# Existence and partial regularity of Legendrian area-minimizing currents

Gerard Orriols  ETH, Rämistrasse 101, 8092 Zürich, Switzerland
<EMAIL_ADDRESS>
(Date: June 13, 2024)

###### Abstract.

We show that Legendrian integral currents in a contact manifold that locally
minimize the mass among Legendrian competitors have a regular set which is
open and dense in their support. We apply this to show existence and partial
regularity of solutions of the Legendrian Plateau problem in the $n$th
Heisenberg group for an arbitrary horizontal $(n-1)$-cycle as prescribed
boundary, and of mass-minimizing Legendrian integral currents in any
$n$-dimensional homology class of a closed contact $(2n+1)$-manifold. In the
case of the Heisenberg group, our result applies to Ambrosio–Kirchheim metric
currents with respect to the Carnot–Carathéodory distance. Our results do not
assume any compatibility between the subriemannian metric and the symplectic
form.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries and notation
  3. 3 Horizontal currents and existence of minimizers
  4. 4 Main regularity theorems
  5. 5 Height bound and Lipschitz approximation
  6. 6 Biharmonic approximation and excess decay
  7. A Metric geometry of the Heisenberg space
  8. B Absence of monotonicity formula in higher dimensions

##  1\. Introduction

The goal of this paper is to lay the foundations of an existence and
regularity theory of area-minimizing currents among Legendrian integral
currents in contact manifolds of arbitrary dimension. The main motivation for
applying the methods of classical Geometric Measure Theory to Legendrian
submanifolds comes from the project initiated by Schoen and Wolfson to develop
a variational theory for the area of Lagrangian submanifolds in symplectic
manifolds [SW99, SW01, SW03].

There are many geometric reasons to study minimizers of the area among
Lagrangian submanifolds. It is natural to ask whether a homology class in a
symplectic manifold which admits a Lagrangian representative contains one of
minimal area which enjoys some partial regularity, as in the unconstrained
case. Moreover, a good understanding of the homology minimization problem for
Lagrangian currents seems to be fundamental before exploring the much more
delicate topic of regularity of Hamiltonian-stationary Lagrangian surfaces and
of existence of global area-minimizers in Hamiltonian isotopy classes, of
which very little is known. See [Oh93] for an introduction to the problem and
some motivating conjectures and [Vit00] for a lower bound on the area for
Hamiltonian isotopy classes in some ambient spaces.

A deeper motivation to study this minimization problem comes from mirror
symmetry and the SYZ conjecture: when the symplectic manifold is Calabi–Yau,
hypothetically there should exist many special Lagrangian submanifolds, which
are calibrated and hence absolute homology minimizers. Therefore one expects
to be able to find them by minimizing area among Lagrangian integral currents
in homology classes which admit them. A very short and elegant computation of
Schoen and Wolfson [SW01] shows that this strategy can be realized, not only
in the Calabi–Yau setting but more generally in Kähler–Einstein manifolds, and
indeed Lagrangian-stationary closed surfaces are automatically minimal (if the
ambient manifold is Calabi–Yau then they must be special Lagrangian). However,
for their argument to work, it is fundamental to know that this object is
regular enough, and in fact Micallef and Wolfson later found examples of
homology classes where singularities cannot be avoided [MW06, Wol05].

It was observed by Schoen and Wolfson, building on an example of Minicozzi
[Min95], that Lagrangian area-minimizing surfaces can be very badly behaved
unless they satisfy an exactness condition. More precisely, Minicozzi showed
that cylinders of the form
$\mathbb{S}^{1}_{\varepsilon}\times\mathbb{R}\subset\mathbb{C}\times\mathbb{C}=\mathbb{R}^{4}$
are area-minimizing among Lagrangian surfaces and, unlike classical area
minimizers, they do not satisfy a monotonicity formula for the area (not even
a universal lower bound for the mass ratio in a ball). Unless one can somehow
control the presence of regions of this type, developing a regularity theory
in this generality seems far out of reach111Note however the a-priori
estimates from [SW99] which do not assume exactness of the surface but rely on
the a-priori square integrability of the mean curvature, which already
excludes many interesting singularities like the Schoen–Wolfson cones..

However, assuming that a smooth Lagrangian surface
$\Sigma^{2}\subset\mathbb{R}^{4}$ is exact (that is, the Liouville form
$\lambda$, which satisfies $\mathrm{d}\lambda=\omega$, restricts to zero on
$\Sigma$), it has a lift $\tilde{\Sigma}$ as a surface in the Heisenberg group
$\mathbb{H}^{2}\simeq\mathbb{R}^{5}$, and the key insight of [SW01] was to
show that this lift satisfies a monotonicity formula assuming just that
$\Sigma$ is stationary under Hamiltonian deformations. This is due to the
larger richness of variations and competitors available in the Legendrian
setting. By extensively using this monotonicity formula and techniques
exclusive of dimension two (such as a conformal parametrization, energy
methods and holomorphic maps), Schoen and Wolfson succeeded in developing an
optimal and rather complete regularity theory for minimizers of the mapping
problem among Legendrian surfaces, and applied it to the Lagrangian setting.
Their main result states that such minimizers are smooth surfaces except at a
finite number of points, which are either of branch type (around which the
parametrization is smooth) or of cone type (around which the parametrization
is Lipschitz). The Hamiltonian-stationary two-dimensional cones are
classified, and among them also the stable ones, but to the best of our
knowledge it is still an open question to determine which of them are
minimizing.

The extension to higher dimensions of the monotonicity formula for
Hamiltonian-stationary Legendrian submanifolds was announced in [SW99, Theorem
3]. We provide a counterexample in Appendix B, showing that a different
approach is needed and that the regularity of Hamiltonian-stationary surfaces
in higher dimensions is more delicate than the regularity of Legendrian
minimizing currents carried out here, where the lower bound in the mass
density will come from an isoperimetric inequality.

Even in dimension two, the monotonicity formula of [SW01] is not explicit, and
its construction goes through the solution of an auxiliary hyperbolic PDE.
Note however that a newer simplified almost monotonicity formula has been
recently found by Rivière [Riv24], still in dimension two, and extended to
parametrized Hamiltonian-stationary varifolds by Rivière and Pigati [PR24].
This monotonicity formula has been applied by Rivière to solve min-max
problems for Legendrian surfaces in Sasakian $5$-manifolds [Riv23], thus
generalizing the pure minimization problem.

In addition to that, the proof of the decay of the excess in [SW01] has the
added difficulty of controlling the parametrization of the surface at each
step, which makes it quite technical. In fact, an attempt to understand and
simplify their arguments with the help of newly available tools was the
starting point of this work.

In contrast to the slower progress in the GMT problem in arbitrary dimensions,
the last few years have seen many new results about the PDE that governs the
local picture for graphs. Since Legendrian graphs are locally lifts of
Lagrangian graphs, and Lagrangian variations agree locally with Hamiltonian
variations, the corresponding Euler–Lagrange equation is precisely the
Hamiltonian-stationary equation. This equation has a very special structure
when the ambient manifold is $\mathbb{C}^{n}$ and has been studied by
Bhattacharya, Chen and Warren [CW19, BW19]. More recently, these authors have
noticed that partial regularity results can be obtained without relying on
this special structure, and therefore hold on arbitrary symplectic manifolds
[BCW23].

Even more recently, Bhattacharya and Skorobogatova [BS23] have shown, still in
the graphical setting, that smallness of the average oscillation of the
Hessian of the potential $u$ in a ball, which corresponds to smallness of the
excess of the graph in a cylinder, is enough to deduce regularity.
Nevertheless they need to know a priori that
$\|\mathrm{D}^{2}u\|_{L^{\infty}}\leq 1-\eta$ to guarantee that the area
functional is convex, and their required smallness of the excess depends on
this $\eta$. Inspired by their theorem, in our main result, 4.7, we remove the
Lipschitz graph assumption and, working with currents, we prove an
$\varepsilon$-regularity theorem for Legendrian area-minimizers analogous to
the classical ones of De Giorgi [De ̵61] and Almgren [Alm68] for minimal
surfaces.

At the same time, as with high-codimension minimal surfaces, the
Euler–Lagrange PDE for graphs lacks an existence theory of weak solutions and
a satisfactory functional space to search for minimizers. The framework of
Legendrian currents developed here provides a solution to these issues and
furthermore allows for boundary data which cannot be expressed as a graph.

###  1.1. Main results

We state here our main existence and regularity results. The relevant notions
will be introduced in the next two sections. For now, note that
$\mathbb{H}^{n}$ denotes the Heisenberg group of topological dimension $2n+1$
with its standard contact structure and metric on the horizontal planes; a
horizontal integral current is just an integral current almost all of whose
tangent planes are isotropic, and a Legendrian current is a horizontal current
of dimension $n$.

######  Theorem 1.1 (Legendrian Plateau problem).

Let $S$ be a compactly supported $(n-1)$-dimensional horizontal integral
current in $\mathbb{H}^{n}$ with $\partial S=0$. Then there exists a compactly
supported Legendrian integral current $T$ with $\partial T=S$ such that
$\mathbf{M}(T)\leq\mathbf{M}(T^{\prime})$ for any other Legendrian integral
current $T^{\prime}$ with $\partial T^{\prime}=S$.

Moreover there exists an open set $\mathcal{U}\subset\mathbb{H}^{n}$ such that
$\operatorname{spt}T\cap\mathcal{U}$ is a real-analytic embedded
$n$-dimensional Legendrian submanifold of $\mathbb{H}^{n}$ and
$\operatorname{spt}T\cap\mathcal{U}$ is dense in
$\operatorname{spt}T\setminus\operatorname{spt}S$.

This theorem can also be stated in terms of Ambrosio–Kirchheim metric currents
with respect to the intrinsic Carnot–Carathéodory distance of
$\mathbb{H}^{n}$, but we have preferred to work with Legendrian
Federer–Fleming currents to take advantage of the homological theory which is
available for them, of the larger literature that uses them, and of their
better compatibility with Lagrangian currents in a symplectic manifold.
Nonetheless we have to use metric currents crucially in a portion of our
arguments, and we have recast our results into that language in A.4.

We also remark that the Legendrian Plateau problem had already been studied by
Minicozzi in his PhD thesis [Min94] and a penalization approach to solve it is
explained in [SW03].

The assumption that the ambient manifold is the Heisenberg group is not
essential here, and the Plateau problem can be solved as well in compact
manifolds or in noncompact manifolds with contolled geometry at infinity. For
closed manifolds we have the following result222Here we assume that the
ambient manifold is smooth—see Section 4 for more precise versions of the
regularity statements under weaker assumptions.:

######  Theorem 1.2 (Legendrian homology minimizers).

Let $(M^{2n+1},\Xi,g)$ be a smooth closed contact manifold with contact
distribution $\Xi$ and a smooth subriemannian metric $g$ defined on $\Xi$. Let
$\mathfrak{a}\in H_{n}(M,\mathbb{Z})$ be a homology class. Then there exists a
Legendrian integral current $T$ representing $\mathfrak{a}$ such that
$\mathbf{M}(T)\leq\mathbf{M}(T^{\prime})$ for any other Legendrian integral
current $T^{\prime}$ in $\mathfrak{a}$.

Moreover there exists an open set $U\subset M$ such that
$\operatorname{spt}T\cap U$ is a smooth embedded $n$-dimensional Legendrian
submanifold of $M$ and $\operatorname{spt}T\cap U$ is dense in
$\operatorname{spt}T$.

The proof, both of the existence and (mainly) of the partial regularity, makes
use of the isoperimetric and coning inequalities for horizontal currents in
$\mathbb{H}^{n}$ which have been recently established by Basso, Wenger and
Young [BWY23] in the context of metric currents. These results give us the
basic ingredients to build up a regularity theory in the absence of a
monotonicity formula, as pioneered by Almgren for currents which minimize an
elliptic functional [Alm68]. In fact, our regularity theorem also works in
great generality, without assuming any compatibility condition between the
metric and the symplectic form (in particular, an almost complex structure may
not exist anywhere), and with minor modifications can be extended to
functionals more general than an anisotropic area.

Although the ideas of the regularity proof were pioneered by Almgren and De
Giorgi, we follow more closely the simplifications introduced by Schoen–Simon
[SS82] and Ambrosio–De Lellis–Schmidt [ADS18] in the proof of the decay of the
excess, approximating our current by (roughly) the graph of the gradient of
the solution of a fourth order linear PDE and using classical $L^{p}$
estimates to control the error terms.

We remark that, whereas the existence theory works as well for horizontal
currents of any dimension $k<n$ (see Theorems 3.12 and 3.15), our regularity
theory relies crucially on the representation of our current as the graph of a
gradient over a Legendrian plane, which is only available when $k=n$. Even if
the surface is given as a graph over a horizontal $k$-plane, we are not aware
of any results in the literature that deal with the corresponding system of
PDE, in contrast with the larger literature on the hamiltonian stationary
equation. In the parametrized case, however, Qiu [Qiu03] successfully extended
the theory of Schoen and Wolfson to isotropic surfaces in any dimension. In
our opinion, the extension of these results to higher ambient dimensions is an
interesting problem.

We believe that a more precise description of the singular set should be
possible, at least in two dimensions and in the Heisenberg group. The key
difficulty here would be to develop a blowup analysis around branch points. It
is natural to conjecture that one would recover the result obtained by Schoen
and Wolfson in the parametrized setting: a locally finite set of singular
points consisting of branch points and Schoen–Wolfson conical singularities.

We also hope to be able to apply our results soon to variational problems for
Lagrangian surfaces; in particular there are some issues to be addressed
regarding the notions of exactness and Legendrian lift for integral currents.
But beyond this application, we believe that our result has many other points
of interest. On one hand, contact geometry has become a subject on its own and
the study of canonical representatives of their homology is a natural and
basic question to study. On the other hand, from the perspective of metric
geometry, we provide one of the very few available partial regularity results
for high codimension minimal surfaces in non-smooth metric spaces (compare
with [ADS18] for Hilbert spaces and with [Sta21] for parametrized two-
dimensional surfaces in $\operatorname{CAT}(0)$ spaces). In addition to that,
the techniques developed here may be applied to other geometric variational
problems in which Legendrian currents appear as unit normal bundles—see the
survey [Ber03] and the references therein for more details.

###  1.2. Structure of the paper

After introducing some preliminary notation and facts in Section 2, the
relevant notions of currents will be presented in Section 3. The existence
part of the Theorems 1.1 and 1.2 is also stated and proved in Section 3, see
3.12 and 3.15 respectively. The regularity part is made precise in Section 4:
in 4.9 we establish that the regular set (where the support of the current is
$C^{1,1/2}$) is dense in the support. Then the higher regularity stated here
follows from 4.11 for smooth manifolds and 4.12 for $\mathbb{H}^{n}$.

The main theorem of the present article, from which the partial regularity is
deduced, is the $\varepsilon$-regularity 4.7. Its proof takes most of the work
and spans Sections 5 and 6. Finally, Appendix A contains all the statements
that we use from metric geometry and serves as a dictionary between our
currents and Carnot–Carathéodory metric currents, and Appendix B contains a
counterexample of the monotonicity of the mass ratios in dimension larger than
two.

###  1.3. Acknowledgments

The author thanks his PhD supervisor, Tristan Rivière, for his support and
encouragement to study the work of Schoen and Wolfson and attempt to simplify
and generalize parts of it. He also wishes to thank Federico Franceschini,
Filippo Gaia, Matilde Gianocca and Tom Ilmanen for inspiring conversations and
for their interest in the work.

##  2\. Preliminaries and notation

###  2.1. The Heisenberg group

Most of our analysis will take place in the Heisenberg group $\mathbb{H}^{n}$,
because thanks to the Darboux theorem (2.1), it is a local model for the
geometry of a contact manifold with good homogeneity properties. Here we
summarize the main geometric and analytical facts that we will use in the
sequel. For the notation, we use mostly the conventions of [Cap+07]; we also
refer to this book for further information and references.

We will often identify $\mathbb{H}^{n}$ with $\mathbb{R}^{2n+1}$ by using the
globally defined exponential coordinates
$(\mathbf{z},\varphi)=(\mathbf{x},\mathbf{y},\varphi)$. Here $\mathbf{z}$
denotes an element of $\mathbb{C}^{n}$, which we will identify with
$\mathbb{R}^{2n}$, and $\mathbf{x},\mathbf{y}$ denote $n$-tuples of real
numbers corresponding to the real and imaginary part of $\mathbf{z}$,
respectively.

In these coordinates, the group operations are

(2.1)
$(\mathbf{x},\mathbf{y},\varphi)\cdot(\mathbf{x}^{\prime},\mathbf{y}^{\prime},\varphi^{\prime})=\left(\mathbf{x}+\mathbf{x}^{\prime},\mathbf{y}+\mathbf{y}^{\prime},\varphi+\varphi^{\prime}+\frac{1}{2}\left(\mathbf{x}\cdot\mathbf{y}^{\prime}-\mathbf{y}\cdot\mathbf{x}^{\prime}\right)\right)$

and

(2.2)
$(\mathbf{x},\mathbf{y},\varphi)^{-1}=(-\mathbf{x},-\mathbf{y},-\varphi).$

The Heisenberg group is naturally a contact manifold with global contact form

(2.3)
$\theta:=\mathrm{d}\varphi-\frac{1}{2}(\mathbf{x}\cdot\mathrm{d}\mathbf{y}-\mathbf{y}\cdot\mathrm{d}\mathbf{x}),$

which defines a hyperplane distribution $\Xi=\Xi_{\mathbb{H}}$ by
$\Xi_{\xi}:=\ker\theta(\xi)$. As all Lie groups, the Heisenberg group is
homogeneous and parallelizable. In particular, left translations allow us to
identify all tangent spaces. Moreover, it can be checked that $\theta$ is
left-invariant and therefore the horizontal bundle $\Xi_{\mathbb{H}}$ can also
be trivialized by using left translations. In what follows, we will use this
observation extensively and identify the horizontal spaces at all points with
$\Xi_{0}$.

We endow $\mathbb{H}^{n}$ with a subriemannian metric $g_{\mathbb{H}}$, which
we will also write as $\langle\cdot,\cdot\rangle$, defined as follows: take
$\langle\cdot,\cdot\rangle$ on $\Xi_{0}$ to be the standard metric of
$\mathbb{C}^{n}=\mathbb{R}^{2n}\simeq\Xi_{0}\subset T_{0}\mathbb{H}^{n}$, and
extend it to the whole $\Xi_{\mathbb{H}}$ in a left-invariant way.

For any horizontal $k$-plane $\pi$ ($1\leq k\leq 2n$) and any smooth function
$f$, we denote by $\nabla^{\pi}f$ the unique vector in $\pi$ such that
$\langle v,\nabla^{\pi}f\rangle=\mathrm{d}f(v)$ for any vector $v\in\pi$. In
the case that $\pi$ is the whole horizontal $2n$-space $\Xi$, we denote
$\nabla^{\pi}f$ as $\nabla^{H}f$. Then $\nabla^{\pi}f$ is the (Riemannian)
orthogonal projection of $\nabla^{H}f$ onto $\pi$ with respect to the
Riemannian metric on $\Xi$.

For the coordinate functions, it is easy to check that

(2.4) $\nabla^{H}x^{i}=\frac{\partial}{\partial
x^{i}}-\frac{1}{2}y^{i}\frac{\partial}{\partial\varphi}\qquad\text{and}\qquad\nabla^{H}y^{i}=\frac{\partial}{\partial
y^{i}}+\frac{1}{2}x^{i}\frac{\partial}{\partial\varphi},$

since all vector fields involved are left-invariant and agree at the origin,
where they form the standard orthonormal basis of $\mathbb{R}^{2n}$. Thus, by
definition, $\\{\nabla^{H}x^{i},\nabla^{H}y^{i}\\}_{i=1,\ldots,n}$ form an
orthonormal basis of $\Xi$ for the standard subriemannian metric.

As is customary, we will work with the Folland–Korányi gauge

(2.5)
$\tau(\mathbf{z},\varphi):=\|(\mathbf{z},\varphi)\|_{\mathrm{FK}}:=\left(|\mathbf{z}|^{4}+16\varphi^{2}\right)^{\frac{1}{4}}.$

Cygan [Cyg81] proved that
$\|\xi\xi^{\prime}\|_{\mathrm{FK}}\leq\|\xi\|_{\mathrm{FK}}+\|\xi^{\prime}\|_{\mathrm{FK}}$
and therefore it defines a left-invariant distance

(2.6)
$\operatorname{dist}_{\mathrm{FK}}(\xi,\xi^{\prime}):=\|\xi^{-1}\xi^{\prime}\|_{\mathrm{FK}}$

whose balls will be denoted by
$\mathcal{B}_{r}:=\\{\xi\in\mathbb{H}^{n}:\|\xi\|_{\mathrm{FK}}<r\\}$. This
distance is (bilipschitz) equivalent to the Carnot–Carathéodory distance, but
for convenience we will use the former most of the time. In particular, it is
smooth away from the origin and enjoys the following useful property: the
function $\tau:\mathbb{H}^{n}\to\mathbb{R}$,
$\tau(\xi)=\|\xi\|_{\mathrm{FK}},$ satisfies

(2.7) $|\nabla^{\pi}\tau|\leq|\nabla^{H}\tau|=\frac{|\mathbf{z}|}{\tau}\leq 1$

for every horizontal $k$-plane $\pi$. This follows from a computation:

$\displaystyle(4\tau^{3}|\nabla^{H}\tau|)^{2}$
$\displaystyle=|\nabla^{H}\tau^{4}|^{2}=|\nabla^{H}|\mathbf{z}|^{4}+16\nabla^{H}\varphi^{2}|^{2}$
$\displaystyle=|\nabla^{H}|\mathbf{z}|^{4}|^{2}+2^{8}|\nabla^{H}\varphi^{2}|^{2}+32\langle\nabla^{H}|\mathbf{z}|^{4},\nabla^{H}\varphi^{2}\rangle$
$\displaystyle=4|\mathbf{z}|^{4}|\nabla^{H}|\mathbf{z}|^{2}|^{2}+2^{10}\varphi^{2}|\nabla^{H}\varphi|^{2}+2^{7}|\mathbf{z}|^{2}\varphi\langle\nabla^{H}|\mathbf{z}|^{2},\nabla^{H}\varphi\rangle$
$\displaystyle=16|\mathbf{z}|^{4}|\mathbf{z}|^{2}+2^{8}\varphi^{2}|\mathbf{z}|^{2}=16|\mathbf{z}|^{2}\tau^{4}.$

###  2.2. Contact manifolds

For simplicity, we will follow [Bla10] and will consider only contact
manifolds admitting a global contact form. All our arguments generalize easily
to manifolds that are not co-orientable. Hence a _contact manifold_ is a
smooth manifold $M^{2n+1}$ equipped with a $1$-form $\theta$ such that
$\theta\wedge(\mathrm{d}\theta)^{n}\neq 0$ everywhere. In fact, we will only
be interested in the $2n$-dimensional plane distribution $\Xi$ given by
$\ker\theta$, so all of the geometric notions discussed below will be
invariant under scaling $\theta$ by a nonvanishing function.

It is clear that the $1$-form $\theta_{\mathbb{H}^{n}}$ from (2.3) defines a
global contact form on $\mathbb{H}^{n}$, since

$\theta_{\mathbb{H}^{n}}\wedge\left(\mathrm{d}\theta_{\mathbb{H}^{n}}\right)^{n}=\theta_{\mathbb{H}^{n}}\wedge\left(\sum_{k}\mathrm{d}x^{k}\wedge\mathrm{d}y^{k}\right)^{n}=n!\,\mathrm{d}\varphi\wedge\mathrm{d}x^{1}\wedge\mathrm{d}y^{1}\wedge\cdots\wedge\mathrm{d}x^{n}\wedge\mathrm{d}y^{n}\neq
0.$

To study the volume of submanifolds, we will need a Riemannian metric $g$
defined on the planes $\Xi$, a so-called subriemannian structure on $M$. Then
we can state a subriemannian version of Darboux’s theorem as follows:

######  Proposition 2.1.

For every point $\xi$ of a contact manifold $(M^{2n+1},\theta)$ there is a
neighborhood $U\ni\xi$, an open set $0\in\mathcal{U}\subset\mathbb{H}^{n}$ and
a diffeomorphism $\phi:U\to\mathcal{U}$ such that
$\theta=\phi^{*}\theta_{\mathbb{H}^{n}}$.

Moreover, if $M$ is compact and $g$ is a Riemannian metric on the contact
distribution $\Xi=\ker\theta$, there exist constants
$0<\lambda\leq\Lambda<\infty$ such that

(2.8) $\lambda\phi^{*}g_{\mathbb{H}^{n}}\leq
g\leq\Lambda\phi^{*}g_{\mathbb{H}^{n}}.$

###### Proof.

The first part is just a restatement of the classical Darboux theorem for
contact manifolds, see for example [Bla10]. The second part just follows from
the compactness of $M$ by covering it with finitely many Darboux charts. ∎

######  Remark 2.2.

Notice that if we assume that the metric $g$ and the symplectic form
$\omega:=\mathrm{d}\theta|_{\Xi}$ are compatible, in the sense that there
exists an almost complex structure $J_{\xi}$ on $\Xi_{\xi}$ smoothly depending
on $\xi$ such that $\omega(u,v)=g(Ju,v)$ for every $u,v\in\Xi$, then we can
make $\lambda$ and $\Lambda$ as close to one as wished by making $U$ smaller.
Indeed, $(\phi^{-1})^{*}\mathrm{d}\theta=\mathrm{d}\theta_{\mathbb{H}}$ is the
standard symplectic form $\omega_{\mathbb{H}}$ on the horizontal plane at
$0\in\mathbb{H}^{n}$, and $(\phi_{*}J)_{0}$ is a complex structure compatible
with it. Therefore extending the scalar product
$\omega_{\mathbb{H}}(\cdot,(\phi_{*}J)_{0}\cdot)$ to a left-invariant
subriemannian metric on $\mathbb{H}^{n}$ makes the resulting space isometric
to the standard subriemannian $(\mathbb{H}^{n},g_{\mathbb{H}})$, and
$D\phi_{\phi^{-1}(0)}$ is a linear isometry onto $\Xi_{0}$.

However we will not need this hypothesis for our regularity result. This makes
the problem more anisotropic and our setting can be compared with that of
Almgren’s regularity theorem for minimizers of elliptic functionals [Alm68,
SS82].

###  2.3. Biharmonic functions

Here we collect some of the classical estimates for solutions to elliptic
fourth order equations with constant coefficients:

######  Proposition 2.3.

Let $(a_{ik}^{jl})_{i,j,k,l=1}^{n}$ be real numbers satisfying the boundedness
condition $|a_{ik}^{jl}|\leq\Lambda_{0}$ and the Legendre ellipticity
condition, $a_{ik}^{jl}\sigma_{ij}\sigma_{kl}\geq\lambda_{0}|\sigma|^{2}$ for
every $n\times n$ symmetric matrix $(\sigma_{ij})$.

Then for any $f\in W^{2,\infty}(B_{R})$ there exists a unique function
$u:B_{R}\to\mathbb{R}$ which is in $W^{2,p}(B_{R})$ for every $1\leq p<\infty$
and solves the Dirichlet problem

(2.9) $\begin{cases}a_{ik}^{jl}\partial_{ijkl}u=0&\text{in }B_{R}\\\
u=f&\text{on }\partial B_{R}\\\ \partial_{\nu}u=\partial_{\nu}f&\text{on
}\partial B_{R}.\end{cases}$

Moreover, $u$ is smooth inside $B_{R}$ and satisfies the interior estimates

(2.10) $\sup_{B_{r}}|\mathrm{D}^{2+k}u|^{2}\leq
C_{k}\frac{1}{(R-r)^{n+2k}}\int_{B_{R}}|\mathrm{D}^{2}u|^{2}\qquad\text{for
any }0<r<R\text{ and any integer }k\geq 0,$ (2.11)
$\sup_{B_{r}}|\mathrm{D}^{2}u-\mathrm{D}^{2}u(0)|^{2}\leq
C\frac{r^{2}}{R^{n}}\int_{B_{R}}|\mathrm{D}^{2}u|^{2}\qquad\text{for any
}0<r<\frac{R}{2},$

the global $L^{p}$ estimate

(2.12) $\int_{B_{R}}|\mathrm{D}^{2}u|^{p}\leq
C_{p}\int_{B_{R}}|\mathrm{D}^{2}f|^{p}\qquad\text{for any }1<p<\infty$

and Agmon’s maximum estimate for the derivative:

(2.13) $\sup_{B_{R}}|\mathrm{D}u|\leq C\sup_{\partial B_{R}}|\mathrm{D}u|\leq
C\sup_{B_{R}}|\mathrm{D}f|.$

Here the constants depend only on $n,\Lambda_{0}/\lambda_{0}$ and their
subindices.

###### Proof.

The existence, uniqueness and interior estimates are classical, see for
example [Agm10]. The estimates up to the boundary follow from [ADN59], and
Agmon’s maximum principle originally appeared in [Agm60]. See also the book
[GGS10] and the references therein. ∎

##  3\. Horizontal currents and existence of minimizers

In order to do calculus of variations with Legendrian submanifolds, as in the
classical Plateau problem, we need to enlarge our class of objects to a space
that includes weaker objects but enjoys better compactness properties. Based
on the success of the Federer–Fleming theory of integral currents [FF60],
which is based on the De Rham complex of differential forms, Franchi,
Serapioni and Serra Cassano introduced in [FSS07] a class of objects in the
Heisenberg group that, in low dimensions, generalizes horizontal submanifolds.
As with Federer–Fleming currents, these objects are defined in duality with a
differential complex introduced by Rumin [Rum94] in any contact manifold.

Here we work in an arbitrary contact manifold $M^{2n+1}$ and take our class of
horizontal currents to be the Rumin currents of dimension at most $n$. In
these dimensions, these currents can be characterized as Federer–Fleming
currents in which the contact form and its exterior derivative restrict to
zero. In the case of the Heisenberg group, integral Rumin currents correspond
to integral metric currents, in the sense of Ambrosio–Kirchheim [AK00], with
respect to the Carnot–Carathéodory metric (see Appendix A for more details).

For Federer–Fleming currents, we follow the conventions of [Sim83]; in
particular we allow our integral currents to have globally infinite mass.

###  3.1. The Rumin complex and horizontal currents

We begin by recalling the Rumin complex in low dimensions. Let $U\subset M$ be
an open set in which $\Xi$ is determined by the $1$-form $\theta$, and let
$0\leq k\leq n$. We define the differential ideal generated by $\Xi$ as

(3.1)
$\mathscr{I}^{k}(U):=\\{\theta\wedge\alpha+\mathrm{d}\theta\wedge\beta:\alpha\in\mathcal{D}^{k-1}(U),\beta\in\mathcal{D}^{k-2}(U)\\}\subset\mathcal{D}^{k}(U).$

Notice that $\mathscr{I}^{k}$ only depends on $\Xi$ and not on $\theta$, and
thus makes sense even if $\Xi$ is not the kernel of a single $1$-form in $U$.
The (lower half of the) Rumin complex is the chain complex

(3.2)
$\mathcal{D}^{0}_{\mathrm{R}}(U)\xrightarrow{\mathrm{d}}\mathcal{D}^{1}_{\mathrm{R}}(U)\xrightarrow{\mathrm{d}}\cdots\xrightarrow{\mathrm{d}}\mathcal{D}^{n-1}_{\mathrm{R}}(U)\xrightarrow{\mathrm{d}}\mathcal{D}^{n}_{\mathrm{R}}(U)$

where

(3.3) $\mathcal{D}^{k}_{\mathrm{R}}(U):=\mathcal{D}^{k}(U)/\mathscr{I}^{k}(U)$

and the differential $\mathrm{d}$ clearly descends to the quotients.

Notice that a linear map $T:\mathcal{D}^{k}_{\mathcal{R}}(U)\to\mathbb{R}$ can
be identified with a map $T:\mathcal{D}^{k}(U)\to\mathbb{R}$ which vanishes on
$\mathscr{I}^{k}$. Thus it is natural to define

######  Definition 3.1 (Horizontal current).

A De Rham current $T\in\mathcal{D}_{k}(U)$ is called _horizontal_ if

(3.4) $T(\theta\wedge\alpha+\mathrm{d}\theta\wedge\beta)=0$

for any $\alpha\in\mathcal{D}^{k-1}(U)$ and $\beta\in\mathcal{D}^{k-2}(U)$. We
denote by $\mathcal{D}^{\mathrm{hor}}_{k}(U)$ the set of horizontal currents
in $U$, and we call them _Legendrian_ when $k=n$.

Horizontal submanifolds (or currents) are sometimes called _isotropic_ in the
literature. It is immediate from the definition that if
$T\in\mathcal{D}^{\mathrm{hor}}_{k}(U)$, then $\partial
T\in\mathcal{D}^{\mathrm{hor}}_{k-1}(U)$. It is also clear that
$\mathcal{D}^{\mathrm{hor}}_{k}(U)$ is closed in $\mathcal{D}_{k}(U)$ with
respect to weak convergence. Next we need a subriemannian notion of mass. We
start with the following definition:

######  Definition 3.2 (Horizontal mass and comass).

Let $\omega\in\bigwedge^{k}T^{*}_{\xi}M$. We define its _horizontal comass_ as

(3.5)
$\|\omega\|^{\mathrm{hor}}_{*}:=\sup\\{\omega(v_{1},\ldots,v_{k}):v_{1},\ldots,v_{k}\in\Xi_{\xi}\subset
T_{\xi}M,|v_{1}\wedge\cdots\wedge v_{k}|_{g}\leq 1\\}.$

For a current $T\in\mathcal{D}^{\mathrm{hor}}_{k}(U)$, we let its _horizontal
mass_ be

(3.6)
$\mathbf{M}^{\mathrm{hor}}(T):=\sup\\{T(\omega):\omega\in\mathcal{D}^{k}(U),\|\omega\|^{\mathrm{hor}}_{*}\leq
1\text{ in }U\\}.$

In order to make use of the Federer–Fleming theory it will be convenient to
extend the metric $g$, only defined on the distribution $\Xi$, to a Riemannian
metric $g_{0}$ on the whole $TM$ and then to embed $(M,g_{0})$ into some
Euclidean space $\mathbb{R}^{L}$ using Nash’s theorem. The following
proposition shows that the relevant notions are independent of this extension.

######  Proposition 3.3.

The horizontal mass enjoys the following properties:

  1. (i)

For any current $T\in\mathcal{D}_{k}$,
$\mathbf{M}(T)\leq\mathbf{M}^{\mathrm{hor}}(T)$ holds.

  2. (ii)

Given $T\in\mathcal{D}_{k}$ with $\mathbf{M}(T)<\infty$, we have
$\mathbf{M}^{\mathrm{hor}}(T)<\infty$ if and only if $T(\theta\wedge\alpha)=0$
for any $\alpha\in\mathcal{D}^{k-1}$. Moreover, in that case
$\mathbf{M}(T)=\mathbf{M}^{\mathrm{hor}}(T)$.

  3. (iii)

A current $T\in\mathcal{D}_{k}$ satisfies
$\mathbf{M}^{\mathrm{hor}}(T)+\mathbf{M}^{\mathrm{hor}}(\partial T)<\infty$ if
and only if $T\in\mathcal{D}^{\mathrm{hor}}_{k}$ and
$\mathbf{M}(T)+\mathbf{M}(\partial T)<\infty$.

In particular, $\mathbf{M}(T)$ and $\mathbf{M}(\partial T)$ do not depend on
the extension $g_{0}$ of $g$ for $T\in\mathcal{D}^{\mathrm{hor}}_{k}$.

###### Proof.

Clearly $\|\omega\|^{\mathrm{hor}}_{*}\leq\|\omega\|_{*}$, so
$\mathbf{M}(T)\leq\mathbf{M}^{\mathrm{hor}}(T)$ for any current $T$, which
proves (i).

We now show (ii): suppose first that $\mathbf{M}^{\mathrm{hor}}(T)<\infty$.
Since $\|\theta\wedge\alpha\|^{\mathrm{hor}}_{*}=0<1$, we have that
$|T(s\theta\wedge\alpha)|\leq\mathbf{M}^{\mathrm{hor}}(T)$ for any $s>0$, so
by letting $s\to\infty$ it follows that $T(\theta\wedge\alpha)=0$.

To prove the opposite implication we will show that
$\mathbf{M}^{\mathrm{hor}}(T)\leq\mathbf{M}(T)$, which together with (i)
yields the equality. First let $X$ be the smooth vector field everywhere
$g_{0}$-orthogonal to $\Xi$ such that $\theta(X)\equiv 1$. Given
$\omega\in\mathcal{D}^{k}$ with $\|\omega\|^{\mathrm{hor}}_{*}\leq 1$,
consider the $k$-form
$\omega^{\prime}:=X\operatorname{\mathbin{\mbox{\Large$\lrcorner$}}}(\theta\wedge\omega)$.

We claim that $\|\omega^{\prime}\|_{*}\leq 1$ everywhere: indeed, fix $\xi\in
M$ and let $v_{1}\wedge\cdots\wedge v_{k}\in\bigwedge^{k}T_{\xi}M$ span the
$k$-plane $V$ with $|v_{1}\wedge\cdots\wedge v_{k}|_{g_{0}}\leq 1$. If $V$ is
not horizontal, by choosing an orthonormal basis of the $(k-1)$-plane
$V\cap\Xi_{\xi}$ and completing it to an orthonormal basis of $V$, we can
suppose that $v_{1},\ldots,v_{k-1}\in\Xi_{\xi}$ and write
$v_{k}=v_{k}^{\prime}+sX$ for some $v_{k}^{\prime}\in\Xi_{\xi}$ and
$s\in\mathbb{R}$. Then, since $v_{1}\wedge\cdots\wedge v_{k-1}\wedge
v_{k}^{\prime}$ is $g_{0}$-orthogonal to $v_{1}\wedge\cdots\wedge
v_{k-1}\wedge sX$, it holds that

$|v_{1}\wedge\cdots\wedge v_{k-1}\wedge
v_{k}^{\prime}|_{g}\leq|v_{1}\wedge\cdots\wedge v_{k}|_{g_{0}}\leq 1,$

which implies that

$(X\operatorname{\mathbin{\mbox{\Large$\lrcorner$}}}\theta\wedge\omega)(v_{1},\ldots,v_{k})=(\theta\wedge\omega)(X,v_{1},\ldots,v_{k})=(\theta\wedge\omega)(X,v_{1},\ldots,v_{k}^{\prime})=\theta(X)\omega(v_{1},\ldots,v_{k}^{\prime}),$

hence

$\omega^{\prime}(v_{1},\ldots,v_{k})\leq|\omega(v_{1},\ldots,v_{k}^{\prime})|\leq|v_{1}\wedge\cdots\wedge
v_{k}^{\prime}|_{g}\leq 1$

and the claim is proven. Now observe that

$\omega^{\prime}=X\operatorname{\mathbin{\mbox{\Large$\lrcorner$}}}(\theta\wedge\omega)=(X\operatorname{\mathbin{\mbox{\Large$\lrcorner$}}}\theta)\wedge\omega-\theta\wedge(X\operatorname{\mathbin{\mbox{\Large$\lrcorner$}}}\omega),$

which implies that for any $T$ satisfying $T(\theta\wedge\cdot)\equiv 0$,

$T(\omega)=\theta(X)T(\omega)-T(\theta\wedge(X\operatorname{\mathbin{\mbox{\Large$\lrcorner$}}}\omega))=T(\omega^{\prime})\leq\mathbf{M}(T)$

and $\mathbf{M}^{\mathrm{hor}}(T)\leq\mathbf{M}(T)$ follows. Finally, (iii) is
immediate from (ii) and the definition of $\mathcal{D}^{\mathrm{hor}}_{k}$. ∎

######  Definition 3.4.

The groups of normal and integral horizontal currents in $U\subset M$ are

(3.7)
$\mathbf{N}^{\mathrm{hor}}_{k}(U):=\mathbf{N}_{k}(U)\cap\mathcal{D}^{\mathrm{hor}}_{k}(U)\qquad\text{and}\qquad\mathbf{I}^{\mathrm{hor}}_{k}(U):=\mathbf{I}_{k}(U)\cap\mathcal{D}^{\mathrm{hor}}_{k}(U).$

The Federer–Fleming compactness theorem [Sim83, Theorem 27.3] immediately
gives:

######  Proposition 3.5.

If $\\{T_{j}\\}\subset\mathbf{I}^{\mathrm{hor}}_{k}(U)$ is a sequence with
$\|T_{j}\|(W)+\|\partial T_{j}\|(W)\leq C(W)$ for every $W\Subset U$, then a
subsequence converges weakly to a current
$T\in\mathbf{I}^{\mathrm{hor}}_{k}(U)$.

It is well known that for a rectifiable current $T$ and $\|T\|$-almost every
point $p$, if $\eta_{p,\rho}$ denotes the map $q\mapsto\frac{q-p}{\rho}$ in
any coordinates around $p$, then
$(\eta_{p,\rho})_{\\#}T\rightharpoonup\Theta(\|T\|,p)T_{p}$ as $\rho\searrow
0$, where $T_{p}$ is the current induced by a plane called the tangent plane
at $p$. For horizontal currents we have the following characterization:

######  Proposition 3.6.

Let $T$ be a rectifiable $k$-current in $U$. Then
$T\in\mathbf{I}^{\mathrm{hor}}_{k}(U)$ if and only if for $\|T\|$-almost every
$p\in U$, the approximate tangent plane $T_{p}$ is isotropic, which means that
both $\theta$ and $\mathrm{d}\theta$ vanish on it.

###### Proof.

If all tangent planes are isotropic, then (3.4) is immediate by the integral
representation formula for $T$. For the reciprocal, we just show that
$\theta_{p}$ vanishes on $T_{p}$ (the computation for $(\mathrm{d}\theta)_{p}$
is similar). Since the statement is local and invariant by diffeomorphisms, we
may suppose that $T$ is a current in a neighborhood of $0\in\mathbb{R}^{m}$
and $p=0$ (in our case, $m=2n+1$, and clearly the statement is much more
general). Let $\alpha\in\mathcal{D}^{k-1}(\mathbb{R}^{m})$ and write, for
short, $\eta_{\rho}:=\eta_{0,\rho}$. Suppose that $\alpha$ is supported in
$B_{R}$. Then

$T_{p}(\theta_{p}\wedge\alpha)=\lim_{\rho\searrow
0}(\eta_{\rho})_{\\#}T(\theta_{p}\wedge\alpha)=\lim_{\rho\searrow
0}T(\eta_{\rho}^{*}\theta_{p}\wedge\eta_{\rho}^{*}\alpha)=\lim_{\rho\searrow
0}T((\eta_{\rho}^{*}\theta_{p}-\theta)\wedge\eta_{\rho}^{*}\alpha)$

and thus

$\displaystyle\left|T_{p}(\theta_{p}\wedge\alpha)\right|$
$\displaystyle\leq\limsup_{\rho\searrow 0}\|T\|(B_{\rho R})\sup_{B_{\rho
R}}|\eta_{\rho}^{*}\theta_{p}-\theta|\sup_{\rho R}|\eta_{\rho}^{*}\alpha|$
$\displaystyle\leq\limsup_{\rho\searrow 0}\|T\|(B_{\rho
R})C\rho^{1-k}\leq\limsup_{\rho\searrow 0}C\rho=0.\qed$

As a consequence, in the coarea formula for horizontal currents we can replace
the gradient of the slicing function by its horizontal gradient. Not only
that, but thanks to the work of Ambrosio and Kirchheim [AK00, AK00a] and the
discussion in Appendix A, we can slice by any function which is Lipschitz with
respect to the intrinsic Carnot–Carathéodory distance. In order to make our
presentation more self-contained, however, we will just state the following
result, which follows from the Euclidean coarea formula and will suffice for
us.

######  Proposition 3.7.

Let $T$ be a horizontal rectifiable current in $\mathbb{H}^{n}$ with finite
mass. Then

(3.8) $\int_{0}^{\infty}\mathbf{M}(\langle
T,\tau,t\rangle)\,\mathrm{d}t\leq\mathbf{M}(T)$

where $\tau:\mathbb{H}\to\mathbb{R}_{\geq 0}$ is the Folland–Korányi gauge
$\tau(\xi)=\|\xi\|_{\mathrm{FK}}$.

###### Proof.

Since $\tau$ is smooth and Lipschitz outside of $\mathcal{B}_{r}$ for any
$r>0$, it follows from (2.7) and the Euclidean coarea formula that

$\int_{r}^{\infty}\mathbf{M}(\langle
T,\tau,t\rangle)\,\mathrm{d}t=\int_{\mathbb{H}^{n}\setminus\mathcal{B}_{r}}|\nabla^{\vec{T}}\tau|\,\mathrm{d}\|T\|\leq\int_{\mathbb{H}^{n}}\,\mathrm{d}\|T\|=\mathbf{M}(T).$

The proposition follows by just letting $r\searrow 0$. ∎

We finally notice that horizontal currents indeed generalize isotropic
submanifolds:

######  Proposition 3.8.

Let $\Sigma^{k}\subset M^{2n+1}$ be a $C^{1}$ embedded isotropic submanifold
with (possibly empty) $C^{1}$ boundary (recall that this means that
$T_{\xi}\Sigma\subset\Xi_{\xi}$ for every $\xi\in\Sigma$). Then
$\llbracket\Sigma\rrbracket\in\mathbf{I}^{\mathrm{hor}}_{k}(M)$.

###### Proof.

We have that for any $\alpha\in\mathcal{D}^{k-1}(M)$,

$\llbracket\Sigma\rrbracket(\theta\wedge\alpha)=\int_{\Sigma}\langle\theta\wedge\alpha,\vec{\tau}_{\Sigma}\rangle\,\mathrm{d}\mathcal{H}^{k}=0$

because $\vec{\tau}_{\Sigma}$ is a wedge product of vectors in $\Xi$. Since
$T_{\xi}(\partial\Sigma)\subset T_{\xi}\Sigma$, it follows that
$\partial\Sigma$ is isotropic too and thus for any
$\beta\in\mathcal{D}^{k-2}(M)$,

$\llbracket\Sigma\rrbracket(\mathrm{d}\theta\wedge\beta)=\llbracket\Sigma\rrbracket(\mathrm{d}(\theta\wedge\beta)+\theta\wedge\mathrm{d}\beta)=\partial\llbracket\Sigma\rrbracket(\theta\wedge\beta)+\llbracket\Sigma\rrbracket(\theta\wedge\mathrm{d}\beta)=\llbracket\partial\Sigma\rrbracket(\theta\wedge\beta)=0.\qed$

######  Remark 3.9.

In light of this example, a natural question is whether any integer-
rectifiable current $T$ which annihilates $\theta$ is automatically
horizontal, that is, also annihilates $\mathrm{d}\theta$. This is in general
false, and a counterexample can be found in [Fu04]. However, as shown in the
same erratum, answer becomes positive if $T$ is an integral current.

######  Theorem 3.10.

Let $T\in\mathbf{I}_{k}(M)$ and suppose that for any
$\alpha\in\mathcal{D}^{k-1}(M)$, $T(\theta\wedge\alpha)=0$. Then
$T\in\mathbf{I}^{\mathrm{hor}}_{k}(M)$.

###### Proof.

This was already proved by Fu in [Fu04], but we prefer to give the short proof
here for the convenience of the reader and to fix some typos. Let $T$ be such
a current and let $E:=\operatorname{spt}(\partial T)$. Then $E$ has locally
finite $\mathcal{H}^{k-1}$ measure, hence $\mathcal{H}^{k}(E)=0$ which implies
that $\|T\|(E)=0$.

Let $\rho:M\to\mathbb{R}$ be the function $\operatorname{dist}(\cdot,E)$ with
respect to any Riemannian metric. Clearly $\rho$ is Lipschitz, so the slices
$\langle T,\rho,r\rangle$ are well-defined integral currents for almost every
$r>0$, and they are oriented by
$\vec{T}\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\tfrac{\mathrm{d}\rho}{|\mathrm{d}\rho|}$.
This implies that $\langle T,\rho,r\rangle(\theta\wedge\beta)=0$ for each
$\beta\in\mathcal{D}^{k-2}(M)$. We compute, for any such $\beta$ and almost
every $r>0$,

$\displaystyle(T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\\{\rho>r\\})(\mathrm{d}\theta\wedge\beta)$
$\displaystyle=(T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\\{\rho>r\\})(\mathrm{d}(\theta\wedge\beta)+\theta\wedge\mathrm{d}\beta)$
$\displaystyle=\partial(T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\\{\rho>r\\})(\theta\wedge\beta)+(T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\\{\rho>r\\})(\theta\wedge\mathrm{d}\beta)$
$\displaystyle=((\partial
T)\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\\{\rho>r\\})(\theta\wedge\beta)-\langle
T,\rho,r\rangle(\theta\wedge\beta)=0.$

Hence, using the fact that $\|T\|(E)=0$ and $\|T\|$ is a Radon measure,

$|T(\mathrm{d}\theta\wedge\beta)|\leq\limsup_{r\searrow
0}|(T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\\{\rho\leq
r\\})(\mathrm{d}\theta\wedge\beta)|\leq\sup_{M}(|\mathrm{d}\theta||\beta|)\limsup_{r\searrow
0}\|T\|\left(\\{\rho\leq r\\}\cap\operatorname{spt}\beta\right)=0.\qed$

###  3.2. Existence for the Plateau problem in the Heisenberg group

We first need a lemma to control the supports of a minimizing sequence.

######  Lemma 3.11.

Let $1\leq k\leq n$, $r>0$ and $m>0$. Then for any current
$T\in\mathbf{I}^{\mathrm{hor}}_{k}(\mathbb{H}^{n})$ with
$\operatorname{spt}\partial T\subset\mathcal{B}_{r}$ and $\mathbf{M}(T)\leq m$
we can find another current
$\hat{T}\in\mathbf{I}^{\mathrm{hor}}_{k}(\mathbb{H}^{n})$ with
$\partial\hat{T}=\partial T$, $\mathbf{M}(\hat{T})\leq\mathbf{M}(T)$ and
$\operatorname{spt}\hat{T}\subset\mathcal{B}_{R}$, where
$R=r+Cm^{\frac{1}{k}}$ for a constant $C=C(n,k)>0$.

###### Proof.

We argue by contradiction. Let $R_{1}=r+\lambda m^{1/k}$, for a dimensional
constant $\lambda>0$ to be determined later, and consider the set

$G=\left\\{t\in[r,R_{1}]:\langle T,\tau,t\rangle\text{ exists and
}\mathbf{M}(\langle T,\tau,t\rangle)\leq\frac{2m}{R_{1}-r}\right\\},$

where $\tau$ is the Folland–Korányi gauge (2.5). Then by (2.7) and the coarea
formula we have that

$\mathcal{L}^{1}([r,R_{1}]\setminus
G)\leq\frac{R_{1}-r}{2m}\int_{[r,R_{1}]}\mathbf{M}(\langle
T,\tau,t\rangle)\,\mathrm{d}t\leq\frac{R_{1}-r}{2m}\mathbf{M}(T)\leq\frac{1}{2}(R_{1}-r),$

so that $\mathcal{L}^{1}(G)\geq\tfrac{1}{2}(R_{1}-r)=\tfrac{1}{2}\lambda
m^{1/k}$. Now for every $t\in G$, consider the current $Q_{t}$ given by the
isoperimetric inequality, which satisfies

$\partial Q_{t}=\langle T,\tau,t\rangle,\qquad\mathbf{M}(Q_{t})\leq
C_{1}\mathbf{M}(\langle
T,\tau,t\rangle)^{\frac{k}{k-1}}\qquad\text{and}\qquad\operatorname{spt}Q_{t}\subset\mathcal{B}_{R},$

where

$R:=R_{1}+C\left(\frac{m}{R_{1}-r}\right)^{\frac{1}{k-1}}=r+\lambda
m^{1/k}+C\lambda^{-\frac{1}{k-1}}m^{1/k}.$

This exists because $\partial\langle T,\tau,t\rangle=-\langle\partial
T,\tau,t\rangle=0$. Then let
$\hat{T}_{t}:=T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathcal{B}_{t}-Q_{t}$,
also supported in $\mathcal{B}_{R}$ and with boundary

$\partial\hat{T}_{t}=\partial(T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathcal{B}_{t})-\partial
Q_{t}=\langle T,\tau,t\rangle+(\partial
T)\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathcal{B}_{t}-\langle
T,\tau,t\rangle=\partial T.$

Now if the lemma were false we must have
$\mathbf{M}(T)<\mathbf{M}(\hat{T}_{t})$. Hence

$\mathbf{M}(T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathcal{B}_{t})+\mathbf{M}(T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathcal{B}_{t}^{c})=\mathbf{M}(T)\leq\mathbf{M}(\hat{T}_{t})\leq\mathbf{M}(T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathcal{B}_{t})+\mathbf{M}(Q_{t})\leq\mathbf{M}(T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathcal{B}_{t})+C_{1}\mathbf{M}(\langle
T,\tau,t\rangle)^{\frac{k}{k-1}}.$

Letting
$g(t):=\mathbf{M}(T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathcal{B}_{t}^{c})$
and using the coarea formula with (2.7), for almost every $t\in G$ we obtain
that

$g(t)=\mathbf{M}(T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathcal{B}_{t}^{c})\leq
C_{1}\mathbf{M}(\langle T,\tau,t\rangle)^{\frac{k}{k-1}}\leq
C_{1}(-g^{\prime}(t))^{\frac{k}{k-1}}.$

or

$\left(-g^{1/k}\right)^{\prime}\geq C^{-1}.$

Since $g^{1/k}$ is monotonically decreasing, we can integrate over all of
$[r,R_{1}]$ and find that

$\displaystyle\frac{1}{2}\lambda m^{1/k}$
$\displaystyle\leq\mathcal{L}^{1}(G)\leq
C\int_{[r,R_{1}]}\left(-g(t)^{1/k}\right)^{\prime}\,\mathrm{d}t\leq
C(g(r)^{1/k}-g(R)^{1/k})$ $\displaystyle\leq Cg(r)^{1/k}\leq
C\mathbf{M}(T)^{1/k}\leq Cm^{1/k},$

which is a contradiction if $\lambda$ is chosen large enough. ∎

Using this lemma the existence part of 1.1 follows directly.

######  Theorem 3.12.

Let $1\leq k\leq n$ and consider a current
$S\in\mathbf{I}^{\mathrm{hor}}_{k-1}(\mathbb{H}^{n})$ with
$\operatorname{spt}S$ compact. Then there exists a current
$T\in\mathbf{I}^{\mathrm{hor}}_{k}(\mathbb{H}^{n})$ also with
$\operatorname{spt}T$ compact such that
$\mathbf{M}(T)\leq\mathbf{M}(T^{\prime})$ for any other current
$T^{\prime}\in\mathbf{I}^{\mathrm{hor}}_{k}(\mathbb{H}^{n})$ with $\partial
T^{\prime}=S$.

###### Proof.

Let $(T_{j})$ be a minimizing sequence, that is,
$T_{j}\in\mathbf{I}^{\mathrm{hor}}_{k}(\mathbb{H}^{n})$, $\partial T_{j}=S$
and

$\mathbf{M}(T_{j})\longrightarrow\inf\\{\mathbf{M}(T^{\prime}):T^{\prime}\in\mathbf{I}^{\mathrm{hor}}_{k}(\mathbb{H}^{n}),\partial
T^{\prime}=S\\}.$

Let $r>0$ be such that $\operatorname{spt}S\subset\mathcal{B}_{r}$. By Lemma
3.11, since $\mathbf{M}(T_{j})$ are bounded, we can improve our sequence to
another minimizing sequence $(\hat{T}_{j})$ with
$\operatorname{spt}\hat{T}_{j}\subset\overline{\mathcal{B}_{R}}$ for some
$R>0$. Then by the compactness theorem and the lower semicontinuity of the
mass, a subsequence converges to a current $T$ belonging to the same class and
supported in $\overline{\mathcal{B}_{R}}$, thus attaining the minimum. ∎

###  3.3. Existence of Legendrian minimizers in a homology class

The existence part of 1.2 follows easily using the compactness theorem once we
can show that horizontal currents exist in any homology class. This is the
content of the following proposition:

######  Proposition 3.13.

Let $(M^{2n+1},\Xi)$ be a closed contact manifold and $1\leq k\leq n$. Then
any homology class $\mathfrak{a}\in H_{k}(M,\mathbb{Z})$ contains a horizontal
integral cycle.

###### Proof.

Let $g$ be any subriemannian metric on $\Xi$. Since $M$ is compact, we can
cover it with finitely many Carnot–Carathéodory geodesic balls
$B_{R}(\xi_{1}),\ldots B_{R}(\xi_{N})$ such that $B_{2R}(\xi_{i})$ admits a
Darboux chart
$\phi_{i}:B_{2R}(\xi_{i})\to\mathcal{U}_{i}\subset\mathbb{H}^{n}$ for each
$i$. Let $0<\lambda\leq\Lambda<\infty$ be such that

$\lambda g_{\mathbb{H}}\leq(\phi_{i}^{-1})^{*}g\leq\Lambda g_{\mathbb{H}},$

so that in particular the maps $\phi_{i}$ preserve distances up to a constant.

Given $\mathfrak{a}\in H_{k}(M,\mathbb{Z})$, represent it by a Lipschitz
polyhedral chain $\bm{\sigma}=\sum_{j}m_{j}\sigma_{j}$, where $m_{j}$ are
positive integers and $\sigma_{j}$ are Lipschitz simplices each contained in
some ball $B_{r}(\xi)$, where $r\leq R/C_{2}$ and $C_{2}$ is a constant to be
determined below (depending only on $n$ and $\Lambda/\lambda$).

We will deform $\bm{\sigma}$ into a horizontal integral current $S$ by
inductively replacing its skeleton using the isoperimetric inequality (A.5).
We first construct, for each $1$-simplex $\tau$ in the $1$-skeleton of
$\bm{\sigma}$, a horizontal integral $1$-current $S_{\tau}$ with the same
boundary $\partial S_{\tau}=\partial\llbracket\tau\rrbracket$, with mass at
most $C^{(1)}r$, and with support still contained in a ball of radius
$C^{(1)}r$ (for example we can take a minimizing Carnot–Carathéodory
geodesic). Moreover we record the existence of a (not necessarily horizontal)
filling, that is, an integral $2$-current $V_{\tau}$ with $\partial
V_{\tau}=S_{\tau}-\llbracket\tau\rrbracket$ and
$\operatorname{diam}(\operatorname{spt}V_{\tau})\leq C^{(1)}r$, which exists
simply because these $1$-currents are supported in a contractible ball.

Then, for each $2$-simplex $\tau$ in the $2$-skeleton of $\bm{\sigma}$, we
construct a horizontal integral $2$-current $S_{\tau}$ whose boundary is
$\partial S_{\tau}=S_{\tau_{0}}-S_{\tau_{1}}+S_{\tau_{2}}$, where the
$1$-simplices $\tau_{0},\tau_{1},\tau_{2}$ are defined by
$\partial\tau=\tau_{0}-\tau_{1}+\tau_{2}$. We construct this horizontal
filling by applying A.5 on $\mathbb{H}^{n}$ and pushing the currents back and
forth by means of the Darboux charts, which preserve horizontality. Note that
the resulting currents have
$\operatorname{diam}(\operatorname{spt}S_{\tau})\leq C^{(2)}r^{2}$ and
$\mathbf{M}(S_{\tau})\leq C^{(2)}r^{2}$, where $C^{(2)}$ depends only on $n$
and $\Lambda/\lambda$. Moreover, since

$\partial
S_{\tau}=S_{\tau_{0}}-S_{\tau_{1}}+S_{\tau_{2}}=\partial(V_{\tau_{0}}-V_{\tau_{1}}+V_{\tau_{2}})+\llbracket\tau_{0}\rrbracket-\llbracket\tau_{1}\rrbracket+\llbracket\tau_{2}\rrbracket=\partial(V_{\tau_{0}}-V_{\tau_{1}}+V_{\tau_{2}}+\llbracket\tau\rrbracket),$

we have that
$S_{\tau}=V_{\tau_{0}}-V_{\tau_{1}}+V_{\tau_{2}}+\llbracket\tau\rrbracket+\partial
V_{\tau}$ for some (not necessarily horizontal) integral $3$-current
$V_{\tau}$ also with $\operatorname{diam}(\operatorname{spt}V_{\tau})\leq
C^{(2)}r$.

This procedure can be iterated and in the $k$-th step, for each $k$-simplex
$\tau$ of $\bm{\sigma}$ with $\partial\tau=\sum_{l=0}^{k}(-1)^{l}\tau_{l}$, we
obtain a $k$-dimensional horizontal integral current $S_{\tau}$ and a
$(k+1)$-dimensional filling $V_{\tau}$ with

$S_{\tau}=\llbracket\tau\rrbracket+\sum_{l=0}^{k}(-1)^{l}\,V_{\tau_{l}}+\partial
V_{\tau}$

because additional terms coming from the fillings of each $\tau_{l}$ cancel
each other, since $\partial\partial\tau=0$. We also need that balls of
diameter at most $C^{(n)}r$ are contained in a Darboux chart; we can guarantee
this by choosing above $C_{2}=C^{(n)}$, which only depends on
$\Lambda/\lambda$ and $n$.

After $n$ steps we will obtain horiontal integral $n$-currents
$S_{\sigma_{j}}$ for each $j$, together with integral $(n+1)$-currents
$V_{\sigma_{j}}$, satisfying
$S_{\sigma_{j}}=\llbracket\sigma_{j}\rrbracket+\sum_{l=0}^{k}(-1)^{l}\,V_{\tau_{jl}}+\partial
V_{\sigma_{j}}$, where $\partial\sigma_{j}=\sum_{l=0}^{k}(-1)^{l}\tau_{jl}$.
Since $\bm{\sigma}=\sum_{j}\sigma_{j}$ is a cycle, its boundary terms cancel
each other and we have

$\sum_{j}S_{\sigma_{j}}=\sum_{j}\llbracket\sigma_{j}\rrbracket+\sum_{l=0}^{k}(-1)^{l}\,V_{\tau_{jl}}+\partial
V_{\sigma_{j}}=\llbracket\bm{\sigma}\rrbracket+\partial\left(\sum_{j}V_{\sigma_{j}}\right),$

which means that the horizontal integral $n$-current
$S:=\sum_{j}S_{\sigma_{j}}$ is a cycle representing $\mathfrak{a}$. ∎

######  Remark 3.14.

The fact that there is no additional homological condition for a class to be
“horizontal” should be compared to the fact that the cohomology of the Rumin
complex is isomorphic to the usual De Rham cohomology. The situation changes
drastically in the symplectic setting.

######  Theorem 3.15.

Let $(M^{2n+1},\Xi,g)$ be a closed subriemannian contact manifold, $1\leq
k\leq n$ and $\mathfrak{a}\in H_{k}(M,\mathbb{Z})$ a homology class. Then
there exists a cycle $T\in\mathbf{I}^{\mathrm{hor}}_{k}(M)$ representing
$\mathfrak{a}$ that minimizes the mass among all such cycles.

###### Proof.

Extend $g$ to a Riemannian metric $g_{0}$ on $M$ and embed the resulting
Riemannian manifold isometrically into some Euclidean space $\mathbb{R}^{L}$.
Let $T_{j}$ be a minimizing sequence in $\mathfrak{a}$; then, by 3.5, after
extracting a subsequence, $T_{j}\to T$ in the flat distance for an integral
current $T\in\mathbf{I}^{\mathrm{hor}}_{k}(M)$. This means that we can write
$T_{j}=T+R_{j}+\partial S_{j}$ with $\mathbf{M}(R_{j}),\mathbf{M}(S_{j})\to
0$. As in [Mor09, Section 5.7], since $\partial T_{j}=\partial T=0$, we have
that $\partial R_{j}=0$, so by the Euclidean isoperimetric inequality,
$R_{j}=\partial Y_{j}$ for an integral $k$-current $Y_{j}$ in $\mathbb{R}^{L}$
with $\operatorname{spt}Y_{j}$ uniformly close to $M$ for $j$ large. Hence we
may retract $Y_{j}$ onto $\tilde{Y}_{j}\in\mathbf{I}_{k}(M)$ and have
$T_{j}=T+\partial(\tilde{Y}_{j}+S_{j})$. In particular, for $j$ large, $T_{j}$
is homologous to $T$ in $M$ and the theorem follows. ∎

###  3.4. General properties of local minimizers

In this section we define local minimizers and prove a strong convergence
theorem. Since these concepts are local, we work with a subriemannian metric
$h$ defined on an open subset $\mathcal{U}$ of the Heisenberg group.

Denote by $\mathscr{L}_{0}^{k}$ the set of oriented isotropic $k$-planes in
$\Xi_{0}$, and set $\mathscr{L}_{0}:=\mathscr{L}_{0}^{n}$, the so-called
oriented Lagrangian Grassmannian. We can identify $\mathscr{L}_{0}^{k}$ with a
subset of $\bigwedge^{k}\Xi_{0}$; note that any
$\vec{\pi}\in\mathscr{L}_{0}^{k}$ has $|\vec{\pi}|=1$. After identifying all
horizontal spaces via left translations, any Lipschitz metric $h$ on
$\Xi|_{\mathcal{U}}$ induces a norm on $\mathscr{L}_{0}^{k}$ for each
$\xi\in\mathcal{U}$ in the usual way:

$|\vec{\pi}|_{h_{\xi}}=\sqrt{h_{\xi}(\vec{\pi},\vec{\pi})}.$

The mass with respect to the metric $h$ then takes the form

(3.9)
$\mathbf{M}^{h}_{\mathcal{U}}(T)=\int_{\mathcal{U}}|\vec{T}(\xi)|_{h_{\xi}}\,\mathrm{d}\|T\|(\xi).$

######  Definition 3.16.

We will say that a current $T\in\mathbf{I}^{\mathrm{hor}}_{k}(\mathbb{H}^{n})$
is _$\mathbf{M}^{h}$ -minimizing in $\mathcal{U}$_ if
$\mathbf{M}^{h}_{\mathcal{U}}(T)\leq\mathbf{M}^{h}_{\mathcal{U}}(T^{\prime})$
whenever $T^{\prime}\in\mathbf{I}^{\mathrm{hor}}_{k}(\mathbb{H}^{n})$ has
$\operatorname{spt}(T^{\prime}-T)\Subset\mathcal{U}$ and $T-T^{\prime}$ is the
boundary of a integral $(k+1)$-current supported in $\mathcal{U}$.

Note that the current whose boundary must be $T-T^{\prime}$ is not required to
be horizontal; in particular, if $k=n$, it will never be so. We add this
condition just to enforce that homology minimizers are mass-minimizing on the
whole manifold. Of course, if $\mathcal{U}$ is contractible, the condition
reduces to $\partial(T-T^{\prime})=0$.

As in the classical setting, sequences of mass minimizing currents have
improved convergence properties. In particular, we will need the following
proposition in the regularity theory. The proof relies crucially on a theorem
Wenger [Wen07] about flat convergence of integral currents in metric spaces
that we adapt in the Appendix.

######  Proposition 3.17.

Let $\mathcal{U}\subset\mathbb{H}^{n}$ be a bounded contractible open set,
$1\leq k\leq n$, and $T_{j}\in\mathbf{I}^{\mathrm{hor}}_{k}(\mathbb{H}^{n})$ a
sequence of currents with $(\partial
T_{j})\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathcal{U}=0$ and
$\sup_{j}\|T_{j}\|(\mathcal{U})<\infty$ that converges weakly to a current
$T$. Suppose that $g_{j}$ is a sequence of subriemannian metrics on
$\Xi|_{\mathcal{U}}$ converging uniformly to a subriemannian metric $g$, and
that $T_{j}$ is $\mathbf{M}^{g_{j}}$-minimizing in $\mathcal{U}$.

Then $T$ is $\mathbf{M}^{g}$-minimizing in $\mathcal{U}$ and
$\|T_{j}\|\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathcal{U}\rightharpoonup\|T\|\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathcal{U}$
as Radon measures in $\mathcal{U}$, i.e. against $C^{0}_{c}(\mathcal{U})$
functions.

###### Proof.

Choose a Lipschitz function $\rho:\mathcal{U}\to[0,1]$ such that the level
sets $\\{\rho>r\\}$ for $r>0$ are compactly contained in $\mathcal{U}$ and
exhaust $\mathcal{U}$. It is a classical fact (see for example [AK00,
Proposition 8.3]) that for almost all $r>0$ there is a subsequence
$T_{j^{\prime}}$ such that all the slices $\langle
T_{j^{\prime}},\rho,r\rangle$ exist, have uniformly bounded masses and

$\langle T_{j^{\prime}},\rho,r\rangle\rightharpoonup\langle
T,\rho,r\rangle\qquad\text{in }\mathcal{D}_{k-1}.$

Moreover we may assume that $\operatorname{spt}\,\langle
T_{j^{\prime}},\rho,r\rangle\subset\\{\rho=r\\}$ and $\langle
T_{j^{\prime}},\rho,r\rangle=-\partial(T_{j^{\prime}}\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\\{\rho>r\\})$
(hence $\partial\langle T_{j^{\prime}},\rho,r\rangle=0$), that the same holds
for $T$, and that $\|T\|(\\{\rho=r\\})=0$. Now A.6 implies that there exist
$S_{j^{\prime}}\in\mathbf{I}^{\mathrm{hor}}_{k}(\mathbb{H}^{n})$ with
$\partial S_{j^{\prime}}=\langle T_{j^{\prime}},\rho,r\rangle-\langle
T,\rho,r\rangle$, with
$\operatorname{spt}S_{j^{\prime}}\subset\mathcal{B}_{s_{j^{\prime}}}(\\{\rho=r\\})$
and with $\mathbf{M}(S_{j^{\prime}})\searrow 0$ and $s_{j^{\prime}}\searrow
0$. In particular, for $j^{\prime}$ large,
$\operatorname{spt}S_{j^{\prime}}\subset\mathcal{B}_{s_{j^{\prime}}}(\\{\rho=r\\})\subset\\{\rho>r/2\\}\Subset\mathcal{U}$.
Now we have that

$\partial S_{j^{\prime}}=\langle T_{j^{\prime}},\rho,r\rangle-\langle
T,\rho,r\rangle=-\partial(T_{j^{\prime}}\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\\{\rho>r\\})+\partial(T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\\{\rho>r\\}),$

so
$(T-T_{j^{\prime}})\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\\{\rho>r\\}-S_{j^{\prime}}$
bounds in $\mathcal{U}$.

Let $Z\in\mathbf{I}^{\mathrm{hor}}_{k}(\mathbb{H}^{n})$ be another horizontal
cycle with $\operatorname{spt}Z\Subset\mathcal{U}$ that bounds a
$(k+1)$-dimensional integral current supported in $\mathcal{U}$. Thus we may
use
$(T-T_{j^{\prime}})\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\\{\rho>r\\}-S_{j^{\prime}}+Z$
to test the minimality of $T_{j^{\prime}}$:

$\mathbf{M}^{g_{j^{\prime}}}_{\mathcal{U}}(T_{j^{\prime}})\leq\mathbf{M}^{g_{j^{\prime}}}_{\mathcal{U}}(T_{j^{\prime}}+(T-T_{j^{\prime}})\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\\{\rho>r\\}-S_{j^{\prime}}+Z).$

It follows that

$\displaystyle\mathbf{M}^{g_{j^{\prime}}}_{\mathcal{U}}(T_{j^{\prime}})$
$\displaystyle\leq\mathbf{M}^{g_{j^{\prime}}}_{\mathcal{U}}(T_{j^{\prime}}\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\\{\rho\leq
r\\}+T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\\{\rho>r\\}+Z-S_{j^{\prime}})$
$\displaystyle\leq\mathbf{M}^{g_{j^{\prime}}}_{\mathcal{U}}(T_{j^{\prime}}\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\\{\rho\leq
r\\})+\mathbf{M}^{g_{j^{\prime}}}_{\mathcal{U}}((T+Z)\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\\{\rho>r\\})+\mathbf{M}^{g_{j^{\prime}}}_{\mathcal{U}}(S_{j^{\prime}})$

and thus

(3.10)
$\mathbf{M}^{g_{j^{\prime}}}_{\mathcal{U}}(T_{j^{\prime}}\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\\{\rho>r\\})\leq\mathbf{M}^{g_{j^{\prime}}}_{\mathcal{U}}((T+Z)\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\\{\rho>r\\})+\mathbf{M}^{g_{j^{\prime}}}_{\mathcal{U}}(S_{j^{\prime}}).$

Letting $j^{\prime}\to\infty$ and using the uniform convergence
$g_{j^{\prime}}\to g$, the lower semicontinuity of the mass, and the fact that
$\mathbf{M}(T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\\{\rho=r\\})=0$,
we obtain

$\mathbf{M}^{g}_{\mathcal{U}}(T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\\{\rho>r\\})\leq\mathbf{M}^{g}_{\mathcal{U}}((T+Z)\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\\{\rho>r\\}),$

from which the $\mathbf{M}^{g}$-minimality of $T$ is clear. To prove the
convergence of the associated measures, we first prove the strict convergence
in the sense of Reshetnyak of the $g$-masses of $T_{j}$ on appropriate open
sets. Choosing $Z=0$ above, for almost every $r>0$, (3.10) gives

$\mathbf{M}^{g_{j^{\prime}}}_{\mathcal{U}}(T_{j^{\prime}}\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\\{\rho>r\\})\leq\mathbf{M}^{g_{j^{\prime}}}_{\mathcal{U}}(T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\\{\rho>r\\})+\mathbf{M}^{g_{j^{\prime}}}_{\mathcal{U}}(S_{j^{\prime}}),$

which together with the uniform convergence $g_{j^{\prime}}\to g$ implies that
for a sequence $\delta_{j^{\prime}}\to 0$,

$\mathbf{M}^{g}_{\mathcal{U}}(T_{j^{\prime}}\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\\{\rho>r\\})\leq\mathbf{M}^{g}_{\mathcal{U}}(T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\\{\rho>r\\})+\delta_{j^{\prime}}.$

Taking the lim sup we obtain that

$\limsup_{j^{\prime}\to\infty}\mathbf{M}^{g}_{\mathcal{U}}(T_{j^{\prime}}\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\\{\rho>r\\})\leq\mathbf{M}^{g}_{\mathcal{U}}(T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\\{\rho>r\\}),$

whereas the opposite inequality

$\liminf_{j^{\prime}\to\infty}\mathbf{M}^{g}_{\mathcal{U}}(T_{j^{\prime}}\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\\{\rho>r\\})\geq\mathbf{M}^{g}_{\mathcal{U}}(T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\\{\rho>r\\})$

holds always thanks to the weak convergence $T_{j^{\prime}}\rightharpoonup T$.
Hence the vector-valued measures333Here we are identifying $\mathbb{H}^{n}$
with $\mathbb{R}^{2n+1}$ and treating the $k$-vectors $\vec{T}_{j}$ as vectors
in the abstract vector space $\bigwedge^{k}\Xi_{0}$.
$T_{j^{\prime}}=\vec{T}_{j^{\prime}}\|T_{j^{\prime}}\|_{g}$ converge strictly
to $T=\vec{T}\|T\|_{g}$ in the open set $\\{\rho>r\\}$. Hence, Reshetnyak’s
continuity theorem (see for example [Rin18, Theorem 10.3]) shows that for any
function $f\in C^{0}_{c}(\\{\rho>r\\})$,

$\int_{\\{\rho>r\\}}f(\xi)\,\mathrm{d}\|T_{j^{\prime}}\|=\int_{\\{\rho>r\\}}\frac{f(\xi)}{|\vec{T}_{j^{\prime}}(\xi)|_{g_{\xi}}}\,\mathrm{d}\|T_{j^{\prime}}\|_{g}\longrightarrow\int_{\\{\rho>r\\}}\frac{f(\xi)}{|\vec{T}(\xi)|_{g_{\xi}}}\,\mathrm{d}\|T\|_{g}=\int_{\\{\rho>r\\}}f(\xi)\,\mathrm{d}\|T\|.$

Finally, given any $f\in C^{0}_{c}(\mathcal{U})$ we can always find some $r>0$
such that $\operatorname{spt}f\Subset\\{\rho>r\\}$, so we have shown that
$\|T_{j^{\prime}}\|\rightharpoonup\|T\|$ for a subsequence, and now a standard
contradiction argument establishes it for the whole sequence. ∎

##  4\. Main regularity theorems

In this section we state the $\varepsilon$-regularity theorem for Legendrian
local area minimizers (4.7) and derive its main consequences for the partial
regularity of area-minimizing Legendrian currents. The proof of 4.7 will be
carried out in the following two sections. Note that from here onwards we
consider only Legendrian currents, that is, $n$-dimensional horizontal
currents.

###  4.1. Geometric considerations

Thanks to the homogeneity of the Heisenberg group, we can develop our
regularity theory around the origin. Fix a Legendrian plane $\pi$ through
$0\in\mathbb{H}^{n}$, namely a horizontal plane $\pi\subset\Xi_{0}$ such that
$\omega|_{\pi}=0$. It will be important to keep in mind that there is a
canonical smooth action of the unitary group $\mathsf{U}(n)$ on
$\mathbb{H}^{n}$ via

(4.1) $U\cdot(\mathbf{z},\varphi)=(U\mathbf{z},\varphi),\qquad
U\in\mathsf{U}(n)$

that preserves all the structures of the Heisenberg group, that is: acts as
group automorphisms, pulls back $\theta$ to itself, and is compatible with the
metric $g_{\mathbb{H}}$, the almost complex structure $J$ and the symplectic
form $\omega$ on $\Xi$. This group action acts transitively on the set of all
Legendrian planes in $\Xi_{0}$. Therefore, up to applying such an
automorphism, there is no loss of generality in supposing that our plane $\pi$
is the plane
$\pi_{0}:=\operatorname{span}\\{\partial_{x^{1}},\ldots,\partial_{x^{n}}\\}$.

Let $\vec{\pi}\in\bigwedge^{n}\Xi_{0}$ be an $n$-vector orienting $\pi$. By a
slight abuse of notation, we will denote also by $\vec{\pi}$ the left-
invariant $n$-vector field that extends $\vec{\pi}$ at the origin. We thus
have the explicit form

(4.2)
$\vec{\pi}_{0}=\vec{\nabla}^{H}x^{1}\wedge\cdots\wedge\vec{\nabla}^{H}x^{n}$

for the plane $\pi_{0}$ with its natural orientation.

There is a canonical projection map
$\Pi:\mathbb{H}^{n}\to\mathbb{R}^{2n}\cong\Xi_{0}$ that “forgets” the last
coordinate and is a group homomorphism: $\Pi(\mathbf{z},\varphi)=\mathbf{z}$.
Then for any Lagrangian plane $\pi\subset\Xi_{0}$ with respect to $\omega$ (we
will call such planes _Legendrian_) we can consider the projections
$\mathbf{p}^{\pi}:\mathbb{H}^{n}\to\pi\cong\mathbb{R}^{n}$, defined as the
composition of $\Pi$ and then the $g_{\mathbb{H}}$-orthogonal projection onto
$\pi$. We will just write $\mathbf{p}$ when the plane $\pi$ is clear. We will
also consider the projection $\mathbf{q}=\mathbf{q}^{\pi}$ onto the orthogonal
of $\pi$ (after applying $\Pi$). For the plane $\pi_{0}$ this has the
coordinate expression

$\mathbf{p}(\mathbf{x},\mathbf{y},\varphi)=\mathbf{x},\qquad\mathbf{q}(\mathbf{x},\mathbf{y},\varphi)=\mathbf{y}.$

It is clear from the expression for the multiplication in $\mathbb{H}^{n}$
that $\Pi:\mathbb{H}^{n}\to\mathbb{R}^{2n}$ is a group homomorphism. Hence the
same is true about $\mathbf{p}^{\pi}$ and $\mathbf{q}^{\pi}$ for any
Legendrian plane $\pi$. For $r>0$ and $x_{0}\in\mathbb{R}^{n}$, let
$B_{r}^{\pi}(x_{0})$ denote the Euclidean open ball
$B_{r}^{\pi}:=\\{x\in\pi:|x-x_{0}|<r\\}\subset\pi$ and define the cylinder
$\mathbf{C}_{r}^{\pi}(x_{0}):=(\mathbf{p}^{\pi})^{-1}(B_{r}^{\pi}(x_{0}))$. We
will omit the superscript $\pi$ when the plane is clear (most of the time it
will be $\pi_{0}$), and also set $B_{r}:=B_{r}(0)$ and
$\mathbf{C}_{r}:=\mathbf{C}_{r}(0)$. We will also denote for short
$\mathbf{C}_{r}^{\pi}(\xi):=\mathbf{C}_{r}^{\pi}(\mathbf{p}^{\pi}(\xi))$ for
$\xi\in\mathbb{H}^{n}$.

As in Section 3 it will be convenient to endow $\mathbb{H}^{n}$ with a left-
invariant Riemannian metric $g_{0}$ such that
$g_{0}|_{\Xi}=g_{\mathbb{H}^{n}}$; this is not absolutely necessary but it
will allow us to recycle standard arguments from the Riemannian setting.
Nevertheless none of our hypotheses or conclusions will involve this auxiliary
metric. In particular, since all the currents that we consider are horizontal,
their mass is independent of $g_{0}$ by 3.3.

Observe that with respect to such a metric $g_{0}$, any of the projections
$\mathbf{p}$ satisfies

(4.3) $|\nabla\mathbf{p}|\leq 1.$

Indeed, it is enough to check this at $0\in\mathbb{H}^{n}$ thanks to the left-
invariance of $g_{0}$ and to the fact that $\mathbf{p}$ is a group
homomorphism. But at $\nabla\mathbf{p}(0)$ is just the projection onto the
$x^{i}$-coordinates, which clearly has norm at most $1$. A similar argument
gives the following: for any oriented horizontal $n$-plane
$\vec{\pi}^{\prime}$ at any point $\xi\in\mathbb{H}^{n}$, its push-forward by
$\mathbf{p}$ is

(4.4)
$\mathbf{p}_{\\#}(\vec{\pi}^{\prime})=\langle\vec{\pi}^{\prime},\vec{\pi}(\xi)\rangle\vec{\pi}.$

As a consequence we get the classical formula for the excess of mass:

######  Proposition 4.1.

Let $T\in\mathbf{I}^{\mathrm{hor}}_{n}(\mathbb{H}^{n})$ and let
$K\subset\mathbb{R}^{n}$ be a measurable set. Then

(4.5)
$\mathbf{M}(T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathbf{p}^{-1}(K))-(\mathbf{p}_{\\#}T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}K)(\vec{\pi})=\frac{1}{2}\int_{\mathbf{p}^{-1}(K)}|\vec{T}-\vec{\pi}|^{2}\,\mathrm{d}\|T\|.$

###### Proof.

Notice that $|\vec{T}-\vec{\pi}|^{2}=2-2\langle\vec{T},\vec{\pi}\rangle$.
Thus, using (4.4),

$\displaystyle\frac{1}{2}\int_{\mathbf{p}^{-1}(K)}|\vec{T}-\vec{\pi}|^{2}\,\mathrm{d}\|T\|$
$\displaystyle=\|T\|(\mathbf{p}^{-1}(K))-\int_{\mathbf{p}^{-1}(B_{r})}\langle\vec{T},\vec{\pi}\rangle\,\mathrm{d}\|T\|$
$\displaystyle=\|T\|(\mathbf{p}^{-1}(K))-(\mathbf{p}_{\\#}T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}K)(\vec{\pi}).\qed$

This leads us to a natural notion of (cylindrical) excess, the quantity that
will lead the regularity theory. Suppose that $T$ is a horizontal integral
current in $\mathbb{H}^{n}$ with $(\partial
T)\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathbf{C}_{r}^{\pi}(x_{0})=0$.
Since
$(\partial\mathbf{p}^{\pi}_{\\#}T)\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}B_{r}^{\pi}(x_{0})=\mathbf{p}^{\pi}_{\\#}((\partial
T)\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathbf{C}_{r}^{\pi}(x_{0}))=0$,
by the constancy theorem we have that
$(\mathbf{p}^{\pi}_{\\#}T)\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}B_{r}^{\pi}(x_{0})=Q\llbracket
B_{r}^{\pi}(x_{0})\rrbracket$ for some integer $Q$. By orienting $\vec{\pi}$
appropriately we can and will suppose that $Q\geq 0$, thus
$\|\mathbf{p}^{\pi}_{\\#}T\|(B_{r}^{\pi}(x_{0}))=Q\omega_{n}r^{n}$, where
$\omega_{n}=\mathcal{L}^{n}(B_{1})$. Thus we can define:

######  Definition 4.2.

The _excess_ of $T$ in the cylinder $\mathbf{C}_{r}^{\pi}(x_{0})$ is the
quantity

(4.6)
$\mathbf{E}(T,\mathbf{C}_{r}^{\pi}(x_{0})):=r^{-n}\left(\|T\|(\mathbf{C}_{r}^{\pi}(x_{0}))-Q\omega_{n}r^{n}\right)=\frac{1}{2}r^{-n}\int_{\mathbf{C}_{r}^{\pi}(x_{0})}|\vec{T}-\vec{\pi}|^{2}\,\mathrm{d}\|T\|.$

###  4.2. Assumptions and the main regularity theorem

Although our regularity result works with more general functionals
(essentially those parametric elliptic functionals defined on
$\mathscr{L}_{0}$ satisfying the ellipticity assumption from [SS82]), for
notational convenience we will restrict ourselves to functionals coming from a
subriemannian metric $h$, as in (3.9). We record here the estimates of $h$
that we will use—all of them correspond to suitable ellipticity conditions for
the associated integrand:

######  Assumption 4.3.

For some $0<\lambda\leq\Lambda<\infty$ and $0<\mu<\infty$, the metric $h$
satisfies the following in $\mathcal{U}$:

(4.7) $\lambda\leq|\vec{\pi}|_{h}\leq\Lambda\qquad\text{for any
}\vec{\pi}\in\mathscr{L}_{0};$ (4.8)
$\lambda^{4}|\vec{\pi}|^{2}|\vec{\varpi}|^{2}\leq
h(\vec{\pi},\vec{\pi})h(\vec{\varpi},\vec{\varpi})-h(\vec{\pi},\vec{\varpi})^{2}\qquad\text{for
any }\vec{\pi},\vec{\varpi}\in\bigwedge\nolimits^{n}(\Xi_{0})\text{ such that
}g_{\mathbb{H}}(\vec{\pi},\vec{\varpi})=0;$ (4.9)
$\lambda|\vec{\pi}-\vec{\varpi}|^{2}\leq|\vec{\pi}|_{h}-\frac{h(\vec{\pi},\vec{\varpi})}{|\vec{\varpi}|_{h}}\qquad\text{for
any }\vec{\pi},\vec{\varpi}\in\mathscr{L}_{0};$ (4.10)
$|h_{\xi}(\vec{\pi})-h_{\zeta}(\vec{\pi})|\leq\mu\operatorname{dist}_{\mathrm{FK}}(\xi,\zeta)\qquad\text{for
any }\vec{\pi}\in\mathscr{L}_{0}\text{ and any }\xi,\zeta\in\mathcal{U}.$

######  Proposition 4.4.

4.3 is satisfied for any subriemannian metric $h$ which is Lipschitz in a
neighborhood of a relatively compact set $\mathcal{U}$.

###### Proof.

(4.7) and (4.10) are clear with some $0<\lambda_{1}\leq\Lambda<\infty$ and
some $0<\mu<\infty$. For (4.8), note that we may assume that
$|\vec{\pi}|=|\vec{\varpi}|=1$. Then, by Cauchy–Schwarz, the right hand side
of (4.8) is positive for any such $(\vec{\pi},\vec{\varpi})$ and the
inequality follows by compactness after possibly making $\lambda_{1}$ smaller.

To show (4.9), first assume that $|\vec{\pi}-\vec{\varpi}|\leq\delta$ for some
$0<\delta\leq\tfrac{1}{2}$ and write
$\vec{\varpi}=(1-t)\vec{\pi}+\vec{\sigma}$ with
$\vec{\sigma}\perp_{g_{\mathbb{H}}}\vec{\pi}$. It is clear that
$(1-t)^{2}+|\vec{\sigma}|^{2}=1$ and
$t^{2}+|\vec{\sigma}|^{2}\leq\tfrac{1}{4}$, hence $0\leq
t\leq|\vec{\sigma}|^{2}$. Consider the expression for $|\vec{\varpi}|_{h}$ and
expand it to second order in $|\vec{\sigma}|$:

$\displaystyle|\vec{\varpi}|_{h}$
$\displaystyle=\sqrt{(1-t)^{2}|\vec{\pi}|_{h}^{2}+|\vec{\sigma}|_{h}^{2}+2(1-t)h(\vec{\pi},\vec{\sigma})}$
$\displaystyle=\sqrt{|\vec{\pi}|_{h}^{2}-(2t-t^{2})|\vec{\pi}|_{h}^{2}+|\vec{\sigma}|_{h}^{2}+2(1-t)h(\vec{\pi},\vec{\sigma})}$
$\displaystyle\geq|\vec{\pi}|_{h}\sqrt{1+\frac{2h(\vec{\pi},\vec{\sigma})-2t|\vec{\pi}|_{h}^{2}+|\vec{\sigma}|_{h}^{2}}{|\vec{\pi}|_{h}^{2}}-C|\vec{\sigma}|^{3}}$
$\displaystyle\geq|\vec{\pi}|_{h}\left(1-t+\frac{h(\vec{\pi},\vec{\sigma})}{|\vec{\pi}|_{h}^{2}}+\frac{|\vec{\sigma}|_{h}^{2}}{2|\vec{\pi}|_{h}^{2}}-\frac{1}{2}\frac{h(\vec{\pi},\vec{\sigma})^{2}}{|\vec{\pi}|_{h}^{4}}-C|\vec{\sigma}|^{3}\right).$

Now use (4.8) to bound the last two terms from below:

$\displaystyle|\vec{\pi}|_{h}|\vec{\varpi}|_{h}$
$\displaystyle\geq(1-t)|\vec{\pi}|_{h}^{2}+h(\vec{\pi},\vec{\sigma})+\frac{|\vec{\pi}|_{h}^{2}|\vec{\sigma}|_{h}^{2}-h(\vec{\pi},\vec{\sigma})^{2}}{2|\vec{\pi}|_{h}^{2}}-C|\vec{\sigma}|^{3}$
$\displaystyle\geq(1-t)|\vec{\pi}|_{h}^{2}+h(\vec{\pi},\vec{\sigma})+\frac{\lambda_{1}^{4}|\vec{\sigma}|^{2}}{2|\vec{\pi}|_{h}^{2}}-C|\vec{\sigma}|^{3}$
$\displaystyle=h(\vec{\pi},\vec{\varpi})+\frac{\lambda_{1}^{4}|\vec{\sigma}|^{2}}{2|\vec{\pi}|_{h}^{2}}(1-C|\vec{\sigma}|)\geq
h(\vec{\pi},\vec{\varpi})+\frac{\lambda_{1}^{4}|\vec{\sigma}|^{2}}{4|\vec{\pi}|_{h}^{2}}$

provided that we choose $\delta$ small enough. In that case,

$\displaystyle|\vec{\pi}|_{h}-\frac{h(\vec{\pi},\vec{\varpi})}{|\vec{\varpi}|_{h}}\geq\frac{\lambda_{1}^{4}|\vec{\sigma}|^{2}}{4|\vec{\pi}|_{h}^{2}|\vec{\varpi}|_{h}^{2}}\geq\frac{\lambda_{1}^{4}}{4\Lambda^{3}}|\vec{\sigma}|^{2}\geq\frac{\lambda_{1}^{4}}{8\Lambda^{3}}(|\vec{\sigma}|^{2}+t^{2})=\frac{\lambda_{1}^{4}}{8\Lambda^{3}}|\vec{\varpi}-\vec{\pi}|^{2}.$

On the other hand, by compactness, (4.9) holds for
$|\vec{\pi}-\vec{\varpi}|\geq\delta$ with some $\lambda^{\prime}>0$, so we are
done by choosing
$\lambda=\min\left\\{\lambda_{1},\lambda^{\prime},\tfrac{\lambda_{1}^{4}}{8\Lambda^{3}}\right\\}$.
∎

######  Remark 4.5.

In the Riemannian setting, given a point $p\in M$, one can choose a good set
of coordinates (for example normal coordinates centered at $p$) such that the
error term in (4.10) with $\xi=0$ is quadratic in
$\operatorname{dist}(\zeta,0)$. This improvement gives directly $C^{1,\alpha}$
regularity of minimizing currents for any $\alpha<1$. In our case however, the
coordinates are additionally required to respect the contact structure, so
obtaining a quadratic error in (4.10) with such a constraint is impossible
without imposing any compatibility condition444In a symplectic manifold
$(W,\omega)$ with a Riemannian metric $g$, a necessary and sufficient
condition for such coordinates to exist at $p\in M$ is that
$\nabla^{g}\omega(p)=0$, where $\nabla^{g}$ is the Levi-Civita connection.
This condition is weaker than Kähler.. This is the reason for the $C^{1,1/2}$
regularity in 4.7 below. Of course, this is not a serious issue since we can
obtain higher regularity provided that the manifold is regular enough.

We will need the following set of hypotheses to state and prove regularity
around $0\in\mathbb{H}^{n}$:

######  Assumption 4.6.

The current $T\in\mathbf{I}^{\mathrm{hor}}_{n}(\mathbb{H}^{n})$ satisfies:

(4.11) $T\text{ is }\mathbf{M}^{h}\text{-minimizing in }\mathcal{U}\text{ (as
in \lx@cref{creftype~refnum}{defn:minimizing}) and
}\operatorname{dist}_{\mathrm{FK}}(\operatorname{spt}T\cap\mathbf{C}_{R/2}^{\pi},\mathbb{H}^{n}\setminus\mathcal{U})\geq\frac{R}{2}$
(4.12) $\partial
T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathbf{C}_{R}^{\pi}=0$
(4.13)
$\mathbf{p}^{\pi}_{\\#}(T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathbf{C}_{R}^{\pi})=Q\llbracket
B_{R}^{\pi}\rrbracket$ (4.14) $\Theta^{n}(\|T\|,\xi)\geq Q\text{ for
}\|T\|\text{-a.e.~{}}\xi\in\mathbf{C}_{R}^{\pi}$ (4.15)
$\mathbf{E}(T,\mathbf{C}_{R}^{\pi})\leq\varepsilon.$

Note that the density $\Theta^{n}(\|T\|,\xi)$ here can be computed either with
respect to a distance coming from a Riemannian metric, or with respect to the
Carnot–Carathéodory distance; they agree $\|T\|$-almost everywhere thanks to
A.1. Before stating the $\varepsilon$-regularity theorem, we need to introduce
an appropriate notion of graph over a Legendrian plane $\pi_{0}$.

If we require that a graph of the form $\\{(x,g(x),f(x)\\}$ is Legendrian, the
following condition must be satisfied:

$\mathrm{d}f=\frac{1}{2}(\mathbf{x}\cdot\mathrm{d}\mathbf{g}-\mathbf{g}\cdot\mathrm{d}\mathbf{x}).$

By considering instead $\tfrac{1}{2}\mathbf{x}\cdot\mathbf{g}-f$, this becomes

$\mathrm{d}\left(\frac{1}{2}(\mathbf{x}\cdot\mathbf{g})-f\right)=\mathbf{g}\cdot\mathrm{d}\mathbf{x},$

hence the function $v(x)=\tfrac{1}{2}\mathbf{x}\cdot\mathbf{g}-f$ recovers
both $f$ and $g$ by the expressions

$f(x)=\frac{1}{2}(\mathbf{x}\cdot\mathrm{D}v(x))-v(x),\qquad
g(x)=\mathrm{D}v(x).$

Note that, in the symplectic $\mathbb{R}^{2n}$, this corresponds to the fact
that a Lagrangian graph is the graph of the gradient of a certain potential
funtion; in the Legendrian case, in addition, we recover the potential from
the last coordinate.

Motivated by this observation, for a $C^{1}$ function
$v:\Omega\subset\pi_{0}\to\mathbb{R}$, we define the map
$\Phi^{v}:\Omega\to\mathbf{p}^{-1}(\Omega)\subset\mathbb{H}^{n}$ as

(4.16)
$\Phi^{v}(x):=\left(x,\,\mathrm{D}v(x),\,\frac{1}{2}\mathbf{x}\cdot\mathrm{D}v(x)-v(x)\right)$

and note that if $v\in C^{1,1}$ then
$\Phi^{v}_{\\#}\llbracket\Omega\rrbracket$ is a Legendrian current, since

$(\Phi^{v})^{*}\theta=\mathrm{d}\left(\frac{1}{2}\mathbf{x}\cdot\mathrm{D}v-v\right)-\frac{1}{2}\left(\mathbf{x}\cdot\mathrm{d}\mathrm{D}v-\mathrm{D}v\cdot\mathrm{d}\mathbf{x}\right)=\mathrm{D}v\cdot\mathrm{d}\mathbf{x}-\mathrm{d}v=0.$

Graphs of this form constitute intrinsic low-dimensional graphs in the sense
of [FSS07]. The mass of such a graph can be computed as follows: it is
immediate to see that

(4.17)
$\mathrm{D}\Phi^{v}(x)e_{i}=\nabla^{H}x^{i}+\sum_{j=1}^{n}\partial_{ij}v(x)\nabla^{H}y^{i},$

where $e_{1},\ldots,e_{n}$ is the standard basis of $\mathbb{R}^{n}$ and
$\\{\nabla^{H}x^{i},\nabla^{H}y^{i}\\}$ is a $g_{\mathbb{H}}$-orthonormal
basis of $\Xi$ adapted to $\pi_{0}$. Thus the $h$-area of $\Phi^{v}$ over any
measurable set $K\subset\mathbb{R}^{n}$ is

(4.18) $\mathbf{M}^{h}(\Phi^{v}_{\\#}\llbracket
K\rrbracket)=\int_{K}\sqrt{\det\nolimits_{ij}(g^{v})_{ij}}\,\mathrm{d}x.$

where $((g^{v})_{ij})$ are the coefficients of the pullback metric
$g^{v}=(\Phi^{v})^{*}h$. In particular, for the metric $g_{\mathbb{H}}$ we
have

(4.19) $\mathbf{M}(\Phi^{v}_{\\#}\llbracket
K\rrbracket)=\int_{K}\sqrt{\det(\mathrm{id}+\mathrm{D}^{2}v\cdot\mathrm{D}^{2}v)}\,\mathrm{d}x.$

We can now state our main regularity theorem.

######  Theorem 4.7 ($\varepsilon$-regularity).

Let $\mathcal{U}\subset\mathbb{H}^{n}$ be an open set, $h$ a subriemannian
metric on $\mathcal{U}$ satisfying 4.3, and
$T\in\mathbf{I}^{\mathrm{hor}}_{n}(\mathbb{H}^{n})$ with
$0\in\operatorname{spt}T$. Then there exist constants
$\bar{\varepsilon},\bar{C}\in(0,1)$, depending only on $n,\Lambda/\lambda$ and
$Q$ such that, if $T$ satisfies 4.6 with $\varepsilon\leq\bar{\varepsilon}$
and $\pi=\pi_{0}$, and moreover $\mu R\leq\bar{\varepsilon}$, then

(4.20)
$T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathbf{C}_{R/72}=Q\,\Phi^{f}_{\\#}\llbracket
B_{R/72}\rrbracket$

for a function $f\in C^{2,1/2}(B_{R/72},\mathbb{R})$ with

(4.21)
$R^{-2}\sup_{B_{R/72}}|f|+R^{-1}\sup_{B_{R/72}}|\mathrm{D}f|+\sup_{B_{R/72}}|\mathrm{D}^{2}f|+R^{1/2}\left[\mathrm{D}^{2}f\right]_{C^{1/2}(B_{R/72})}\leq\bar{C}\left(\mathbf{E}(T,\mathbf{C}_{R}^{\pi_{0}})+\mu
R\right)^{1/2}.$

In particular,
$T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathbf{C}_{R/72}$ is a
$C^{1,1/2}$ Legendrian graph over $\pi_{0}$.

The Proof of 4.7 will occupy Sections 5 and 6. In the rest of this section we
present the main consequences of 4.7.

###  4.3. Partial regularity

4.7 motivates the following definition: we say that a point
$\xi\in\operatorname{spt}T\setminus\operatorname{spt}\partial T$ is regular if
$\operatorname{spt}T$ is a $C^{1,1/2}$ submanifold in a neighborhood of $\xi$;
otherwise we call it singular. This determines a partition
$\operatorname{spt}T\setminus\operatorname{spt}\partial
T=\operatorname{Reg}T\cup\operatorname{Sing}T$.

For a Radon measure $\nu$ on $\mathbb{H}^{n}$, we define its $n$-dimensional
density with respect to the Carnot–Carathéodory distance as

(4.22) $\Theta^{n}_{d_{\mathrm{CC}}}(\nu,\xi):=\lim_{r\searrow
0}\frac{\nu(\mathscr{B}_{r}(\xi))}{\omega_{n}r^{n}}$

whenever the limit exists. Here
$\mathscr{B}_{r}(\xi):=\\{\zeta\in\mathbb{H}^{n}:d_{\mathrm{CC}}(\zeta,\xi)<r\\}$
denotes a ball with respect to the Carnot–Carathéodory distance. A.1, in
particular Equation (A.3), implies that for a horizontal integral current
$T\in\mathbf{I}^{\mathrm{hor}}_{n}(\mathbb{H}^{n})$ and for $\|T\|$-almost
every $\xi$, $\Theta^{n}_{d_{\mathrm{CC}}}(\|T\|,\xi)$ exists and agrees with
the usual density $\Theta^{n}_{d_{0}}(\|T\|,\xi)$ computed with respect to a
Riemannian distance $d_{0}$. As above, when its precise value does not matter
on sets of $\|T\|$-measure zero, we will not specify with respect to which
distance it is computed.

######  Theorem 4.8.

Let $\mathcal{U}\subset\mathbb{H}^{n}$ be an open set, $h$ a subriemannian
metric on $\mathcal{U}$ satisfying 4.3, and
$T\in\mathbf{I}^{\mathrm{hor}}_{n}(\mathbb{H}^{n})$ a
$\mathbf{M}^{h}$-minimizing current in $\mathcal{U}$. Suppose that a point
$\xi\in\mathcal{U}$ has a neighborhood $\mathcal{V}\subset\mathcal{U}$ such
that $\operatorname{spt}\partial T\cap\mathcal{V}=\varnothing$,
$\Theta^{n}(\|T\|,\zeta)\geq Q$ for $\|T\|$-a.e. $\zeta\in\mathcal{V}$ and

$(\delta_{1/\rho_{j}}\circ\ell_{\xi^{-1}})_{\\#}T\rightharpoonup
Q\llbracket\vec{\pi}_{0}\rrbracket\qquad\text{in
}\mathcal{D}_{n}(\mathbb{H}^{n})\text{ for a sequence }\rho_{j}\searrow 0,$

where $Q=\Theta^{k}_{d_{\mathrm{CC}}}(\|T\|,\xi)$ is a positive integer and
$\vec{\pi}_{0}$ is a Legendrian plane. Then $\xi\in\operatorname{Reg}(T)$.

###### Proof.

Let $\eta_{\xi,\rho_{j}}(\zeta)=\delta_{1/\rho_{j}}(\ell_{\xi^{-1}}(\zeta))$
and consider the currents $T_{j}:=(\eta_{\xi,\rho_{j}})_{\\#}T\rightharpoonup
Q\llbracket\vec{\pi}_{0}\rrbracket$. Elementary scaling considerations show
that these are local $\mathbf{M}^{h_{j}}$-minimizers on
$\mathcal{V}_{j}:=\eta_{\xi,\rho_{j}}(\mathcal{V})$ for the metrics
$h_{j}:=h\circ(\eta_{\xi,\rho_{j}}^{-1})$ (here we are viewing the metrics as
$\mathrm{Sym}^{2}(\Xi_{0}^{*})$-valued functions). Note that being an
$\mathbf{M}^{h}$-minimizer is invariant under multiplication of $h$ by a
constant.

It is clear that the sets $\mathcal{V}_{j}$ exhaust $\mathbb{H}^{n}$, that the
metrics $h_{j}$ converge to $h_{\xi}$ locally uniformly (since $h$ is in fact
Lipschitz) and that, in their domains, the metrics $h_{j}$ satisfy 4.3 with
the same constants $\Lambda,\lambda$ as $h$ and with $\mu_{j}=\mu\rho_{j}\to
0$.

We claim that for well chosen $1<R<2$, $6<r<7$ and $j$ large enough, the
currents
$\tilde{T}_{j}:=T_{j}\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathcal{B}_{r}$
satisfy the assumptions of 4.7 with $\mathcal{U}:=\mathcal{B}_{5}$. Since
$T_{j}$ agrees with $\tilde{T}_{j}$ in a neighborhood of $0$, this will
establish that $0$ is a regular point of $T_{j}$ and hence $\xi$ is a regular
point of $T$.

First of all, the condition
$\mu_{j}R\leq\bar{\varepsilon}=\bar{\varepsilon}(n,\Lambda/\lambda,Q)$ is
clear for $j$ large. Since $T_{j}$ is a sequence of minimizers with uniformly
bounded masses in $\mathcal{B}_{10}$ (thanks to the existence and finiteness
of $\Theta^{k}_{d_{\mathrm{CC}}}(\|T\|,\xi)$), by 3.17 we have that
$\|T_{j}\|\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathcal{B}_{10}\rightharpoonup
Q\mathcal{H}^{n}\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\pi_{0}\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathcal{B}_{10}$
in the sense of measures. In particular, testing with the compact set
$\\{|\mathbf{y}|+|\varphi|\geq 1\\}\cap\overline{\mathcal{B}_{9}}$ we have
that

$\|T_{j}\|\left(\\{|\mathbf{y}|+|\varphi|\geq
1\\}\cap\overline{\mathcal{B}_{9}}\right)\xrightarrow{j\to\infty}0.$

Now applying Lemma 5.1 below we deduce that, for $j$ large,

(4.23) $\operatorname{spt}T_{j}\cap\\{|\mathbf{y}|+|\varphi|\geq
2\\}\cap\overline{\mathcal{B}_{8}}=\varnothing,$

since any point $\zeta$ in the latter set must satisfy
$c\leq\|T_{j}\|(\mathcal{B}_{1/100}(\zeta))\leq\|T_{j}\|(\\{|\mathbf{y}|+|\varphi|\geq
1\\}\cap\overline{\mathcal{B}_{9}})$ for a constant $c>0$ independent of $j$.

We are ready to check the Assumptions 4.6 for $\tilde{T}_{j}$: to show (4.11),
it is clear that $\tilde{T}_{j}$ is a minimizer in $\mathcal{B}_{5}$ since
$\mathcal{B}_{5}\subset\mathcal{B}_{r}$. Moreover, given a point
$\zeta\in\operatorname{spt}\tilde{T}_{j}\cap\mathbf{C}_{R/2}$, since
$\zeta\in\operatorname{spt}T_{j}\cap\overline{\mathcal{B}_{8}}$, by (4.23) we
must have that $|\mathbf{y}|+|\varphi|<2$, in particular
$|\mathbf{x}|,|\mathbf{y}|,|\varphi|\leq 2$ and then a computation shows that
$\|\zeta\|_{\mathrm{FK}}\leq 4<5-\tfrac{R}{2}$, so the additional condition
about the support in (4.11) is satisfied too.

For (4.12), notice that
$\operatorname{spt}(\partial\tilde{T}_{j})\subset\operatorname{spt}T_{j}\cap\partial\mathcal{B}_{r}\subset\operatorname{spt}T_{j}\cap\overline{\mathcal{B}_{8}}$.
Hence any point
$\zeta\in\operatorname{spt}(\partial\tilde{T}_{j})\cap\mathbf{C}_{R}$ must
have, on one hand $|\mathbf{x}|<R<2$, and on the other hand, again by (4.23),
$|\mathbf{y}|,|\varphi|<2$, which again implies that
$\|\zeta\|_{\mathrm{FK}}\leq 4<r$. Thus such a point cannot exist and this
proves (4.12).

So far we have that
$(\partial\tilde{T}_{j})\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathbf{C}_{R}=0$;
this implies that
$\partial(\mathbf{p}_{\\#}\tilde{T}_{j})\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}B_{R}=\mathbf{p}_{\\#}(\partial\tilde{T}_{j}\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathbf{C}_{R})=0$
and therefore, by the constancy theorem, we can write
$\mathbf{p}_{\\#}(\tilde{T}_{j}\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathbf{C}_{R})=(\mathbf{p}_{\\#}\tilde{T}_{j})\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}B_{R}=Q_{j}\llbracket
B_{R}\rrbracket$ for a sequence of integers $Q_{j}$. On the other hand, if we
choose an appropriate $r$, we have that

(4.24)
$\tilde{T}_{j}=T_{j}\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathcal{B}_{r}\rightharpoonup
Q\llbracket\vec{\pi}_{0}\rrbracket\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathcal{B}_{r}=Q\llbracket
B_{r}\rrbracket$

and, if in addition we choose a suitable $R$, it also holds that
$\tilde{T}_{j}\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathbf{C}_{R}\rightharpoonup
Q\llbracket
B_{r}\rrbracket\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathbf{C}_{R}=Q\llbracket
B_{R}\rrbracket$. Then $Q_{j}\llbracket
B_{R}\rrbracket=\mathbf{p}_{\\#}(\tilde{T}_{j}\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathbf{C}_{R})\rightharpoonup
Q\llbracket B_{R}\rrbracket$, which implies that eventually $Q_{j}=Q$ and
(4.13) holds.

Trivially (4.14) is inherited from the assumption for $T$ on $\mathcal{V}$.
Finally by (4.24) and 3.17 we have that
$\mathbf{M}(\tilde{T}_{j}\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathbf{C}_{R})\to\mathbf{M}(Q\llbracket
B_{r}\rrbracket\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathbf{C}_{R})=Q\omega_{n}R^{n}$
and therefore

$\mathbf{E}(\tilde{T}_{j},\mathbf{C}_{R})=R^{-n}\left(\mathbf{M}(\tilde{T}_{j},\mathbf{C}_{R})-Q\omega_{n}R^{n}\right)\leq\bar{\varepsilon}$

for $j$ large, which is (4.15). ∎

######  Corollary 4.9.

Let $\mathcal{U},h$ and $T$ be as in 4.8. Then $\operatorname{Reg}(T)$ is
dense in $\operatorname{spt}T\setminus\operatorname{spt}\partial
T\cap\mathcal{U}$.

###### Proof.

Let $\mathcal{V}\subset\mathcal{U}\setminus\operatorname{spt}\partial T$ be
any open set with $\operatorname{spt}T\cap\mathcal{V}\neq\varnothing$, and let
$Q\geq 1$ be the essential minimum of $\Theta^{n}(\|T\|,\cdot)$ on
$\operatorname{spt}T\cap\mathcal{V}$, that is, the greatest positive integer
$Q$ such that $\Theta^{n}(\|T\|,\xi)\geq Q$ for $\|T\|$-almost every $\xi$ in
$\mathcal{V}$. It is clear that the set of
$\xi\in\mathcal{V}\cap\operatorname{spt}T$ such that $\Theta^{n}(\|T\|,\xi)=Q$
has positive $\|T\|$-measure, and hence by (A.3) and A.3 in the Appendix we
may find some such $\xi$ with

$(\delta_{1/\rho}\circ\ell_{\xi^{-1}})_{\\#}T\rightharpoonup
Q\llbracket\vec{T}(\xi)\rrbracket\qquad\text{in
}\mathcal{D}_{n}(\mathbb{H}^{n})\text{ as }\rho\searrow 0$

and $Q=\Theta^{n}_{d_{\mathrm{CC}}}(\|T\|,\xi)$. Thus the hypothesis of 4.8
are satisfied and therefore $\xi\in\operatorname{Reg}(T)$. This shows that any
nonempty relatively open subset of
$\operatorname{spt}T\cap\mathcal{U}\setminus\operatorname{spt}\partial T$
intersects $\operatorname{Reg}T$, i.e. $\operatorname{Reg}T$ is dense in
$\operatorname{spt}T\cap\mathcal{U}\setminus\operatorname{spt}\partial T$. ∎

In particular, we get almost everywhere regularity under the additional (very
strong) assumption of multiplicity one. This includes nevertheless all
Legendrian currents which are graphs of gradients of $C^{1,1}$ functions,
without any a priori bound on their $W^{2,\infty}$ norm, and even of $W^{2,n}$
functions. This answers partially a question from [BS23]. Our argument however
assumes that the current is minimizing with respect to competitors that are
not necessarily graphical, so we do not see how to use our results to show
almost everywhere regularity of $W^{2,n}$ minimizing solutions of the
Hamiltonian stationary equation.

######  Corollary 4.10.

Let $\mathcal{U},h$ and $T$ be as in 4.8. If in addition,
$\Theta^{n}(\|T\|,\cdot)=1$ $\|T\|$-almost everywhere, then
$\mathcal{H}^{n}(\operatorname{Sing}T)=0$.

###### Proof.

This follows, as in 4.9, by applying 4.8 in conjunction with A.3 with $Q=1$,
but now we can do this at $\|T\|$-almost every point. ∎

###  4.4. Higher regularity

Higher regularity around points where $\operatorname{spt}T$ is $C^{1,1/2}$
follows from the work of [BCW23]. Here we assume that the ambient metric is
smooth, as they do; intermediate results for $C^{k,\alpha}$ metrics should
follow by analyzing their proofs.

######  Theorem 4.11.

Suppose that $\mathcal{U}$, $T$ and $h$ are as in 4.7, and moreover $h$ is
smooth. Then for any regular point $\xi\in\operatorname{Reg}(T)$,
$\operatorname{spt}T$ is in fact a smooth Legendrian submanifold in a
neighborhood of $\xi$.

###### Proof.

Let $\xi\in\operatorname{Reg}(T)$, so that $T=Q\Phi^{f}_{\\#}\llbracket
B_{r}^{\pi_{0}}\rrbracket$ for some $f:C^{2,1/2}(B_{r},\mathbb{R})$ on a
Legendrian plane $\pi_{0}$. It is clear that for any test function $\psi\in
C^{\infty}_{c}(B_{r})$ and $t\in\mathbb{R}$ close enough to zero, the currents
$Q\Phi^{f+t\psi}_{\\#}\llbracket B_{r}\rrbracket$ are admissible competitors
for the $\mathbf{M}^{h}$-minimality of $T$. Therefore by (4.18)

$\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0}\mathbf{M}^{h}\left(\Phi^{f+t\psi}_{\\#}\llbracket
B_{r}\rrbracket\right)=\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0}\int_{B_{r}}F(x,f,\mathrm{D}f,\mathrm{D}^{2}f)\,\mathrm{d}x=0$

where we have set
$F(x,f,\mathrm{D}f,\mathrm{D}^{2}f):=\sqrt{\det\nolimits_{ij}\left(g^{f+t\psi})_{ij}\right)}\,\mathrm{d}x=0$
with $g^{v}:=(\Phi^{v})^{*}h$. This kind of functionals satisfy the
requirements555 The authors show this precisely in [BCW23, Section 3] assuming
that $h$ does not depend on $f$ and that $h$ is compatible with the symplectic
structure. However, their proof of higher regularity, based on difference
quotients, is very robust and works as well in this setting. We remark that,
in our main case of interest, when the contact manifold is a fibration over a
sympletic manifold with a compatible metric, their computations apply
directly.  of the higher regularity theory developed in [BCW23] and in
particular their Theorem 2.3 implies that $f$ is smooth. ∎

######  Corollary 4.12.

If $T\in\mathbf{I}^{\mathrm{hor}}_{n}(\mathbb{H}^{n})$ is a Legendrian local
mass minimizer for the standard metric, then $\operatorname{Reg}(T)$ is a
real-analytic Legendrian submanifold.

###### Proof.

This follows from the work of Morrey [Mor58] as in [CW19]. ∎

##  5\. Height bound and Lipschitz approximation

In this section we make the first steps towards the proof of 4.7 following
[SS82]. We will work with a fixed Legendrian plane
$\pi=\pi_{0}=\operatorname{span}\\{\partial_{x^{1}},\ldots,\partial_{x^{n}}\\}$.
Here we are only assuming that $h$ satisfies (4.7) from the assumptions.

###  5.1. Area control and oscillation bounds

The following lower bound on the mass of a minimizing current is well known in
spaces which enjoy isoperimetric inequalities. However here we will need to
adapt the proof to be able to control the support of the fillings.

######  Lemma 5.1 (density lower bound).

Let $T\in\mathbf{I}^{\mathrm{hor}}_{n}(\mathbb{H}^{n})$ be
$\mathbf{M}^{h}$-minimizing in $\mathcal{U}$, and let $\xi\in\mathcal{U}$ and
$r>0$ be such that $\mathcal{B}_{r}(\xi)\subset\mathcal{U}$ and $(\partial
T)\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathcal{B}_{r}(\xi)=0$.
Then

(5.1) $\|T\|(\mathcal{B}_{r}(\xi))\geq cr^{n}$

for a constant $c=c(n,\Lambda/\lambda)>0$.

###### Proof.

We may assume that $\xi=0$ and let $\tau(\zeta)=\|\zeta\|_{\mathrm{FK}}$. As
in the proof of Lemma 3.11, let

$G:=\left\\{0<s<\frac{r}{2}:\langle T,\tau,s\rangle\text{ exists and
}\mathbf{M}(\langle T,\tau,s\rangle)\leq\frac{4m}{r}\right\\}.$

where $m:=\|T\|(\mathcal{B}_{r/2})$. We may suppose that

(5.2) $2C_{1}^{n-1}m<\left(\frac{r}{2}\right)^{n},$

where $C_{1}$ denotes the constant from the isoperimetric inequality (A.5),
otherwise we are done. By the coarea formula (3.7),

$\mathcal{L}^{2}([0,\tfrac{r}{2}]\setminus
G)\leq\frac{r}{4m}\int_{0}^{\frac{r}{2}}\mathbf{M}(\langle
T,\tau,s\rangle)\,\mathrm{d}s\leq\frac{r}{4m}\|T\|(\mathcal{B}_{r/2})=\frac{1}{4}r,$

hence $\mathcal{L}^{1}(G)\geq\tfrac{1}{4}r$. For each $s\in G$ we may apply
A.5 and obtain a current $Q_{s}$ with

$\partial Q_{s}=\langle T,\tau,s\rangle,\qquad\mathbf{M}(Q_{s})\leq
C_{1}\mathbf{M}(\langle T,\tau,s\rangle)^{\frac{n}{n-1}}$

and additionally
$\operatorname{spt}Q_{s}\subset\mathcal{B}_{r}\subset\mathcal{U}$ thanks to
(5.2). Now for
$T_{s}=Q_{s}+T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathcal{B}_{s}^{c}$
we have that

$\partial T_{s}=\langle
T,\tau,s\rangle+\partial(T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathcal{B}_{s}^{c})=\partial(T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathcal{B}_{s})-(\partial
T)\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathcal{B}_{s}+\partial(T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathcal{B}_{s}^{c}),=\partial
T$

so $T_{s}$ is a valid competitor for $\mathbf{M}^{h}$ and we get

$\|T\|(\mathcal{B}_{s})\leq\lambda^{-1}\mathbf{M}^{h}(T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathcal{B}_{s})\leq\lambda^{-1}\mathbf{M}^{h}(Q_{s})\leq\frac{\Lambda}{\lambda}\mathbf{M}(Q_{s})\leq\frac{\Lambda}{\lambda}C_{1}\mathbf{M}(\langle
T,\tau,s\rangle)^{\frac{n}{n-1}}.$

Letting $g(s):=\|T\|(\mathcal{B}_{s})$ and using again the coarea formula, we
see that

$g(s)^{\frac{n-1}{n}}\leq Cg^{\prime}(s).$

Thus, since $g(s)>0$ for every $s>0$, we have that
$g^{\frac{1}{n}}(s)^{\prime}=\frac{1}{n}g^{\prime}(s)g(s)^{\frac{1-n}{n}}\geq
c$ for every $s\in G$. Thanks to the monotonicity of $g(s)^{1/n}$ we may
integrate and get

$g^{\frac{1}{n}}(r)\geq
g^{\frac{1}{n}}\left(\tfrac{r}{2}\right)\geq\int_{G}\left(g^{\frac{1}{n}}(s)\right)^{\prime}\,\mathrm{d}s\geq
c\mathcal{L}^{1}(G)\geq cr.\qed$

As a consequence we obtain a bound on the vertical oscillation of the current,
as in [Fed69, Lemma 5.3.4] or in the Appendix of [SS82]. The bound for the
$\mathbf{q}$ components is standard, but since the proof of the bound for
$\varphi$ uses the same strategy, we have decided to present them together for
completeness.

######  Lemma 5.2 (height bounds).

There are constants $\varepsilon_{0},C>0$ depending only on
$n,\Lambda/\lambda$ and $Q$ such that, if $T$ satisfies 4.6 at scale $R=1$
with $\varepsilon\leq\varepsilon_{0}$, then

(5.3)
$\sup_{\xi_{1},\xi_{2}\in\operatorname{spt}T\cap\mathbf{C}_{1/2}}|\mathbf{q}(\xi_{1})-\mathbf{q}(\xi_{2})|\leq
C\mathbf{E}(T,\mathbf{C}_{1})^{\frac{1}{2n}}.$

Moreover, if $\mathbf{q}(\xi_{0})=0$ for some
$\xi_{0}\in\operatorname{spt}T\cap\mathbf{C}_{1/2}$, then also

(5.4)
$\sup_{\xi_{1},\xi_{2}\in\operatorname{spt}T\cap\mathbf{C}_{1/4}}|\varphi(\xi_{1})-\varphi(\xi_{2})|\leq
C\mathbf{E}(T,\mathbf{C}_{1})^{\frac{1}{n(n+1)}}.$

###### Proof.

Denote $E:=\mathbf{E}(T,\mathbf{C}_{1})$ and choose
$\varepsilon_{0}\leq\tfrac{1}{2}Q\omega_{n}$ so that

(5.5)
$\|T\|(\mathbf{C}_{1})=\mathbf{M}(\mathbf{p}_{\\#}(T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathbf{C}_{1}))+E\leq\frac{3}{2}Q\omega_{n}.$

Let $f:\mathbb{H}^{n}\to\mathbb{R}$ be a smooth function and for $-\infty\leq
t_{1}\leq t_{2}\leq+\infty$ define

$W(t_{1},t_{2}):=\\{\xi\in\mathbf{C}_{1}:t_{1}<f(\xi)<t_{2}\\}$

and $T_{t}:=T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}W(t,\infty)$ for
$t\in\mathbb{R}$. Let $t_{0}$ be a median of $f$, in the sense that

$\max\\{\|T\|(W(-\infty,t_{0})),\|T\|(W(t_{0},\infty))\\}\leq\frac{1}{2}\|T\|(\mathbf{C}_{1}).$

We proceed in three steps.

  1. Step 1.

We show that whenever $t_{1}>t_{0}$ is such that $\mathbf{M}(T_{t_{1}})\geq
2E$,

(5.6) $(t_{1}-t_{0})\mathbf{M}(T_{t_{1}})^{\frac{n-1}{n}}\leq
C\int_{W(t_{0},t_{1})}\left|\nabla^{\vec{T}}f\right|\,\mathrm{d}\|T\|$

for a constant $C=C(n,\Lambda/\lambda,Q)>0$. First observe that
$\mathbf{M}(T_{t_{0}})\leq\tfrac{3}{4}Q\omega_{n}$. By the standard coarea
formula and (4.3) we have

$\displaystyle\mathcal{L}^{n}\\{x\in
B_{1}:\Theta^{n}(\|\mathbf{p}_{\\#}T_{t}\|,x)>0\\}$
$\displaystyle\leq\mathcal{H}^{n}\\{\xi\in\mathbf{C}_{1}:\Theta^{n}(\|T_{t}\|,\xi)>0\\}$
$\displaystyle=\mathcal{H}^{n}\\{\xi\in\mathbf{C}_{1}:\Theta^{n}(\|T_{t}\|,\xi)\geq
Q\\}\leq\frac{1}{Q}\mathbf{M}(T_{t})\leq\frac{1}{Q}\mathbf{M}(T_{t_{0}})\leq\frac{3}{4}\omega_{n}.$

Notice that here is the only place where we use the assumption (4.14). Thus,
thanks to the fact that $\mathcal{L}^{n}\\{x\in
B_{1}:\Theta^{n}(\|\mathbf{p}_{\\#}T_{t}\|,x)=0\\}\geq\tfrac{1}{4}\omega_{n}$,
we may apply the Poincaré–Sobolev inequality (see [Fed69, 5.3.2]) to the
integer-valued BV functions corresponding to (the signed multiplicity of)
$\mathbf{p}_{\\#}T_{t}$:

$\|\mathbf{p}_{\\#}T_{t}\|(B_{1})^{\frac{n-1}{n}}\leq
C\|\partial\mathbf{p}_{\\#}T_{t}\|(B_{1})=C\|\mathbf{p}_{\\#}\partial
T_{t}\|(B_{1})=C\|\mathbf{p}_{\\#}\partial T_{t}\|(B_{1}).$

Now for $\mathcal{L}^{1}$-almost every $t>t_{0}$ we may apply (4.3) and use
slicing to deduce

$\|\mathbf{p}_{\\#}T_{t}\|(B_{1})^{\frac{n-1}{n}}\leq
C\|\mathbf{p}_{\\#}\partial T_{t}\|(B_{1})\leq C\|\partial
T_{t}\|(\mathbf{C}_{1})=C\|\langle T,f,t\rangle\|(\mathbf{C}_{1}).$

On the other hand, by 4.1,

$\|T_{t}\|(\mathbf{C}_{1})\leq\|\mathbf{p}_{\\#}T_{t}\|(B_{1})+\frac{1}{2}\int_{\mathbf{C}_{1}}|\vec{T}-\vec{\pi}_{0}|^{2}\,\mathrm{d}\|T_{t}\|\leq\|\mathbf{p}_{\\#}T_{t}\|(B_{1})+E.$

Assuming also that $t<t_{1}$, we have that
$\|T_{t}\|(\mathbf{C}_{1})\geq\|T_{t_{1}}\|(\mathbf{C}_{1})\geq 2E$, thus
$E\leq\|\mathbf{p}_{\\#}T_{t}\|(B_{1})$ and we get

$\|T_{t}\|(\mathbf{C}_{1})^{\frac{n-1}{n}}\leq\left(2\|\mathbf{p}_{\\#}T_{t}\|(B_{1})\right)^{\frac{n-1}{n}}\leq
C\|\langle T,f,t\rangle\|(\mathbf{C}_{1}).$

Integrating gives finally the desired estimate:

$(t_{1}-t_{0})\|T_{t_{1}}\|(\mathbf{C}_{1})^{\frac{n-1}{n}}\leq
C\int_{t_{0}}^{t_{1}}\|\langle
T,f,t\rangle\|(\mathbf{C}_{1})\,\mathrm{d}t=C\int_{W(t_{0},t_{1})}\left|\nabla^{\vec{T}}f(\xi)\right|\,\mathrm{d}\|T\|(\xi).$

  2. Step 2.

We now show the classical height bound (5.3). Choose
$f(\mathbf{x},\mathbf{y},\varphi)=y^{k}$ for a fixed $k\in\\{1,\ldots,n\\}$.
We claim that there exists $\overline{t}\geq t_{0}$ such that
$\|T\|(W(\overline{t},\infty))\leq\sqrt{E}$ and $\overline{t}-t_{0}\leq
CE^{\frac{1}{2n}}$. If $\|T\|(W(t_{0},\infty))\leq\sqrt{E}$ we may just choose
$\overline{t}=t_{0}$, otherwise define $\overline{t}$ to be the supremum of
all $t_{1}>t_{0}$ such that $\|T\|(W(t_{1},\infty))\geq\sqrt{E}$. Then it is
easy to see that $\|T\|(W(\overline{t},\infty))\leq\sqrt{E}$. To get the bound
on $\overline{t}-t_{0}$, we apply (5.6) for all such $t_{1}$ and pass to the
limit $t_{1}\nearrow\overline{t}$:

(5.7) $(\overline{t}-t_{0})E^{\frac{n-1}{2n}}\leq
C\int_{W(t_{0},\overline{t})}\left|\nabla^{\vec{T}}y^{k}\right|\,\mathrm{d}\|T\|$

(note that this is justified since $\varepsilon_{0}\leq\tfrac{1}{4}$
guarantees that $\sqrt{E}\geq 2E$). We now claim that

(5.8) $\left|\nabla^{\vec{T}}y^{k}\right|^{2}\leq
1-|\langle\vec{T},\vec{\pi}_{0}\rangle|^{2}.$

Recall that $\vec{\pi}_{0}=\nabla^{H}x^{1}\wedge\cdots\wedge\nabla^{H}x^{n}$
and $\\{\nabla^{H}x^{i},\nabla^{H}y^{i}\\}$ are a global orthonormal frame of
$\Xi$. Thus

$1=|\vec{T}|^{2}\geq|\langle\vec{T},\nabla^{H}x^{1}\wedge\cdots\wedge\nabla^{H}x^{n}\rangle|^{2}+\sum_{\mathbf{e}}\left|\left\langle\vec{T},\nabla^{H}y^{k}\wedge\mathbf{e}\right\rangle\right|^{2}$

where the sum is over the orthonormal basis of $\bigwedge^{n-1}(\Xi)$ induced
by our orthonormal frame. On the other hand,

$|\nabla^{\vec{T}}y^{k}|^{2}=|\vec{T}\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathrm{d}y^{k}|^{2}=\sum_{\mathbf{e}}|\langle\vec{T}\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathrm{d}y^{k},\mathbf{e}\rangle|^{2}=\sum_{\mathbf{e}}|\langle\vec{T},\nabla^{H}y^{k}\wedge\mathbf{e}\rangle|^{2}$

and (5.8) follows. Putting this together with the standard inequality
$1-\langle\vec{T},\vec{\pi}_{0}\rangle^{2}\leq|\vec{T}-\vec{\pi}_{0}|^{2}$ we
have

$\left|\nabla^{\vec{T}}y^{k}\right|^{2}\leq|\vec{T}-\vec{\pi}_{0}|^{2}.$

Using this after applying Cauchy–Schwarz and the bound (5.5) on (5.7), we get

$(\overline{t}-t_{0})E^{\frac{1}{2}-\frac{1}{2n}}\leq
C\left(\int_{W(t_{0},\overline{t})}\left|\nabla^{\vec{T}}y^{k}\right|^{2}\,\mathrm{d}\|T\|\right)^{1/2}\leq
CE^{\frac{1}{2}},$

from which the estimate $\overline{t}-t_{0}\leq CE^{\frac{1}{2n}}$ follows.
Finally let $\xi\in\operatorname{spt}T\cap\mathbf{C}_{1/2}$, suppose that
$y^{k}(\xi)>\overline{t}$ and let
$r=\min\left\\{\tfrac{1}{2},y^{k}(\xi)-\overline{t}\right\\}$. Since
$\mathcal{B}_{r}(\xi)\subset W(\overline{t},\infty)$, the lower bound (5.1)
shows that

$cr^{n}\leq\|T\|(W(\overline{t},\infty))\leq\sqrt{E}\leq\sqrt{\varepsilon_{0}}.$

By choosing $\varepsilon_{0}$ small enough we can guarantee that
$r<\tfrac{1}{2}$; in particular, $r=y^{k}(\xi)-\overline{t}$ and it follows
that $y^{k}(\xi)\leq\overline{t}+CE^{\frac{1}{2n}}\leq
t_{0}+CE^{\frac{1}{2n}}$. Repeating the same argument for the level sets of
$f$ below $t_{0}$ the oscillation bound for $y^{k}$ follows.

  3. Step 3.

We finally show the bound for $\varphi$. In order to use the bound from Step 2
we will have to apply Step 1 in the smaller cylinder $\mathbf{C}_{1/2}$; we
can do this thanks to the scale invariance of the hypotheses (see 5.3 below).
In this case we choose $f(\mathbf{x},\mathbf{y},\varphi):=\varphi$ and,
arguing as before, we will find $\overline{t}\geq t_{0}$ such that

$\overline{t}-t_{0}\leq
CE^{\frac{1}{n(n+1)}}\qquad\text{and}\qquad\|T\|(\mathbf{C}_{1/2}\cap\\{\varphi>\overline{t}\\})\leq
E^{\frac{1}{2(n+1)}}.$

To see this, we argue as above and define $\overline{t}$ as the supremum of
those $t_{1}$ for which the second inequality does not hold. Then using (5.6)
(after making $\varepsilon_{0}$ small enough) and passing to the limit
$t_{1}\nearrow\overline{t}$ we get

$(\overline{t}-t_{0})E^{\frac{n-1}{2n(n+1)}}\leq
C\int_{\mathbf{C}_{1/2}\cap\\{t_{0}<\varphi<\overline{t}\\}}\left|\nabla^{\vec{T}}\varphi\right|\,\mathrm{d}\|T\|.$

Since $T$ is horizontal, $\|T\|$-almost everywhere it holds that

$\nabla^{\vec{T}}\varphi=\frac{1}{2}\left(\mathbf{x}\cdot\nabla^{\vec{T}}\mathbf{y}-\mathbf{y}\cdot\nabla^{\vec{T}}\mathbf{x}\right),$

hence
$|\nabla^{\vec{T}}\varphi|\leq\frac{1}{2}\sum_{k}|x^{k}||\nabla^{\vec{T}}y^{k}|+|y^{k}||\nabla^{\vec{T}}x^{k}|$.
Now (5.3) together with the additional hypothesis on $\operatorname{spt}T$
implies that $|y^{k}|\leq CE^{\frac{1}{2n}}$ for every point in
$\operatorname{spt}T\cap\mathbf{C}_{1/2}$, and moreover we have the trivial
bounds $|x^{k}|\leq 1$ and $|\nabla^{\vec{T}}x^{k}|\leq 1$. Therefore

$\displaystyle(\overline{t}-t_{0})E^{\frac{n-1}{2n(n+1)}}$ $\displaystyle\leq
C\int_{\mathbf{C}_{1/2}\cap\\{t_{0}<\varphi<\overline{t}\\}}\left(E^{\frac{1}{2n}}+\sum_{k}|\nabla^{\vec{T}}y^{k}|\right)\,\mathrm{d}\|T\|$
$\displaystyle\leq
CE^{\frac{1}{2n}}+C\sum_{k}\left(\int_{\mathbf{C}_{1/2}}|\nabla^{\vec{T}}y^{k}|^{2}\,\mathrm{d}\|T\|\right)^{1/2}$
$\displaystyle\leq CE^{\frac{1}{2n}}+CE^{\frac{1}{2}}$ $\displaystyle\leq
CE^{\frac{1}{2n}}$

and the bound $\overline{t}-t_{0}\leq CE^{\frac{1}{n(n+1)}}$ follows. Finally
consider a point
$\xi_{0}\in\operatorname{spt}T\cap\mathbf{C}_{1/4}\cap\\{\varphi>\overline{t}\\}$
with coordinates $(\mathbf{x}_{0},\mathbf{y}_{0},\varphi_{0})$ and let
$0<r\leq\tfrac{1}{4}$. Given a point
$\xi=(\mathbf{x},\mathbf{y},\varphi)\in\operatorname{spt}T\cap\mathcal{B}_{r}(\xi_{0})$,
we have that

$\xi_{0}^{-1}\xi=(\mathbf{x}^{\prime},\mathbf{y}^{\prime},\varphi^{\prime})=\left(\mathbf{x}-\mathbf{x}_{0},\mathbf{y}-\mathbf{y}_{0},\varphi-\varphi_{0}+\frac{1}{2}(-\mathbf{x}_{0}\cdot\mathbf{y}+\mathbf{y}_{0}\cdot\mathbf{x})\right)\in\mathcal{B}_{r}(0),$

so it is clear that $|\mathbf{x}|\leq\tfrac{1}{2}$ and, again thanks to (5.3),

$\varphi>\varphi_{0}-\frac{1}{2}(-\mathbf{x}_{0}\cdot\mathbf{y}+\mathbf{y}_{0}\cdot\mathbf{x})-\frac{r^{2}}{4}\geq\varphi_{0}-\frac{1}{2}(|\mathbf{x}_{0}||\mathbf{y}|+|\mathbf{y}_{0}||\mathbf{x}|)-\frac{r^{2}}{4}\geq\varphi_{0}-\overline{C}E^{\frac{1}{2n}}-\frac{r^{2}}{4}\geq\overline{t}$

whenever $0<r\leq r_{0}$, where $r_{0}$ is the solution of
$\tfrac{1}{4}r_{0}^{2}+\overline{C}E^{\frac{1}{2n}}=\varphi_{0}-\overline{t}$.
Arguing as in the end of Step 2, let
$r=\min\left\\{\tfrac{1}{4},r_{0}\right\\}$ and apply (5.1) to
$\mathcal{B}_{r}(\xi_{0})$, noting that
$\mathcal{B}_{r}(\xi_{0})\cap\operatorname{spt}T\subset\mathbf{C}_{1/2}\cap\\{\varphi>\overline{t}\\}$:

$cr^{n}\leq\|T\|(\mathcal{B}_{r}(\xi_{0}))=\|T\|(\mathcal{B}_{r}(\xi_{0})\cap\operatorname{spt}T)\leq\|T\|(\mathbf{C}_{1/2}\cap\\{\varphi>\overline{t}\\})\leq
CE^{\frac{1}{2(n+1)}}.$

After making $\varepsilon_{0}$ smaller if necessary, it follows that
$r<\tfrac{1}{4}$, so $r_{0}=r\leq CE^{\frac{1}{2n(n+1)}}$ and finally

$\varphi_{0}-t_{0}\leq(\varphi_{0}-\overline{t})+(\overline{t}-t_{0})\leq\frac{1}{4}r_{0}^{2}+\overline{C}E^{\frac{1}{2n}}+CE^{\frac{1}{n(n+1)}}\leq
CE^{\frac{1}{n(n+1)}}$

The oscillation bound now follows by arguing analogously below $t_{0}$. ∎

By using the scaling automorphisms of $\mathbb{H}^{n}$ we get with little
effort the following scaled version of the height estimates:

######  Corollary 5.3.

For the same constants $\varepsilon_{0},C>0$ as in Lemma 5.2 the following
holds: suppose that $T$ satisfies 4.6 for some $R>0$ and some
$\varepsilon\leq\varepsilon_{0}$, and that
$\\{\mathbf{q}=0\\}\cap\operatorname{spt}T\cap\mathbf{C}_{R/2}\neq\varnothing$.
Then

(5.9)
$\sup_{\xi_{1},\xi_{2}\in\operatorname{spt}T\cap\mathbf{C}_{R/2}}|\mathbf{q}(\xi_{1})-\mathbf{q}(\xi_{2})|\leq
C\mathbf{E}(T,\mathbf{C}_{R})^{\frac{1}{2n}}R$

and

(5.10)
$\sup_{\xi_{1},\xi_{2}\in\operatorname{spt}T\cap\mathbf{C}_{R/4}}|\varphi(\xi_{1})-\varphi(\xi_{2})|\leq
C\mathbf{E}(T,\mathbf{C}_{R})^{\frac{1}{n(n+1)}}R^{2}.$

###### Proof.

Let $\delta_{R}:\mathbb{H}^{n}\to\mathbb{H}^{n}$ be the automorphism
$\delta_{R}(\mathbf{z},\varphi)=(R\mathbf{z},R^{2}\varphi)$. If $T$ satisfies
4.6 for some $R>0$, then $\tilde{T}:=(\delta_{R^{-1}})_{\\#}T$ is also in
$\mathbf{I}^{\mathrm{hor}}_{n}(\mathbb{H}^{n})$ and satisfies the same
assumptions with $R=1$, except possibly for (4.11). This is clear for (4.12),
(4.13) and for (4.14) (since $\delta_{R}$ is a diffeomorphism). To get (4.15)
we use the definition

$\mathbf{E}((\delta_{R^{-1}})_{\\#}T,\mathbf{C}_{1})=\|(\delta_{R^{-1}})_{\\#}T\|(\mathbf{C}_{1})-Q\omega_{n}=R^{-n}\|T\|(\mathbf{C}_{R})-Q\omega_{n}=\mathbf{E}(T,\mathbf{C}_{R});$

here we have made use of the formula for the push-forward of currents,

$\|(\delta_{R^{-1}})_{\\#}T\|(\mathbf{C}_{1})=\int_{\mathbf{C}_{R}}\left|\left.{\wedge}^{n}(\delta_{R^{-1}})_{\\#}\right|_{\xi}(\vec{T})\right|\,\mathrm{d}\|T\|(\xi)=R^{-n}\int_{\mathbf{C}_{R}}\,\mathrm{d}\|T\|=R^{-n}\|T\|(\mathbf{C}_{R}),$

since the tangent map $(\delta_{R^{-1}})_{\\#}$ restricts to
$\Xi_{\xi}\longrightarrow\Xi_{R^{-1}\xi}$ as multiplication by $R^{-1}$ (under
the identification of all tangent planes to $\mathbb{H}^{n}$ via left
translations).

The scaled back version of (4.11) asserts that $\tilde{T}$ is
$\mathbf{M}^{\tilde{h}}$-minimizing in
$\tilde{\mathcal{U}}:=\delta_{R^{-1}}(\mathcal{U})$ for the metric
$\tilde{h}:=\delta_{R}^{*}h$. This follows from a similar computation: Since
the quotient $\Lambda/\lambda$ is unaltered for $\tilde{h}$, we may apply the
estimates (5.3) and (5.4) to $\tilde{T}$. Then (5.9) and (5.10) follow by
simply scaling back. ∎

###  5.2. Approximation by the graph of a Lipschitz gradient

In order to embed most of the support of our minimizing current into such a
graph, we will need an appropriate version of the Whitney–Glaeser $C^{1,1}$
extension theorem:

######  Theorem 5.4.

Let $G\subset\overline{B_{R}}\subset\mathbb{R}^{n}$ be any set,
$L_{0},L_{1},L_{2}>0$ given constants and $f:G\to\mathbb{R}$,
$\mathbf{g}:G\to\mathbb{R}^{n}$ given functions with $|f|\leq L_{0}$ and
$|\mathbf{g}|\leq L_{1}$ on $G$. Suppose that the following Whitney conditions
are satisfied: for any $x,y\in G$,

(5.11) $|\mathbf{g}(x)-\mathbf{g}(y)|\leq
L_{2}|x-y|\quad\text{and}\quad|f(x)-f(y)-\mathbf{g}(y)\cdot(\mathbf{x}-\mathbf{y})|\leq
L_{2}|x-y|^{2}.$

Then there exists a function $\tilde{f}\in C^{1,1}(\overline{B_{R}})$ such
that $\tilde{f}=f$ and $\mathrm{D}\tilde{f}=\mathbf{g}$ on $G$, and which
satisfies the following bounds:

(5.12) $\|\tilde{f}\|_{L^{\infty}}\leq
CL_{0},\qquad\|\mathrm{D}\tilde{f}\|_{L^{\infty}}\leq
CL_{0}+CRL_{2},\qquad\|\mathrm{D}^{2}\tilde{f}\|_{L^{\infty}}\leq CL_{2}$

for a constant $C=C(n)>0$.

###### Proof.

This follows from the proof of Theorem VI.2.4 from [Ste70]. Notice that, in
the definition of the extension through a sum on Whitney cubes, we do not need
to restrict ourselves to cubes near $G$ since we are working on a bounded
domain. Moreover, we may assume that $G$ is closed if we extend $f$ and
$\mathbf{g}$ by continuity. The more precise, scale-invariant estimates given
here follow by closely examining the proof. ∎

Now the approximation of $T$ by a Lipschitz graph follows the strategy of
[SS82].

######  Lemma 5.5 (Lipschitz approximation).

Suppose that a current $T\in\mathbf{I}^{\mathrm{hor}}_{n}(\mathbb{H}^{n})$
satisfies 4.6 with $\varepsilon\leq\varepsilon_{0}$ and has
$0\in\operatorname{spt}T$. Then for every $0<\gamma\leq 1$ there exists a
$C^{1,1}$ function $v:B_{R/9}\to\mathbb{R}$ with the following properties:

(5.13) $\sup_{B_{R/9}}|\mathrm{D}^{2}v|\leq\gamma,$ (5.14)
$\sup_{B_{R/9}}|\mathrm{D}v|\leq
C\left(\mathbf{E}(T,\mathbf{C}_{R})^{\frac{1}{2n}}+\gamma\right)R,\qquad\sup_{B_{R/9}}|v|\leq
C\mathbf{E}(T,\mathbf{C}_{R})^{\frac{1}{n(n+1)}}R^{2},$ (5.15)
$T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathbf{p}^{-1}(K)=T^{v}\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathbf{p}^{-1}(K)\text{
for a set }K\subset B_{R/9}\text{ with }\mathcal{L}^{n}(B_{R/9}\setminus
K)\leq C\gamma^{-n(n+1)}R^{n}\mathbf{E}(T,\mathbf{C}_{R})$

and

(5.16) $\|T-T^{v}\|(\mathbf{C}_{R/9})\leq
C\gamma^{-n(n+1)}R^{n}\mathbf{E}(T,\mathbf{C}_{R}),$

where $T^{v}$ is the current $T^{v}:=Q(\Phi^{v})_{\\#}\llbracket
B_{R/9}\rrbracket$ and the constant $C$ depends only on $n$, $\Lambda/\lambda$
and $Q$.

Moreover we have the following estimate for the Dirichlet energy of $v$:

(5.17) $\int_{B_{R/9}}|\mathrm{D}^{2}v|^{2}\,\mathrm{d}x\leq
C\gamma^{-n(n+1)}R^{n}\mathbf{E}(T,\mathbf{C}_{R}).$

###### Proof.

Fix $0<\eta<\varepsilon_{0}$ to be determined later and consider

(5.18) $G_{\gamma}:=\left\\{x\in
B_{R/9}:\mathbf{E}(T,\mathbf{C}_{r}(x))\leq\eta\;\forall
r\in(0,8R/9)\right\\}.$

For each $x\in B_{R/9}\setminus G_{\gamma}$ we can find $0<r<8R/9$ such that
$\mathbf{E}(T,\mathbf{C}_{r}(x))>\eta$. Vitali’s covering Lemma gives us a
countable disjoint subcollection of such balls
$B_{r_{1}}(x_{1}),B_{r_{2}}(x_{2}),\ldots\subset B_{R}$ such that
$\\{B_{5r_{i}}(x_{i})\\}$ cover $B_{R/9}\setminus G_{\gamma}$. Then

(5.19) $\mathcal{L}^{n}(B_{R/9}\setminus
G_{\gamma})\leq\sum_{i}\omega_{n}(5r_{i})^{n}\leq\frac{\omega_{n}5^{n}}{\eta}\sum_{i}\frac{1}{2}\int_{\mathbf{C}_{r_{i}}(x_{i})}|\vec{T}-\vec{\pi}_{0}|^{2}\,\mathrm{d}\|T\|\leq\frac{\omega_{n}5^{n}}{\eta}R^{n}\mathbf{E}(T,\mathbf{C}_{R}).$

On the other hand, let
$\xi_{1},\xi_{2}\in\operatorname{spt}T\cap\mathbf{p}^{-1}(G_{\gamma})$ and
denote $\xi_{i}=(\mathbf{x}_{i},\mathbf{y}_{i},\varphi_{i})$. Consider the
current $\tilde{T}:=(\ell_{\xi_{1}^{-1}})_{\\#}T$, so that for any
$r\in(4|\mathbf{x}_{1}-\mathbf{x}_{2}|,8R/9)$ we have
$0,\xi_{1}^{-1}\xi_{2}\in\operatorname{spt}\tilde{T}\cap\mathbf{C}_{r/4}$ and
$\mathbf{E}(\tilde{T},\mathbf{C}_{r})=\mathbf{E}(T,\mathbf{C}_{r}(x_{1}))\leq\eta$.
The coordinates of the translated point are

$\xi_{1}^{-1}\xi_{2}=\left(\mathbf{x}_{2}-\mathbf{x}_{1},\mathbf{y}_{2}-\mathbf{y}_{1},\varphi_{2}-\varphi_{1}+\frac{1}{2}(-\mathbf{x}_{1}\cdot\mathbf{y}_{2}+\mathbf{y}_{1}\cdot\mathbf{x}_{2})\right)$

and thus the height bound from 5.3 gives

$|\mathbf{y}_{2}-\mathbf{y}_{1}|\leq
C\eta^{\frac{1}{2n}}|\mathbf{x}_{2}-\mathbf{x}_{1}|\qquad\text{and}\qquad\left|\varphi_{2}-\varphi_{1}+\frac{1}{2}(-\mathbf{x}_{1}\cdot\mathbf{y}_{2}+\mathbf{y}_{1}\cdot\mathbf{x}_{2})\right|\leq
C\eta^{\frac{1}{n(n+1)}}|\mathbf{x}_{2}-\mathbf{x}_{1}|^{2}$

after letting $r\searrow 4|\mathbf{x}_{2}-\mathbf{x}_{1}|$. In particular, if
$\mathbf{x}_{1}=\mathbf{x}_{2}$ it follows that
$\mathbf{y}_{1}=\mathbf{y}_{2}$ and $\varphi_{1}=\varphi_{2}$, so
$\operatorname{spt}T\cap\mathbf{p}^{-1}(x)$ consists of exactly one
point666There is at least one point since otherwise
$\operatorname{dist}_{\mathrm{FK}}(\operatorname{spt}T,\mathbf{p}^{-1}(x))>0$
and thus
$\mathbf{M}(\mathbf{p}_{\\#}T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}B_{s}(x))=0$
for some small $s>0$, contradicting the assumption that $Q\geq 1$. for every
$x\in G_{\gamma}$. Hence we may define functions
$f:G_{\gamma}\to\mathbb{R},\mathbf{g}:G_{\gamma}\to\mathbb{R}^{n}$ as follows:
if $(\mathbf{x},\mathbf{y},\varphi)$ are the coordinates of the unique point
in $\operatorname{spt}T\cap\mathbf{p}^{-1}(x)$, we set
$f(x):=\tfrac{1}{2}(\mathbf{x}\cdot\mathbf{y})-\varphi$ and
$\mathbf{g}(x):=\mathbf{y}$.

Let us show that we can apply 5.4 with $L_{0}=CE^{\frac{1}{n(n+1)}}R^{2}$,
$L_{1}=CE^{\frac{1}{2n}}R$ and $L_{2}=C\eta^{\frac{1}{n(n+1)}}$, where we are
using $E:=\mathbf{E}(T,\mathbf{C}_{R})$ for short. Since
$0\in\operatorname{spt}T$ and $E\leq\varepsilon_{0}$, the bounds $|f|\leq
L_{0}$ and $|\mathbf{g}|\leq L_{1}$ are clear. For the conditions involving
$L_{2}$, just notice that

$\displaystyle|f(x_{1})-f(x_{2})-\mathbf{g}(x_{2})\cdot(\mathbf{x}_{1}-\mathbf{x}_{2})|$
$\displaystyle=\left|\frac{1}{2}(\mathbf{x}_{1}\cdot\mathbf{y}_{1})-\varphi_{1}-\frac{1}{2}(\mathbf{x}_{2}\cdot\mathbf{y}_{2})+\varphi_{2}-\mathbf{y}_{2}\cdot(\mathbf{x}_{1}-\mathbf{x}_{2})\right|$
$\displaystyle=\left|\varphi_{2}-\varphi_{1}+\frac{1}{2}(-\mathbf{x}_{1}\cdot\mathbf{y}_{2}+\mathbf{y}_{1}\cdot\mathbf{x}_{2})\right|+\frac{1}{2}|(\mathbf{y}_{1}-\mathbf{y}_{2})\cdot(\mathbf{x}_{1}-\mathbf{x}_{2})|$
$\displaystyle\leq
C\eta^{\frac{1}{n(n+1)}}|\mathbf{x}_{1}-\mathbf{x}_{2}|^{2}+C\eta^{\frac{1}{2n}}|\mathbf{x}_{1}-\mathbf{x}_{2}|^{2}$
$\displaystyle\leq
C\eta^{\frac{1}{n(n+1)}}|\mathbf{x}_{1}-\mathbf{x}_{2}|^{2}.$

whereas the condition for $\mathbf{g}$ is clear. Thus we obtain a function
$v\in C^{1,1}(B_{R},\mathbb{R})$ satisfying the bounds

$\|v\|_{L^{\infty}}\leq
CE^{\frac{1}{n(n+1)}}R^{2},\qquad\|\mathrm{D}v\|_{L^{\infty}}\leq
CE^{\frac{1}{2n}}R+C\eta^{\frac{1}{n(n+1)}}R,\qquad\|\mathrm{D}^{2}v\|_{L^{\infty}}\leq
C\eta^{\frac{1}{n(n+1)}}$

and such that $\operatorname{spt}T\cap\mathbf{p}^{-1}(x)=\\{\Phi^{v}(x)\\}$
for every $x\in G_{\gamma}$. Choosing $\eta=c\gamma^{n(n+1)}$ for $c$ small
enough proves (5.12) and (5.14).

Now note that for $\mathcal{L}^{n}$-a.e. $x\in G_{\gamma}$, the slice $\langle
T,\mathbf{p},x\rangle$ is a $0$-dimensional integral current supported in
$\mathbf{p}^{-1}(x)\cap\operatorname{spt}T=\\{\Phi^{v}(x)\\}$ and satisfies
$\mathbf{p}_{\\#}\langle T,\mathbf{p},x\rangle=Q\llbracket x\rrbracket$, so we
must have $\langle T,\mathbf{p},x\rangle=Q\llbracket\Phi^{v}(x)\rrbracket$.
Therefore, for a set $K\subset G_{\gamma}$ with
$\mathcal{L}^{n}(G_{\gamma}\setminus K)$ we have that
$T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathbf{p}^{-1}(K)=T^{v}\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathbf{p}^{-1}(K)$,
which together with (5.19) and our choice of $\eta$ gives (5.15). From this we
get (5.16) as a consequence of the formula for the area (4.19) and (4.5):

$\displaystyle\|T-T^{v}\|(B_{R/9})$
$\displaystyle\leq\|T-T^{v}\|(\mathbf{p}^{-1}(K))+\|T\|(\mathbf{p}^{-1}(B_{R/9}\setminus
K))+\|T^{v}\|(\mathbf{p}^{-1}(B_{R/9}\setminus K))$
$\displaystyle=\|T\|(\mathbf{p}^{-1}(B_{R/9}\setminus
K))+Q\int_{B_{R/9}\setminus
K}\sqrt{\det(\mathrm{id}+(\mathrm{D}^{2}v)^{2})}\,\mathrm{d}\mathcal{L}^{n}$
$\displaystyle\leq C\mathcal{L}^{n}(B_{R/9}\setminus
K)+\frac{1}{2}\int_{\mathbf{p}^{-1}(B_{R/9}\setminus
K)}|\vec{T}-\vec{\pi}_{0}|^{2}\,\mathrm{d}\|T\|$ $\displaystyle\leq
C\mathcal{L}^{n}(B_{R/9}\setminus
G_{\gamma})+R^{n}\mathbf{E}(T,\mathbf{C}_{R})$ $\displaystyle\leq
C\gamma^{-n(n+1)}R^{n}\mathbf{E}(T,\mathbf{C}_{R}).$

Finally (5.17) is a consequence of the following two estimates: on one hand,

$\int_{B_{R/9}\setminus K}|\mathrm{D}^{2}v|^{2}\leq
C\gamma^{2}\cdot\gamma^{-n(n+1)}R^{n}\mathbf{E}(T,\mathbf{C}_{R})\leq
C\gamma^{-n(n+1)}R^{n}\mathbf{E}(T,\mathbf{C}_{R}),$

and on the other hand, again using (4.19), (4.5) and the Taylor expansion of
the area integrand

$\sqrt{\det(\mathrm{id}+A^{2})}\geq 1+c|A|^{2}\qquad\text{for }|A|\leq 1,$

we find

$\displaystyle\int_{K}|\mathrm{D}^{2}v|^{2}$ $\displaystyle\leq
C\int_{K}\left(\sqrt{\det(\mathrm{id}+(\mathrm{D}^{2}v)^{2})}-1\right)=\frac{C}{Q}\left(\|T^{v}\|(\mathbf{p}^{-1}(K))-Q\mathcal{L}^{n}(K)\right)$
$\displaystyle=\frac{C}{Q}\left(\|T\|(\mathbf{p}^{-1}(K))-Q\mathcal{L}^{n}(K)\right)\leq
CR^{n}\mathbf{E}(T,\mathbf{C}_{R})\leq
C\gamma^{-n(n+1)}R^{n}\mathbf{E}(T,\mathbf{C}_{R}).\qed$

######  Remark 5.6.

By using interpolation, we can obtain a better bound on the derivative of the
$C^{1,1}$ function $v$ even when $\gamma$ is relatively large. Indeed, the
estimate

(5.20) $\sup_{B_{R/9}}|\mathrm{D}v|\leq
C\mathbf{E}(T,\mathbf{C}_{R})^{\frac{1}{2n(n+1)}}R$

follows from interpolating between the bounds for $\|v\|_{L^{\infty}}$ and
$\|\mathrm{D}^{2}v\|_{L^{\infty}}$ from Lemma 5.5 (if necessary by considering
a $C^{1,1}$ extension on a larger ball) and using that $\gamma\leq 1$.

##  6\. Biharmonic approximation and excess decay

The main goal of this section is to prove the following lemma, which will be
the main ingredient in the proof of 4.7:

######  Lemma 6.1 (excess improvement).

There exists a constant $C_{*}=C_{*}(n,\Lambda/\lambda,Q)$ with the following
property: for any $0<\rho\leq\tfrac{1}{72}$ there is $\varepsilon_{\rho}>0$
such that if $T\in\mathbf{I}^{\mathrm{hor}}_{n}(\mathbb{H}^{n})$ satisfies 4.6
with $\pi_{0}$ in place of $\pi$ and $\varepsilon\leq\varepsilon_{\rho}$, and
in addition $0\in\operatorname{spt}T$ and $\mu R\leq\varepsilon_{\rho}$, then
there exists an oriented Legendrian plane $\vec{\pi}_{1}\in\mathscr{L}_{0}$
such that

(6.1) $|\vec{\pi}_{1}-\vec{\pi}_{0}|^{2}\leq
C_{*}\left(\mathbf{E}(T,\mathbf{C}_{R}^{\pi_{0}})+\mu R\right)$

and

(6.2)
$\mathbf{E}(T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathbf{C}_{R/2}^{\pi_{0}},\mathbf{C}_{\rho
R}^{\pi_{1}})\leq
C_{*}\left(\rho^{2}\mathbf{E}(T,\mathbf{C}_{R}^{\pi_{0}})+\rho^{-n}R\mu\right).$

The strategy to prove Lemma 6.1 is inspired by [SS82] but actually uses some
technical simplifications of [ADS18] that allow us to avoid regularization and
instead use classical $L^{p}$ estimates for solutions to elliptic fourth order
constant coefficient equations in a ball. In particular, our choice of a good
radius comes from [ADS18], but since our setting is fundamentally anisotropic,
their strategy does not work for us and instead we use two distinct Lipschitz
approximations as in [SS82].

The idea (when $h$ is the standard Heisenberg metric) is to approximate the
current $T$ by the graph of $\Phi^{u}$, where $u$ is the biharmonic function
whose boundary data matches that of a Lipschitz approximation $\Phi^{v}$ on a
suitably chosen ball. As already mentioned, our method actually works for more
general functionals than those induced by metrics, but for simplicity we will
restrict to these.

Before starting will need some computations from multilinear algebra. For any
$C^{1,1}$ function $w$, the $n$-vector orienting $T^{w}$ is

(6.3)
$\vec{T}^{w}=\frac{(\Phi^{w})_{\\#}\vec{\pi}_{0}}{\left|(\Phi^{w})_{\\#}\vec{\pi}_{0}\right|}=\frac{(\Phi^{w})_{\\#}\vec{\pi}_{0}}{\mathcal{J}(\Phi^{w})},$

where we have introduced the Jacobian

(6.4) $\mathcal{J}(\Phi^{w})=\left|(\Phi^{w})_{\\#}\vec{\pi}_{0}\right|$

and

(6.5)
$\begin{split}(\Phi^{w})_{\\#}\vec{\pi}_{0}&=(\Phi^{w})_{\\#}(e_{1}\wedge\cdots\wedge
e_{n})=((\Phi^{w})_{\\#}e_{1})\wedge\cdots\wedge((\Phi^{w})_{\\#}e_{n})\\\
&=\left(\nabla^{H}x^{1}+\sum_{j=1}^{n}\partial_{1j}w\nabla^{H}y^{j}\right)\wedge\cdots\wedge\left(\nabla^{H}x^{n}+\sum_{j=1}^{n}\partial_{nj}w\nabla^{H}y^{j}\right)\\\
&=\vec{\pi}_{0}+\sum_{i,j=1}^{n}\partial_{ij}w\,\vec{\pi}_{i}^{j}+\mathcal{O}^{\perp}(|\mathrm{D}^{2}w|^{2}+|\mathrm{D}^{2}w|^{n}).\end{split}$

Here we have introduced the $n$-vectors

$\vec{\pi}_{i}^{j}=\nabla^{H}x^{1}\wedge\cdots\wedge\nabla^{H}x^{i-1}\wedge\nabla^{H}y^{j}\wedge\nabla^{H}x^{i+1}\wedge\cdots\wedge\nabla^{H}x^{n}$

and the notation $\mathcal{O}^{\perp}(f)$ indicates a term of size
$\mathcal{O}(f)$ that belongs to the $g_{\mathbb{H}}$-orthogonal of
$\operatorname{span}\\{\vec{\pi}_{0},\vec{\pi}_{i}^{j}\\}\subset\bigwedge^{n}(\Xi)$.
It follows that

$1\leq\mathcal{J}(\Phi^{w})^{2}\leq
1+C|\mathrm{D}^{2}w|^{2}+C|\mathrm{D}^{2}w|^{2n}$

and hence

(6.6) $1\leq\mathcal{J}(\Phi^{w})\leq
1+C(|\mathrm{D}^{2}w|^{2}+|\mathrm{D}^{2}w|^{n}).$

Therefore, if $|\mathrm{D}^{2}w|\leq 1$, by expanding in (6.3) we find that

(6.7)
$|\vec{T}^{w}-\vec{\pi}_{0}|^{2}=|\mathrm{D}^{2}w|^{2}\left(1+\mathcal{O}(|\mathrm{D}^{2}w|^{2})\right).$

In general we have the estimate

(6.8)
$\frac{1}{|\Phi^{w}_{\\#}\vec{\pi}_{0}|_{h_{0}}}=\frac{1}{|\vec{\pi}_{0}|_{h_{0}}}\left(1-\sum_{i,j}\frac{\partial_{ij}(w)\,h_{0}(\vec{\pi}_{i}^{j},\vec{\pi}_{0})}{|\vec{\pi}_{0}|_{h_{0}}^{2}}+\mathcal{O}(|\mathrm{D}^{2}w|^{2})\right).$

Indeed, for $|\mathrm{D}^{2}w|\leq 1$ this comes from the Taylor expansion
$|\Phi^{w}_{\\#}\vec{\pi}_{0}|_{h_{0}}^{2}=|\vec{\pi}_{0}|_{h_{0}}^{2}+2\sum_{ij}\partial_{ij}w\,h_{0}(\vec{\pi}_{0},\vec{\pi}_{i}^{j})+\mathcal{O}(|\mathrm{D}^{2}w|^{2})$,
whereas for $|\mathrm{D}^{2}w|\geq 1$ this is trivial given the lower bound

(6.9)
$|\Phi^{w}_{\\#}\vec{\pi}_{0}|_{h_{0}}\geq\lambda|\Phi^{w}_{\\#}\vec{\pi}_{0}|\geq\lambda.$

We also have the expression

(6.10) $\mathcal{J}(\Phi^{w})=\sqrt{\det(\mathrm{id}+(\mathrm{D}^{2}w)^{2})},$

which follows from (6.5) using the usual formula for the Jacobian of a linear
map.

###### Proof of Lemma 6.1.

It is clear that we may suppose $R=1$ and
$E:=\mathbf{E}(T,\mathbf{C}_{1})\leq\varepsilon_{0}$. We fix two parameters
$0<\delta\ll\eta\ll 1$ to be determined later but depending explicitly on $n$
only, and let $v_{0}$ and $v_{\delta}$ be the functions $v_{0},v_{\delta}\in
C^{1,1}(B_{1/9},\mathbb{R})$ provided by Lemma 5.5 with the choices $\gamma=1$
and $\gamma=E^{\delta}$, respectively. We record the estimates for the energy
of $v_{0}$ and $v_{\delta}$ coming from (5.17):

(6.11) $\int_{B_{1/9}}|\mathrm{D}^{2}v_{0}|^{2}\,\mathrm{d}x\leq CE,$ (6.12)
$\int_{B_{1/9}}|\mathrm{D}^{2}v_{\delta}|^{2}\,\mathrm{d}x\leq
CE^{1-n(n+1)\delta}.$

It will be convenient to denote the $1$-Lipschitz approximation
$L:=T^{v_{0}}=Q(\Phi^{v_{0}})_{\\#}\llbracket B_{1/9}\rrbracket$ and also, for
any $r>0$,
$T_{r}:=T\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathbf{C}_{r}$ and
$L_{r}:=L\operatorname{\mathbin{\mbox{\Large$\llcorner$}}}\mathbf{C}_{r}$.

  1. Step 1.

_The fourth order linear PDE._ We define the coefficients

$\tilde{a}_{ik}^{jl}:=h_{0}(\vec{\pi}_{i}^{j},\vec{\pi}_{k}^{l}),\qquad
b_{i}^{j}:=h_{0}(\vec{\pi}_{0},\vec{\pi}_{i}^{j})\qquad\text{and}\qquad
a_{ik}^{jl}:=\tilde{a}_{ik}^{jl}-\frac{b_{i}^{j}b_{k}^{l}}{|\vec{\pi}_{0}|_{h_{0}}^{2}},$

where the indices $i,j,k,l$ range from $1$ to $n$. It follows from (4.8) that
$(a_{ik}^{jl})$ satisfies the Legendre ellipticity condition: given a
symmetric matrix $(\sigma_{ij})$, we have that

$\displaystyle a_{ik}^{jl}\sigma_{ij}\sigma_{kl}$
$\displaystyle=\frac{1}{|\vec{\pi}_{0}|_{h_{0}}^{2}}(h_{0}(\vec{\pi}_{i}^{j},\vec{\pi}_{k}^{l})h_{0}(\vec{\pi}_{0},\vec{\pi}_{0})-h_{0}(\vec{\pi}_{0},\vec{\pi}_{i}^{j})h_{0}(\vec{\pi}_{0},\vec{\pi}_{k}^{l}))\sigma_{ij}\sigma_{kl}$
$\displaystyle=\frac{1}{|\vec{\pi}_{0}|_{h_{0}}^{2}}(h_{0}(\sigma_{ij}\vec{\pi}_{i}^{j},\sigma_{kl}\vec{\pi}_{k}^{l})h_{0}(\vec{\pi}_{0},\vec{\pi}_{0})-h_{0}(\vec{\pi}_{0},\sigma_{ij}\vec{\pi}_{i}^{j})h_{0}(\vec{\pi}_{0},\sigma_{kl}\vec{\pi}_{k}^{l}))$
$\displaystyle=\frac{1}{|\vec{\pi}_{0}|_{h_{0}}^{2}}(h_{0}(\vec{\sigma},\vec{\sigma})h_{0}(\vec{\pi}_{0},\vec{\pi}_{0})-h_{0}(\vec{\pi}_{0},\vec{\sigma})^{2})\geq\frac{\lambda^{4}}{\Lambda^{2}}|\vec{\sigma}|^{2},$

where we have used
$\vec{\sigma}=\sum_{ij}\sigma_{ij}\vec{\pi}_{i}^{j}\perp_{g_{\mathbb{H}}}\vec{\pi}_{0}$.
We remark that these coefficients arise in the second order Taylor expansion
of the function
$(\sigma_{ij})\mapsto\left|\vec{\pi}_{0}+\sigma_{ij}\vec{\pi}_{i}^{j}\right|_{h_{0}}$.

Now for a radius $\tfrac{1}{18}<\sigma<\tfrac{1}{9}$ to be fixed soon, we
consider the solution $u:B_{\sigma}\to\mathbb{R}$ of

(6.13) $\begin{cases}a_{ik}^{jl}\partial_{ijkl}u=0&\text{in }B_{\sigma}\\\
u=v_{\delta}&\text{on }\partial B_{\sigma}\\\
\partial_{\nu}u=\partial_{\nu}v_{\delta}&\text{on }\partial
B_{\sigma}\end{cases}$

provided by 2.3, and recall that $u\in W^{2,p}(B_{\sigma})$ for every $1\leq
p<\infty$ and $u\in C^{\infty}_{\mathrm{loc}}(B_{\sigma})$.

  2. Step 2.