits only integral minimizers. If
$b_{d}=0$, we are done. Now suppose $b_{d}>0$ and $v\geq\nu.$ For non-torsion
$[\Sigma]\in H_{d}(M,\mathbb{Z})$ that occurs as pre-image of modulo $v$, in
the notation of Section 2.1, write

$\displaystyle[\Sigma]=\sum_{j}\alpha_{j}[\Sigma_{j}]+\beta_{j}[\tau_{j}],$

with all $\alpha_{j}\in[-\frac{v}{2},\frac{v}{2}]\cap\mathbb{Z}$ and all
$\beta_{j}\in[-N,N]\cap\mathbb{Z}.$ Then by Lemma 2.8, we have

$\displaystyle\inf_{\mathbb{Z}/v\mathbb{Z}}\mathbf{M}([\Sigma\mod
v])\leq\inf_{\mathbb{Z}}\mathbf{M}([\Sigma])\leq
D\left\lVert\Sigma\right\rVert\leq
D\sum_{j}|\alpha_{j}|\left\lVert\Sigma_{j}\right\rVert.$

Now apply Lemma 2.4 to deduce that for any mod $v$ area-minimizing current in
$[\Sigma]$ and any point $p\in M,$ we have

$\displaystyle\theta_{p}<C\sum_{j}|\alpha_{j}|\left\lVert\Sigma_{j}\right\rVert\leq
Cb_{d}\sup_{j}\left\lVert\Sigma_{j}\right\rVert\max_{j}|\alpha_{j}|.$

Thus, if

(3.1)
$\displaystyle\max_{j}|\alpha_{j}|<\frac{v}{2Cb_{d}\sup_{j}\left\lVert\Sigma_{j}\right\rVert},$

then every area-minimizing mod $v$ current in $[\Sigma\mod v]$ is integral, by
Lemma 2.5.

Note that asymptotically we have

(3.2)
$\displaystyle\\#\bigg{\\{}(\alpha_{1},\cdots,\alpha_{j},\cdots)|\max_{j}|\alpha_{j}|<\frac{v}{2Cb_{d}\sup_{j}\left\lVert\Sigma_{j}\right\rVert}\bigg{\\}}=\bigg{(}\frac{v}{2Cb_{d}\sup_{j}\left\lVert\Sigma_{j}\right\rVert}\bigg{)}^{b_{d}}+O(v^{b_{d}-1})$

Note that the right-hand side of the inequality 3.1 is independent of the
coefficients $\beta_{j}.$ In other words, the same estimate holds for any
other homology class $[\Sigma^{\prime}]$ so that $[\Sigma^{\prime}]-[\Sigma]$
is torsion. Thus, counting all non-torsion $[\Sigma]$ asymptotically with
(3.1) amounts to multiplying (3.2) with $\\#G\otimes\mathbb{Z}/v\mathbb{Z}$,
minus the number of torsion classes.

Consequently, breaking down the numerator in the theorem into non-torsion and
torsion classes, we have

$\displaystyle\liminf_{v}\frac{\\#\\{H_{d}(M,\mathbb{Z}/v\mathbb{Z})\ni[\Pi]\textnormal{
having only integral area-
minimizers}\\}}{\\#\\{H_{d}(M,\mathbb{Z}/v\mathbb{Z})\ni[\Pi]\textnormal{
admitting integral representatives}\\}}$ $\displaystyle\geq$
$\displaystyle\liminf_{v}\frac{\Bigg{(}\bigg{(}\frac{v}{2Cb_{d}\sup_{j}\left\lVert\Sigma_{j}\right\rVert}\bigg{)}^{b_{d}}+O(v^{b_{d}-1})-1\Bigg{)}\\#G\otimes\mathbb{Z}/v\mathbb{Z}+\\#G\otimes\mathbb{Z}/v\mathbb{Z}}{v^{b_{d}}\times\\#G\otimes\mathbb{Z}/v\mathbb{Z}}$
$\displaystyle=$
$\displaystyle\frac{1+O(v^{-1})}{(2Cb_{d}\sup_{j}\left\lVert\Sigma_{j}\right\rVert)^{b_{d}}}>0.$

We are done.

###  3.2. Proof of Corollary 1

First, note that $\inf_{\mathbb{Z}/v\mathbb{Z}}\mathbf{M}_{g}([\Sigma\mod v])$
and $\inf_{\mathbb{Z}}\mathbf{M}_{g}([\Sigma])$ depend continuously on the
metric. For the proof, one can use a straightforward adaptation of Lemma 3.7
of [36] with $\mathbb{Z}$ and $\mathbb{Z}/v\mathbb{Z}$ coefficient deformation
theorems. We need to work a bit harder for
$\inf_{\mathbb{Z}/v\mathbb{Z}}\mathbf{M}_{g}([\Sigma\mod v]).$

######  Lemma 3.1.

$\inf_{\mathbb{Z}/v\mathbb{Z}}\mathbf{M}_{g}([\Sigma\mod v])$ depends
continuously on the metric $g.$

###### Proof.

First, let us prove that in any fixed $g$ the
$\inf_{\mathbb{Z}}\mathbf{M}_{g}([\Sigma\mod v])$ is always achieved on some
class $[\Sigma]+v[\Pi].$ (Note bullet (1) of our corollary follows directly
from this.) We argue by contradiction. Suppose not, then there exists a
sequence of distinct homology classes $\Pi_{l}$ so that a sequence of area-
minimizing integral current $T_{l}$ in $[\Sigma]+v[\Pi_{l}]$ has strictly
decreasing mass. Since the torsion subgroup of $H_{d}(M,\mathbb{Z})$ is
finite, one can choose a torsion-free subsequence $[\Pi_{l}]$, not relabeled.
In the notation of Section 2.2, write

$[\Sigma]+v[\Pi_{l}]=\sum_{j_{l}}\alpha_{j_{l}}[\Sigma_{j_{l}}]+\beta_{j_{l}}[\tau_{j_{l}}].$

Since $[\Pi_{l}]$ is an infinite distinct sequence of non-torsion classes,
without loss of generality, one can suppose that $\alpha_{1_{l}}\to\infty.$ By
Lemma 2.7, this implies that

(3.3)
$\displaystyle\mathbf{M}(T_{l})\geq\left\lVert[\Sigma]+v[\Pi_{l}]\right\rVert\geq
C^{-1}\sum_{j_{l}}|\alpha_{j_{l}}|\left\lVert\Sigma_{j_{l}}\right\rVert\geq
C^{-1}|\alpha_{1_{l}}|\left\lVert\Sigma_{1_{l}}\right\rVert\to\infty,$

a contradiction.

Next, let us prove that for any metric $g$, there exists an open set
$\Omega_{g}$ in the space of Riemannian metrics, so that for any
$h\in\Omega_{g},$ $\inf_{\mathbb{Z}}\mathbf{M}_{h}([\Sigma\mod v])$ is
achieved on some fixed finite collection of homology classes, regardless of
$h$. By continuous dependence of $\left\lVert\cdot\right\rVert_{h}$ on the
metric (Lemma 3.7 in [36]) and by Lemma 2.7, there exist an open set
$\Omega_{g}\ni g$ so that for any $h\in\Omega_{g}$, we have
$\left\lVert\Sigma\right\rVert_{h}\geq
C\sum_{j}|\alpha_{j}|\left\lVert\Sigma_{j}\right\rVert_{h}$ with some $C>0$ in
the notation of decomposition in Section 2.2. Shrink $\Omega_{g}$ if necessary
so that
$\sup_{j}\sup_{h\in\Omega_{g}}\left\lVert\Sigma_{j}\right\rVert_{h}<\infty.$
Now arguing by contradiction, we deduce that there is a sequence of metrics
$g_{j}\to g$ and distinct non-torsion homology classes $[\Pi_{j}]$ so that
$\inf_{\mathbb{Z}}\mathbf{M}_{g_{j}}([\Sigma\mod v])$ is achieved on
$[\Sigma]+v[\Pi_{j}].$ The same estimate as in (3.3) in the previous paragraph
gives the contradiction, since we have uniform control on $C$ and
$\left\lVert\Sigma_{j}\right\rVert_{h}$ near $g.$

To sum it up, near any fixed metric $g$, there are $k$ different homology
classes $[\Pi_{l}]$ so that

$\displaystyle\inf_{\mathbb{Z}}\mathbf{M}_{h}([\Sigma\mod
v])=\min_{j}\inf_{\mathbb{Z}}\mathbf{M}_{h}([\Sigma]+v[\Pi_{j}]).$

Our lemma then follows from the continuous dependence of
$\inf_{\mathbb{Z}}\mathbf{M}_{h}([\Sigma]+v[\Pi_{l}])$ on the metric and
finiteness of $k.$ ∎

Now we know that both sides of (1.1) depend continuously on the metric. Thus
our corollary follows directly from the definition of $K_{[\Sigma\mod v]}$ and
Theorem 1.2.

##  4\. Proof of Theorem 1.5 and Theorem 1.6

###  4.1. Proof of Theorem 1.5

By Lemma 2.2, there is a smooth connected non-orientable submanifold $Q$ of
$M$, so that

$\displaystyle[\Sigma\mod v]=[\frac{v}{2}Q\mod v].$

Apply Lemma 2.3 with $t=\frac{v}{2},$ we get a smooth metric $g$ on $M,$ so
that $\frac{v}{2}Q$ is area-minimizing mod $v$ in $h$. Moreover, there is a
neighborhood $U$ that deformation retracts onto $Q$, so that any stationary
varifold not contained in $U$ must have an area at least
$R\frac{v}{2}\mathbf{M}(Q).$ Now, let $T$ be an area-minimizing integral
current in $[\Sigma]$. If $T$ is not contained in $U,$ we are done. If $T$ is
contained in $U,$ the $d$-th integral homology of $U$ is the same as the
$d$-th integral homology of $Q$, by homotopy invariance. Thus,
$H_{d}(U,\mathbb{Z})=0$ by non-orientability of $Q$. This is a contradiction,
as $T\not=0$ in integral homology.

###  4.2. Proof of Theorem 1.6

Let $S^{d-1}$ be the unit sphere in $\mathbb{R}^{d}\subset\mathbb{R}^{d+c}.$
Set

$\displaystyle\Gamma_{0}=S^{d-1}\times\\{0\\}+S^{d-1}\times\\{(\epsilon,0,\cdots,0)\\}+\cdots+S^{d-1}\times\\{((v-1)\epsilon,\cdots,0)\\}.$

Note that

$\displaystyle\Gamma_{0}-vS^{d-1}=\partial\sum_{j=1}^{v}S^{d-1}\times[0,j\epsilon].$

Now, we can do successive connected sums to make $\Gamma_{0}$ into a connected
submanifold $\Gamma.$ By making the necks of connected sums with radius at
most $\epsilon$, we can make sure that $\Gamma$

$\displaystyle\Gamma-\Gamma_{0}=\partial C,$

where $C$ has area at most $O(\epsilon^{d}).$

In other words $\Gamma-vS^{d-1}=\partial
D=\partial(C+\sum_{j=1}^{v}S^{d-1}\times[0,j\epsilon])$ for some $D$ with area
at most $O(\epsilon).$ This implies that any area-minimizing integral current
$T$ with $\partial T=\Gamma$ must have mass in
$[v\mathbf{M}(S^{d-1})-O(\epsilon),v\mathbf{M}(S^{d-1})+O(\epsilon)].$

On the other hand,

$\displaystyle\partial(C+\Sigma_{j=1}^{v}S^{d-1}\times[0,j\epsilon])\mod
v=\Gamma-vS^{d-1}\mod v=\Gamma.$

Thus, any area-minimizing mod $v$ current $T^{\prime}$ with $\partial
T^{\prime}=\Gamma\mod v$ must have an area at most $O(\epsilon).$ Taking
$\epsilon$ small, we are done.

## References

  * [1] William K. Allard, First Variation of a Varifold, Annals of Mathematics, Vol. 95, No. 3 (May 1972), pp. 417-491
  * [2] William K. Allard, First Variation of a Varifold: Boundary Behavior, Annals of Mathematics, Vol. 101, No. 3 (May 1975), pp. 418-446
  * [3] Frederick J. Almgren, Jr. Almgren’s big regularity paper. Q-valued functions minimizing Dirichlet’s integral and the regularity of area-minimizing rectifiable currents up to codimension 2. With a preface by Jean E. Taylor and Vladimir Scheffer. World Scientific Monograph Series in Mathematics, 1. World Scientific Publishing Co., Inc.
  * [4] V. M. Buchstaber, Modules of differentials of the Atiyah-Hirzebruch spectral sequence, I Math. USSR Sb. 7:2 (1969), 299–313; 12:1 (1970), 59–75.
  * [5] S. Buoncristiano; M. Dedò, On resolving singularities and relating bordism to homology, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (1980), Volume: 7, Issue: 4, page 605-624
  * [6] J. Dadok, R. Harvey and F. Morgan, Calibrations on $\mathbb{R}^{8}$, Transactions of the American Mathematical Society, May 1988, Vol. 307, No. 1 (May, 1988), pp. 1-40
  * [7] Camillo De Lellis,Allard’s interior regularity theorem: an invitation to stationary varifolds, Nonlinear analysis in geometry and applied mathematics. Part 2, 23–49. Harv. Univ. Cent. Math. Sci. Appl. Ser. Math., 2 International Press, Somerville, MA, 2018
  * [8] C. De Lellis; G. De Philippis; J. Hirsch; A. Massaccesi, On the boundary behavior of mass-minimizing integral currents, available at https://www.math.ias.edu/delellis/node/148,
  * [9] Camillo De Lellis, Jonas Hirsch, Andrea Marchese, Salvatore Stuvard Regularity of area minimizing currents mod p, Geometric and Functional Analysis, 30, pages1224–1336 (2020)
  * [10] Camillo De Lellis, Jonas Hirsch, Andrea Marchese, Luca Spolaor, Salvatore Stuvard, Area minimizing hypersurfaces modulo p: a geometric free-boundary problem, preprint available at https://arxiv.org/abs/2105.08135
  * [11] Camillo De Lellis, Jonas Hirsch, Andrea Marchese, Luca Spolaor, Salvatore Stuvard, Fine structure of the singular set of area minimizing hypersurfaces modulo p, preprint available at https://arxiv.org/abs/2201.10204
  * [12] Camillo De Lellis, Paul Minter, Anna Skorobogatova, The Fine Structure of the Singular Set of Area-Minimizing Integral Currents III: Frequency 1 Flat Singular Points and Uniqueness of Tangent Cones, available at https://arxiv.org/abs/2304.11553
  * [13] C. De Lellis; E. Spadaro, Regularity of area-minimizing currents I: $L^{p}$ gradient estimates, Geom. Funct. Anal. 24 (2014), no. 6,
  * [14] C. De Lellis; E. Spadaro,  Regularity of area-minimizing currents II: center manifold, Ann. of Math. (2) 183 (2016), no. 2, 499–575.
  * [15] C. De Lellis; E. Spadaro, Regularity of area-minimizing currents III: blow-up, Ann. of Math. (2) 183 (2016), no. 2, 577–617. Geom. Funct. Anal. 24 (2014), no. 6, 1831–1884.
  * [16] C. De Lellis; E. Spadaro; L. Spolaor, Uniqueness of tangent cones for 2-dimensional almost minimizing currents, Comm. Pure Appl. Math. 70, 1402-1421
  * [17] C. De Lellis; E. Spadaro; L. Spolaor, Regularity theory for 2-dimensional almost minimal currents I: Lipschitz approximation, Trans. Amer. Math. Soc. 370 (2018), no. 3, 1783–1801
  * [18] C. De Lellis; E. Spadaro; L. Spolaor, Regularity theory for 2-dimensional almost minimal currents II: branched center manifold, Ann. PDE 3 (2017), no. 2, Art. 18, 85 pp.
  * [19] C. De Lellis; E. Spadaro; L. Spolaor, Regularity theory for 2-dimensional almost minimal currents III: blowup To appear in Jour. Diff. Geom.
  * [20] J. Dadok, R. Harvey and F. Morgan, Calibrations on $\mathbb{R}^{8}$, Transactions of the American Mathematical Society, May 1988, Vol. 307, No. 1 (May, 1988), pp. 1-40
  * [21] Paul E. Ehrlich, Continuity properties of the injectivity radius function, Compositio Math.29(1974), 151–178.
  * [22] Herbert Federer, Geometric Measure Theory Springer, New York, 1969.
  * [23] Herbert Federer, The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension. Bull. Amer. Math. Soc. 76 (1970), 767–771.
  * [24] Herbert Federer, Real flat chains, cochains and variational problems, Indiana Univ. Math. J. 24 (1974), 351–407.
  * [25] Herbert Federer, Wendell Fleming, H. Normal and integral currents Ann. of Math. (2) 72 (1960), 458–520.
  * [26] Michael Freedman, personal communications
  * [27] Michael Freedman, Matthew Headrick, Bit threads and holographic entanglement Comm. Math. Phys. 352 (2017), no. 1, 407–438.
  * [28] Alfred Gray Tubes. Second edition. With a preface by Vicente Miquel. Progress in Mathematics, 221. Birkhäuser Verlag, Basel, 2004.
  * [29] Robert Hardt, Leon Simon,  Area minimizing hypersurfaces with isolated singularities J. Reine Angew. Math.362(1985), 102–129.
  * [30] Reese Harvey; H. Blaine Lawson, Jr. Calibrated geometries. Acta Math. 148 (1982),
  * [31] Reese Harvey; H. Blaine Lawson, Jr. Calibrated foliations (foliations and mass-minimizing currents). Amer. J. Math. 104 (1982), no. 3, 607–633.
  * [32] Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002.
  * [33] Gary Lawlor, A sufficient criterion for a cone to be area-minimizing, Mem. Amer. Math. Soc. 91 (1991), no. 446, vi+111 pp.
  * [34] John M. Lee, Introduction to smooth manifolds. Second edition. Graduate Texts in Mathematics, 218. Springer, New York, 2013.
  * [35] Zhenhua Liu, Homologically area-minimizing surfaces with non-smoothable singularities, preprint available at arxiv.org/abs/2206.08315
  * [36] Zhenhua Liu, Homologically area-minimizing surfaces that cannot be calibrated, preprint available at arxiv.org/abs/2310.19860
  * [37] Pertti Mattila, Geometry of sets and measures in Euclidean spaces Fractals and rectifiability, Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. xii+343 pp.
  * [38] Bruno Martelli, answer to Mathoverflow question https://mathoverflow.net/questions/462979/impossibility-of-realizing-codimension-1-homology-classes-by-embedded-non-orient
  * [39] William H. Meeks III, Julie Patrusky, Representing codimension-one homology classes by embedded submanifolds, Pacific J. Math. 68 (1977), no. 1, 175–176.
  * [40] Paul Minter and Neshan Wickramasekera , A structure theory for stable codimension 1 integral varifolds with applications to area minimising hypersurfaces mod $p$, J. Amer. Math. Soc. Published electronically: October 3, 2023
  * [41] Frank Morgan, On the singular structure of two-dimensional area minimizing surfaces in $R^{n}$. Math. Ann. 261 (1982), no. 1
  * [42] Frank Morgan, Examples of unoriented area-minimizing surfaces. Trans. AMS 283 (1984), 225-237
  * [43] Frank Morgan, Calibrations modulo $\nu$. Adv. in Math. 64 (1987), no. 1, 32–50.
  * [44] Frank Morgan, A regularity theorem for minimizing hypersurfaces modulo v, Trans. Amer. Math. Soc. 297 (1986), no. 1, 243–253.
  * [45] Frank Morgan, personal communications.
  * [46] Aaron Naber, Daniele Valtorta, The singular structure and regularity of stationary varifolds, J. Eur. Math. Soc., Volume 22, Issue 10, 2020
  * [47] Dana Mackenzie, Sufficient conditions for a pair of n-planes to be area-minimizing. Math. Ann. 279 (1987), no. 1, 161–164.
  * [48] Barrett O’Neill, Semi-Riemannian geometry. With applications to relativity. Pure and Applied Mathematics, 103. Academic Press, Inc.
  * [49] Leon Simon, Lectures on Geometric Measure Theory, Proceedings for the Centre for Mathematical Analysis, Australian National University, Canberra, 1983.
  * [50] Rene Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86.
  * [51] C. T. C. Wall, Determination of the cobordism ring, Ann. of Math. (2) 72 (1960), 292–311.
  * [52] C. T. C. Wall,Differential topology Cambridge Stud. Adv. Math., 156 Cambridge University Press, Cambridge, 2016, viii+346 pp.
  * [53] Brian White, A regularity theorem for minimizing hypersurfaces modulo p, Volume 44 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 1986.
  * [54] Brian White, Stratification of minimal surfaces, mean curvature flows, and harmonic maps, J. Reine Angew. Math. 488 (1997), 1–35.
  * [55] Hassler Whitney, Differentiable manifolds. Ann. of Math. (2) 37 (1936), no. 3, 645–680.
  * [56] Robert Young, Quantitative nonorientability of embedded cycles , Duke Math. J. 167(1): 41-108
  * [57] Yongsheng Zhang, On extending calibration pairs. Adv. Math. 308 (2017), 645–670.
  * [58] Yongsheng Zhang, On realization of tangent cones of homologically area-minimizing compact singular submanifolds. J. Differential Geom. 109 (2018), no. 1, 177–188.