# Isotropic surfaces and moment map flow

François Jauberteau  François Jauberteau, Laboratoire Jean Leray, Université
de Nantes<EMAIL_ADDRESS>and  Yann Rollin  Yann Rollin,
Laboratoire Jean Leray, Université de Nantes<EMAIL_ADDRESS>
###### Abstract.

We consider the moduli space of isotropic maps from a closed surface $\Sigma$
to a symplectic affine space and construct a Kähler moment map geometry, on a
space of differential forms on $\Sigma$, such that the isotropic maps
correspond to certain zeroes of the moment map. The moment map geometry
induces a modified moment map flow, whose fixed point set correspond to
isotropic maps. This construction can be adapted to the polyhedral setting. In
particular, we prove that the polyhedral modified moment map flow induces a
strong deformation retraction from the space of polyhedral maps onto the space
of polyhedral isotropic maps.

###### Key words and phrases:

polyhedral maps, piecewise linear symplectic geometry, Lagrangian tori,
isotropic tori, isotropic immersions, Kähler moment map, moment map flow

###### 2010 Mathematics Subject Classification:

Primary 5299, 53D12; Secondary 39A14, 39A70, 47B39, 53D50, 53D20, 53D30

Supported by IRL CRM-CNRS 3457, UQAM, CIRGET, UMR6629, ANR-21-CE40-0017, CHL
11-LABX-0020-01

##  1\. Introduction

###  1.1. Motivations

_Symplectic geometry_ is the natural mathematical framework for _Hamiltonian
mechanics_. Gromov showed that certain symplectic properties are _flexible_ ,
thanks to _convex integration_ , resulting in _h-principle_ theorems [8],
whereas others are _rigid_ , due to the existence of _pseudoholomorphic
curves_ [9]. In particular the Gromov-Lees theorem [9, 13, 5] establishes an
h-principle for the problem of _isotropic immersions_ in a symplectic
manifold. As a corollary, it is possible to approximate every immersed
submanifold of an affine symplectic space by isotropic immersed submanifolds,
with respect to the $C^{0}$ topology.

Our research started as we were conducting some numerical experiments for
Lagrangian surfaces and their _mean curvature flow_ ; numerical simulations
naturally involve piecewise linear geometry and we quickly realized that the
only explicit examples of closed piecewise linear Lagrangian were the polygons
in ${\mathbb{C}}$ and some polygonal versions of the Clifford torus in
${\mathbb{C}}^{m}$. Assessing the state of the art, we noticed that very
little is known about piecewise linear symplectic geometry. Then, we turned to
the existence problem for piecewise linear isotropic submanifolds in affine
symplectic spaces [11, 17] and obtained the following approximation theorem:

######  Theorem 1.1.1 ([17]).

Let $f:\Sigma\looparrowright V$ be a smooth isotropic immersion, where
$\Sigma$ is a surface diffeomorphic to a quotient torus, and $(V,\omega_{V})$
is a symplectic affine space. Then, there exists a family of piecewise linear
maps $f_{N}:\Sigma\to V$ for $N\in{\mathbb{N}}$, with the following
properties:

  1. (1)

$f_{N}:\Sigma\looparrowright V$ is a topological immersion;

  2. (2)

$f_{N}$ converges toward $f$ in $C^{1}$-norm.

As a corollary, every smoothly immersed torus in $V$ can be approximated in
$C^{0}$-norm by isotropic piecewise linear immersed tori.

Surprisingly, the original proof of Theorem 1.1.1 relies on an _infinite
dimensional Kähler moment map geometry_ and the _fixed point principle_
developed in [11]. However, an alternate proof using only on _soft techniques_
was given subsequently by Etourneau [6], with significant technical
simplifications.

Several authors have been considering problems of piecewise linear symplectic
geometry, including Gratza [7] and Bertelson-Distexhe [3]. Panov also
introduced a notion of _polyhedral Kähler_ geometry in [16]. The space of
piecewise linear symplectic maps of a $4$-torus is also shown to be related to
a _hyperKähler moment map geometry_ in [18]. It seems that classical results
of smooth symplectic differential geometry are curiously challenging to
generalize to the piecewise linear setting. Perhaps this is a manifestation of
symplectic rigidity ? To show how much has to be accomplished in the field,
here is a short list of challenging open questions:

  1. (1)

The _local Darboux theorem_ , well known in the smooth setting, turns out to
be a conjecture for piecewise linear symplectic manifolds of dimension at
least $4$ (cf. [3] for a definition of piecewise linear symplectic manifolds).

  2. (2)

The _deformation theory of Lagrangian submanifolds_ is well described in the
smooth setting, thanks to the Lagrangian neighborhood theorem [14]. However,
there is no analogue of such a deformation theory in the case of piecewise
linear Lagrangian submanifolds of a symplectic affine space. This motivates
the work and partial answers of [11, 17].

  3. (3)

The _local structure_ of the infinite dimensional Lie group of symplectic
diffeomorphisms is well understood [2]. However, very little is known about
the local (and global) structure of the space of piecewise linear symplectic
maps, even in the case of a $4$-dimensional torus [18].

The above questions seem hard settle for an obvious reason: the Moser’s trick
generally fails in the piecewise linear setting, although it may be be used
with significant efforts, in the case of piecewise linear volume forms for
instance [3] for instance. All these complications lead us to speculate
whether some _exotica_ arise in the context of piecewise linear symplectic
geometry.

A Kähler moment map geometry inspired by Donaldson [4] was introduced in [11]
to construct the approximations $f_{N}$ of Theorem 1.1.1. The construction
starts from a Kähler surface $(\Sigma,g_{\Sigma},J_{\Sigma},\omega_{\Sigma})$
and a Hermitian affine space $V$ with symplectic form $\omega_{V}$. These
structures induce a _formal_ Kähler structure
$({\mathscr{M}},{\mathcal{G}},{\mathcal{J}},\Omega)$ with Kähler form
$\Omega$, on the moduli space ${\mathscr{M}}=C^{\infty}(\Sigma,V)$. The group
of Hamiltonian diffeomorphisms $\mathrm{Ham}(\Sigma,\omega_{\Sigma})$ acts by
precomposition on the moduli space space ${\mathscr{M}}$ and preserves the
formal Kähler structure. In fact the action of
$\mathrm{Ham}(\Sigma,\omega_{\Sigma})$ on ${\mathscr{M}}$ is formally
Hamiltonian, with moment map given by

$\mu^{D}(f)=-\frac{f^{*}\omega_{V}}{\omega_{\Sigma}},$

understood as an element of the Lie algebra of
$\mathrm{Ham}(\Sigma,\omega_{\Sigma})$, identified to the space of smooth
function $C_{0}^{\infty}(\Sigma,{\mathbb{R}})$ orthogonal to constants. Thus,
zeroes of the moment map $\mu^{D}$ correspond to isotropic maps
$f\in{\mathscr{M}}$. A detailed presentation of this moment map geometry is
given in [11].

A finite dimensional approximation $\phi_{N}$ of the energy functional $\phi$
of the moment map $\mu^{D}$ is considered in [11]. The downward gradient flow
of $\phi_{N}$ provides a flow of piecewise linear surfaces. A numerical
version of the flow and numerical experiments carried out in [11] provided
some effective examples of piecewise linear Lagrangian tori in
${\mathbb{C}}^{2}$. Overall, this flow of piecewise linear surfaces has been
interesting from an experimental perspective. However, it is not satisfactory
from a mathematical point of view, for several reasons: we could not prove
that the flow is convergent, although it seems well behaved numerically.
Furthermore, the flow does not come from a finite dimensional moment map
picture and its geometrical interpretation is somewhat unclear.

As a response to these objections, a new moment map geometry and its
corresponding flow are introduced in this paper. They do not have any of the
above issues: the new moment map geometry can be immediately adapted to the
polyhedral setting and much stronger mathematical results are obtained. In
particular, Theorem E is a Duistermaat type theorem, which shows that the flow
has the nicest possible behavior.

###  1.2. Statement of results

Let $V$ be a Hermitian affine space, with underlying vector space $\vec{V}$
and and _symplectic form_ $\omega_{V}$. We consider a _smooth closed oriented
surface_ $\Sigma$, endowed with a _Riemannian metric_ $g_{\Sigma}$, a
_compatible almost complex structure_ $J_{\Sigma}$ and a corresponding _Kähler
form_ $\omega_{\Sigma}=g_{\Sigma}(J_{\Sigma},\cdot,\cdot)$.

The _moduli space of smooth map_ $f:\Sigma\to V$ is denoted ${\mathscr{M}}$.
Formally ${\mathscr{M}}$ is an affine space with $\Omega^{0}(\Sigma,\vec{V})$
as the underlying vector space. The _differential_ of a smooth map
$f\in{\mathscr{M}}$ defines an exact $\vec{V}$-valued $1$-form $df$ and we
have a linear differential operator

$d:{\mathscr{M}}\to{\mathscr{F}}=\Omega^{1}(\Sigma,\vec{V}).$

The image of ${\mathscr{M}}$ by $d$ is the space of exact $1$-forms denoted
${\mathscr{F}}_{0}$. We show that the moduli space ${\mathscr{F}}$ carries a
natural Euclidean $L^{2}$-inner product ${\mathcal{G}}$, defined by Formula
(2.3), and a compatible almost complex structure ${\mathcal{J}}$, defined by
Formula (2.5). The corresponding Kähler form is denoted
$\Omega={\mathcal{G}}({\mathcal{J}}\cdot,\cdot)$.

Formula (2.6) defines an action of the gauge group
${\mathbb{T}}^{\mathbb{C}}=C^{\infty}(\Sigma,{\mathbb{C}}^{*})$ on
${\mathscr{F}}$. The action of the real subgroup
${\mathbb{T}}=C^{\infty}(\Sigma,{\mathbb{R}})$ is merely the action by complex
multiplication on $\vec{V}$-valued differential $1$-forms and we have the
following result:

######  Theorem A.

Let $\Sigma$ be a smooth closed surface endowed with a Kähler structure
$(\Sigma,g_{\Sigma},J_{\Sigma},\omega_{\Sigma})$ and $V$, a Hermitian affine
space.

The moduli space ${\mathscr{F}}=\Omega^{1}(\Sigma,\vec{V})$ carries a natural
Kähler structure $({\mathscr{F}},{\mathcal{G}},{\mathcal{J}},\Omega)$ and an
action of the gauge group
${\mathbb{T}}^{\mathbb{C}}=C^{\infty}(\Sigma,{\mathbb{C}}^{*})$.

The almost complex structure ${\mathcal{J}}$ is invariant under the action of
${\mathbb{T}}^{\mathbb{C}}$. The action of ${\mathbb{T}}^{\mathbb{C}}$ is the
complexification of the ${\mathbb{T}}$-action, where
${\mathbb{T}}=C^{\infty}(\Sigma,S^{1})$ is the real subgroup.

The ${\mathbb{T}}$-action preserves the Kähler structure
$({\mathscr{F}},{\mathcal{J}},{\mathcal{G}},\Omega)$ and is Hamiltonian. The
map $\mu:{\mathscr{F}}\to\mathfrak{t}$, where $\mathfrak{t}$ is the Lie
algebra of ${\mathbb{T}}$, given by

$\mu(F)=-\frac{F^{*}\omega_{V}}{\omega_{\Sigma}},$

is a moment map. In other words, $\mu$ is ${\mathbb{T}}$-invariant and, for
every $\zeta\in\mathfrak{t}$,

$D\langle\\!\langle\mu,\zeta\rangle\\!\rangle=-\iota_{X_{\zeta}}\Omega,$

where $X_{\zeta}$ is the vector field on ${\mathscr{F}}$ defined by the
infinitesimal action of $\zeta$ on ${\mathscr{F}}$.

######  Corollary B.

With the assumptions of Theorem A, a smooth map $f:\Sigma\to V$ is isotropic
if, and only if,

$\mu(df)=0.$

######  Remark 1.2.1.

In the finite dimensional setting, the famous Kempf-Ness theorem relates the
existence of zeroes of a Kähler moment map in a complexified orbit with an
algebro-geometric notion of stability [15]. In view Theorem A and Corollary B,
it is tempting to try to extend this theory in our case: the existence of
isotropic maps should be related to some kind of algebraic condition. The
answer turns out to be somewhat trivial for the total moduli space
${\mathscr{F}}$, as discussed at §2.12. However, we are mainly interested in
the zeroes of the moment map that belong to the subspace
${\mathscr{F}}_{0}\subset{\mathscr{F}}$, as they are related to isotropic maps
by Corollary B. Unfortunately, ${\mathscr{F}}_{0}$ is not invariant under the
gauge group action and it is not clear how to adapt the classical theory from
this point.

We consider the _energy functional_ of the moment map

$\phi:{\mathscr{F}}\to{\mathbb{R}}$

given by

$\phi(F)=\frac{1}{2}\|\mu(F)\|^{2}_{L^{2}}.$

By construction, $\phi$ is non negative and the vanishing locus of $\phi$
agrees with the vanishing locus of the moment map. Our idea is to interpret
the functional $\phi$ as a _Morse-Bott_ function on the moduli space
${\mathscr{F}}$, in the spirit of Atiyah-Bott [1], who considered the case of
the Yang-Mills functional. The expectation is that the flow is going to
produce a _strong deformation retraction_ of ${\mathscr{M}}$ onto the space of
isotropic maps. However, it is not clear whether $\phi$ satisfies the Morse-
Bott condition in any reasonable sense. Another difficulty is that the
subspace ${\mathscr{F}}_{0}$ is not gauge invariant and may not be preserved
by the usual Morse-Bott gradient flow. We get around this issue by defining
the _modified moment map flow_ , as the downward gradient flow of the
restricted functional $\phi:{\mathscr{F}}_{0}\to{\mathbb{R}}$. More precisely,
the modified moment map flow along ${\mathscr{F}}_{0}$ is given by the
evolution equation

$\boxed{\frac{\partial F}{\partial t}=-\nabla^{\circ}\phi(F),}$

where $F\in{\mathscr{F}}_{0}$ and $\nabla^{\circ}\phi$ is the gradient of the
restricted functional. Our first theorem shows that the flow is geometrically
relevant and is a well posed problem from an analytical point of view:

######  Theorem C.

The fixed point locus of the modified moment map flow on ${\mathscr{F}}_{0}$,
in other words the critical set of $\phi:{\mathscr{F}}_{0}\to{\mathbb{R}}$,
agrees with the vanishing locus of $\phi:{\mathscr{F}}_{0}\to{\mathbb{R}}$.
Furthermore,

  1. (1)

the modified moment maps flow has the short time existence property and

  2. (2)

the $L^{2}$-norm is non increasing along the flow.

######  Remarks 1.2.2.

  1. (1)

The definition of the _short time existence property_ involves the use of
Hölder completions of the moduli space ${\mathscr{F}}$. A technical version of
the short time existence is given at Theorem 3.1.3.

  2. (2)

Unfortunately, the modified moment map flow does not seem to have any nice
regularizing properties, like parabolic flows for instance. Consequently, the
long time existence of the flow is unclear.

  3. (3)

In the case where $\dim_{\mathbb{R}}V=4$, we show at Corollary 3.2.3 that
$\phi:{\mathscr{F}}_{0}\to{\mathbb{R}}$ is a Morse-Bott function, in a
neighborhood of every _monomorphism_ (cf. Definition 2.10.2) of its vanishing
locus.

The purpose of §4 is to extend all the above constructions to the _polyhedral_
setting. The concepts needed for stating the results are quickly introduced
below and the reader may refer directly to §4 for the exact definitions.

A _polyhedral surface_ $\Sigma$ is a topological surface endowed with a
triangulation ${\mathscr{T}}$ and a _polyhedral metric_ $g_{\Sigma}$. An
_orientation_ of $\Sigma$ induces a _polyhedral Kähler form_ $\omega_{\Sigma}$
and a _polyhedral almost complex structure_ $J_{\Sigma}$ adapted to
$g_{\Sigma}$.

A _polyhedral map_ with respect to a triangulation ${\mathscr{T}}$ is a
continuous map $f:\Sigma\to V$ such that the restriction of $f$ to every
simplex of the triangulation is an _affine map_. The moduli space of
polyhedral maps $f:\Sigma\to V$ is denoted ${\mathscr{M}}({\mathscr{T}})$.

A polyhedral map is generally not differentiable. However, the restriction of
a polyhedral map to any simplex of the triangulation has a well defined
tangent map. Accordingly, a polyhedral map $f:\Sigma\to V$ is called a
_polyhedral isotropic map_ , if the pullback of $\omega_{V}$ by $f$,
restricted to every simplex of the triangulation, vanishes identically.

The existence of polyhedral isotropic _immersions_ in
${\mathscr{M}}({\mathscr{T}})$ is an open question. The proof of Theorem 1.1.1
proceeds by introducing a particular sequence of triangulations
${\mathscr{T}}_{N}$, in the case of the $2$-torus $\Sigma$, with a large
number of simplices of order ${\mathscr{O}}(N^{2})$ and stepsize of order
${\mathscr{O}}(N^{-1})$. The piecewise linear approximations $f_{N}$ are in
fact polyhedral isotropic maps that belong to
${\mathscr{M}}({\mathscr{T}}_{N})$. However, the topology of the space of
polyhedral isotropic immersions in ${\mathscr{M}}({\mathscr{T}})$ remains
completely mysterious for a fixed triangulation ${\mathscr{T}}$ of $\Sigma$.

Returning to the general case of a oriented polyhedral surface $\Sigma$, we
pursue the analogy with the smooth setting. The space vector space
${\mathscr{F}}({\mathscr{T}})$ is the space of families
$F=(F_{\sigma})_{{\mathscr{K}}_{2}}$, where $\sigma$ belongs to the set of
facets ${\mathscr{K}}_{2}$ of the triangulation ${\mathscr{T}}$ and
$F_{\sigma}$ is a constant $\vec{V}$-valued $1$-form on $\sigma$. Because the
restriction of a polyhedral map to every simplex of the triangulation is
differentiable, there is a natural differential map

$d:{\mathscr{M}}({\mathscr{T}})\to{\mathscr{F}}({\mathscr{T}})$

and its image is denoted ${\mathscr{F}}_{0}({\mathscr{T}})$.

We also define the group ${\mathbb{T}}^{\mathbb{C}}({\mathscr{T}})$ as the
space of families $\lambda=(\lambda_{\sigma})_{\sigma\in{\mathscr{K}}_{2}}$,
where $\lambda_{\sigma}:\sigma\to{\mathbb{C}}^{*}$ is a constant function. The
real subgroup of families $\lambda$ with $\lambda_{\sigma}\in S^{1}$ is
denoted ${\mathbb{T}}({\mathscr{T}})$. The group
${\mathbb{T}}^{\mathbb{C}}({\mathscr{T}})$ acts on
${\mathscr{F}}({\mathscr{T}})$ by Formula (4.2). As in the smooth case, we
construct a Euclidean inner product ${\mathcal{G}}$, a compatible almost
complex structure ${\mathcal{J}}$ and a corresponding Kähler form $\Omega$ on
${\mathscr{F}}({\mathscr{T}})$. Thus we have a Kähler structure
$({\mathscr{F}}({\mathscr{T}}),{\mathcal{G}},{\mathcal{J}},\Omega)$ with an
action of ${\mathbb{T}}^{\mathbb{C}}({\mathscr{T}})$ and we can state our next
result:

######  Theorem D.

Theorem A and Corollary B hold in the polyhedral setting.

As in the smooth setting, we define an energy of the moment map
$\phi:{\mathscr{F}}({\mathscr{T}})\to{\mathbb{R}}$ by
$\phi(F)=\frac{1}{2}\|\mu(F)\|^{2}_{L^{2}}$. The downward gradient of the
functional $\phi$ restricted to ${\mathscr{F}}_{0}({\mathscr{T}})$ is called
the _polyhedral modified moment map flow_.

The moduli spaces ${\mathscr{M}}({\mathscr{T}})$ and
${\mathscr{F}}({\mathscr{T}})$ are finite dimensional and much stronger result
than Theorem C are expected, as the polyhedral moment map flow is an ordinary
differential equation. This is indeed the case, and we obtain the following
Duistermaat type theorem:

######  Theorem E.

For $F\in{\mathscr{F}}_{0}({\mathscr{T}})$, the polyhedral modified moment map
flow admits a unique solution $F_{t}\in{\mathscr{F}}_{0}({\mathscr{T}})$,
defined for $t\in[0,+\infty)$, such that $F_{0}=F$. Furthermore, $F_{t}$
admits a limit $F_{\infty}\in{\mathscr{F}}_{0}({\mathscr{T}})$ as $t$ goes to
$+\infty$, with the property that $\phi(F_{\infty})=0$.

The extended flow

$\Theta:[0,+\infty]\times{\mathscr{F}}_{0}({\mathscr{T}})\to{\mathscr{F}}_{0}({\mathscr{T}})$

defined by $\Theta(t,F)=F_{t}$ for $t\in[0,+\infty)$ and
$\Theta(+\infty,F)=F_{\infty}$ has the following properties:

  1. (1)

The fixed point locus of the flow $\Theta$ is the vanishing set of
$\phi:{\mathscr{F}}_{0}({\mathscr{T}})\to{\mathbb{R}}$.

  2. (2)

The flow defines a strong deformation retraction of
${\mathscr{F}}_{0}({\mathscr{T}})$ onto the vanishing locus of
$\phi:{\mathscr{F}}_{0}({\mathscr{T}})\to{\mathbb{R}}$.

  3. (3)

There are non trivial flow lines converging toward $0$. More precisely, there
exists $F\in{\mathscr{F}}_{0}({\mathscr{T}})\setminus 0$, such that
$\Theta(+\infty,F)=0$.

  4. (4)

The flow $\Theta$ has exponential convergence rate in a neighborhood of every
regular point of the vanishing locus of
$\phi:{\mathscr{F}}_{0}({\mathscr{T}})\to{\mathbb{R}}$.

The flow $\Theta$ on ${\mathscr{F}}_{0}$ can be lifted as a flow
$\hat{\Theta}$ on ${\mathscr{M}}({\mathscr{T}})$ and we have the following
corollary:

######  Corollary F.

There exists a unique map

$\hat{\Theta}:[0,+\infty]\times{\mathscr{M}}({\mathscr{T}})\to{\mathscr{M}}({\mathscr{T}})$

such that

  1. (1)

$d\circ\hat{\Theta}_{t}=\Theta_{t}\circ d$ and

  2. (2)

$\hat{\Theta}_{0}=\mathrm{id}$ on ${\mathscr{M}}({\mathscr{T}})$,

where we used the notation $\hat{\Theta}_{t}=\hat{\Theta}(t,\cdot)$. The map
$\hat{\Theta}$ is the flow of the evolution equation given by Formula (4.4)
and defines a strong deformation retraction of ${\mathscr{M}}({\mathscr{T}})$
onto the subspace of polyhedral isotropic maps.

The existence of regular points (cf. Definition 4.10.1) of the vanishing set
of $\phi:{\mathscr{F}}_{0}({\mathscr{T}})\to{\mathbb{R}}$ is not well
understood in comparison to the smooth setting (cf. §3.2). However, we show
that the approximating scheme of [11, 17] provides examples of regular points:

######  Theorem G.

In the case where $\Sigma$ is a $2$-torus and $f:\Sigma\looparrowright V$ is a
smooth isotropic immersion, let $f_{N}\in{\mathscr{M}}({\mathscr{T}}_{N})$ be
the Jauberteau-Rollin-Tapie isotropic polyhedral immersions of Theorem 1.1.1
approximating $f$. Then $F_{N}=df_{N}\in{\mathscr{F}}_{0}({\mathscr{T}}_{N})$
is a regular point of the vanishing locus of
$\phi:{\mathscr{F}}_{0}({\mathscr{T}}_{N})\to{\mathbb{R}}$ for every
sufficiently large $N$.

###  1.3. Open problems and future research

In view of the approximation scheme given by Theorem 1.1.1, it seems sensible
to expect that some type Gromov-Lees theorem should apply to the polyhedral
setting, provided some flexibility for the choice of triangulation
${\mathscr{T}}$ of $\Sigma$. If this is indeed the case, we expect the
following consequence:

######  Conjecture H.

Let $\Sigma$ be a surface endowed with a triangulation ${\mathscr{T}}$. Then,
up to passing to a subdivision of ${\mathscr{T}}$, the space of polyhedral
isotropic immersions in ${\mathscr{M}}({\mathscr{T}})$ is homotopically
equivalent to the space of smooth isotropic immersions of $\Sigma$ in $V$.

For triangulation with lower complexity, an adapted Morse-Bott cohomology
theory for $\phi:{\mathscr{F}}_{0}({\mathscr{T}})\to{\mathbb{R}}$ should lead
to topological invariants for the space of non constant polyhedral isotropic
maps. Theorem E shows that the polyhedral modified moment map flow is
extremely well behaved, which is a strong incentive to develop a Morse-Bott
theory, with a renormalized flow briefly discussed at §4.11.

A numerical version of the polyhedral modified moment map flow is currently
being developed for testing purposes of the Morse-Bott theory and for
producing effective examples of polyhedral isotropic immersed surfaces. We
sketch the relevant mathematical ingredients of a computer program that
produces approximate solutions of the flow at §4.12. The code is to be
released very soon [10].

###  1.4. Acknowledgments

A part of this research was carried out in 2023-2024, thanks to the support of
the CNRS International Research Lab CRM in Montreal. Yann Rollin was hosted by
the CIRGET at UQAM and wishes to thank all his colleagues for creating such an
enjoyable and stimulating mathematical environment, in particular, Antonio
Alfieri, Vestislav Apostolov, Steven Boyer, Charles Cifarelli, Olivier Collin,
Alexandra Haedrich, Julien Keller, Abdellah Lahdili, Steven Lu, Duncan McCoy,
Frédéric Rochon and Carlo Scarpa. We also thank Mélanie Bertelson and Vincent
Borrelli, for some useful discussions.

###### Contents

  1. 1 Introduction
    1. 1.1 Motivations
    2. 1.2 Statement of results
    3. 1.3 Open problems and future research
    4. 1.4 Acknowledgments
  2. 2 Kähler moment map
    1. 2.1 Target space
    2. 2.2 Source space
    3. 2.3 Fiberwise structures on tensor bundles
    4. 2.4 Moduli spaces and differentials
    5. 2.5 Euclidean structure and Hodge theory
    6. 2.6 Kähler structures on the moduli space
    7. 2.7 An involution
    8. 2.8 Gauge group action
    9. 2.9 Infinitesimal gauge group action
    10. 2.10 Symplectic density
    11. 2.11 Hamiltonian action
    12. 2.12 Stability and isotropic maps
    13. 2.13 Energy of the moment map
  3. 3 The modified moment map flow
    1. 3.1 Basic properties of the flow
    2. 3.2 Energy as a Morse-Bott function
  4. 4 Polyhedral isotropic surfaces
    1. 4.1 Definition of polyhedral surfaces
    2. 4.2 Whitney cohomology
    3. 4.3 Orientation and Kähler structure
    4. 4.4 Polyhedral maps and differentials
    5. 4.5 Moduli space structure in the polyhedral setting
    6. 4.6 Polyhedral symplectic density
    7. 4.7 Hamiltonian gauge group action
    8. 4.8 Polyhedral modified moment map flow
    9. 4.9 A Duistermaat theorem
    10. 4.10 Regular points of the moduli space
    11. 4.11 Topology of the moduli space
    12. 4.12 Numerical flow

##  2\. Kähler moment map

This section is devoted to the description of an infinite dimensional moment
map geometry, which provides an interpretation of smooth isotropic maps as
some particular zeroes of a moment map.

###  2.1. Target space

Let $V$ be an _affine space_ and $\vec{V}$, its underlying vector space. We
assume that $\vec{V}$ is a _complex vector space_ , of complex dimension
$m\geq 2$, endowed with a _Hermitian inner product_ $h_{V}$,
anti-${\mathbb{C}}$-linear in the first variable. For every
$v_{1},v_{2}\in\vec{V}$, the decomposition

$h_{V}(v_{1},v_{2})=g_{V}(v_{1},v_{2})+i\omega_{V}(v_{1},v_{2}),$

into the real and imaginary parts of $h_{V}$ provides a _Euclidean inner
product_ $g_{V}$ and a _symplectic form_ $\omega_{V}$. Multiplication by $i$
defines a linear endomorphism $i:\vec{V}\to\vec{V}$, also called an _almost
complex structure_. By definition the almost complex structure $i$ is
_compatible_ with $g_{V}$ and $\omega_{V}$, in the sense that

$g_{V}(iv_{1},iv_{2})=g_{V}(v_{2},v_{2})\quad\mbox{ and
}\quad\omega_{V}(v_{1},v_{2})=g_{V}(iv_{1},v_{2})$

for every $v_{1},v_{2}\in\vec{V}$.

###  2.2. Source space

Let $\Sigma$ be a smooth closed and oriented surface, endowed with a
Riemannian metric $g_{\Sigma}$. We also assume that $\Sigma$ is connected in
all this paper, for simplicity of notations. We denote by $\omega_{\Sigma}$
the _volume form_ of $g_{\Sigma}$ compatible with the orientation. The
corresponding _almost complex structure_ $J_{\Sigma}\in\mathrm{End}(T\Sigma)$
is defined as a fiberwise rotation of $T\Sigma\to\Sigma$ with angle
$+\frac{\pi}{2}$ according to the orientation. By construction, $J_{\Sigma}$
is compatible with $g_{\Sigma}$ and $\omega_{\Sigma}$ is the associated
_Kähler form_ , in the sense that

$g_{\Sigma}(J_{\Sigma}\eta_{1},J_{\Sigma}\eta_{2})=g_{\Sigma}(\eta_{1},\eta_{2})\quad\mbox{
and
}\quad\omega_{\Sigma}(\eta_{1},\eta_{2})=g_{\Sigma}(J_{\Sigma}\eta_{1},\eta_{2}),$

for every $x\in\Sigma$ and $\eta_{1},\eta_{2}\in T_{x}\Sigma$. In conclusion,
$\Sigma$ is endowed with a Kähler structure

$(\Sigma,g_{\Sigma},J_{\Sigma},\omega_{\Sigma}).$

###  2.3. Fiberwise structures on tensor bundles

The bundle $T^{*}\Sigma\to\Sigma$ is identified to $T\Sigma\to\Sigma$ using
the duality induced by the Riemannian metric $g_{\Sigma}$. Thus
$T^{*}\Sigma\to\Sigma$ and all the tensor bundles are endowed with an induced
fiberwise Euclidean inner product denoted $g_{\Sigma}$ as well.

The structure of complex vector space on $\vec{V}$ induces a canonical
structure of complex vector bundle on
$\Lambda^{n}\Sigma\otimes\vec{V}\to\Sigma$, where the tensor product is taken
with respect to ${\mathbb{R}}$. Furthermore, $h_{V}$ and $g_{\Sigma}$ induce a
fiberwise Hermitian inner product on
$\Lambda^{n}\Sigma\otimes\vec{V}\to\Sigma$, denoted $h$, and defined by the
following property: for every $x\in\Sigma$,
$\beta_{1},\beta_{2}\in\Lambda^{n}_{x}\Sigma$ and $v_{1},v_{2}\in\vec{V}$,
then

$h(\beta_{1}\otimes v_{1},\beta_{2}\otimes
v_{2})=g_{\Sigma}(\beta_{1},\beta_{2})h_{V}(v_{1},v_{2}).$

The complex structure $J_{\Sigma}$ acts on $\Lambda^{n}\Sigma\to\Sigma$, by
composition on the right: for every $x\in\Sigma$,
$\beta\in\Lambda_{x}^{n}\Sigma$ and $\eta_{1},\cdots,\eta_{n}\in T_{x}\Sigma$,
we put

$(\beta\circ
J_{\Sigma})(\eta_{1},\cdots,\eta_{n})=\beta(J_{\Sigma}\eta_{1},\cdots,J_{\Sigma}\eta_{n}).$

This action is isometric, in the sense that for every
$\beta_{1},\beta_{2}\in\Lambda_{x}^{n}\Sigma$, we have
$g_{\Sigma}(\beta_{1},\beta_{2})=g_{\Sigma}(\beta_{1}\circ
J_{\Sigma},\beta_{2}\circ J_{\Sigma})$

The action of $J_{\Sigma}$ extends canonically to
$\Lambda^{n}\Sigma\otimes\vec{V}\to\Sigma$ and the Hermitian product $h$ is
$J_{\Sigma}$-invariant, in the sense that, for every
$\beta_{1},\beta_{2}\in\Lambda^{n}_{x}\Sigma\otimes\vec{V}$, we have

$h(\beta_{1},\beta_{2})=h(\beta_{1}\circ J_{\Sigma},\beta_{2}\circ
J_{\Sigma}).$

We consider the Riemannian metric $g$ given by the real part of $h$. Then
$J_{\Sigma}$ and i act isometrically on
$\Lambda^{n}\Sigma\otimes\vec{V}\to\Sigma$, in the sense that

$g(\beta_{1},\beta_{2})=g(\beta_{1}\circ J_{\Sigma},\beta_{2}\circ
J_{\Sigma}),\quad g(i\beta_{1},i\beta_{2})=g(\beta_{1},\beta_{2}).$

We define a fiberwise almost complex structure on
$T^{*}\Sigma\otimes\vec{V}\to\Sigma$, by the formula

$J\cdot F=-F\circ J_{\Sigma}.$ (2.1)

By definition, for every $F_{1},F_{2}\in T^{*}_{x}\Sigma\otimes\vec{V}$, we
have

$g(iF_{1},iF_{2})=g(J\cdot F_{1},J\cdot F_{2})=g(F_{1},F_{2}).$

In particular, the formula

$\omega(F_{1},F_{2})=g(J\cdot F_{1},F_{2})$ (2.2)

defines a fiberwise symplectic form on the bundle
$T^{*}\Sigma\otimes\vec{V}\to\Sigma$.

###  2.4. Moduli spaces and differentials

We consider the moduli space of smooth $\vec{V}$-valued differential $1$-forms

${\mathscr{F}}=\Omega^{1}(\Sigma,\vec{V})$

and the moduli space of smooth maps

${\mathscr{M}}=C^{\infty}(\Sigma,V).$

Notice that ${\mathscr{M}}$ is an affine space with
$\Omega^{0}(\Sigma,\vec{V})$ as the underlying vector space. For every smooth
map, $f:M\to V$, the tangent map $f_{*}:T\Sigma\to TV=V\times\vec{V}$ can be
regarded as a differential

$df:T\Sigma\to\vec{V}$

given by $df=\pi_{2}\circ f_{*}$, where $\pi_{2}:TV\to\vec{V}$ is the second
canonical projection. Hence we have a differential operator between moduli
spaces

${\mathscr{M}}\stackrel{{\scriptstyle d}}{{\longrightarrow}}{\mathscr{F}}.$

The image of $d$ is the subspace of _exact_ $\vec{V}$-valued differential
forms, denoted ${\mathscr{F}}_{0}\subset{\mathscr{F}}$. Furthermore, $d$ is
injective up to translations by constant map in $\Omega^{0}(\Sigma,\vec{V})$,
identified to $\vec{V}$. Thus, $d$ induces a bijection

${\mathscr{M}}/\vec{V}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}{\mathscr{F}}_{0}.$

###  2.5. Euclidean structure and Hodge theory

The fiberwise Euclidean inner product $g$ on the vector bundle
$\Lambda^{n}\Sigma\otimes\vec{V}\to\Sigma$ induces an $L^{2}$-Euclidean inner
product on $\Omega^{n}(\Sigma,\vec{V})$, given by

${\mathcal{G}}(\beta_{1},\beta_{2})=\int_{\Sigma}g(\beta_{1},\beta_{2})\;\omega_{\Sigma},\quad\mbox{for
$\beta_{1},\beta_{2}\in\Omega^{n}(\Sigma,\vec{V})$.}$ (2.3)

For simplicity, we will often use the notations

$\langle\beta_{1},\beta_{2}\rangle=g(\beta_{1},\beta_{2}),\quad\langle\\!\langle\beta_{1},\beta_{2}\rangle\\!\rangle={\mathcal{G}}(\beta_{1},\beta_{2})$

and

$|\beta|=\sqrt{g(\beta,\beta)},\quad\|\beta\|_{L^{2}}=\sqrt{{\mathcal{G}}(\beta,\beta)}.$

The formal adjoint $d^{\star}$ of $d$ is defined by the property

${\mathcal{G}}(\beta_{1},d\beta_{2})={\mathcal{G}}(d^{\star}\beta_{1},\beta_{2})\quad\mbox{for
$\beta_{1}\in\Omega^{n+1}(\Sigma,\vec{V})$ and
$\beta_{2}\in\Omega^{n}(\Sigma,\vec{V})$}$

and the _Laplace operator_ $\Delta$ is given by the formula

$\Delta=dd^{\star}+d^{\star}d.$

The classical _Hodge theory_ extends to $\vec{V}$-valued differential forms.
In particular, we obtain a ${\mathcal{G}}$-orthogonal projection onto the
space of $\vec{V}$-valued exact $1$-forms denoted

$\Pi:{\mathscr{F}}\to{\mathscr{F}}_{0}.$

_Hölder spaces_ are better suited for elliptic operators, as the Laplace
operator $\Delta$. Recall that the Riemannian metric $g_{\Sigma}$ and the
fiberwise inner product $g$ induce a $C^{k,\nu}$-_Hölder norm_ for tensor
fields over $\Sigma$, denoted $\|\cdot\|_{k,\nu}$, where $k\in{\mathbb{N}}$ is
the number of derivatives and $\nu\in(0,1)$ is the Hölder regularity exponent
for the $k$-th derivative (cf. [12] for an explicit definition).

Hölder norms define Hölder completed spaces. In particular, we denote by
${\mathscr{F}}^{k,\nu}$ and ${\mathscr{M}}^{k,\nu}$ the completion of
${\mathscr{F}}$ and ${\mathscr{M}}$. For every $F\in{\mathscr{F}}^{k,\nu}$
with $k\geq 0$, the _Hodge decomposition_ theorem states that $F$ admits a
${\mathcal{G}}$-orthogonal decomposition

$F=F_{h}+\Delta G,$

where $G\in{\mathscr{F}}^{k+2,\nu}$ and $F_{h}$ is a smooth harmonic form. In
particular, we can define an orthogonal _Hodge projection_
$\Pi:{\mathscr{F}}^{k,\nu}\to{\mathscr{F}}^{k,\nu}_{0}$ by

$\Pi(F)=dd^{\star}G$

of $F$ onto its exact component. By Proposition 2.5.1, the projector $\Pi$ is
continuous with respect to Hölder topology. This result, which is an immediate
consequence of Hodge theory, is a crucial argument for the proof of Theorem C,
via the Cauchy-Lipschitz theorem (cf. Theorem 3.1.3). We provide a proof of
the proposition for the sake of self-containedness.

######  Proposition 2.5.1.

For $k\geq 0$ and $\nu\in(0,1)$, the orthogonal projection
$\Pi:{\mathscr{F}}^{k,\nu}\to{\mathscr{F}}_{0}^{k,\nu}$ onto the exact
component of a differential form is a continuous linear map with respect to
the $C^{k,\nu}$-norm. In other words, there exists a real constant $c>0$ such
that

$\|\Pi(F)\|_{{k,\nu}}\leq c\|F\|_{{k,\nu}},\quad\mbox{for every
$F\in{\mathscr{F}}^{k,\nu}$.}$

###### Proof.

For $F\in{\mathscr{F}}^{k,\nu}$, the Hodge decomposition theorem shows that

$F=F_{h}+\Delta G,$ (2.4)

where $G\in{\mathscr{F}}^{k+2,\nu}$ is a $1$-form orthogonal to harmonic forms
and $F_{h}$ is the harmonic part of $F$.

A $C^{k,\nu}$-estimate for $F$ provides a control on the $C^{k-2,\nu}$-norm of
$\Delta F$. By Formula (2.4), we have $\Delta F=\Delta^{2}G$, since $F_{h}$ is
harmonic. Hence the $C^{k,\nu}$-norm of $F$ control the $C^{k-2,\nu}$-norm of
$\Delta^{2}G$. The operator $\Delta$ is selfajoint, hence $\Delta G$ is
orthogonal to the kernel of $\Delta$. Then, elliptic Schauder estimates
provide a control on the $C^{k,\nu}$-norm of $\Delta G$. Since $G$ was chosen
orthogonal to harmonic forms, we deduce a $C^{k+2,\alpha}$ control on $G$ by
the Schauder estimates. In conclusion, there exists a universal constant
$c_{1}>0$, such that

$\|G\|_{{k+2,\nu}}\leq c_{1}\|F\|_{{k,\nu}}.$

Finally, the $C^{k+2,\nu}$-norm of $G$ controls the $C^{k,\nu}$-norm of
$dd^{\star}G$ and we conclude that there exists a universal constant $c>0$
that satisfies the proposition. ∎

###  2.6. Kähler structures on the moduli space

The space of differential forms $\Omega^{n}(\Sigma,\vec{V})$ has a structure
of module over $C^{\infty}(\Sigma,{\mathbb{C}})$, acting by complex
multiplication on $\vec{V}$-valued forms. More precisely

$(\lambda F)_{x}=\lambda(x)F_{x}$

for every $\lambda\in C^{\infty}(\Sigma,{\mathbb{C}})$,
$F\in\Omega^{n}(\Sigma,\vec{V})$ and $x\in\Sigma$.

The space of differential $1$-forms ${\mathscr{F}}$ admits an alternate almost
complex structure

${\mathcal{J}}:{\mathscr{F}}\to{\mathscr{F}},$

defined by

$({\mathcal{J}}\cdot F)_{x}\cdot\eta=(JF_{x})\cdot\eta=-F_{x}\circ
J_{\Sigma}\cdot\eta,$ (2.5)

where $J$ is defined by Formula (2.1), for every $x\in\Sigma$ and $\eta\in
T_{x}\Sigma$. By construction, the almost complex structure ${\mathcal{J}}$ is
compatible with the Eulidean $L^{2}$-inner product ${\mathcal{G}}$ on
${\mathscr{F}}$. The corresponding Kähler form $\Omega$ deduced from
${\mathcal{G}}$ and ${\mathcal{J}}$ is given by

$\Omega(\dot{F}_{1},\dot{F}_{2})={\mathcal{G}}({\mathcal{J}}\dot{F}_{1},F_{2})$

for every $\dot{F}_{1},\dot{F}_{2}\in{\mathscr{F}}$. Equivalently

$\Omega(\dot{F}_{1},\dot{F}_{2})=\int_{\Sigma}\omega(\dot{F}_{1},\dot{F}_{2})\omega_{\Sigma},$

where $\omega(\dot{F}_{1},\dot{F}_{2})=g(J\cdot\dot{F}_{1},\dot{F}_{2})$.

In conclusion, we have a natural Kähler structure

$({\mathscr{F}},{\mathcal{G}},{\mathcal{J}},\Omega)$

on the moduli space ${\mathscr{F}}$.

###  2.7. An involution

The complex vector space ${\mathscr{F}}$ admits several almost complex
structures: the almost complex structure $i:{\mathscr{F}}\to{\mathscr{F}}$,
corresponding to the multiplication by $i$ and the almost complex structure
${\mathcal{J}}$ described above. By construction, the two almost complex
structure $i$ and ${\mathcal{J}}$ commute. Therefore, the endomorphism

${\mathscr{R}}:{\mathscr{F}}\to{\mathscr{F}}$

defined by

${\mathscr{R}}F=i{\mathcal{J}}\cdot F$

is a _linear isometric involution_ of ${\mathscr{F}}$, which commutes with $i$
and ${\mathcal{J}}$. We obtain a ${\mathcal{G}}$-orthogonal decomposition

${\mathscr{F}}={\mathscr{F}}^{+}\oplus{\mathscr{F}}^{-}$

where ${\mathscr{F}}^{\pm}$ are the eigenspaces associated to the eigenvalues
$\pm 1$ of ${\mathscr{R}}$. More explicitely, we have

${\mathscr{F}}^{+}=\\{F\in{\mathscr{F}},F\circ J_{\Sigma}=iF\\}\quad\mbox{ and
}$ ${\mathscr{F}}^{-}=\\{F\in{\mathscr{F}},F\circ J_{\Sigma}=-iF\\}.$

In other words, ${\mathscr{F}}^{+}$ (resp. ${\mathscr{F}}^{-}$) is the
subspace of complex (resp. anti-complex) morphisms $F:T\Sigma\to\vec{V}$.
Hence, every $F\in{\mathscr{F}}$ admits a unique orthogonal decomposition

$F=F^{+}+F^{-},$

where $F^{\pm}\in{\mathscr{F}}^{\pm}$. By definition, we have

${\mathscr{R}}F=F^{+}-F^{-}.$

###  2.8. Gauge group action

We consider the infinite dimensional complex Lie group

${\mathbb{T}}^{\mathbb{C}}=C^{\infty}(\Sigma,{\mathbb{C}}^{*}),$

with trivial Lie algebra

$\mathfrak{t}^{\mathbb{C}}=C^{\infty}(\Sigma,{\mathbb{C}}).$

We define an action of ${\mathbb{T}}^{\mathbb{C}}$ on ${\mathscr{F}}$ as
follows: given $\lambda\in{\mathbb{T}}^{\mathbb{C}}$ and $F\in{\mathscr{F}}$,
we put

$\lambda\cdot F=\bar{\lambda}^{-1}F^{+}+\lambda F^{-},$ (2.6)

where $F=F^{+}+F^{-}$ according to the splitting
${\mathscr{F}}={\mathscr{F}}^{+}\oplus{\mathscr{F}}^{-}$ and $\bar{\lambda}$
denotes the complex conjugate. By construction the action of
${\mathbb{T}}^{\mathbb{C}}$ preserves the almost complex structure
${\mathcal{J}}$ for the following obvious reason:

######  Lemma 2.8.1.

Each subspace ${\mathscr{F}}^{\pm}$ is ${\mathcal{J}}$ invariant and the
action of ${\mathbb{T}}^{\mathbb{C}}$ on ${\mathscr{F}}$ is
${\mathcal{J}}$-linear.

The group ${\mathbb{T}}^{\mathbb{C}}$ contains the real subgroup

${\mathbb{T}}=C^{\infty}(\Sigma,S^{1}),$

where $S^{1}\subset{\mathbb{C}}$ is the unit circle. If
$\lambda\in{\mathbb{T}}$, then $\bar{\lambda}^{-1}=\lambda$, and group action
is merely given by complex multiplication:

$\lambda\cdot F=\lambda F,\quad\mbox{ for every }\lambda\in{\mathbb{T}}.$

###  2.9. Infinitesimal gauge group action

For $\zeta\in\mathfrak{t}^{\mathbb{C}}=C^{\infty}(\Sigma,{\mathbb{C}})$ we
define an exponential map

$\exp:\mathfrak{t}^{\mathbb{C}}\to{\mathbb{T}}^{\mathbb{C}},$

by

$\exp(\zeta)=e^{i\zeta},$

so that the space of real valued functions
$\mathfrak{t}=C^{\infty}(\Sigma,{\mathbb{R}})$ is identified to the Lie
algebra of ${\mathbb{T}}$. As usual, the infinitesimal action of
$\zeta\in\mathfrak{t}^{\mathbb{C}}$ is the vector field $X_{\zeta}$ on the
moduli space ${\mathscr{F}}$, defined by

$X_{\zeta}(F)=\left.\frac{d}{dt}\right|_{t=0}\exp(t\zeta)\cdot F.$

Formula (2.6) shows that

$X_{\zeta}(F)=i\bar{\zeta}F^{+}+i\zeta F^{-}.$ (2.7)

In particular, if $\zeta\in\mathfrak{t}$, we have

$X_{\zeta}(F)=i\zeta F.$ (2.8)

If, on the contrary, $\zeta\in i\mathfrak{t}$ is a purely imaginary function,
we have

$X_{\zeta}(F)=-i\zeta{\mathscr{R}}F.$ (2.9)

By (2.8) and (2.9), the infinitesimal action of $\zeta\in\mathfrak{t}$
satisfies

${\mathcal{J}}\cdot X_{\zeta}(F)={\mathcal{J}}\cdot i\zeta
F=\zeta{\mathscr{R}}F=-i(i\zeta){\mathscr{R}}F=X_{i\zeta}(F),$

and we deduce the following result:

######  Lemma 2.9.1.

The action of ${\mathbb{T}}^{\mathbb{C}}$ on ${\mathscr{F}}$ is the
${\mathcal{J}}$-complexification of the action of ${\mathbb{T}}$.

In addition, the gauge group ${\mathbb{T}}$ acts isometrically on
${\mathscr{F}}$. Indeed for every $\lambda\in{\mathbb{T}}$ and
$F\in{\mathscr{F}}$,

$\|\lambda\cdot F\|^{2}_{L^{2}}=\|\lambda F\|^{2}_{L^{2}}=\|F\|^{2}_{L^{2}}.$

In conclusion, we have the following result:

######  Proposition 2.9.2.

The action of ${\mathbb{T}}$ on ${\mathscr{F}}$ is linear and preserves the
Kähler structure $({\mathscr{F}},{\mathcal{G}},{\mathcal{J}},\Omega)$.

###  2.10. Symplectic density

The map

${\mu:{\mathscr{F}}}$${\mathfrak{t}=C^{\infty}(\Sigma,{\mathbb{R}})}$${F}$${-\frac{1}{2}g(F,{\mathscr{R}}F).}$
(2.10)

can be interpreted as a _symplectic density_ , according to the following
lemma:

######  Lemma 2.10.1.

The following formulas hold

$\mu(F)=-\frac{1}{2}\omega(iF,F)=-\frac{1}{2}(|F^{+}|^{2}-|F^{-}|^{2})=-\frac{F^{*}\omega_{V}}{\omega_{\Sigma}},$
(2.11)

for every $F\in{\mathscr{F}}$, where the pullback is defined by
$(F^{*}\omega_{V})(\eta_{1},\eta_{2})=\omega_{V}(F(\eta_{1}),F(\eta_{2}))$,
for every $x\in\Sigma$ and $\eta_{1},\eta_{2}\in T_{x}\Sigma$.

###### Proof.

Using the fact that
$-\frac{1}{2}g({\mathscr{R}}F,F)=-\frac{1}{2}g(i{\mathcal{J}}F,F)=-\frac{1}{2}g({\mathcal{J}}iF,F)=-\frac{1}{2}\omega(iF,F)$
we deduce the first identity.

Using the decomposition $F=F^{+}+F^{-}$, we write
$g({\mathscr{R}}F,F)=g(F^{+}-F^{-},F^{+}+F^{-})$. Using the fact the $F^{+}$
and $F^{-}$ are pointwise $g$-orthogonal, we deduce that
$g({\mathscr{R}}F,F)=|F^{+}|^{2}-|F^{-}|^{2}$, which proves the second
identity.

Let $x\in\Sigma$ and $(e_{1},e_{2})$ be a $g_{\Sigma}$-orthonormal oriented
basis of $T_{x}\Sigma$. In particular $\omega_{\Sigma}(e_{1},e_{2})=1$ and
$J_{\Sigma}e_{1}=e_{2}$. Hence $({\mathcal{J}}\cdot
F)(e_{1})=-F(J_{\Sigma}e_{1})=-F(e_{2})$. Similarly $({\mathcal{J}}\cdot
F)(e_{2})=F(e_{1})$. Hence ${\mathscr{R}}F(e_{1})=-iF(e_{2})$ and
${\mathscr{R}}F(e_{2})=iF(e_{1})$. By definition

$\displaystyle\mu(F)(x)$ $\displaystyle=-\frac{1}{2}g({\mathscr{R}}F,F)(x)$
$\displaystyle=-\frac{1}{2}g_{V}({\mathscr{R}}F(e_{1}),F(e_{1}))-\frac{1}{2}g_{V}({\mathscr{R}}F(e_{2}),F(e_{2}))$
$\displaystyle=-\frac{1}{2}(-g_{V}(iF(e_{2}),F(e_{1}))+g_{V}(iF(e_{1}),F(e_{2})))$
$\displaystyle=-\omega_{V}(F(e_{1}),F(e_{2}))$
$\displaystyle=-\frac{(F^{*}\omega_{V})(e_{1},e_{2})}{\omega_{\Sigma}(e_{1},e_{2})},$

which proves the last identity of the lemma. ∎

We introduce some h-principle terminology before stating an immediate
corollary below.

######  Definition 2.10.2.

A differential form ${\mathscr{F}}$ is called isotropic if
$F:T\Sigma\to\vec{V}$ maps every tangent space to an isotropic subspace of $V$
or, equivalently, if $F^{*}\omega_{V}=0$. If the restriction of
$F:T\Sigma\to\vec{V}$ to every tangent space is injective, we say that $F$ is
a monomorphism.

######  Corollary 2.10.3.

Let $F$ be an element of ${\mathscr{F}}$. The following properties are
equivalent:

  1. (1)

$F\in{\mathscr{F}}$ is isotropic

  2. (2)

$\mu(F)=0\in\mathfrak{t}$ .

  3. (3)

$|F^{+}|=|F^{-}|$ identically on $\Sigma$.

If $F\in{\mathscr{F}}_{0}$, we deduce the following interpretation for
isotropic maps:

######  Corollary 2.10.4.

For every map $f\in{\mathscr{M}}$, the following properties are equivalent:

  1. (1)

$f$ is an isotropic map.

  2. (2)

$F=df\in{\mathscr{F}}_{0}$ is isotropic.

  3. (3)

The map $f$ satisfies the equation $\mu(df)=0$.

In particular, the space of isotropic maps modulo $\vec{V}$ agrees with the
zeroes of $\mu$ in ${\mathscr{F}}_{0}$ via the bijection
$d:{\mathscr{M}}/\vec{V}\to{\mathscr{F}}_{0}$.

###  2.11. Hamiltonian action

We now show that $\mu$ is indeed a moment map:

######  Theorem 2.11.1.

The action of ${\mathbb{T}}$ on ${\mathscr{F}}$ is Hamiltonian with moment map
$\mu$. More precisely, $\mu$ is ${\mathbb{T}}$-invariant and

$D\langle\\!\langle\mu,\zeta\rangle\\!\rangle=-\iota_{X_{\zeta}}\Omega$

for every $\zeta\in\mathfrak{t}$.

###### Proof.

The invariance of $\mu$ is clear, by definition. The proof of the theorem
starts with a computation:

######  Lemma 2.11.2.

For every $F,\dot{F}\in{\mathscr{F}}$, we have

$D\mu|_{F}\cdot\dot{F}=-g({\mathscr{R}}F,\dot{F}).$

###### Proof.

By bilinearity, $2D\mu|_{F}\cdot
F=-g({\mathscr{R}}F,\dot{F})-g(F,{\mathscr{R}}\dot{F})$ and he lemma follows
from the fact that ${\mathscr{R}}$ is pointwise $g$-selfadjoint. ∎

For $\zeta\in\mathfrak{t}$, we have $X_{\zeta}(F)=i\zeta F$ by Formula (2.8)
and it follows that

$\displaystyle\Omega(X_{\zeta}(F),\dot{F})$ $\displaystyle=\Omega(i\zeta
F,\dot{F})$ $\displaystyle=\langle\\!\langle{\mathcal{J}}i\zeta
F,\dot{F}\rangle\\!\rangle$
$\displaystyle=\langle\\!\langle\zeta{\mathscr{R}}F,\dot{F}\rangle\\!\rangle$
$\displaystyle=\int_{\Sigma}g(\zeta{\mathscr{R}}F,\dot{F})\omega_{\Sigma}$
$\displaystyle=-\int_{\Sigma}\zeta D\mu|_{F}\cdot\dot{F}\omega_{\Sigma}$
$\displaystyle=-\langle\\!\langle
D\mu|_{F}\cdot\dot{F},\zeta\rangle\\!\rangle,$

which proves the theorem. ∎

###### Proof of Theorem A and Corollary B.

The restatement of the constructions carried out at §2, together with Theorem
2.11.1 and Lemma 2.10.1 prove Theorem A. Corollary B is a restatement of
Corollary 2.10.4. ∎

###  2.12. Stability and isotropic maps

The Kempf-Ness theory, for Kähler moment map geometry, relates the symplectic
reduction with geometric invariant theory. This point of view seems appealing
in our case, where ${\mathscr{F}}$ is acted on by the complex gauge group
${\mathbb{T}}^{\mathbb{C}}$ and the $\mu$ is a moment map for ${\mathbb{T}}$.
The question of existence of a zero of the moment map in a
${\mathbb{T}}^{\mathbb{C}}$-orbit is rather trivial: for simplicity, we define
the ${\mathbb{T}}^{\mathbb{C}}$-invariant subspace of _generic_ differentials
${\mathscr{F}}_{gen}$ as the subspace of nowhere vanishing differential forms.
Then we have the following result

######  Lemma 2.12.1.

For every $F\in{\mathscr{F}}_{gen}$, the following properties are equivalent:

  1. (1)

$F^{+}\in{\mathscr{F}}_{gen}$ and $F^{-}\in{\mathscr{F}}_{gen}$.

  2. (2)

There exists $\lambda\in{\mathbb{T}}^{\mathbb{C}}$ such that $\mu(\lambda\cdot
F)=0$.

In particular, the orbit of $F\in{\mathscr{F}}^{\pm}\setminus 0$ does not
contain any zero of the moment map.

###### Proof.

For $\lambda:\Sigma\to{\mathbb{R}}\setminus 0$ we have

$\mu(\lambda\cdot
F)=-\frac{1}{2}(\lambda^{-2}|F^{+}|^{2}-\lambda^{2}|F^{-}|^{2})$

and the lemma is obvious. ∎

However, we are mostly interested in the zeroes of the moment map in
${\mathscr{F}}_{0}$, as they are differentials of isotropic maps by Corollary
B. It would be interesting to understand how the space of isotropic maps in
${\mathscr{M}}$ is related to some notion of geometric stability on
${\mathscr{F}}$. Unfortunately, the image ${\mathscr{F}}_{0}=d({\mathscr{M}})$
is not ${\mathbb{T}}$-invariant and it is not clear how to obtain an analogue
of Kempf-Ness theory from this point.

###  2.13. Energy of the moment map

We consider the energy of the moment map $\mu$, given by the functional

${\phi:{\mathscr{F}}}$${{\mathbb{R}}}$${F}$${\phi(F)=\frac{1}{2}\|\mu(F)\|^{2}_{L^{2}}}$
(2.12)

Obviously, $\phi$ is non negative and

$\phi^{-1}(0)=\mu^{-1}(0)$

which is to say that the vanishing locus is the space of isotropic
differential forms in ${\mathscr{F}}$. By Corollary B, the vanishing locus of
$\phi:{\mathscr{F}}_{0}\to{\mathbb{R}}$ is identified to the subspace of
isotropic maps in ${\mathscr{M}}$ modulo the action of $\vec{V}$ by
translations via the correspondence
$d:{\mathscr{M}}/\vec{V}\to{\mathscr{F}}_{0}$.

We prove various formulas about the differential and the gradient of the
functional $\phi$ on ${\mathscr{F}}$:

######  Proposition 2.13.1.

For every $F,\dot{F}\in{\mathscr{F}}$, we have

$D\phi|_{F}\cdot\dot{F}=-\langle\\!\langle\mu(F),g({\mathscr{R}}F,\dot{F})\rangle\\!\rangle,$
$D\phi|_{F}\cdot F=4\phi(F),$

and the gradient of the functional $\phi:{\mathscr{F}}\to{\mathbb{R}}$ is
given by the formula,

$\nabla\phi(F)=-\mu(F){\mathscr{R}}F.$

or

$\nabla\phi=-{\mathcal{J}}Z,$ (2.13)

where $Z$ is the vector field on ${\mathscr{F}}$ defined by

$Z(F)=X_{\mu(F)}(F).$

###### Proof.

By definition of $\phi(F)=\frac{1}{2}\|\mu(F)\|^{2}$ hence

$\left.D\phi\right|_{F}\cdot\dot{F}=\langle\\!\langle\mu(F),D\mu|_{F}\cdot\dot{F}\rangle\\!\rangle.$

By Lemma 2.11.2, we have

$\left.D\phi\right|_{F}\cdot\dot{F}=-\langle\\!\langle\mu(F),g({\mathscr{R}}F,\dot{F})\rangle\\!\rangle.$

In particular, for $\dot{F}=F$, we have $D\phi|F\cdot
F=-\langle\\!\langle\mu(F),g({\mathscr{R}}F,F)\rangle\\!\rangle=2\langle\\!\langle\mu(F),\mu(F)\rangle\\!\rangle=2\|\mu(F)\|^{2}=4\phi(F)$.

Now

$\displaystyle D\phi|_{F}\cdot\dot{F}$
$\displaystyle=\langle\\!\langle\mu(F),g({\mathscr{R}}F,\dot{F})\rangle\\!\rangle$
$\displaystyle=\int_{\Sigma}\mu(F)g({\mathscr{R}}F,\dot{F})\omega_{\Sigma}$
$\displaystyle=\int_{\Sigma}g(\mu(F){\mathscr{R}}F,\dot{F})\omega_{\Sigma}$
$\displaystyle=\langle\\!\langle\mu(F){\mathscr{R}}F,\dot{F}\rangle\\!\rangle$

and we deduce that

$\nabla\phi(F)=-\mu(F){\mathscr{R}}F.$

In particular
${\mathcal{J}}\nabla\phi(F)=-\mu(F){\mathcal{J}}{\mathscr{R}}F=\mu(F)iF=X_{\mu(F)}(F)$
and we conclude that

$\nabla\phi=-{\mathcal{J}}Z$

where $Z$ is the vector field on ${\mathscr{F}}$ defined by
$Z(F)=X_{\mu(F)}(F)$. ∎

######  Corollary 2.13.2.

The following properties are equivalent for $F\in{\mathscr{F}}_{0}$

  1. (1)

$F$ is a zero of $\phi:{\mathscr{F}}_{0}\to{\mathbb{R}}$.

  2. (2)

$F$ is a zero of $\mu$ in ${\mathscr{F}}_{0}$.

  3. (3)

$F$ is isotropic.

  4. (4)

$F$ is a critical point of $\phi:{\mathscr{F}}_{0}\to{\mathbb{R}}$.

###### Proof.

We know that $(2)\Leftrightarrow(3)$ by Corollary 2.10.4 and we prove that
$(1)\Rightarrow(2)\Rightarrow(4)\Rightarrow(1)$. If $F$ is a zero of
$\phi:{\mathscr{F}}_{0}\to{\mathbb{R}}$, it is a zero of $\mu$, by definition
of $\phi$. If $\mu(F)=0$, then $D\phi|_{F}=0$ by Proposition 2.13.1, hence $F$
is a critical point of $\phi:{\mathscr{F}}\to{\mathbb{R}}$. This implies that
$F$ is also a critical point of the restricted functional
$\phi:{\mathscr{F}}_{0}\to{\mathbb{R}}$. If $F$ is a critical point of the
restriction, then $D\phi|_{F}\cdot F=0$. By Proposition 2.13.1, we deduce that
$0=D\phi|_{F}\cdot F=4\phi(F)$ and we conclude that $\phi(F)=0$. ∎

##  3\. The modified moment map flow

The subspace of exact differentials ${\mathscr{F}}_{0}\subset{\mathscr{F}}$ is
not stable under the ${\mathbb{T}}$-action. Hence, the gradient $\nabla\phi$
is generally not tangent to ${\mathscr{F}}_{0}$. The restriction of the
functional $\phi:{\mathscr{F}}_{0}\to{\mathbb{R}}$ admits a gradient vector
field $\nabla^{\circ}\phi$ related to $\nabla\phi$ by the formula

$\nabla^{o}\phi(F)=\Pi(\nabla\phi(F)).$

We define the _modified moment map flow_ by the evolution equation

$\frac{\partial F}{\partial t}=-\nabla^{o}\phi(F)$ (3.1)

along ${\mathscr{F}}_{0}$ and, more generally, along the Hölder completion
${\mathscr{F}}_{0}^{k,\nu}$.

###  3.1. Basic properties of the flow

An interesting feature of the flow is the following decay property:

######  Theorem 3.1.1.

Let $F_{t}$ be a solution of the modified moment map flow, for $t$ in some
interval of $I$. Then

$\frac{\partial}{\partial t}\|F_{t}\|^{2}_{L^{2}}=-8\phi(F_{t}).$ (3.2)

In particular $t\mapsto\|F_{t}\|_{L^{2}}$ is non increasing along $I$.

###### Proof.

We have

$\displaystyle\frac{1}{2}\frac{\partial}{\partial t}\|F_{t}\|_{L^{2}}^{2}$
$\displaystyle=\langle\\!\langle\frac{\partial F_{t}}{\partial
t},F_{t}\rangle\\!\rangle$
$\displaystyle=-\langle\\!\langle\nabla^{o}\phi(F_{t}),F_{t}\rangle\\!\rangle$
$\displaystyle=-\langle\\!\langle\nabla\phi(F_{t}),F_{t}\rangle\\!\rangle$
$\displaystyle=-4\phi(F_{t}).$

∎

Another important essential property is that the stationary points of the
modified moment map flow are exactly the zeroes of $\phi$.

######  Proposition 3.1.2.

The fixed point set of the modified moment map flow agrees with the vanishing
set of $\phi:{\mathscr{F}}_{0}\to{\mathbb{R}}$.

###### Proof.

This is an immediate consequence of Corollary 2.13.2. ∎

The modified moment map flow has the short time existence property according
to the following theorem:

######  Theorem 3.1.3.

For every $k\geq 0$, $\nu\in(0,1)$ and $F\in{\mathscr{F}}_{0}^{k,\nu}$, there
exists $\varepsilon>0$ and a unique solution of the modified moment map flow
$F_{t}\in{\mathscr{F}}_{0}^{k,\nu}$, defined for
$t\in[-\varepsilon,\varepsilon]$ such that $F_{0}=F$.

###### Proof.

The map $F\mapsto\nabla\phi(F)$ is polynomial of order $3$ in the coefficients
of $F$ by Proposition 2.13.1. In particular, it is locally Lipschitz with
respect to the $C^{k,\nu}$-norm. By Proposition 2.5.1, the Hodge projector
$\Pi:{\mathscr{F}}^{k,\nu}\to{\mathscr{F}}_{0}^{k,\nu}$ is continuous for the
$C^{k,\nu}$-norm. Therefore the maps
$F\mapsto\nabla^{\circ}\phi(F)=\Pi\nabla\phi(F)$ is locally Lipschitz on
${\mathscr{F}}_{0}^{k,\nu}$. The Cauchy-Lipschitz theorem applies and the
theorem is proved. ∎

###### Proof of Theorem C.

The result is just a restatement of Theorem 3.1.3, Proposition 3.1.2 and
Theorem 3.1.1. ∎

###  3.2. Energy as a Morse-Bott function

By Proposition 2.13.1, we have

$D\phi|_{F}\cdot\dot{F}_{1}=-\langle\\!\langle\mu(F),g({\mathscr{R}}F,\dot{F}_{1})\rangle\\!\rangle$

for every $F,\dot{F}_{1}\in{\mathscr{F}}$ and second variation of $\phi$ with
respect to some $\dot{F}_{2}\in{\mathscr{F}}$ is given by

$D^{2}\phi|_{F}\cdot(\dot{F}_{1},\dot{F}_{2})=\langle\\!\langle
D\mu|_{F}\cdot\dot{F}_{1},D\mu|_{F}\cdot\dot{F}_{2}\rangle\\!\rangle-\langle\\!\langle\mu(F),g({\mathscr{R}}\dot{F}_{2},\dot{F}_{1})\rangle\\!\rangle.$

In particular, if $\mu(F)=0$, we have

$D^{2}\phi|_{F}\cdot(\dot{F}_{1},\dot{F}_{2})=\langle\\!\langle
D\mu|_{F}\cdot\dot{F}_{1},D\mu|_{F}\cdot\dot{F}_{2}\rangle\\!\rangle$ (3.3)

which proves the following lemma:

######  Lemma 3.2.1.

Let $F\in{\mathscr{F}}$ be an isotropic differential. Then, $D^{2}\phi|_{F}$
is a non negative bilinear form. Furthermore, the kernel of $D^{2}\phi|_{F}$
is given by the kernel of the linear map $\ker D\phi|_{F}$.

According to the above lemma, the functional
$\phi:{\mathscr{F}}_{0}\to{\mathbb{R}}$ seems to behave like a Morse-Bott
function. The rest of this section is devoted to proving that this is indeed
the case, in a neighborhood of a monomorphism.

Let $s_{0}$ be the $0$-section of the bundle $T^{*}\Sigma\to\Sigma$. The map

$\Phi:\mathrm{Diff}_{k,\nu}(\Sigma)\times\Omega^{1}_{k,\nu}(\Sigma)\to
C^{k,\nu}(\Sigma,T^{*}\Sigma)$

defined by $\Phi(\varphi,s)=s\circ\varphi$ for some $k\geq 1$ is smooth.
Furthermore $\Phi$ defines a local diffeomorphism between a neighborhood of
$(\mathrm{id},s_{0})$ and a neighborhood of $s_{0}$ in
$C^{k,\nu}(\Sigma,T^{*}\Sigma)$.

It is a well known fact that $T^{*}\Sigma$ is endowed with a canonical
symplectic form $\omega_{T^{*}}=d\Lambda$, where $\Lambda$ is the _Liouville
form_ on $T^{*}\Sigma$. Furthermore, a map $h:\Sigma\to T^{*}\Sigma$,
sufficiently close to $s_{0}$ in $C^{1}$-norm is isotropic if, and only if,
$h=\Phi(\phi,s)$ for some closed differential $1$-form $s$ [14].

It follows that the subspace of isotropic maps is a submanifold of
$C^{k,\nu}(\Sigma,T^{*}\Sigma)$ in a neighborhood of $s_{0}$. Furthermore, the
subspace of isotropic maps is locally diffeomorphic via $\Psi$ to the
submanifold $\mathrm{Diff}_{k,\nu}(\Sigma)\times\Omega^{1,c}_{k,\nu}(\Sigma)$
in a neighborhood of $(\mathrm{id},s_{0})$, where
$\Omega^{1,c}_{k,\nu}(\Sigma)$ is the subspace of closed forms.

We now specialize to the particular case where $V$ has real dimension $4$. Let
$f:\Sigma\looparrowright V$ be an isotropic immersion. Then, the image
$M=f(\Sigma)$ is known an _immersed Lagrangian surface_ of $V$. By the
Lagrangian neighborhood theorem [14], there exists a neighborhood
${\mathscr{U}}$ of the zero section $s_{0}$ in $T^{*}\Sigma$ and a
neighborhood ${\mathscr{V}}$ of $M$ in $V$, together with a smooth immersion
$\theta:{\mathscr{U}}\to{\mathscr{V}}$, such that $\theta\circ s_{0}=f$ and
$\theta^{*}\omega_{V}=\omega_{T^{*}}$.

Every map $\tilde{f}:\Sigma\to V$, sufficiently close to $f$ $C^{1}$-norm, is
given by a map $h:\Sigma\to T^{*}\Sigma$ sufficiently close to $s_{0}$, such
that $\tilde{f}=\theta\circ h$. Furthemore, $\tilde{f}$ is isotropic if, and
only if, $h$ is isotropic since $\theta$ preserves the symplectic forms. We
put $\Psi(\varphi,s)=\theta\circ s\circ\varphi$ and deduce the following
result:

######  Theorem 3.2.2.

We assume that $\dim_{\mathbb{R}}V=4$ and let $f:\Sigma\looparrowright V$ be
an isotropic immersion with Hölder regularity $C^{k,\nu}$, for some $k\geq 1$.
There exists smooth map

$\Psi:\mathrm{Diff}_{k,\nu}(\Sigma)\times\Omega^{1}_{k,\nu}(\Sigma)\to
C^{k,\nu}(\Sigma,V)$

defined in a neighborhood of $\mathrm{Diff}_{k,\nu}(\Sigma)\times\\{s_{0}\\}$,
such that

  1. (1)

$\Psi(\mathrm{id},s_{0})=f$

  2. (2)

$\Psi$ is $\mathrm{Diff}_{k,\nu}(\Sigma)$-equivariant

  3. (3)

$\Psi$ is a local diffeomorphism in a neighborhood of $(\mathrm{id},s_{0})$

  4. (4)

The map $\Psi$ restrict as a local diffeomorphism at $(\mathrm{id},s_{0})$
between $\mathrm{Diff}_{k,\nu}(\Sigma)\times\Omega^{1,c}_{k,\nu}(\Sigma)$ the
isotropic deformations of $f$.

In particular, the subspace of isotropic deformations in a neighborhood is a
submanifold of $C^{k,\nu}(\Sigma)$ in a sufficiently small open neighborhood
of $f$.

For $k\geq 1$, the differential $d$ induces a diffeomorphism

${\mathscr{M}}^{k,\alpha}\to{\mathscr{F}}^{k-1,\alpha}_{0}\times V$

given by $f\mapsto(df,f(x_{0}))$, where $x_{0}\in\Sigma$ is some fixed marked
point. The above diffeormorphism restrics to a diffeomorphism between
isotropic maps and istropic differentials and we deduce the following
corollary:

######  Corollary 3.2.3.

Assuming $\dim_{\mathbb{R}}V=4$ and $k\geq 0$, let
$F\in{\mathscr{F}}_{0}^{k,\nu}$ be an isotropic monomorphism, for some $k\geq
0$. Then the subspace of isotropic differentials is a submanifold of
${\mathscr{F}}_{0}^{k,\nu}$ in a neighborhood of $F$. Furthemore, the Hessian
of $\phi$ is positive in transverse direction to the submanifold of isotropic
differential near $F$.

##  4\. Polyhedral isotropic surfaces

In this section, we adapt all the smooth constructions of §2 and §3 to the
_polyhedral setting_.

###  4.1. Definition of polyhedral surfaces

A _triangulation_ of a surface $\Sigma$ is a triple
${\mathscr{T}}=(\Sigma,{\mathscr{K}},\ell)$, where:

  1. (1)

$\Sigma$ is a topological surface,

  2. (2)

${\mathscr{K}}$ is a locally finite _simplicial complex_ , contained in some
ambient affine space $A$ and

  3. (3)

$\ell:|{\mathscr{K}}|\to\Sigma$ is a homeomorphism, where $|{\mathscr{K}}|$ is
the topological space associated to the simplicial complex.

Given a triangulation ${\mathscr{T}}$, every simplex $\sigma\in{\mathscr{K}}$
defines a subset $\ell(\sigma)\subset\Sigma$, homeomorphic to $\sigma$. It is
often convenient to think of $\sigma$ as a domain in $\Sigma$, using the
homeomorphism $\ell$ and we will take the liberty to drop the reference to
$\ell$ in our notations.

Every simplex $\sigma$ of an affine space $A$ spans an affine subspace of $A$,
with underlying vector space denoted $\vec{\sigma}$, also called the _tangent
direction of the simplex_ $\sigma$.

Let $g_{\Sigma}=(g_{\sigma})_{\sigma\in{\mathscr{K}}}$, be a family, where
$g_{\sigma}$ is a Euclidean inner product on $\vec{\sigma}$. Suppose that for
every $\sigma_{1},\sigma_{2}\in{\mathscr{K}}$ with
$\sigma_{2}\subset\sigma_{1}$, the metric $g_{\sigma_{2}}$ agrees with the
restriction of $g_{\sigma_{1}}$ to $\vec{\sigma}_{2}$. In such a case, we say
that $g_{\Sigma}$ is a _polyhedral metric_ on $\Sigma$.

######  Definition 4.1.1.

A topological surface $\Sigma$ endowed with a triangulation
${\mathscr{T}}=(\Sigma,{\mathscr{K}},\ell)$ and a polyhedral metric
$g_{\Sigma}$ is called a _polyhedral surface_.

The polyhedral metric $g_{\Sigma}$ provides a flat Riemannian metric
$g_{\sigma}$ on each simplex $\sigma\in{\mathscr{K}}$. As in the smooth case
discussed at §2.3, the metric $g_{\sigma}$ induces a fiberwise Euclidean inner
product, denoted $g_{\sigma}$ as well, on all the _tensor bundles_ over the
simplex $\sigma$. Together with $g_{V}$, we deduce a Euclidean fiberwise inner
product for the bundles of $\vec{V}$-valued forms on $\sigma$, also denoted
$g_{\sigma}$. For simplicity of notation, these inner products are sometimes
denoted $\langle\cdot,\cdot\rangle$ and the corresponding norm is denoted
$|\cdot|$.

###  4.2. Whitney cohomology

The theory of Whitney forms and Whitney cohomology [21] is a generalisation,
in the piecewise linear setting, of smooth differential forms and de Rham
cohomology. The spaces of smooth differential forms $\Omega^{n}(\sigma)$ on a
simplex $\sigma\in{\mathscr{K}}$ are compatible with pullbacks in the
following sense: for every $\sigma_{1},\sigma_{2}\in{\mathscr{K}}$ with
$\sigma_{2}\subset\sigma_{1}$ and $\beta_{1}\in\Omega^{n}(\sigma_{1})$, the
pullback $\beta_{2}$ of $\beta_{1}$ on $\sigma_{2}$ is a smooth differential
form as well.

We denote by $\Omega^{n}(\Sigma,{\mathscr{T}})$ the space of families of
differential form $\beta=(\beta_{\sigma})_{\sigma\in{\mathscr{K}}}$, where
each $\beta_{\sigma}\in\Omega^{n}(\Sigma)$. Given
$\beta=(\beta_{\sigma})\in\Omega^{n}(\Sigma,{\mathscr{T}})$, suppose that for
every $\sigma_{1},\sigma_{2}\in{\mathscr{K}}$, with
$\sigma_{2}\subset\sigma_{1}$, the pull back of $\beta_{\sigma_{1}}$ agrees
with $\beta_{\sigma_{2}}$. Then we say that $\beta$ satifies the _Whitney
condition_ , or that $\beta$ is a _Whitney form_. The space of Whitney forms
is denoted $\Omega_{w}^{n}(\Sigma,{\mathscr{T}})$.

There is a natural exterior derivative
$d:\Omega^{n}(\Sigma,{\mathscr{T}})\to\Omega^{n+1}(\Sigma,{\mathscr{T}})$
given by $d\beta=(d\beta_{\sigma})_{\sigma\in{\mathscr{K}}}$. Furthermore, the
spaces of Whitney forms are preserved by $d$ and we obtain the Whitney complex

$d:\Omega^{n}_{w}(\Sigma,{\mathscr{T}})\to\Omega_{w}^{n+1}(\Sigma,{\mathscr{T}})$

which defines the Whitney cohomology denoted
$H^{n}_{w}(\Sigma,{\mathbb{R}},{\mathscr{T}})$. The constructions readily
extend to the case of $\vec{V}$-valued differential forms and we obtain the
Whitney cohomology spaces $H^{n}_{w}(\Sigma,\vec{V},{\mathscr{T}})$. Similarly
to the de Rham cohomology, the Whitney cohomology agrees with the usual
cohomology spaces $H^{n}(\Sigma,{\mathbb{R}})\simeq
H^{n}_{w}(\Sigma,{\mathbb{R}},{\mathscr{T}})$ and $H^{n}(\Sigma,\vec{V})\simeq
H^{n}_{w}(\Sigma,\vec{V},{\mathscr{T}})$. In particular, the Whitney
cohomology does not depend on the choice of triangulation ${\mathscr{T}}$.

###  4.3. Orientation and Kähler structure

An _orientation_ of the simplicial complex ${\mathscr{K}}$ of a polyhedral
surface $\Sigma$ is also called an orientation of $\Sigma$. An _oriented
polyhedral surface carries_ a canonical area form
$\omega_{\Sigma}=(\omega_{\sigma})_{\sigma\in{\mathscr{K}}}\in\Omega^{2}_{w}(\Sigma,{\mathscr{T}})$
characterized by

  1. (1)

$\omega_{\sigma}$ is an exterior $2$-form on $\vec{\sigma}$.

  2. (2)

For every $\sigma\in{\mathscr{K}}_{2}$, the volume form $\omega_{\sigma}$ is
compatible with the orientation of the facet and $|\omega_{\sigma}|=1$, with
respect to $g_{\sigma}$.

  3. (3)

If $\sigma\in{\mathscr{K}}$ is an edge or a vertex then $\omega_{\sigma}=0$.

The combination of $\omega_{\sigma}$ and $g_{\sigma}$ defines a family of
almost complex structures
$J_{\Sigma}=(J_{\sigma})_{\sigma\in{\mathscr{K}}_{2}}$ on the facets of
${\mathscr{K}}$, where $J_{\sigma}:\vec{\sigma}\to\vec{\sigma}$ is an almost
complex structure on $\vec{\sigma}$, with the property that
$\omega_{\sigma}=g_{\sigma}(J_{\sigma}\cdot,\cdot)$.

In conclusion, an oriented polyhedral surface $\Sigma$ is endowed with the
structures $({\mathscr{T}},g_{\Sigma},J_{\Sigma},\omega_{\Sigma})$, referred
to as a _polyhedral Kähler structure_ , a notion introduced by Dmitri Panov in
[16], where he investigates the $4$ dimensional case.

######  Remark 4.3.1.

A polyhedral Kähler surface $\Sigma$ as above admits a smooth structure [16,
19, 20] such that

  1. (1)

$g_{\Sigma}$ and $\omega_{\Sigma}$ are smooth away from the vertices of the
triangulation.

  2. (2)

The metric may have conical singularities at the vertices.

  3. (3)

The almost complex structure $J_{\Sigma}$ is smooth on $\Sigma$.

In presence of conical singularities, the map $\ell:\sigma\to\Sigma$ is not
smooth at the vertices. In particular, the smooth differential forms on
$\Sigma$ do not necessarily induce smooth families of Whitney forms in
$\Omega^{n}_{w}(\Sigma,{\mathscr{T}})$.

###  4.4. Polyhedral maps and differentials

We continue our constructions and analogies with the smooth setting, when
$\Sigma$ is a closed oriented polyhedral surface.

A map $f:\Sigma\to V$, or more generally a map to some affine space, is called
_polyhedral_ with respect to ${\mathscr{T}}$ if:

  1. (1)

the map $f$ is continuous and

  2. (2)

the restriction of $f$ to every simplex of the triangulation ${\mathscr{T}}$
is an affine map.

The space of polyhedral maps is denoted

${\mathscr{M}}({\mathscr{T}})=\\{f:\Sigma\to V,\mbox{ $f$ is
${\mathscr{T}}$-polyhedral}\\}.$

The restriction of a polyhedral map $f$ to every simplex
$\sigma\in{\mathscr{T}}$ is affine, hence differentiable. However, $f$ is
generally not differentiable at every point of $\Sigma$. Nevertheless, we can
define the differential $df$ as a family

$df=(df|_{\sigma})_{\sigma\in{\mathscr{K}}}.$

By construction, $df$ is a Whitney form
$df\in\Omega^{1}_{w}(\Sigma,\vec{V},{\mathscr{T}})$. The moduli space
${\mathscr{M}}({\mathscr{T}})$ is an affine space with underlying vector space
contained in $\Omega^{0}_{w}(\Sigma,{\mathscr{T}},\vec{V})$ and it follows
that $df$ is an _exact_ Whitney form.

These observations motivate the introduction of a space of constant
$\vec{V}$-valued differentials, closely related to
$\Omega^{1}(\Sigma,\vec{V},{\mathscr{T}})$:

${\mathscr{F}}({\mathscr{T}})=\\{(F_{\sigma})_{\sigma\in{\mathscr{K}}_{2}},F_{\sigma}\in\vec{\sigma}^{*}\otimes\vec{V}\\},$

where $\vec{\sigma}^{*}$ is the dual of the tangent direction $\vec{\sigma}$.
An element of $\vec{\sigma}^{*}\otimes\vec{V}$ is a linear map
$F_{\sigma}:\vec{\sigma}\to\vec{V}$ and this map can be regarded as a constant
$\vec{V}$-valued differential $1$-form on $\sigma$.

However, there are no compatibility conditions a priori along the edges of the
triangulation. We introduce the subspace of _Whitney forms_ which is a
slightly different condition since $\sigma\in{\mathscr{K}}_{2}$: an element
$F=(F_{\sigma})_{\sigma\in{\mathscr{K}}_{2}}\in{\mathscr{F}}({\mathscr{T}})$
satisfies the _Whitney condition_ if for every
$\sigma_{1},\sigma_{2}\in{\mathscr{K}}_{2}$ with a common edge $\sigma_{3}$,
the pullbacks of $F_{\sigma_{1}}$ and $F_{\sigma_{2}}$ agree along
$\sigma_{3}$. In this case, $F$ is called a _Whitney form_ and we define:

${\mathscr{F}}_{w}({\mathscr{T}})=\\{F\in{\mathscr{F}}({\mathscr{T}}),\quad
F\mbox{ is a Whitney form}\\}.$

Thanks to the Whitney condition, a Whitney form
$F\in{\mathscr{F}}_{w}({\mathscr{T}})$ defines via the pullback an extended
family $(F_{\sigma})$ for $\sigma\in{\mathscr{K}}$ which is understood as an
element of $\Omega^{1}_{w}(\Sigma,\vec{V},{\mathscr{T}})$. Hence, we have a
canonical embedding

${\mathscr{F}}_{w}({\mathscr{T}})\hookrightarrow\Omega^{1}_{w}(\Sigma,\vec{V},{\mathscr{T}}).$

The elements $F\in{\mathscr{F}}_{w}({\mathscr{T}})$ are families of constant
differential forms along the vertices of ${\mathscr{T}}$, hence they are
_closed_. Thus, every $F$ defines a Whitney cohomology class denoted $[F]\in
H^{1}_{w}(\Sigma,\vec{V})$. We define the subspace of exact forms in
${\mathscr{F}}_{w}({\mathscr{T}})$ by

${\mathscr{F}}_{0}({\mathscr{F}})=\\{F\in{\mathscr{F}}_{w}({\mathscr{T}}),\quad[F]=0\\}.$

By construction, the differential $d$ defines a map

$d:{\mathscr{M}}({\mathscr{T}})\to{\mathscr{F}}_{0}({\mathscr{T}}).$

Furthermore the map $d$ induced an homeomorphism

$d:{\mathscr{M}}({\mathscr{T}})/\vec{V}\to{\mathscr{F}}_{0}({\mathscr{T}}).$

where $\vec{V}$ acts by translations on ${\mathscr{M}}({\mathscr{T}})$.

###  4.5. Moduli space structure in the polyhedral setting

The space ${\mathscr{F}}({\mathscr{T}})$ carries a Kähler structure defined
similarly to the smooth setting. For $F=(F_{\sigma})$ and
$H=(H_{\sigma})\in{\mathscr{F}}({\mathscr{T}})$, we put

${\mathcal{G}}(F,H)=\sum_{\sigma\in{\mathscr{K}}_{2}}\int_{\sigma}\langle
F_{\sigma},H_{\sigma}\rangle\omega_{\sigma}.$

The integrands of the above integrals are constant on each simplex. Hence each
term of the above integral is equal to $\langle
F_{\sigma},H_{\sigma}\rangle\mathrm{area}(\sigma)$, where
$\mathrm{area}(\sigma)$ is the area of $\sigma$ with respect to the Euclidean
metric $g_{\sigma}$. As in the smooth case, we will use the notation
$\langle\\!\langle F,H\rangle\\!\rangle={\mathcal{G}}(F,H)$ and
$\|F\|_{L^{2}}$ for the norm deduced from ${\mathcal{G}}$.

An almost complex structure ${\mathcal{J}}$ on ${\mathscr{F}}({\mathscr{T}})$
is given by

$({\mathcal{J}}F)_{\sigma}=-F_{\sigma}\circ J_{\sigma}.$

By construction ${\mathcal{J}}$ is compatible with ${\mathcal{G}}$ and we
obtain a Kähler form

$\Omega={\mathcal{G}}({\mathcal{J}}\cdot,\cdot).$

Finally, we have defined a flat Kähler structure

$({\mathscr{F}}({\mathscr{T}}),{\mathcal{G}},{\mathcal{J}},\Omega)$

on the moduli space.

As in the smooth case, the almost complex structure $i$ acts by multiplication
on $\vec{V}$-valued forms and we obtain a linear involution ${\mathscr{R}}$ on
${\mathscr{F}}({\mathscr{T}})$, defined by

${\mathscr{R}}F=i{\mathcal{J}}F.$

The involution gives an orthogonal splitting into eigenspaces

${\mathscr{F}}({\mathscr{T}})={\mathscr{F}}^{+}({\mathscr{T}})\oplus{\mathscr{F}}^{-}({\mathscr{T}})$

and we introduce the ${\mathcal{G}}$-orthogonal projection onto the subspace
of exact differentials

$\Pi:{\mathscr{F}}({\mathscr{T}})\to{\mathscr{F}}_{0}({\mathscr{T}}).$

###  4.6. Polyhedral symplectic density

We denote denote by $\mathfrak{t}^{\mathbb{C}}({\mathscr{T}})$ the space of
maps $\zeta:{\mathscr{K}}_{2}\to{\mathbb{C}}$. Equivalently, an element
$\zeta$ can be understood as a family of constant maps
$(\zeta_{\sigma})_{\sigma\in{\mathscr{K}}_{2}}$ on each facet $\sigma$ of the
triangulation. We also denote by $\mathfrak{t}({\mathscr{T}})$ the space of
real valued maps $\zeta:{\mathscr{K}}_{2}\to{\mathbb{R}}$ and we define

$\mu:{\mathscr{F}}({\mathscr{T}})\to\mathfrak{t}({\mathscr{T}})$

by the formula

$\mu(F)(\sigma)=-\frac{1}{2}\langle({\mathscr{R}}F)_{\sigma},F_{\sigma}\rangle.$

We can make sense of the pullback $F^{*}\omega_{V}$ as a family of
differential $2$-forms along each facet defined by

$(F^{*}_{\sigma}\omega_{V})(\eta_{1},\eta_{2})=\omega_{V}(F_{\sigma}\cdot\eta_{1},F_{\sigma}\cdot\eta_{2})$

for every $\eta_{1},\eta_{2}\in\vec{\sigma}$, and
$\sigma\in{\mathscr{K}}_{2}$. As in the smooth case, the map $\mu$ is a
symplectic density, in the sense of the following lemma:

######  Lemma 4.6.1.

For every $F\in{\mathscr{F}}({\mathscr{T}})$, we have the identity

$\mu(F)(\sigma)=-\frac{F_{\sigma}^{*}\omega_{V}}{\omega_{\sigma}}$ (4.1)

for every facet $\sigma\in{\mathscr{K}}_{2}$.

This motivates the following definition:

######  Definition 4.6.2.

A differential $F\in{\mathscr{F}}({\mathscr{T}})$ is called isotropic if
$F_{\sigma}:\vec{\sigma}\to V$ is isotropic for every
$\sigma\in{\mathscr{K}}_{2}$. A polyhedral map
$f\in{\mathscr{M}}({\mathscr{T}})$ is called isotropic if the pullback of
$\omega_{V}$ by $f$ vanishes along every simplex of the triangulation.

In particular, by Lemma (4.6.1), we have

######  Lemma 4.6.3.

A differential $F\in{\mathscr{F}}({\mathscr{T}})$ is isotropic if, and only
if, $\mu(F)=0$. A polyhedral map $f\in{\mathscr{M}}({\mathscr{T}})$ is
isotropic if, and only if, $F=df$ is isotropic.

###  4.7. Hamiltonian gauge group action

We define the complex gauge group ${\mathbb{T}}^{\mathbb{C}}({\mathscr{T}})$
as the space of non vanishing complex valued functions
$\lambda:{\mathscr{K}}_{2}\to{\mathbb{C}}^{*}$. Alternatively, $\lambda$ can
be thought of as a family of constant functions $(\lambda_{\sigma})$ along
each facet of the triangulation. The group
${\mathbb{T}}^{\mathbb{C}}({\mathscr{T}})$ acts on
${\mathscr{F}}({\mathscr{T}})$ via the formula

$\lambda\cdot F=\bar{\lambda}^{-1}F^{+}+\lambda F^{-},$ (4.2)

where $F^{\pm}$ are the component of $F$ given by the splitting
${\mathscr{F}}({\mathscr{T}})={\mathscr{F}}^{+}({\mathscr{T}})\oplus{\mathscr{F}}^{-}({\mathscr{T}})$.
The real subgroup ${\mathbb{T}}({\mathscr{T}})$ is the the space of functions
$\lambda:{\mathscr{K}}_{2}\to S^{1}$ and acts by complex multiplication on
${\mathscr{F}}({\mathscr{T}})$. The exponential map

$\exp:\mathfrak{t}^{\mathbb{C}}({\mathscr{T}})\to{\mathbb{T}}^{\mathbb{C}}({\mathscr{T}})$

defined by

$\exp\zeta=e^{i\zeta},$

shows that $\mathfrak{t}^{\mathbb{C}}({\mathscr{T}})$ is the Lie algebra of
${\mathbb{T}}^{\mathbb{C}}({\mathscr{T}})$ and identifies the Lie algebra of
the subgroup $\mathfrak{t}({\mathscr{T}})$ with $\mathfrak{t}({\mathscr{T}})$.
The infinitesimal action of $\mathfrak{t}^{\mathbb{C}}({\mathscr{T}})$ on
${\mathscr{F}}({\mathscr{T}})$ is given by

$X_{\zeta}(F)=i\bar{\zeta}F^{+}+i\zeta F^{-}.$

As in the smooth case, the construction has the following nice properties:

######  Proposition 4.7.1.

The ${\mathbb{T}}^{\mathbb{C}}({\mathscr{T}})$-action on
${\mathscr{F}}({\mathscr{T}})$ perserves the almost complex structure
${\mathcal{J}}$. Furthermore, the
${\mathbb{T}}^{\mathbb{C}}({\mathscr{T}})$-action is the
${\mathcal{J}}$-complexification of the ${\mathbb{T}}({\mathscr{T}})$-action.
The ${\mathbb{T}}$-action preserves the metric ${\mathcal{G}}$ and the
symplectic form $\Omega$ as well. Furthermore the
${\mathbb{T}}({\mathscr{T}})$-action is Hamiltonian with moment map
$\mu:{\mathscr{F}}({\mathscr{T}})\to\mathfrak{t}({\mathscr{T}})$ given by
Formula (4.1). More precisely, $\mu$ is
${\mathbb{T}}({\mathscr{T}})$-invariant and for every
$\zeta\in\mathfrak{t}({\mathscr{T}})$, we have

$D\langle\\!\langle\mu,\zeta\rangle\\!\rangle=-\iota_{X_{\zeta}}\Omega.$

###### Proof.

The proof is formally identical to the smooth setting. ∎

###  4.8. Polyhedral modified moment map flow

By analogy with the smooth setting, we consider the energy of the moment map

$\phi:{\mathscr{F}}({\mathscr{T}})\to{\mathbb{R}}$

defined by

$\phi(F)=\frac{1}{2}\|\mu(F)\|^{2}_{L^{2}}.$

By contruction, $\phi$ is non negative and its vanishing set is the subspace
of isotropic differentials in ${\mathscr{F}}({\mathscr{T}})$. We denote by
$\nabla\phi$ the gradient of $\phi:{\mathscr{F}}\to{\mathbb{R}}$ with respect
to the metric ${\mathcal{G}}$. The gradient of the restricted functional
$\phi:{\mathscr{F}}_{0}({\mathscr{T}})\to{\mathbb{R}}$ is denoted
$\nabla^{\circ}\phi$ and satisfies $\nabla^{\circ}\phi(F)=\Pi(\nabla\phi(F))$.
Then we define the polyhedral modified moment map flow by the evolution
equation along ${\mathscr{F}}_{0}({\mathscr{T}})$:

$\frac{\partial F}{\partial t}=-\nabla^{\circ}\phi(F).$ (4.3)

As in the smooth case, the formal identities are the same and we can prove:

######  Proposition 4.8.1.

The critical points of the restricted functional
$\phi:{\mathscr{F}}_{0}({\mathscr{T}})\to{\mathbb{R}}$, in other words the
fixed points of the modified moment map flow, agree with its vanishing locus.

For every $F\in{\mathscr{F}}_{0}({\mathscr{T}})$, there exists a polyhedral
map $f\in{\mathscr{M}}({\mathscr{T}})$ such that $F=df$, unique up to the
action of $\vec{V}$ with the property that $f$ is isotropic if, and only if,
$\phi(F)=0$.

######  Remark 4.8.2.

The Second statement of the proposition implies that the modified moment map
flow defines a flow for polyhedral maps as well: we fix a point
$x_{0}\in\Sigma$ and a point $v_{0}\in V$. For every
$F\in{\mathscr{F}}_{0}({\mathscr{F}})$, we define the integral
$f=\chi(F)\in{\mathscr{M}}({\mathscr{T}})$ as the unique map such that $df=F$
and $f(x_{0})=v_{0}$. Any solution $F_{t}$ of the modified moment map flow
defines a family $f_{t}=\chi(F_{t})$ solution of the evolution equation

$\boxed{\frac{\partial f}{\partial t}=-\chi\circ\nabla^{\circ}\phi(df)}$ (4.4)

and vice-versa. The fixed points of this flow are, by definition, the
isotropic polyhedral maps.

######  Proposition 4.8.3.

A solution of the polyhedral modified moment map flow
$F_{t}\in{\mathscr{F}}_{0}({\mathscr{T}})$, defined for $t$ in some interval
satisfies

$\frac{\partial}{\partial t}\|F\|^{2}_{L^{2}}=-8\phi(F_{t}).$

In particular, the $L^{2}$-norm is non increasing along the interval.

###### Proof.

The ingredients of the proof are the same as in Theorem 3.1.1, for the smooth
setting. In the polyhedral context, we have the identity

$D\phi|_{F}\cdot F=4\phi(F)$ (4.5)

identical to the one in Proposition 2.13.1 and the rest of the argument is
identical. ∎

Proposition 4.8.3 has the following strong consequence:

######  Corollary 4.8.4.

For every $F\in{\mathscr{F}}_{0}({\mathscr{T}})$, there exists a unique
solution $F_{t}$ defined for $t\in[0,+\infty)$ and such that $F_{0}=F$.
Furthermore, $\|F_{t}\|_{L^{2}}$ is bounded by $\|F\|_{L^{2}}$.

###### Proof.

The short time existence is immediate Proposition 4.8.3 and classical ODE
theory. The decay of the $L^{2}$-norm implies that a solution of the flow
cannot blow up in finite time, which implies the long time existence. ∎

###  4.9. A Duistermaat theorem

In fact, the polyhedral modified moment map flow converges and provides a
strong deformation retraction onto the space of polyhedral isotropic maps.

######  Theorem 4.9.1.

For every $F\in{\mathscr{F}}_{0}({\mathscr{T}})$, the solution $F_{t}$ of the
modified moment map flow, defined for $t\in[0,+\infty)$ with $F_{0}=F$
converges toward a limit $F_{\infty}$ such that $\phi(F_{\infty})=0$. Thus, we
can define an extended flow

$\Theta:[0,+\infty]\times{\mathscr{F}}_{0}({\mathscr{T}})\to{\mathscr{F}}_{0}({\mathscr{T}})$

by $\Theta(t,F)=F_{t}$ and $\Theta(+\infty,F)=F_{\infty}$. Furthermore, the
map $\Theta$ is a strong deformation retraction onto the vanishing locus of
$\phi:{\mathscr{F}}_{0}({\mathscr{T}})\to{\mathbb{R}}$.

###### Proof.

The proof is essentially contained in the analogous result [18, Theorem
7.8.4], where a downward gradient flow is also studied in a finite dimensional
situation. Although the space ${\mathscr{F}}_{0}({\mathscr{T}})$, the
functional $\phi:{\mathscr{F}}_{0}({\mathscr{T}})\to{\mathbb{R}}$ and the
Euclidean metric ${\mathcal{G}}$ are completely different in [18], they share
similar formal properties:

  1. (1)

The vector space is finite dimensional.

  2. (2)

The functional $\phi$ is a polynomial function.

  3. (3)

The functional $\phi$ is non negative.

  4. (4)

The critical points of $\phi$ agree with its vanishing set.

  5. (5)

The solutions of the downward gradient flow satisfy the decay property of
Proposition 4.8.3 .

Properties (1) and (5) imply the long time existence of the flow proved at
Corollary 4.8.4. Property (2) implies that $\phi$ is analytic. In particular
the Łojaziewicz inequality can be applied to $\phi$, which is crucial to prove
the convergence of the flow lines. The interested reader may check that only
these properties are used in the proof of [18, Theorem 7.8.4] and that they
imply the congergence and continuity of the flow. ∎

###  4.10. Regular points of the moduli space

We obtain much stronger result for the behavior of the modified moment map
flow in the polyhedral setting, as proved by Theorem 4.9.1. On the contrary,
the regularity of the space of isotropic maps is much easier to study in the
smooth setting, at least in a neighborhood of an immersion, as showed at §3.2.
However, some partial results can be obtained in the polyhedral case, as we
are going to see in this section.

######  Definition 4.10.1.

A differential form $F\in{\mathscr{F}}_{0}({\mathscr{T}})$ is called regular,
if the differential of the restricted moment map
$\mu:{\mathscr{F}}_{0}({\mathscr{T}})\to\mathfrak{t}({\mathscr{T}})$ has
constant rank in a neighborhood of $F$.

By definition, the vanishing locus of
$\phi:{\mathscr{F}}_{0}({\mathscr{T}})\to{\mathbb{R}}$ is a submanifold of
${\mathscr{F}}_{0}({\mathscr{T}})$ in a neighborhood of a regular point $F$.
Furthermore, the following theorem shows that $\phi$ behaves like a Morse-Bott
function on such a neighborhood:

######  Theorem 4.10.2.

Let $F\in{\mathscr{F}}_{0}({\mathscr{T}})$ be a regular point of the vanishing
locus of $\phi$. Then, there exists an open neighborhood
$U\subset{\mathscr{F}}_{0}({\mathscr{T}})$ of $F$, such that:

  1. (1)

the vanishing locus of $\phi:U\to{\mathbb{R}}$ is a submanifold of $U$;

  2. (2)

the Hessian of $\phi$ is positive definite in direction transverse to the
vanishing locus;

  3. (3)

$U$ is invariant under the modified moment map flow $\Theta$, which has an
exponential convergence rate.

###### Proof.

The Hessian of $\phi$ is calculated in the smooth setting at §3.2 and is given
a point $F$ such $\phi(F)=0$ by Formula (3.3). The same calculations show that
the formula holds in the polyhedral setting as well and obtain

$D^{2}\phi|_{F}\cdot(\dot{F}_{1},\dot{F}_{2})=\langle\\!\langle
D\mu|_{F}\cdot\dot{F}_{1},D\mu|_{F}\cdot\dot{F}_{2}\rangle\\!\rangle$

a any point $F$ of the vanishing locus of
$\phi:{\mathscr{F}}_{0}({\mathscr{T}})\to{\mathbb{R}}$. If $F$ is a regular,
the vanishing locus of $\phi:{\mathscr{F}}_{0}({\mathscr{T}})\to{\mathbb{R}}$
is a submanifold of ${\mathscr{F}}_{0}({\mathscr{T}})$ in a neighborhood of
$F$ and the above formula shows that the Hessian of $\phi$ is definite
positive in transverse directions to the submanifold. In conclusion, $\phi$
behaves like a Morse-Bott function in a neighborhood of $F$ and its vanishing
set is a stable critical component. The statements of the theorem are
classical facts of ODE theory under these assumptions. ∎

Constructing regular points of the moduli space is not a simple problem. We
are going to show that Theorem 1.1.1 provides a construction of regular
points. In [11], a quotient $2$-torus $\Sigma={\mathbb{R}}^{2}/\Gamma$ is
considered together with a smooth isotropic immersion
$f:\Sigma\looparrowright{\mathbb{R}}$. Refinements of stepsize
${\mathscr{O}}(N^{-1})$ of the lattice $\Gamma$ provide a family of
quadrangulations ${\mathcal{Q}}_{N}$ of $\Sigma$, with $N^{2}$ facets. The
quadrangulations can be completed into triangulations ${\mathscr{T}}_{N}$ by
replacing each quadrilateral of ${\mathcal{Q}}_{N}$ with a pyramid. The main
construction of [11] provides a sequence of polyhedral isotropic maps
$f_{N}\in{\mathscr{M}}({\mathscr{T}}_{N})$, for every $N$ sufficiently large,
such that $f_{N}$ approximates $f$ in $C^{1}$-norm by [17]. The construction
of $f_{N}$ relies on an effective version of the fixed point principle applied
to the family of symplectic densities

$\mu_{N}\circ
d:{\mathscr{M}}({\mathscr{T}}_{N})\to\mathfrak{t}({\mathscr{T}}_{N}),$

where
$\mu_{N}:{\mathscr{F}}_{0}({\mathscr{T}}_{N})\to\mathfrak{t}({\mathscr{T}}_{N})$
is the moment map. The fact that the cohomology class of $\omega_{V}$ vanishes
implies that $\mu_{N}\circ d$ takes values in the subspace of
$\zeta\in\mathfrak{t}({\mathscr{T}}_{N})$ orthogonal to constant functions.
Hence $\mu_{N}\circ d$ can never be a submersion, but it may have maximal rank
with corank one. A careful examination of the proof in [11] allows to make the
following observation:

######  Lemma 4.10.3.

For every $N$ sufficiently large, the differential of the map $\mu_{N}\circ
d:{\mathscr{M}}({\mathscr{T}}_{N})\to\mathfrak{t}({\mathscr{T}}_{N})$ has
maximal rank at $f_{N}$, where $f_{N}$ is the Jauberteau-Rollin-Tapie
approximation of $f$. In particular,
$F_{N}=df_{N}\in{\mathscr{F}}_{0}({\mathscr{T}}_{N})$ is a regular point of
the vanishing set of
$\phi:{\mathscr{F}}_{0}({\mathscr{T}}_{N})\to{\mathbb{R}}$.

We can now complete the proofs of the main theorems given in the introduction.

###### Proof of Theorem G.

The result is an immediate consequence of Lemma 4.10.3. ∎

###### Proof of Theorem E.

The result is a combination of Proposition 4.8.1, Theorem 4.9.1 and Theorem
4.10.2. The statement $(3)$ will be proved at the end of §4.11. ∎

###### Proof of Corollary F.

The differential
$d:{\mathscr{M}}({\mathscr{T}})\to{\mathscr{F}}_{0}({\mathscr{T}})$ admits a
unique right inverse

$\chi:{\mathscr{F}}_{0}({\mathscr{T}})\to{\mathscr{M}}({\mathscr{T}})$

defined by the condition that $d\circ\chi=\mathrm{id}$ and $f=\chi(F)$
satisfies $f(x_{0})=v_{0}$ for some marked points $x_{0}\in\Sigma$ and
$v_{0}\in V$. We consider the continuous map

$\hat{\Theta}(t,f)=\chi\circ\Theta(t,df)+(f(x_{0})-v_{0}).$

By construction

$\hat{\Theta}:[0,+\infty]\times{\mathscr{M}}({\mathscr{T}})\to{\mathscr{M}}({\mathscr{T}})$

defines a strong deformation retraction of ${\mathscr{M}}({\mathscr{T}})$ onto
the subspace of isotropic polyhedral maps. It is easy to see that
$\hat{\Theta}$ is the flow of the evolution equation on
${\mathscr{M}}({\mathscr{T}})$ defined at Remark 4.8.2. In particular
$\Theta_{t}\circ d=d\circ\hat{\Theta}_{t}$, which proves the corollary. ∎

###  4.11. Topology of the moduli space

The homotopy of the space of smooth isotropic immersions
$f:\Sigma\looparrowright V$ can be reduced to a matter of algebraic topology,
thanks to the Gromov-Lees theorem [9, 13]: every isotropic immersion $f$
defines an exact differential form $F=df\in\Omega^{1}(\Sigma,\vec{V})$ which
is by definition an isotropic monomorphism. The Gromov-Lees theorem
establishes an h-principle, which shows that the differential
$d:{\mathscr{M}}/\vec{V}\to{\mathscr{F}}_{0}$ induces a homotopy equivalence
from the space of isotropic immersions to the space of isotropic
monomorphisms. Computing the homotopy of the latter space turns out to be a
basic problem of algebraic topology, related to the Grassmanian of isotropic
planes in $\vec{V}$.

Unfortunately, the Gromov-Lees theory does not extend immediately to the
polyhedral setting. Thus, most topological properties for the space of
polyhedral isotropic immersions are open questions although the partial
results of [11, 17, 6] lead us to state Conjecture H.

The map
$d:{\mathscr{M}}({\mathscr{T}})/\vec{V}\to{\mathscr{F}}_{0}({\mathscr{T}})$ is
a homeomorphism, that identifies the subspace of polyhedral isotropic maps
modulo $\vec{V}$ with the space of exact isotropic differentials. The latter
space is a cone in ${\mathscr{F}}_{0}({\mathscr{T}})$, which implies that it
is contractible. Therefore, the space of polyhedral isotropic maps is
homotopically trivial. The same property holds in the smooth setting, which is
why the problem of immersions is considered instead. There are several issues
here to formulate analogous questions in the polyhedral setting:

  1. (1)

A polyhedral map $f\in{\mathscr{M}}({\mathscr{T}})$ which is a topological
immersion define a monomorphism $F=df\in{\mathscr{F}}_{0}({\mathscr{T}})$.
However the converse is generally not true and some local injectivity
condition along the skeleton of the triangulation should be added to obtain a
correspondence.

  2. (2)

The space of monomorphisms in ${\mathscr{F}}_{0}({\mathscr{T}})$ does not seem
to be preserved by the modified moment map flow $\Theta$. From this point, the
prospect of using the flow to investigate the homotopy type of the space of
polyhedral isotropic immersions seems rather low.

However the space of non constant polyhedral isotropic maps may already
contain some non trivial topology. Furthemore, this subspace modulo $\vec{V}$
is identified to ${\mathscr{F}}_{0}({\mathscr{T}})\setminus 0$ via $d$.
Finally, ${\mathscr{F}}_{0}({\mathscr{T}})\setminus 0$ is invariant under the
gauge group action and invariant under the modified moment map flow, for
finite time.

######  Question 4.11.1.

What is the homotopy type of the space of non constant polyhedral isotropic
maps $f\in{\mathscr{M}}({\mathscr{T}})$ ?

The space of isotropic exact differentials in
${\mathscr{F}}_{0}({\mathscr{T}})\setminus 0$ is invariant by scaling. In
particular, it is homotopically equivalent to the vanishing set of the
restriction $\phi:\SS({\mathscr{T}})\to{\mathbb{R}}$, where
$\SS({\mathscr{T}})$ is the unit sphere in ${\mathscr{F}}_{0}({\mathscr{T}})$.

The idea to tackle Question 4.11.1 is to interpret
$\phi:\SS({\mathscr{T}})\to{\mathbb{R}}$ as a Morse-Bott function to obtain
some dynamical interaction between its critical components. By definition the
critical set of $\phi:\SS({\mathscr{T}})\to{\mathbb{R}}$ contains its
vanishing set, but there may be other critical points, called the _solitons_.
More explicitly, a soliton $F\in{\mathscr{F}}_{0}({\mathscr{T}})$ is a
solution of the equation $\nabla^{\circ}\phi(F)=\kappa F$ for some
$\kappa\in{\mathbb{R}}$. The constant $\kappa$ is determined by Formula 4.5
and we obtain the soliton equation:

$\boxed{\|F\|^{2}_{L^{2}}\nabla^{\circ}\phi(F)=4\phi(F).}$ (4.6)

Solitons are by definition the fixed points of the downward gradient flow of
the restricted functional $\phi:\SS({\mathscr{T}})\to{\mathbb{R}}$ defined for
$F\in\SS({\mathscr{T}})$ by

$\boxed{\frac{\partial F}{\partial t}=4\phi(F)-\nabla^{\circ}\phi(F)}$ (4.7)

and called the _renormalized_ polyhedral flow.

Many questions are open at this stage for the renormalized flow and the
solitons:

  1. (1)

Is $\phi:\SS({\mathscr{T}})\to{\mathbb{R}}$ a Morse-Bott type function, in
some sense, perhaps up to the choice a refinement of the triangulation and a
generic polyhedral metric ?

  2. (2)

What are the intrinsic geometrical properties of polyhedral maps
$f\in{\mathscr{M}}({\mathscr{T}})$ such that $F=df$ is a soliton ?

  3. (3)

Does the functional $\phi:\SS({\mathscr{T}})\to{\mathbb{R}}$ define a Morse-
Bott cohomology theory ?

We can prove that solitons never reduce to the vanishing locus of $\phi$.

######  Lemma 4.11.2.

There exists solitons $G\in{\mathscr{F}}_{0}({\mathscr{T}})$ such that
$\phi(G)\neq 0$.

###### Proof.

Clearly, $\phi:\SS({\mathscr{T}})\to{\mathbb{R}}$ is not the zero map, as it
is easy to construct polyhedral maps $f\in{\mathscr{M}}({\mathscr{T}})$ which
are not isotropic. In particular $\phi(df)>0$ in this case. We conclude that
$\phi$ admits a maximum at some point $G\in\SS({\mathscr{T}})$ with
$\phi(G)>0$. If follows that $G$ is a critical point of
$\phi:\SS({\mathscr{T}})\to{\mathbb{R}}$, which is to say a soliton.
Furthermore, $\phi(G)>0$ implies that $G$ is not isotropic. ∎

######  Remark 4.11.3.

We introduce a family $F_{t}\in{\mathscr{F}}_{0}({\mathscr{T}})$ given by
$F_{t}=r(t)G$, where $G\in{\mathscr{F}}_{0}({\mathscr{T}})$ is a soliton such
that $\phi(G)>0$ and

$r(t)=\frac{1}{\sqrt{8(t-t_{0})\phi(G)}}$

is defined on the interval $(t_{0},+\infty)$, for some choice of
$t_{0}\in{\mathbb{R}}$. It is readily checked that $F_{t}$ is a solution of
the polyhedral modified moment map flow. We notice that $F_{t}$ defines a non
trivial solution of flow, with the property that
$\displaystyle\lim_{t\to+\infty}F_{t}=0$.

###### Proof of Theorem E, item (3).

The existence of non isotropic solitons in $\SS({\mathscr{T}})$ provides non
trivial solutions of the polyhedral modified moment map flow as discussed
above. This completes the proof of the theorem. ∎

###  4.12. Numerical flow

The polyhedral modified moment map flow is an ordinary differential evolution
equation. A numerical version of the flow can be implemented on a computer. In
this section we outline all the ingredients for such a computer program. The
code is being developed currently and is going to be released very soon [10].

For simplicity, we specialize to the case where $\Sigma$ is a quotient
$2$-torus. Let $\Gamma$ be the lattice

$\Gamma=\gamma_{1}{\mathbb{Z}}\oplus\gamma_{2}{\mathbb{Z}}\subset{\mathbb{C}}$

where $\gamma_{1}=1$ $\gamma_{2}=e^{i\frac{\pi}{3}}$. The surface $\Sigma$,
defined by

$\Sigma={\mathbb{C}}/\Gamma,$

is endowed with the flat Kähler structure
$(\Sigma,g_{\Sigma},J_{\Sigma},\omega_{\Sigma})$ deduced from the canonical
flat Kähler structure of ${\mathbb{C}}$. The rhombus in ${\mathbb{C}}$ with
vertices $0,\gamma_{1},\gamma_{1}+\gamma_{2}$ and $\gamma_{2}$ is a
fundamental domain for the action of $\Gamma$. This rhombus is composed of two
equilateral triangles and each triangle has area $\frac{\sqrt{3}}{4}$. Using
the $\Gamma$-action, we obtain the familiar tiling of ${\mathbb{C}}$ by
equilateral triangles:

Figure 1. Triangulation of ${\mathbb{C}}$ by equilateral triangles

This tiling is a simplicial decomposition of ${\mathbb{C}}$ and can be
regarded as a triangulation ${\mathscr{T}}^{\prime}$ of ${\mathbb{C}}$. Let
$N$ be a positive integer. We define the lattice

$\Gamma_{N}=N^{-1}\Gamma,$

understood as the set of vertices of the rescaled triangulation
${\mathscr{T}}^{\prime}_{N}$ of ${\mathscr{T}}^{\prime}$, by a factor
$N^{-1}$. By Definition, the triangulation ${\mathscr{T}}^{\prime}_{N}$ is
$\Gamma$-invariant. Hence, ${\mathscr{T}}^{\prime}_{N}$ defines a
triangulation of the quotient $\Sigma$, denoted ${\mathscr{T}}_{N}$, with each
triangle of area $\frac{\sqrt{3}}{4N^{2}}$.

We consider the particular space $V={\mathbb{C}}^{2}$ endowed with its
canonical structure of Hermitian affine space and the coordinates
$(x_{1}+ix_{2},x_{3}+ix_{4})\in V$, where $x_{i}\in{\mathbb{R}}$. The
symplectic form $\omega_{V}$ is given in coordinates by the formula

$\omega_{V}=dx_{1}\wedge dx_{2}+dx_{3}\wedge dx_{4}$

and the almost complex structure $i:\vec{V}\to\vec{V}$, given by the
multiplication by $i$ satisfies the identities

$i\frac{\partial}{\partial x_{1}}=\frac{\partial}{\partial x_{2}}\quad\mbox{
and }\quad i\frac{\partial}{\partial x_{3}}=\frac{\partial}{\partial x_{4}}.$
(4.8)

Every $F\in{\mathscr{F}}({\mathscr{T}}_{N})$ is given by a collection
$F=(F_{\sigma})_{\sigma\in{\mathscr{K}}_{2}}$ where
$F_{\sigma}\in\vec{\sigma}^{*}\otimes\vec{V}$. We use the canoncial
coordinates $z=u_{1}+iu_{2}\in{\mathbb{C}}$ on the universal cover
${\mathbb{C}}$ of $\Sigma$. By definition we have

$-du_{1}\circ J_{\Sigma}=du_{2}\mbox{ and }-du_{2}\circ J_{\Sigma}=-du_{1}$
(4.9)

The $du_{i}\otimes\frac{\partial}{\partial x_{j}}$ provide a
$g_{\sigma}$-orthonormal basis of $\vec{\sigma}^{*}\otimes\vec{V}$ for each
facet of the triangulation ${\mathscr{T}}_{N}$. Hence, $F_{\sigma}$ admits a
decomposition

$F_{\sigma}=\sum_{\begin{subarray}{c}1\leq i\leq 2\\\ 1\leq j\leq
4\end{subarray}}F_{\sigma ij}du_{i}\otimes\frac{\partial}{\partial x_{j}}$
(4.10)

where $F_{\sigma ij}\in{\mathbb{R}}$. In particular, for
$F,H\in{\mathscr{F}}({\mathscr{T}}_{N})$, we have

$\langle\\!\langle F,H\rangle\\!\rangle=\sum_{\begin{subarray}{c}1\leq i\leq
2\\\ 1\leq j\leq 4\\\ \sigma\end{subarray}}F_{\sigma ij}H_{\sigma
ij}\frac{\sqrt{3}}{4N^{2}}$

where the sum is taken over all the facets $\sigma$ of ${\mathscr{T}}_{N}$.

The space ${\mathscr{F}}_{0}({\mathscr{T}}_{N})$ is the subspace of exact
$\vec{V}$-valued Whitney forms in ${\mathscr{F}}({\mathscr{T}}_{N})$. We need
to construct an orthonormal basis of this subspace to compute the matrix of
the projector
$\Pi:{\mathscr{F}}({\mathscr{T}}_{N})\to{\mathscr{F}}_{0}({\mathscr{T}}_{N})$.

We consider the affine canonical base of $V$ given by the points $e_{0}=0$ and
$e_{j}=e_{0}+\frac{\partial}{\partial x_{j}}$ for $1\leq j\leq 4$. For each
vertex $\sigma_{0}$ of ${\mathscr{T}}_{N}$ and $1\leq j\leq 4$, we define the
polyhedral function $f^{\sigma_{0}j}:\Sigma\to V$ given by

$f^{{\sigma_{0}}j}(\sigma_{0}^{\prime})=\left\\{\begin{array}[]{ll}e_{j}&\mbox{if
$\sigma_{0}=\sigma^{\prime}_{0}$,}\\\
e_{0}&\mbox{otherwise.}\end{array}\right.$

for every vertex $\sigma_{0}^{\prime}$ of ${\mathscr{T}}_{N}$. The above
function extends uniquely as a polyhedral function
$f^{\sigma_{0}j}\in{\mathscr{M}}({\mathscr{T}}_{N})$ and defines the exact
differential

$F^{\sigma_{0}j}=df^{\sigma_{0}j}\in{\mathscr{F}}_{0}({\mathscr{T}}_{N}).$

The family $F^{\sigma_{0}j}$ for $\sigma_{0}\in{\mathscr{K}}_{0}$ and $1\leq
j\leq 4$ spans ${\mathscr{F}}_{0}({\mathscr{T}}_{N})$, by definition. The only
relations between the $F^{\sigma j}$ are the $4$ relations given by

$\sum_{\sigma_{0}}F^{\sigma_{0}j}=0$

for $1\leq j\leq 4$, where the sum is taken over all the vertices of the
triangulation ${\mathscr{T}}_{N}$. We obtain the following lemma:

######  Lemma 4.12.1.

For a fixed vertex $\sigma^{\prime}_{0}$ of the triangulation
${\mathscr{T}}_{N}$, The family

$F^{\sigma_{0}j}\in{\mathscr{F}}_{0}({\mathscr{T}}_{N}),$

where $1\leq j\leq 4$ and $\sigma_{0}$ belongs to the vertices of
${\mathscr{T}}_{N}$ with $\sigma_{0}\neq\sigma^{\prime}_{0}$, is a basis of
${\mathscr{F}}_{0}({\mathscr{T}}_{N})$.

However, the family $F^{\sigma_{0}j}$ is not orthonormal and we have to use
the Gram-Schmidt algorithm. The differential $F^{\sigma_{0}j}$ can be computed
explicitly, working on the universal cover ${\mathbb{C}}$ of $\Sigma$, and
assuming $\sigma_{0}=0$ for simplicity. There are six triangles of
${\mathscr{T}}^{\prime}_{N}$ with vertex $\sigma_{0}=0$. We can compute the
differential of $f^{\sigma_{0}j}$ on each triangle. In particular on the
triangle $\sigma_{1}$ with vertices $0$, $\gamma_{1}/N$ and $\gamma_{2}/N$ we
have

$F^{\sigma_{0}j}_{\sigma_{1}}=\left(-Ndu_{1}-\frac{N}{\sqrt{6}}du_{2}\right)\otimes\frac{\partial}{\partial
x_{j}}.$ (4.11)

The differential on all the other triangles with vertex $\sigma_{0}=0$ is
obtained by rotations of angle $\frac{\pi}{3}$. For every other triangle
$\sigma$ not in the star of $\sigma_{0}$, we have
$F_{\sigma}^{\sigma_{0}j}=0$. In particular, on the triangle $\sigma_{2}$ with
vertices $0$, $-\gamma_{1}/N$ and $e^{2i\frac{\pi}{3}}/N$, we have the formula

$F^{\sigma_{0}j}_{\sigma_{2}}=\left(Ndu_{1}-\frac{N}{\sqrt{6}}du_{2}\right)\otimes\frac{\partial}{\partial
x_{j}}.$ (4.12)

######  Lemma 4.12.2.

The family of Whitney forms
$F^{\sigma_{0}j}\in{\mathscr{F}}_{0}({\mathscr{T}}_{N})$ described above
satisfies the identities

$\displaystyle\langle\\!\langle F^{\sigma_{0}j},$ $\displaystyle
F^{\sigma_{0}^{\prime}j^{\prime}}\rangle\\!\rangle=$
$\displaystyle\left\\{\begin{array}[]{cl}\frac{7\sqrt{3}}{4}&\mbox{ if
$i=i^{\prime}$ and $\sigma_{0}=\sigma_{0}^{\prime}$,}\\\
-\frac{5}{4\sqrt{3}}&\mbox{if $i=i^{\prime}$ and $\sigma_{0}$ and
$\sigma_{0}^{\prime}$ are joined by an edge,}\\\
0&\mbox{otherwise.}\end{array}\right.$

###### Proof.

The third case is clear, since the families $(F_{\sigma}^{\sigma_{0}j})$ and
$(F_{\sigma}^{\sigma_{0}^{\prime}j^{\prime}})$ have disjoint supports for
$j\neq j^{\prime}$ or if $\sigma_{0}$ and $\sigma^{\prime}_{0}$ do not belong
to the same facet of ${\mathscr{T}}_{N}$.

We assume now that, on the contrary, $j=j^{\prime}$ and $\sigma_{0}$ and
$\sigma^{\prime}_{0}$ belong to the same facet. Then either
$\sigma_{0}=\sigma^{\prime}_{0}$ or $\sigma_{0}$ and $\sigma^{\prime}_{0}$ are
the two ends of an edge in ${\mathscr{T}}_{N}$.

In the first case, working on the universal cover and assuming $\sigma_{0}=0$,
Formula (4.11) shows that
$|F^{\sigma_{0}j}_{\sigma_{1}}|^{2}=\frac{7N^{2}}{6}$. Using the fact that the
area of a facet is $\frac{\sqrt{3}}{4N^{2}}$ and that there are $6$ facets in
the star of $\sigma_{0}$, we find

$\|F^{\sigma_{0}j}\|^{2}_{L^{2}}=\frac{7\sqrt{3}}{4}.$

In the second case, working on the universal cover, we may assume that
$\sigma_{0}^{\prime}=\frac{\gamma_{1}}{N}$. By Formulas (4.11) and (4.12), we
deduce that $\langle
F^{\sigma_{0}j}_{\sigma_{1}},F^{\sigma_{0}^{\prime}j}_{\sigma_{1}}\rangle=-\frac{5N^{2}}{6}$.
There is a second facet of ${\mathscr{T}}_{N}$ in the common support of
$F^{\sigma_{0}j}$ and $F^{\sigma_{0}^{\prime}j}$. By symmetry, the inner
product is the same on the second facet. Since the area of each triangle is
equal to $\frac{\sqrt{3}}{4N^{2}}$, we obtain

$\langle\\!\langle
F^{\sigma_{0}j},F^{\sigma_{0}^{\prime}j}\rangle\\!\rangle=-\frac{5N^{2}}{6}\cdot
2\cdot\frac{\sqrt{3}}{4N^{2}}=-\frac{5}{4\sqrt{3}}.$

∎

The family $F^{\sigma_{0}j}$ provides a basis of
${\mathscr{F}}_{0}({\mathscr{T}}_{N})$ thanks to Lemma 4.12.1 and the Gram-
Schmidt method gives an algorithm to construct an orthonomal basis on
${\mathscr{F}}_{0}({\mathscr{T}}_{N})$ from this data. Furthermore, the Gram-
Schmidt method is pretty straightforward regarding computer implementation
thanks to the explicit formulas of Lemma 4.12.2.

Eventually, the matrix of the endomorphism

${\mathscr{R}}:{\mathscr{F}}_{N}({\mathscr{F}}_{N})\to{\mathscr{F}}({\mathscr{T}}_{N})$

can be explicitely computed in the coordinates given by (4.10), using Formula
(4.9) and Formula (4.8) as ${\mathscr{R}}F=-iF\circ J_{\Sigma}$, by
definition. Thus, we have an algorithm to compute
$\mu(F)=-\frac{1}{2}\langle{\mathscr{R}}F,F\rangle$ as well as
$\nabla\phi(F)=-\mu(F){\mathscr{R}}F$. We also have a matrix representation
for $\Pi$ via Gram-Schmidt and we can compute
$\nabla^{\circ}\phi(F)=\Pi(\nabla(F))$. It is now easy to find approximate
solutions of the polyhedral modified moment map flow (4.3) via the Euler
method. Similarly we can apply the Euler approximation method to the
polyhedral renormalized flow (4.7).

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