# Unstable minimal surfaces in symmetric spaces of non-compact type

Nathaniel Sagman  and  Peter Smillie

###### Abstract.

We prove that if $\Sigma$ is a closed surface of genus at least 3 and $G$ is a
split real semisimple Lie group of rank at least $3$ acting faithfully by
isometries on a symmetric space $N$, then there exists a Hitchin
representation $\rho:\pi_{1}(\Sigma)\to G$ and a $\rho$-equivariant unstable
minimal map from the universal cover of $\Sigma$ to $N$. This follows from a
new lower bound on the index of high energy minimal maps into an arbitrary
symmetric space of non-compact type. Taking $G=\textrm{PSL}(n,\mathbb{R})$,
$n\geq 4$, this disproves the Labourie conjecture.

##  1\. Introduction

###  1.1. Hitchin representations

Let $\Sigma_{g}$ be a closed orientable surface of genus $g\geq 2$, $G$ the
adjoint form of a split real simple Lie group, and
$\textrm{Rep}(\pi_{1}(\Sigma_{g}),G)$ the representation variety. There is a
unique irreducible embedding $i_{G}:\textrm{PSL}(2,\mathbb{R})\to G$. Given a
Fuchsian representation $\rho$ from $\pi_{1}(\Sigma_{g})$ to
$\textrm{PSL}(2,\mathbb{R})$, $i_{G}\circ\rho$ defines a representation to
$G$. The Hitchin component, consisting of the Hitchin representations, is the
connected component of $\textrm{Rep}(\pi_{1}(\Sigma_{g}),G)$ containing
representations of the form $i_{G}\circ\rho$. Understood to generalize
Teichmüller space, Hitchin components are the first and most important
examples of higher Teichmüller spaces.

In his seminal paper [20], Hitchin found using Higgs bundles that the Hitchin
component is homeomorphic to a cell, just like Teichmüller space. Around the
same time, Choi-Goldman showed that the Hitchin component for
$G=\textrm{PSL}(3,\mathbb{R})$ parametrizes geometric structures on surfaces:
it is the space of marked convex $\mathbb{RP}^{2}$ structures on $\Sigma_{g}$
[6]. About a decade later, Labourie [23, Theorem 1.5] and Fock-Goncharov [15]
independently proved that Hitchin representations are discrete and faithful.
Now known to share many dynamical properties with Fuchsian representations,
Hitchin representations—and more generally Anosov representations into non-
compact real Lie groups—have become fruitful objects of study. For more
information we recommend the survey [41].

Let $K$ be a maximal compact subgroup of $G$, so that $G/K$ with its Killing
metric is a Riemannian symmetric space. Given an irreducible representation
$\rho:\pi_{1}(\Sigma_{g})\to G$, Donaldson [11] and Corlette [9] proved that
for every Riemann surface structure $S$ on $\Sigma_{g}$ with universal cover
$\tilde{S}$, there exists a unique $\rho$-equivariant harmonic map
$h:\tilde{S}\to G/K$. Labourie showed that for Hitchin representations, there
is a choice of Riemann surface structure such that the harmonic map $h$ is
minimal [24]. Labourie’s minimal surfaces are stable minima for the
equivariant area functional.

Labourie conjectured that for $G=\textrm{PSL}(n,\mathbb{R})$, every Hitchin
representation $\rho:\pi_{1}(\Sigma_{g})\to G$ determines a unique
$\rho$-equivariant minimal map into the symmetric space. It is natural to
extend this conjecture to all Hitchin representations into split real simple
Lie groups. Labourie’s existence theory applies even more generally to all
representations in higher Teichmuüller spaces. These include maximal and
$\Theta$-positive representations (see [3] and [18]). The uniqueness question
has been posed for these representations, and is also referred to in some
places as the Labourie Conjecture. To distinguish, we call the uniqueness
conjecture for all higher Teichmüller spaces the Generalized Labourie
Conjecture.

The uniqueness of the minimal map for $\mathrm{PSL}(2,\mathbb{R})$ follows
more or less from the definition of a Fuchsian representation. Labourie noted
that for $\mathrm{PSL}(3,\mathbb{R})$, uniqueness had been established by
Loftin [26]. Labourie then proved the generalized conjecture for all split
real simple $G$ of rank $2$ [25] (this adds $\textrm{PSp}(4,\mathbb{R})$ and
$G_{2}$), and Collier-Tholozan-Toulisse proved it for maximal representations
into Hermitian Lie groups of rank $2$ [8]. See also the work of Collier [7],
Alessandrini-Collier [1], and Nie [31].

The Labourie Conjecture and its generalization have attracted significant
interest in higher Teichmüller theory. Whenever the uniqueness conjecture
holds for Hitchin representations into a split real Lie group $G$, so in
particular for all such $G$ of rank $2$, the Hitchin component admits a
natural complex structure. More precisely, there is a mapping class group
equivariant parametrization as the total space of an explicit holomorphic
vector bundle over Teichmüller space (see [24, section 2] and [25, section
1]). This is preferable to Hitchin’s original parametrization, which depends
on a choice of complex structure on $\Sigma_{g}$ and doesn’t enjoy mapping
class group symmetry. For spaces of maximal representations in rank $2$,
similar equivariant parametrizations as complex manifolds have been obtained
in [7] and [1], although the complex structures are more complicated.

As far as we’re aware, the Generalized Labourie Conjecture was believed to be
true until March 2021, when Marković demonstrated the existence of a product
of Fuchsian representations into $\textrm{PSL}(2,\mathbb{R})^{3}$ with
multiple minimal surfaces in the corresponding product of closed hyperbolic
surfaces [28] (see also [27]). This gives a counterexample to the Generalized
Labourie Conjecture for maximal representations into Hermitian Lie groups of
rank at least $3$. In recent joint work with Marković, we prove that every
unstable equivariant minimal surface in $\mathbb{R}^{n}$ gives rise to an
unstable minimal surface in a product of $n$ closed hyperbolic surfaces [29].

In this paper we settle Labourie’s original conjecture. We first prove that
the index of a high energy minimal map into a non-compact symmetric space of
rank at least $3$ is bounded below by the index of a related equivariant
minimal map into $\mathbb{R}^{n}$. Specializing to split real Lie groups, we
use the new lower bound to deduce that the generalized conjecture fails for
Hitchin representations into every split real group of rank at least $3$.

For split real $G$ splitting into simple factors as $G=G_{1}\times\dots\times
G_{m},$ we use Hitchin to mean that each component representation to $G_{i}$
is Hitchin. In the semisimple case the associated symmetric space can have
different invariant metrics leading to different meanings of “minimal,” and
our theorem is true for any invariant metric.

######  Theorem A.

Let $G$ be the adjoint form of a split real semisimple Lie group with rank at
least $3$ acting faithfully by isometries on a symmetric space $N$. For every
$g\geq 3,$ there exists a Hitchin representation $\rho:\pi_{1}(\Sigma_{g})\to
G$ and an unstable $\rho$-equivariant minimal map into $N$. In particular,
there are at least two equivariant minimal surfaces in $N$.

We restrict to $g\geq 3$ because there are no unstable equivariant minimal
surfaces in $\mathbb{R}^{n}$ of genus $2$; we expect the result to extend to
$g=2$ (see [29, Theorem D] for $G=\textrm{PSL}(2,\mathbb{R})^{3}$). By general
principles, we can use Theorem A to prove the following corollary.

######  Corollary A.

For every $g\geq 3$ and pair $G,N$ as in Theorem A, there exists a Hitchin
representation $\rho:\pi_{1}(\Sigma_{g})\to G$ with at least two area
minimizing $\rho$-equivariant minimal maps into $N$.

We remark that there are other candidates for an equivariant and holomorphic
parametrization of the Hitchin components. See [16] and [32].

###  1.2. The case $\textrm{PSL}(2,\mathbb{R})^{n}$

Before describing the proof of Theorem A, we explain our previous work with
Marković [29]. On a fixed Riemann surface structure $S$ over $\Sigma_{g}$ with
universal cover $\tilde{S}$, every holomorphic quadratic differential $\phi$
on $S$ corresponds to a unique Fuchsian representation to
$\textrm{PSL}(2,\mathbb{R})$ and an equivariant harmonic map from $\tilde{S}$
to the hyperbolic space $\mathbb{H}^{2}$ (see [20], [40], and [42]). The
quadratic differential $\phi$ also corresponds to a unique equivariant
harmonic maps to an $\mathbb{R}$-tree: in one direction, the leaf space
$(T,d)$ for the vertical foliation of $\phi$ is an $\mathbb{R}$-tree, and the
projection map from $\tilde{S}\to(T,d)$ is harmonic. Given $\phi$, we consider
the ray of representations with harmonic maps determined by $R\phi,$ $R>0.$
After suitably rescaling the metric on the target $\mathbb{H}^{2}$, the maps
converge in some sense as $R\to\infty$ to a harmonic map to the
$\mathbb{R}$-tree for $\phi$ (see [43] for details).

If quadratic differentials $\phi_{1},\dots,\phi_{n}$ sum to $0$, the
equivariant map to $(\mathbb{H}^{2})^{n}$ determined by
$(R\phi_{1},\dots,R\phi_{n})$ is minimal and converges as $R\to\infty$ to a
minimal map in a product of $\mathbb{R}$-trees. We prove that every unstable
minimal surface for $\textrm{PSL}(2,\mathbb{R})^{n}$ converges along the ray
to an unstable minimal surface in a product of trees, and conversely every
unstable minimal surface in a product of trees is approximated by unstable
$\textrm{PSL}(2,\mathbb{R})^{n}$-minimal surfaces [29, Theorem B2].

To produce unstable minimal surfaces in products of trees, we observed that
when all the $\phi_{i}^{\prime}s$ are squares, each $\mathbb{R}$-tree folds
onto a copy of $\mathbb{R}$, and the minimal surface in the product maps onto
an equivariant minimal surface in $\mathbb{R}^{n}$. We prove that instability
in these two contexts is equivalent [29, Theorem C]. We then noted that
unstable equivariant minimal surfaces in $\mathbb{R}^{n}$ are abundant for
$n\geq 3$: the most natural example is the lift of an unstable minimal surface
in the $n$-torus to $\mathbb{R}^{n}$, and, in fact, any non-planar equivariant
minimal surface in dimension $3$ is unstable (see [29, section 5.3]).

Finally, we found that minimal surfaces in $\mathbb{R}^{n}$ are at the heart
of the problem: in the general setting where we start with $n$ quadratic
differentials summing to $0$, giving an equivariant minimal map into a product
of trees, there is a branched covering on which the differentials lift to
squares. If the corresponding equivariant minimal map from the branched cover
into $\mathbb{R}^{n}$ can be destabilized through equivariant variations that
are invariant under the Deck group of the branched covering, then the minimal
surface in the original product of $\mathbb{R}$-trees is unstable.

###  1.3. The idea of the proof

Let $G$ be a semisimple real Lie group of real rank $r$ with finite center and
no compact factors. Parreau has compactified
$\textrm{Rep}(\pi_{1}(\Sigma_{g}),G)$, with boundary objects corresponding to
$\pi_{1}(\Sigma_{g})$-actions on rank $r$ buildings [33]. We approach the
boundary along Hitchin rays, which generalize our construction for
$\mathrm{PSL}(2,\mathbb{R})^{n}$ by way of $G$-Higgs bundles. It is
conjectured in [21] that, with generic hypothesis, along these Hitchin rays
the harmonic maps converge in an appropriate sense to a harmonic map to a
building. Theorem B2 from [29] suggests that any counterexample to the
Labourie conjecture should come from an unstable minimal surface in a
building. It’s then feasible to imagine that these minimal surfaces behave
like minimal surfaces in $\mathbb{R}^{r}$. Perhaps after passing to a branched
covering, the apartments of the buildings can fold onto each other in a way
that produces equivariant minimal surfaces in $\mathbb{R}^{r}$. This geometric
picture gives an intuitive reason to expect stability in rank $2$ and
instability in rank at least $3$.

We now describe the Hitchin rays in more detail for $G$ isogenous to
$\mathrm{PGL}(n,\mathbb{R})$, in which case the $G$-Higgs bundles are
essentially classical Higgs bundles as defined by Hitchin. Note that the rank
is $n-1$. A Higgs bundle $(E,\overline{\partial}_{E},\phi)$ is a holomorphic
vector bundle $(E,\overline{\partial}_{E})$ with a holomorphic section
$\phi\in H^{0}(S,\textrm{End}(E)\otimes\mathcal{K})$ called the Higgs field.
By the non-abelian Hodge correspondence, every harmonic map to a
$\textrm{PGL}(n,\mathbb{C})$-symmetric space (including harmonic maps to the
$\mathrm{PGL}(n,\mathbb{R})$ subspace) arises from a unique polystable Higgs
bundle. Although the Higgs bundle is a tractable holomorphic object, the
harmonic map is not easy to understand from the Higgs bundle data. Indeed, the
harmonic map is related to the Higgs bundle through Hitchin’s self-duality
equations, which form a system of $n^{2}$ coupled non-linear PDE’s.
Accordingly, the precise analysis of convergence to harmonic maps to buildings
isn’t fully fleshed out.

A Higgs bundle $(E,\overline{\partial}_{E},\phi)$ is generically semisimple if
the Higgs field is semisimple on the complement of a finite set. Fortunately,
for such Higgs bundles it is possible to extract from Hitchin’s equations
certain scalar subsolutions, and then to use the maximum principle to
understand the behavior of the harmonic map as $R\to\infty$. This is carried
out by Mochizuki, using estimates of Simpson, who calls the resulting
phenomenon asymptotic decoupling [30]. He essentially proves and convergence
to a map to a building in open sets that should map into a single apartment.

Every generically semisimple Higgs bundle has an associated branched covering
space called the cameral cover ([21, Section 3], [10]) and an equivariant
minimal map $\tilde{f}$ from a covering space of the cameral cover to
$\mathbb{R}^{n-1}$. We extrapolate from Mochizuki’s analysis that off the
branch locus, the intrinsic data (the induced metric and part of the second
fundamental form) of the minimal map to the symmetric space associated to the
Higgs bundles $(E,\overline{\partial}_{E},R\phi)$ converges to the intrinsic
data of the map $\tilde{f}$ to $\mathbb{R}^{n-1}$. Unfortunately, the precise
convergence to buildings at points where the Higgs field is not semisimple is
yet to be understood. The local picture at these points has been worked out
only for minimal surfaces associated to $\textrm{PSL}(3,\mathbb{R})$ and
$\textrm{PSp}(4,\mathbb{R})$ Hitchin representations [12], [37].

Happily, the difficulties at non-semisimple points are readily overcome: by
the log cut-off trick, any destabilizing variation of a minimal surface can be
perturbed to a destabilizing variation supported on the complement of any
discrete set. The upshot of Mochizuki’s work and this last observation is that
we can maneuver around the technicalities of buildings, and treat high energy
minimal maps as if they approximate minimal surfaces in $\mathbb{R}^{n-1}$.
With this guiding principle, we prove in Theorem B that if the limiting
minimal surface in $\mathbb{R}^{n-1}$ is unstable under variations equivariant
by a certain action, then so are the high energy minimal surfaces in the
symmetric space. This instability result is the technical step, although the
computation becomes simple and natural if we work in the setting of harmonic
bundles and vary by well-chosen gauge transformations.

Finally, we show that for any equivariant minimal map $\tilde{f}$ into
$\mathbb{R}^{n-1}$, one can find a Higgs bundle for a Hitchin representation
to $\textrm{PSL}(n,\mathbb{R})$ whose limiting minimal map is isometric to
$\tilde{f}$. Briefly, one can easily turn the Weierstrass data for any minimal
surface into a Cartan-valued holomorphic $1$-form for
$\mathfrak{sl}(n,\mathbb{C})$, and evaluating the Hitchin fibration on this
form defines a point in the Hitchin base. Applying a Hitchin section to this
point returns the desired Higgs bundle. To conclude the argument, one picks
$\tilde{f}$ to be any unstable equivariant minimal surface in
$\mathbb{R}^{n-1}$, which is possible only for $n\geq 4$. The Hitchin section
is defined for all split real $G$, and we use this to prove Theorem A in
general.

###  1.4. The index of minimal maps

We now give a careful statement of Theorem B. To keep things classical in the
introduction, we focus mainly on the $\mathrm{PGL}(n,\mathbb{C})$ case.

Let $(E,\overline{\partial}_{E},\phi)$ be a generically semisimple stable
Higgs bundle. There exists $m\leq n$ and a finite subset $B$ such that for
every $p\in S-B$, there is a neighbourhood $U\subset S-B$ of $p$ on which the
Higgs bundle decomposes holomorphically as $E|_{U}=\oplus_{i=1}^{m}E_{i}$, and
$\phi$ acts on each $E_{i}$ by multiplication by a holomorphic $1$-form
$\phi_{i}$. Let $\textrm{End}^{0}(E)\subset\textrm{End}(E)$ be the subbundle
of traceless endomorphisms. Ranging over a covering of $S-B$, the locally
defined projections to the $E_{i}$ span a flat holomorphic subbundle
$F_{\phi}^{0}\subset\textrm{End}^{0}(E)|_{S-B}$ with parallel metric and real
structure, which we call the toral bundle of $\phi$. $\phi$ itself is a
holomorphic $F_{\phi}^{0}$-valued $1$-form. Using the real structure on
$F_{\phi}^{0}$, $\phi$ further defines a flat Riemannian affine bundle
$M_{\phi}$ over $S$, which we call the apartment bundle and comes equipped
with a harmonic section $f$ whose holomorphic derivative is $\phi$. This is
constructed in section 3.2.

The cameral cover is a minimal degree branched covering $\tau:C\to S$ on which
the monodromy of $F_{\phi}^{0}$ trivializes. On $C-\tau^{-1}(B)$, $\phi$
admits globally defined eigen-$1$-forms $\phi_{1},\dots,\phi_{m}$, which then
extend to all of $C$ by the removable singularities theorem. Lifting to a
covering space $\tilde{C}$ and integrating the real parts gives a harmonic map
to $\mathbb{R}^{m}$

$\tilde{f}=(\tilde{f}_{1},\dots,\tilde{f}_{m}),\hskip
2.84526pt\tilde{f}_{i}(z)=\sqrt{r_{i}}\textrm{Re}\int_{z_{0}}^{z}\phi_{i},$

where $r_{i}=\mathrm{rank}(E_{i})$, unique up to translation and equivariant
with respect to a representation to $\mathbb{R}^{m}$. It’s tidier to work with
the equivalent harmonic section of the apartment bundle $M_{\phi}$, but it’s
helpful to keep this geometric picture in mind.

Using Hitchin and Simpson’s side of the non-abelian Hodge correspondence, the
Higgs bundle produces a surface group representation
$\rho:\pi_{1}(\Sigma_{g})\to\textrm{PGL}(n,\mathbb{C})$ and a
$\rho$-equivariant harmonic map $h$ from $\tilde{S}$ to the symmetric space of
$\textrm{PGL}(n,\mathbb{C})$. This is equivalent to the data of a bundle $E$
with projectively flat connection $D$ and harmonic metric $H$. If we rescale
the Higgs field via the $\mathbb{R}^{+}$-action
$R\cdot(E,\overline{\partial}_{E},\phi)=(E,\overline{\partial}_{E},R\phi)$,
then we get new pairs of representations and maps $(\rho_{R},h_{R})$, or
equivalently triples $(E_{R},D_{R},H_{R})$. While our results concern the maps
$h_{R},$ for the problem at hand it’s easier to compute with the metrics
$H_{R}$.

For each $R$, the map $h_{R}$ is minimal if and only if $f$ is. Assuming
minimality of $f$, let $\textrm{Ind}(H_{R})$ be the equivariant index for the
area of $h_{R}$, and $\textrm{Ind}(f)$ the index of $f$ as a section of
$M_{\phi}$, which is the same as the equivariant index for the area of
$\tilde{f}$. The indices are rigorously defined in section 3.2.

######  Theorem B for $\textrm{PGL}(n,\mathbb{C})$.

$\liminf_{R\to\infty}\textrm{Ind}(H_{R})\geq\textrm{Ind}(f)$.

Inspired by the situation for $\mathrm{PSL}(2,\mathbb{R})^{n}$, we conjecture
that $\textrm{Ind}(H_{R})$ is non-decreasing with $R$ and
$\lim_{R\to\infty}\textrm{Ind}(H_{R})=\textrm{Ind}(f)$.

We turn now to the most general case of a real semisimple Lie group $G$ acting
on a symmetric space $N$, and a generically semisimple stable $G$-Higgs bundle
$(P,A^{0,1},\phi)$. Here $(P,A^{0,1})$ is a holomorphic $K^{\mathbb{C}}$
bundle and $\phi$ is a holomorphic $1$-form valued in
$P\times_{K^{\mathbb{C}}}\mathfrak{p}^{\mathbb{C}}$. As in the classical case,
there is an $\mathbb{R}^{+}$-family of representations and harmonic maps, or
$G$-harmonic bundles $(M_{R},h_{R})$, in which $M_{R}$ is the bundle with
fiber $N$ assoicated to $P$, which inherits a flat connection, and $h_{R}$ is
a harmonic section.

The right generalization of the bundle $F^{0}_{\phi}$ to this setting is what
we call the $G$-toral bundle of $\phi$, whose fiber is the center of the
centralizer of $\phi$ in $P\times_{K^{\mathbb{C}}}\mathfrak{p}^{\mathbb{C}}$.
As in the $\mathrm{PGL}(n,\mathbb{C})$ case, this is really a vector bundle
away from some bad set $B$, and its rank is at most the rank of $G$. From the
toral bundle, we build the $G$-apartment bundle $M^{G}_{\phi}$, together with
its minimal section $f$ (section 3.4.1). Then we prove

######  Theorem B.

$\liminf_{R\to\infty}\textrm{Ind}(h_{R})\geq\textrm{Ind}(f)$.

The proof almost follows directly from the case of
$\mathrm{PGL}(n,\mathbb{C})$ by using the adjoint representation of $G$. The
only minor complication arises from our insistence, in the semisimple case, of
allowing an arbitrary invariant metric on the symmetric space $N$.

######  Remark 1.1.

By the work of Toledo [39], the associated energy functional on Teichmüller
space is plurisubharmonic. Since the energy index is equal to the area index
[14, Theorem 3.4], $\textrm{Ind}(h_{R})$ is at most $3g-3$.

###  1.5. Acknowledgements

We would like to thank Nicolas Tholozan, Eugen Rogozinnikov, and Max
Riestenberg for helpful conversations, and Mike Wolf in particular for
pointing out the connection between [28] and minimal surfaces in
$\mathbb{R}^{n}$.

##  2\. Preliminaries

###  2.1. Hermitian metrics

For a complex vector space $V$, we write $V^{\vee}$ for its dual and
$\overline{V}$ for its complex conjugate. Similarly, for a linear map of
complex vector spaces $f:V\to W,$ we write $f^{\vee}:W^{\vee}\to V^{\vee}$ for
the transpose and $\overline{f}:\overline{V}\to\overline{W}$ for the
conjugate. A Hermitian metric $H$ on $V$ is a linear isomorphism
$H:V\to\overline{V}^{\vee}$ such that $H=\overline{H}^{\vee}$ and the
associated sesquilinear form is positive definite. If $f\in\textrm{End}(V)$
and $H$ is a Hermitian metric on $V$, then the adjoint with respect to $H$ is
$f^{*_{H}}=H^{-1}\overline{f}^{\vee}H:V\to V$.

Let $\mathbb{P}\mathrm{Met}(V)$ be the real projectivization of the space of
Hermitian metrics on $V$. We will use $[H]$ to denote the equivalence class of
a metric $H$ in $\mathbb{P}\mathrm{Met}(V)$. The group $G=\mathrm{PGL}(V)$
acts transitively on $\mathbb{P}\mathrm{Met}(V)$ via

(1) $g\cdot[H]=[\overline{\hat{g}^{-1}}^{\vee}H\hat{g}^{-1}]$

where $\hat{g}$ is any lift of $g$ to $\mathrm{GL}(V)$. The stabilizer of a
class $[H]$ is a maximal compact subgroup $K_{H}\subset G$, which exhibits
$\mathbb{P}\mathrm{Met}(V)$ as a symmetric space of non-compact type.

Let $\mathfrak{g}$ be the Lie algebra of $G$, which we identify with the
traceless endomorphisms $\textrm{End}^{0}(V)$. Let
$\mathrm{Hom}_{\mathbb{R}}(V,\overline{V}^{\vee})$ be the real subspace of
$\mathrm{Hom}(V,\overline{V}^{\vee})$ consisting of maps equal to their
conjugate transpose. For any $[H]\in\mathbb{P}\mathrm{Met}(V)$, the tangent
space $T_{[H]}\mathbb{P}\mathrm{Met}(V)$ is the quotient of
$\textrm{Hom}_{\mathbb{R}}(V,\overline{V}^{\vee})$ by the line $[H]$. If
$X\in\mathfrak{g}$, then the derivative of the action $\eqref{eqn: action}$ is
given by

$X\cdot[H]=[-\overline{X}^{\vee}H-HX]\in T_{[H]}\mathbb{P}\mathrm{Met}(V).$

If $Y\in\textrm{End}(V)$, write $Y^{0}$ for its traceless part. There is a
$\mathfrak{g}$-valued $1$-form $\varphi$ on $\mathbb{P}\mathrm{Met}(V)$
defined by

(2) $\varphi_{[H]}=-\frac{1}{2}(H^{-1}dH)^{0}.$

for any lift $H$ to $\mathrm{Hom}_{\mathbb{R}}(V,\overline{V}^{\vee})$.
Combining the last two equations, we find

(3) $\varphi_{[H]}(X\cdot[H])=\frac{1}{2}(X^{*_{H}}+X).$

Note that the operator $*_{H}$ depends only on the projective equivalence
class $[H]$.

Let $\mathrm{End}_{H}^{0}\subset\mathfrak{g}$ be the space of $H$-self-adjoint
endomorphisms of $V$. Then $\varphi_{[H]}(X\cdot[H])$ is the projection of $X$
to $\mathrm{End}_{H}^{0}$, and it follows that $\varphi$ defines an
isomorphism from $T\mathbb{P}\mathrm{Met}(V)$ to the bundle over
$\mathbb{P}\mathrm{Met}(V)$ whose fiber over a metric $H$ is
$\mathrm{End}_{H}^{0}$.

The Killing metric on $\mathbb{P}\mathrm{Met}(V)$ is defined to be

(4) $2n\operatorname{\textrm{tr}}(\varphi\otimes\varphi).$

In the sequel, we will omit the tensor product $\otimes$ from our notation,
writing this as $2n\operatorname{\textrm{tr}}(\varphi^{2})$, which is to be
read as a bilinear form. This metric is $G$-invariant, and every $G$-invariant
metric on $\mathbb{P}\mathrm{Met}(V)$ is a constant multiple of the Killing
one.

To simplify the statement of the following proposition, we use $\varphi$ to
identify $T\mathbb{P}\mathrm{Met}(V)$ with the subbundle
$\mathrm{End}_{H}^{0}$ of the trivial $\mathfrak{g}$ bundle over
$\mathbb{P}\mathrm{Met}(V)$ whose fiber over $[H]$ is the subspace
$\mathrm{End}_{H}^{0}$. The Lie bracket and differential $d$ are defined on
this trivial $\mathfrak{g}$ bundle.

######  Proposition 2.1 (Theorem X.2.6 in [22]).

For sections $X,Y,Z$ of $\mathrm{End}_{H}^{0}$, implicitly identified with
vector fields on the symmetric space, the Levi-Civita connection $\nabla^{lc}$
is given by

(5) $\nabla_{X}^{lc}Y=d_{X}Y-[X,Y].$

and curvature of $\nabla^{lc}$ is given by

$R(X,Y)Z=-[[X,Y],Z]$

######  Remark 2.2.

Although the metric is canonical only up to a constant, the Levi-Civita
connection and curvature are independent of this choice.

###  2.2. Harmonic maps, bundles, and minimal surfaces

Let $S$ be a possibly non-compact Riemann surface and $h:S\to N$ a smooth map
from $S$ to a Riemannian manifold $N$. Let $\mathcal{K}$ be the canonical
bundle of $S$, and let $\partial h$ be the $(1,0)$-part of the derivative of
$h$, which is a section of $h^{*}TN\otimes_{\mathbb{R}}\mathcal{K}$. Let
$\operatorname{\overline{\partial}}$ be the $(0,1)$-part of the connection on
$h^{*}TN\otimes_{\mathbb{R}}\mathbb{C}$ induced from the Levi-Civita
connection of $N$, as well as its natural extension to
$h^{*}TN\otimes_{\mathbb{R}}\mathbb{C}$-valued forms.

######  Definition 2.3.

The map $h$ is harmonic if $\overline{\partial}\partial h=0$.

If $\nu=\langle\cdot,\cdot\rangle$ is the Riemannian metric on $N$, the Hopf
differential is the section of $\mathcal{K}^{2}$ defined by $\langle\partial
h,\partial h\rangle$. We call $h$ conformal if $\langle\partial h,\partial
h\rangle=0$. We eschew words like almost conformal and branched conformal,
instead calling a classically conformal map a conformal immersion.

######  Definition 2.4.

A harmonic and conformal map is called a minimal map.

The area density of $h$ is the $(1,1)$-form on $S$ given by

(6) $a_{h}=\sqrt{\langle\partial
h,\overline{\partial}h\rangle^{2}-|\langle\partial h,\partial h\rangle|^{2}}$

where the expression inside the square-root is interpreted as a real section
of $\mathcal{K}^{2}\otimes\overline{\mathcal{K}}^{2}$. If $S$ is compact, we
write the area of $h(S)$ as

(7) $A(h)=\int_{S}a_{h}.$

Analogously, we define the energy density $e_{h}=\langle\partial
h,\operatorname{\overline{\partial}}h\rangle$ and total energy
$E(h)=\int_{S}e_{h}$. For arbitrary $S$, $h$ is harmonic if and only if it is
a critical point of $E$ with respect to compactly supported variations on all
compact subsurfaces, and if $h$ is harmonic, then it is minimal if and only if
it is a critical point of $A$ in this sense. In particular, a minimal map
defines a minimal surface in the classical sense if it is an immersion and an
embedding.

######  Remark 2.5.

A minimal map $h$ is allowed to be constant. If it is non-constant, then since
$\partial h$ is a holomorphic section of a holomorphic vector bundle, its
vanishing locus is discrete. By conformality, $h$ is an immersion wherever
$\partial h$ is nonzero. Therefore, a minimal map is an immersion away from a
discrete set.

####  2.2.1. Flat Riemannian bundles

Most of the time, for us $S$ will be a compact Riemann surface. We are not
interested in minimal maps from $S$ per se, but rather in minimal sections of
flat Riemannian bundles over $S$, which we now define. Let $M\to S$ be a fiber
bundle. A flat connection on $M$ is a foliation transverse to the fibers of
$M$ of complementary dimension, which induces isomorphisms between nearby
fibers, and hence local flat trivializations $M|_{U}=U\times M_{z}$, where
$z\in U\subset S$ and $M_{z}$ is the fiber over $z$. A Riemannian structure on
$M$ is a Riemannian metric on the fibers. $M$ is a flat Riemannian bundle if
it has both a flat connection and a Riemannian structure, and the induced
identifications between the fibers are isometries.

Let $M\to S$ be a flat Riemannian bundle, and let $h$ be a section. We say
that $h$ is minimal (or harmonic or conformal respectively) if the
corresponding map to $M_{z}$ is minimal (resp. harmonic, conformal) for every
local flat trivialization $M|_{U}=U\times M_{z}$. Since these notions are
invariant under isometry, it suffices to check them in a set of
trivializations which covers $S$.

If we don’t want to choose trivializations, we can also give a more abstract
definition of minimality. Let $T^{\mathrm{vert}}M$ be the vertical tangent
bundle of $M$. The metric on $M$ induces a metric on $T^{\mathrm{vert}}M$, and
the flat connection on $M$ lets us lift the Levi-Civita connection on the
fibers of $M$ to a connection on $T^{\mathrm{vert}}M$, which we also call the
Levi-Civita connection. If $h$ is a section of $M$, we simply interpret
$\partial h$ as a $1$-form valued in $h^{*}T^{\mathrm{vert}}M$ and
$\operatorname{\overline{\partial}}$ in terms of the Levi-Civita connection on
$T^{\mathrm{vert}}M$. Having done so, the original definitions of harmonic and
conformal still make sense. Naturally, the formulas (6) and (7) still define
the area density and area for sections of flat Riemannian bundles, and minimal
sections are still critical points of area. The condition that $h$ is an
immersion is potentially ambiguous; the correct condition is that the the area
form $a_{h}$ (defined with respect to the vertical tangent bundle) is
nonvanishing. It remains true that a minimal section is an immersion away from
a discrete set.

####  2.2.2. Bundles and equivariance

If $\tilde{S}$ is a universal covering space of $S$, and $\tilde{M}$ is the
pullback of $M$ to $\tilde{S}$, the flat connection gives canonical
isomorphisms between each fiber of $\tilde{M}$, each of which is furthermore
identified with the space of flat sections of $\tilde{M}$. Let
$\Gamma^{\mathrm{flat}}(\tilde{M})$ be this space. Viewing it as the space of
flat sections, the Deck group of $\tilde{S}$ acts on
$\Gamma^{\mathrm{flat}}(\tilde{M})$ by pullback. Call this action $\rho$.

If $h$ is a section of $M$ over $S$, we can pull it back to a section
$\tilde{h}$ of $\tilde{M}$. By identifying each fiber of $\tilde{M}$ with the
space of flat sections, we can view $\tilde{h}$ as a map
$\tilde{h}:\tilde{S}\to\Gamma^{\mathrm{flat}}(\tilde{M})$. This map will be
equivariant by the action $\rho$. Conversely, $\rho$-equivariant maps from
$\tilde{S}$ to $\Gamma^{\mathrm{flat}}(\tilde{M})$ descend to sections of $M$
on $S$. Clearly $h$ is minimal if and only if $\tilde{h}$ is.

####  2.2.3. Harmonic and minimal bundles

Let $E$ be a complex vector bundle over $S$. A connection $D$ on $E$ is called
projectively flat if its curvature is equal to $\omega\mathrm{Id}$, where
$\omega$ is a smooth $2$-form. Since the identity is in the center of
$\mathrm{End}(E)$, a projectively flat connection on $E$ induces a flat
connection on $\mathrm{End}(E)$.

If $(E,D)$ is a projectively flat bundle, we define the projectivized bundle
of metrics $\mathbb{P}\mathrm{Met}(E)$, whose fiber at a point $z$ is
$\mathbb{P}\mathrm{Met}(E(z))$. A choice of invariant metric $\nu$ on
$\mathbb{P}\mathrm{Met}(E)$ makes it into a flat Riemannian bundle. Unless we
specify $\nu$, we will always take the Killing metric (4). The projectively
connection $D$ defines a flat connection on $\mathbb{P}\mathrm{Met}(E)$. In
this way, $\mathbb{P}\mathrm{Met}(E)$ has the structure of a flat Riemannain
bundle.

The main object of study in this paper will be triples $(E,D,H)$, where $E$ is
a vector bundle over $S$, $D$ a projectively flat connection, and $H$ a smooth
Hermitian metric. Given such a triple, let $[H]$ be the corresponding section
of $\mathbb{P}\mathrm{Met}(E)$. The first part of the following definition is
standard, but the second part is not.

######  Definition 2.6.

$(E,D,H)$ is a harmonic bundle if the associated section $[H]$ of
$\mathbb{P}\mathrm{Met}(E)$ is harmonic. If $[H]$ is moreover minimal, we call
$(E,D,H)$ a minimal harmonic bundle.

Note that the definition of minimal would be equivalent had we chosen any
other scaling of the metric on $\mathbb{P}\mathrm{Met}(E)$, and that the
definition of harmonic does not reference the metric at all, only the Levi-
Civita connection.

In the introduction, we mentioned the existence theorem for harmonic maps of
Donaldson [11] and Corlette [9]. Rephrasing in terms of harmonic metrics, this
gives one side of the non-abelian Hodge correspondence. A representation
$\rho:\pi_{1}(S_{g})\to\textrm{PGL}(n,\mathbb{C})$ is called reductive if the
Zariski closure of the image subgroup is a reductive subgroup of
$\textrm{PGL}(n,\mathbb{C})$. It is irreducible if it is not contained in a
non-trivial parabolic subgroup.

######  Theorem 2.7 (Non-abelian Hodge correspondence I).

Let $(E,D)$ be a projectively flat vector bundle of rank $n$ over a closed
surface $S$, and suppose that the holonomy representation is reductive. Then
there exists a harmonic metric $H$ on $(E,D)$. If the holonomy is irreducible,
then $[H]$ is unique.

We will often write that $(E,D,H)$ is determined by NAH I, even though only
$[H]$ is uniquely defined; since we only ever use the operator $*_{H}$, which
depends only on $[H]$, we hope the reader will excuse this laziness.

###  2.3. Higgs bundles

We owe an interpretation of the definitions 2.3 and 2.4 of harmonic and
minimal in the context of triples $(E,D,H)$. First we need a good description
of $[H]^{*}T^{\mathrm{vert}}\mathbb{P}\mathrm{Met}(E)$. This is provided by
the endomorphism-valued $1$-form $\varphi$ defined in (2). On each fiber
$E(z)$, $\varphi$ is an $\mathrm{End}^{0}(E(z))$-valued $1$-form on
$\mathbb{P}\mathrm{Met}(E(z))$, which for each
$[H^{\prime}]\in\mathbb{P}\mathrm{Met}(E(z))$, identifies
$T_{[H^{\prime}]}\mathbb{P}\mathrm{Met}(E(z))$ with
$\mathrm{End}^{0}_{H^{\prime}}(E(z))$, the space of $H^{\prime}$-self-adjoint
endomoprhisms of $E(z)$. Note the complexification of
$\mathrm{End}^{0}_{H^{\prime}}(E)$ is just $\mathrm{End}^{0}(E)$, so $\varphi$
identifies the complexification of the tangent space at each point
$[H^{\prime}]$ with $\mathrm{End}^{0}(E)$.

In particular, if $(E,D,H)$ is a flat bundle with a metric, and $[H]$ is the
corresponding section of $\mathbb{P}\mathrm{Met}(E)$, then the pullback
$\varphi_{H}:=[H]^{*}\varphi$ of $\varphi$ by the section $[H]$ identifies
$[H]^{*}T^{\mathrm{vert}}\mathbb{P}\mathrm{Met}(E)$ with
$\mathrm{End}^{0}_{H}(E)$, and therefore identifies the space of variations of
the section $[H]$ with the space of sections of $\mathrm{End}^{0}_{H}(E)$ over
$S$. Since $\varphi_{H}$ is a an $\mathrm{End}(E)$-valued $1$-form, we can
define a connection on $E$ by

$\nabla^{H}=D-\varphi_{H}$

This induces the connection $D-[\varphi_{H},\cdot]$ on $\mathrm{End}(E)$,
which we also write $\nabla^{H}$. If $X$ and $Y$ are sections of
$\mathrm{End}_{H}^{0}(E)$ (which we implicitly identify with
$[H]^{*}T^{\mathrm{vert}}(M)$), then using equation (3) and the fact that $X$
is already equal to its $H$-self-adjoint part, we have

$\nabla^{H}_{X}Y=D_{X}Y-[X,Y].$

Comparing this with the formula (5) in Proposition 2.1, we conclude

######  Proposition 2.8.

The subbundle $End^{0}_{H}(E)$ of traceless $H$-self-adjoint endomorphisms is
invariant by $\nabla^{H}$, and the restriction of $\nabla^{H}$ to this bundle
is the pullback by the section $[H]$ of the Levi-Civita connection on
$T^{\mathrm{vert}}\mathbb{P}\mathrm{Met}(E)$.

Proposition 2.8 allows us to express harmonicity of $(E,D,H)$ in terms of the
connection $\nabla^{H}$. Using the Riemann surface structure on $S$, let
$\phi$ be the $(1,0)$-part of $\varphi_{H}$, and let
$\operatorname{\overline{\partial}}_{E}$ be the $(0,1)$-part of the connection
$\nabla^{H}$. Recall also that the Killing metric on $\mathrm{End}^{0}_{H}(E)$
is given by $\langle X,Y\rangle=2n\operatorname{\textrm{tr}}(XY)$. Definitions
2.3 and 2.4 translate to the result below.

######  Proposition 2.9.

$(E,D,H)$ is a harmonic bundle if
$\operatorname{\overline{\partial}}_{E}\phi=0$. It is a minimal bundle if in
addition $\operatorname{\textrm{tr}}(\phi^{2})=0$.

Since we’re working on a Riemann surface, the Koszul-Malgrange theorem
guarantees the existence of a complex structure on $E$ with del-bar operator
$\overline{\partial}_{E}$. This motivates the introduction of Higgs bundles.

######  Definition 2.10.

A Higgs bundle on $S$ is the data $(E,\overline{\partial}_{E},\phi),$ where
$(E,\overline{\partial}_{E})$ is a holomorphic vector bundle of rank $n$ and
$\phi$ is a holomorphic section of $\textrm{End}(E)\otimes\mathcal{K}$ called
the Higgs field.

In this context, Theorem 2.7 says that every projectively flat bundle $(E,D)$
with reductive holonomy gives the data of a Higgs bundle. Actually, this
passage from projectively flat bundles to Higgs bundles is generically
reversible. Assume now that $S$ is a closed Riemann surface. If $\deg(E)$ is
the degree of $E$ and $\textrm{rank}(E)$ the rank, the slope of a complex
vector bundle $E$ over $S$ is the quantity

$\mu(E)=\frac{\textrm{deg}(E)}{\textrm{rank}(E)}.$

######  Definition 2.11.

A Higgs bundle $(E,\overline{\partial}_{E},\phi)$ is stable if for all
$\Phi$-invariant holomorphic subbundles $E^{\prime}\subset E$ of positive
rank, we have $\mu(E^{\prime})<\mu(E)$. It is polystable if it is a direct sum
of stable Higgs bundles all of the same slope.

The proposition below is clear from the definitions, and we record it for
future use.

######  Proposition 2.12.

If $(E,\overline{\partial}_{E},\phi)$ is stable, then for
$\gamma\in\mathbb{C}^{*},$ $(E,\overline{\partial}_{E},\gamma\phi)$ is stable.

The converse to Theorem 2.7 is due to Hitchin [20] and Simpson [35]. Given a
Hermitian metric $H$ on a holomorphic bundle $(E,\overline{\partial}_{E})$,
let $\partial^{H}$ be the unique $(1,0)$-connection with the property that
$\overline{\partial}_{E}+\partial^{H}$ is $H$-unitary.

######  Theorem 2.13 (Non-abelian Hodge correspondence II).

If a Higgs bundle $(E,\overline{\partial}_{E},\phi)$ is stable, then there
exists a Hermitian metric $H$ on $E$ such that
$(E,\overline{\partial}_{E}+\partial^{H}+\phi+\phi^{*_{H}})$ is a projectively
flat bundle. The harmonic section $[H]$ of $\mathbb{P}\mathrm{Met}(E)$ is
unique.

Throughout the paper, we use the abbreviation NAH for the non-abelian Hodge
correspondence.

######  Remark 2.14.

If $(E,\operatorname{\overline{\partial}}_{E},\phi)$ is a direct sum of stable
Higgs bundles $(E_{i},\operatorname{\overline{\partial}}_{E_{i}},\phi_{i})$
(i.e., is polystable), then NAH II gives a harmonic bundle structure to each
$E_{i}$, and hence a harmonic map $[H]=\prod_{i}[H_{i}]$ into
$M=\prod_{i}\mathbb{P}\mathrm{Met}(E_{i})$. There is an $n-1$ dimensional
space of natural embeddings of $\prod_{i}\mathbb{P}\mathrm{Met}(E_{i})$ into
$\mathbb{P}\mathrm{Met}(E)$ corresponding to different relative scalings of
the $n$ factors. By choosing an embedding, we can construct a harmonic bundle
$(E,D,H)$.

It is also sometimes useful to not embed $M$ into $\mathbb{P}\mathrm{Met}(E)$,
but rather to view it as a flat Riemannian bundle of symmetric spaces in its
own right, with a harmonic section $[H]$. Note that $M$ itself has an
$n$-dimensional family of invariant metrics $\nu$ corresponding to choosing a
rescaling of the Killing metric on each factor. This is related to but not the
same as the family of embeddings; for instance, there is a single metric in
this $n$-dimensional family which makes all of the embeddings of $M$ into
$\mathbb{P}\mathrm{Met}(E)$ isometric. Note that the definition of a minimal
section of $M$ depends on a choice $\nu$ of invariant metric on its fibers.

######  Definition 2.15.

We will say that a polystable Higgs bundle is minimal with respect to an
invariant metric $\nu$ if the associated harmonic map to $M$ is minimal with
respect to $\nu$.

###  2.4. Symmetric spaces and $G$-Higgs bundles

This subsection can be skipped if the reader is only interested in Theorem B
for $\mathrm{PGL}(n,\mathbb{C})$ or Theorem A for
$\mathrm{PSL}(n,\mathbb{R})$.

####  2.4.1. Symmetric spaces of non-compact type

To us, a symmetric space of non-compact type is a connected and simply
connected Riemannian manifold with an inversion symmetry about each point,
whose de Rham decomposition contains only non-compact symmetric spaces, and no
factors of $\mathbb{R}$. The isometry group of a symmetric space of non-
compact type is always semi-simple with no compact factors and trivial center.
Such groups are our primary interest, but in order to make it easy to apply
the results to other standard settings, we allow certain subgroups and non-
faithful actions as well.

######  Definition 2.16.

A Lie group $G$ is admissible if it is semisimple with finite center and no
non-compact factors. An admissible pair $(G,K)$ is an admissible Lie group $G$
together with a maximal compact subgroup $K$.

######  Definition 2.17.

An action of an admissible Lie group $G$ is essentially faithful if its kernel
is discrete.

If $(G,K)$ is an admissible pair, we will always write
$\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ for the corresponding Lie algebra
decomopsition. Let $N$ be the pointed space $G/K$. Using a generalization of
the $\mathfrak{g}$-valued 1-form $\varphi$, one can show that the
$G$-invariant metric on $N$ are in bijection with the $K$-invariant positive
definite bilinear forms on $\mathfrak{p}$. Any choice of metric makes $N$ into
a symmetric space of non-compact type with an essentially faithful transitive
$G$-action.

######  Definition 2.18.

An admissible triple $(G,K,\nu)$ is an admissible pair $(G,K)$ together with a
$K$-invariant positive definite bilinear form $\nu$ on $\mathfrak{p}$.

If $N$ is a symmetric space of non-compact type, and $h$ a point of $N$, this
determines canonically a Lie algebra $\mathfrak{g}$ of Killing fields and a
decomposition $\mathfrak{g}=\mathfrak{k}_{h}+\mathfrak{p}_{h}$, with
$\mathfrak{k}_{h}$ the Killing fields vanishing at $h$, as well as the metric
$\nu$. The pair $(G=\mathrm{Aut}(N),K=\mathrm{Stab}_{G}(h))$ is admissible
with these Lie algebras, but $N$ does not uniquely determine the Lie group
$G$.

######  Remark 2.19.

If $G$ is simple, then $\mathfrak{g}$ has a unique invariant bilinear form up
to scale. For the purpose of studying minimal surfaces in $N$, scaling the
metric is unimportant. However, if $N$ is reducible, then different scalings
of its different factors determine geniunely different notions of minimal
surfaces. Thus, in the reducible case $N$ contains an important additional
piece of information that $G$ does not.

Theorem X.2.6 of [22] proves that the formulas for the curvature and
connection of $\mathbb{P}\mathrm{Met}(E)$ in Proposition 2.1 continue to hold
for $N$. Observe that these formulas only depend on $G$, and not on the choice
invariant bilinear form. Alternatively, we can deduce them by isometrically
embedding $N$ as a totally geodesic subspace of a product of symmetric spaces
of the type $\mathbb{P}\mathrm{Met}$.

######  Proposition 2.20.

If $(G,K,\nu)$ is an admissible triple, there is a representation
$\sigma=\prod_{i}\sigma_{i}:G\to\prod_{i}\mathrm{SL}_{n_{i}}(\mathbb{C})$
sending $K$ to $\prod_{i}\mathrm{SU}(n_{i})$, and constants $a_{i}$, such that
the induced map
$(G/K,\nu)\to\prod_{i}(\mathbb{P}\mathrm{Met}(\mathbb{C}^{n_{i}}),a_{i}\operatorname{\textrm{tr}})$
is a totally geodesic isometry.

###### Proof.

This is essentially Theorem 2.6.5 of [13]. The $K$-invariant form $\nu$ on
$\mathfrak{p}$ extends uniquely to a $G$-invariant form on $\mathfrak{g}$
which we also call $\nu$. It also determines a positive definite form
$\nu_{K}$ on $\mathfrak{g}$, which is equal to $\nu$ on $\mathfrak{p}$ and
$-\nu$ on $\mathfrak{k}$. Take an orthonormal basis of $\mathfrak{g}$ with
respect to the form $\nu_{K}$, and use the adjoint representation to map $G$
to to $\mathrm{SL}({\mathrm{dim}(\mathfrak{g}),\mathbb{C})}$. Since the
adjoint representation preserves the splitting of $\mathfrak{g}$ into simple
pieces $\mathfrak{g}_{i}$, it actually maps to
$\prod_{i}\mathrm{SL}(n_{i},\mathbb{R})$, where
$n_{i}=\mathrm{dim}(\mathfrak{g}_{i})$. Including
$\mathrm{SL}(n_{i},\mathbb{R})$ into $\mathrm{SL}(n_{i},\mathbb{C})$ gives the
required representation. ∎

The most important invariant of a symmetric space $N$ of non-compact type is
its rank, which is the largest dimensional subspace of its tangent space at
any point on which the sectional curvature vanishes. Equivalently, this is the
dimension of a maximal abelian subalgebra of $\mathfrak{g}$ contained in
$\mathfrak{p}$. We call such an algebra a maximal toral subalgebra of
$\mathfrak{p}$, using the word toral to emphasize that it necessarily consists
of semisimple elements.

Let us now additionally fix a maximal toral subalgebra $\mathfrak{a}$ of
$\mathfrak{p}$. The Weyl group of $\mathfrak{a}$ (sometimes called the
restricted Weyl group of $\mathfrak{a}$) is the normalizer of $\mathfrak{a}$
in $K$ modulo its centralizer. The roots of $\mathfrak{a}$ (sometimes called
restricted roots) are the nonzero characters of $\mathfrak{a}$ that arise in
its adjoint action on $\mathfrak{g}$. We reserve the word Cartan subalgebra
for a (complex) maximal toral subalgebra of the complexification
$\mathfrak{g}^{\mathbb{C}}$. The real Lie algebra $\mathfrak{g}$ is called
split if the complexification of a maximal toral subalgebra of $\mathfrak{p}$
is a Cartan subalgebra of $\mathfrak{g}^{\mathbb{C}}$. A Lie group $G$ is
called split if its Lie algebra is split.

Let $K^{\mathbb{C}}\subset G^{\mathbb{C}}$ and
$\mathfrak{a}^{\mathbb{C}}\subset\mathfrak{p}^{\mathbb{C}}\subset\mathfrak{g}^{\mathbb{C}}$
be complexifications of $K$, $\mathfrak{a}$, and $\mathfrak{p}$ respectively.
Let $\mathcal{O}(\mathfrak{p})^{K}$ the ring of $K$-invariant complex
polynomials on $\mathfrak{p}$, and $\mathcal{O}(\mathfrak{a})^{W}$ the ring of
$W$-invariant polynomials on $\mathfrak{a}$. These are the same as the ring of
$K^{\mathbb{C}}$ invariant polynomials on $\mathfrak{p}^{\mathbb{C}}$ and
$W$-invariant poynomials on $\mathfrak{a}^{\mathbb{C}}$ respectively. The
Chevalley restriction theorem gives an isomorphism
$\mathcal{O}(\mathfrak{p})^{K}=\mathcal{O}(\mathfrak{a})^{W}$, and says that
$\mathcal{O}(\mathfrak{p})^{K}$ is free on $r$ generators, where $r$ is the
dimension of $\mathfrak{a}$, i.e., the rank of $N$. For example, if
$G=\mathrm{PGL}(n,\mathbb{R})$, so that $\mathfrak{p}$ consists of symmetric
matrices, then the elementary symmetric polynomials $p_{i}$, $i=2,\ldots,n$
applied to the eigenvalues is a minimal set of generators for
$\mathrm{O}(\mathfrak{p})^{K}$.

An element $X$ of $\mathfrak{p}^{\mathbb{C}}$ is called regular semisimple if
it is semisimple and contained in a unique Cartan subalgebra
$\mathfrak{a}_{X}^{\mathbb{C}}$ of $\mathfrak{p}^{\mathbb{C}}$. For
$X\in\mathfrak{a}^{\mathbb{C}}$, $X$ is regular semisimple if and only if no
root of $\mathfrak{a}$ vanishes on $X$. We will use the following standard
lemma in section 4.

######  Lemma 2.21.

If $X\in\mathfrak{p}^{\mathbb{C}}$ is regular semisimple,
$Y\in\mathfrak{p}^{\mathbb{C}}$ is arbitrary, and every invariant polynomial
$p\in\mathcal{O}(\mathfrak{p})^{K}$ has $p(X)=p(Y)$, then $X=\mathrm{Ad}_{k}Y$
for some $k$ in $K$. If $k$ and $k^{\prime}$ are two elements of $K$
conjugating $Y$ to $X$, and $Z\in\mathfrak{p}^{\mathbb{C}}$ commutes with $Y$,
then $\mathrm{Ad}_{k}Z=\mathrm{Ad}_{k^{\prime}}Z$.

####  2.4.2. $G$-Higgs bundles

Let $N$ be a symmetric space of non-compact type, and let
$G=\mathrm{Isom}(N)$. Let $M$ be a flat Riemannian bundle over $S$, each of
whose fibers is isometric to $N$. If $\tilde{S}$ is a universal cover of $S$,
then $M$ defines, up to conjugacy, a holonomy representation from the Deck
group of $\tilde{S}$ to $G$. Similar to the $\textrm{PGL}(n,\mathbb{C})$ case,
the bundle is reductive if the Zariski closure of any representative of the
conjugacy class gives a reductive subgroup of $G$. It is irreducible if no
representative is contained in a non-trivial parabolic subgroup of $G$.
Corlette’s existence result on harmonic maps is formulated in this setting
[9].

######  Theorem 2.22 ($G$-NAH I).

There exists a harmonic section $h:S\to M$ if and only if the holonomy of $M$
is reductive. If the holonomy is irreducible, then $h$ is unique.

Let $Q$ be the bundle whose fiber at $z\in S$ is the $G$-torsor of isometries
from $N$ to $M_{z}$. Then $Q$ is a $G$-principle bundle which inherits a flat
connection $A$ (an equivariant $\mathfrak{g}$-valued $1$-form) from the flat
connection on $M$. If we fix once and for all a point $h_{0}$ in $N$ with
stabilizer $K$ and corresponding decomposition
$\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$, then a section $h$ of $M$ gives a
reduction $Q^{h}$ of the structure group of $Q$ to $K$, whose fiber at $z$ is
the subset of isometries sending $h_{0}$ to $h(z)$. The pullback of the
vertical tangent bundle of $M$ by the section $h$ is therefore identified with
$Q^{h}\times_{K}\mathfrak{p}$, with $K$ acting on $\mathfrak{p}$ by
conjugation. Write $\nabla^{h}$ for the pullback of the Levi-Civita connection
by $h$ to $Q^{h}\times_{K}\mathfrak{p}$, given by Proposition 2.1. The
harmonicity of $h$ is equivalent to the holomorphicity of $\partial h$ with
respect to the $(0,1)$-part of $\nabla^{h}$. Similarly, the conformality of
$h$ with respect to the metric $\nu$ on $N$ is equivalent to the condition
$\nu((\partial h)^{2}):=\nu(\partial h\otimes\partial h)=0$.

######  Definition 2.23.

We say that $(M,h)$ is a $G$-harmonic bundle if $h$ is harmonic. It is a
$G$-minimal bundle with respect to a metric $\nu$ if $h$ is minimal with
respect to $\nu$.

This leads us to a definition of a $G$-Higgs bundle. Let $(G,K)$ be any
admissible pair.

######  Definition 2.24.

A $G$-Higgs bundle is a pair $(P,A^{0,1},\phi),$ where $P$ is a holomorphic
principal $K^{\mathbb{C}}$ bundle on $S$ with compatible $(0,1)$-connection
$A^{0,1}$, and $\phi$ is a holomorphic section of
$\textrm{ad}\mathfrak{p}^{\mathbb{C}}\otimes\mathcal{K}$ called the Higgs
field.

Here we abbreviate
$\mathrm{ad}\mathfrak{p}^{\mathbb{C}}=P\times_{K^{\mathbb{C}}}\mathfrak{p}^{\mathbb{C}}$.

The other direction of NAH is also true in this context. A reduction $P^{c}$
of the structure group of $P$ to $K\subset K^{\mathbb{C}}$ defines a real
structure on the Lie algebra $\mathrm{ad}\mathfrak{k}^{\mathbb{C}}$, which we
write as $*_{c}$. The Chern connection of $A^{0,1}$ with respect to this
reduction is $A^{0,1}+(A^{0,1})^{*_{c}}$, and its curvature $F^{c}$ is a real
$\mathrm{ad}\mathfrak{k}^{\mathbb{C}}$-valued $2$-form.

In addition, $P^{c}$ defines a real structure on
$\mathrm{ad}\mathfrak{p}^{\mathbb{C}}$. The connection
$\theta=A^{0,1}+(A^{0,1})^{*_{c}}+\phi+\phi^{*_{c}}$ is flat if and only if
the curvature $F^{c}$ satisfies

$F^{c}-[\phi,\phi^{*_{c}}]=0$

Stability and polystability for $G$-Higgs bundles are defined analogously to
the $\textrm{PGL}(n,\mathbb{C})$ case (see [17]). The analog of Proposition
2.12 remains true for $G$-Higgs bundles. The following is proved in [5].

######  Theorem 2.25 ($G$-NAH II).

Let $(P,A^{0,1},\phi)$ be a stable $G$-Higgs bundle. There exists a unique
reduction of the structure group $P^{c}$ such that
$(P^{c}\times_{K}G,A^{0,1}+(A^{0,1})^{*_{c}}+\phi+\phi^{*_{c}})$ is a flat
$G$-principal bundle ($h$ therefore defines a harmonic section of the
associated flat bundle).

######  Definition 2.26.

A stable $G$-Higgs bundle $(P,A^{0,1},\phi)$ is called minimal with respect to
an invariant bilinear form $\nu$ on the symmetric space $G/K$ if
$\nu(\phi^{2})=0$.

Let’s give a word on the relation to ordinary Higgs bundles. First, suppose
that $H$ is a complex Lie group. Since we only define $G$-Higgs bundles for
real Lie groups $G$, to make sense of an $H$-Higgs bundle, we must first
restrict scalars from $\mathbb{C}$ to $\mathbb{R}$. The complexification of
$G=\mathrm{Res}_{\mathbb{C}/\mathbb{R}}H$ is isomorphic to
$H\times\overline{H}$. With repsect to any Cartan involution, both
$\mathfrak{k}^{\mathbb{C}}$ and $\mathfrak{p}^{\mathbb{C}}$ are isomorphic to
the Lie algebra $\mathfrak{h}$. In this way, we recover the usual notion of an
$H$-Higgs bundle for a complex group $H$.

Next, if $(P,A^{0,1},\phi)$ is a $\mathrm{PGL}(n,\mathbb{C})$-Higgs bundle,
the obstruction to holomorphically extending the structure group of $P$ to
$\mathrm{GL}(n,\mathbb{C})$ is a class in $H^{2}(S,\mathcal{O}^{*})$, where
$\mathcal{O}^{*}$ is the sheaf of invertible holomorphic functions. This space
is trivial. We have to choose an extension, but having done so, we get a
classical Higgs bundle with $E$ the associated vector bundle. A reduction of
the structure group from $\mathrm{PGL}(n,\mathbb{C})$ to $\mathrm{PU}(n)$
determines a Hermitian metric on $E$ up to scaling by a real-valued function
on $S$.

Finally, if $G$ and $G^{\prime}$ are two real semisimple groups, a
homomorphism $\sigma:G\to G^{\prime}$ turns a $G$-Higgs bundles into a
$G^{\prime}$-Higgs bundle. If $(P,A^{0,1},\phi)$ is stable, then
$(\sigma(P),\sigma(\phi))$ will be polystable, but may not be stable (see
[4]). By the discussion on polystability in 2.3, the failure of stability is
not a big concern.

Putting together the previous three paragraphs, this gives a way of
associating a classical Higgs bundle to a $G$-Higgs bundle provided we are
given a representation of $G$ into $\mathrm{PGL}_{n}(\mathbb{C})$. We will use
this association in order to apply the asymptotic decoupling estimates to
$G$-Higgs bundles, under the assumption that the representation is essentially
faithful.

##  3\. Convergence of the second variation along the $\mathbb{R}^{+}$-flow

The purpose of this section is to prove Theorem B.

###  3.1. The second variation of area

Before studying the second variation of area for maps to symmetric spaces, we
give a general formula for the second variation of area for a minimal
immersion to a Riemannian manifold. Let $h:S\times(-\epsilon,\epsilon)\to N$
be a one-parameter variation of a minimal map. We write $h$ for the initial
map and hope there is no confusion. Let $\nabla$ be the connection on
$S\times(-\epsilon,\epsilon)$ obtained by pulling back the Levi-Civita
connection from $N$. We define sections $\dot{h}$ and $\ddot{h}$ of
$h^{*}TN|_{S\times\\{0\\}}$ by

$\dot{h}=dh\Big{(}\frac{\partial}{\partial t}\Big{)}|_{t=0},\hskip
2.84526pt\ddot{h}=\Big{(}\nabla_{\frac{\partial}{\partial
t}}dh\Big{(}\frac{\partial}{\partial t}\Big{)}\Big{)}|_{t=0}.$

Similarly, we have $1$-forms with values in
$h^{*}TM\otimes_{\mathbb{R}}\mathbb{C}$,

$\partial\dot{h}=\Big{(}\nabla^{1,0}dh\Big{(}\frac{\partial}{\partial
t}\Big{)}\Big{)}|_{t=0},\hskip
2.84526pt\overline{\partial}\dot{h}=\Big{(}\nabla^{0,1}dh\Big{(}\frac{\partial}{\partial
t}\Big{)}\Big{)}|_{t=0}.$

Let $a(t)$ be the area form induced on $S$ from the map $h$ at time $t$, and
let $\ddot{a}=\frac{d^{2}}{dt^{2}}\big{|}_{t=0}a(t)$. In this context, the
standard second variation of area formula takes the following form. We remark
that we make no assumption about $\dot{h}$ being a normal variation.

######  Proposition 3.1 (Second variation of area).

If $h$ is a minimal immersion at time 0, then the second derivative of the
induced area form at time 0 is given by

(8)
$\frac{1}{2}\ddot{a}=\langle\partial\dot{h},\overline{\partial}\dot{h}\rangle+\langle
R(\dot{h},\partial
h)\dot{h},\overline{\partial}h\rangle-\frac{2|\langle\partial\dot{h},\partial
h\rangle|^{2}}{\langle\partial
h,\overline{\partial}h\rangle}+\frac{1}{2}d(\langle\ddot{h},\overline{\partial}h\rangle-\langle\partial
h,\ddot{h}\rangle)$

This should be a standard formula. We give a proof for the reader’s
convenience.

###### Proof.

From equation (6) we have $a(t)=\sqrt{\langle\partial
h_{t},\overline{\partial}h_{t}\rangle^{2}-|\langle\partial h_{t},\partial
h_{t}\rangle|^{2}}$. Since $h$ is conformal, $\langle\partial h_{t},\partial
h_{t}\rangle$ is $O(t)$, so to find the second derivative in $t$ we can Taylor
expand:

(9) $\begin{split}a(t)=\langle\partial
h_{t},\overline{\partial}h_{t}\rangle-\frac{|\langle\partial h_{t},\partial
h_{t}\rangle|^{2}}{2\langle\partial
h,\overline{\partial}h\rangle^{2}}+O(t^{3})\end{split}$

Since the Levi-Civita connection is torsion-free, we get that

$\displaystyle\Big{(}\nabla_{\frac{\partial}{\partial t}}\partial
h_{t}\Big{)}|_{t=0}$ $\displaystyle=\partial\dot{h}$
$\displaystyle\Big{(}\nabla_{\frac{\partial}{\partial
t}}\nabla_{\frac{\partial}{\partial t}}\partial h_{t}\Big{)}|_{t=0}$
$\displaystyle=R(\dot{h},\partial h)\dot{h}+\partial\ddot{h}$

and similarly for $\overline{\partial}h_{t}$. We compute each term separately.

$\displaystyle\frac{d^{2}}{dt^{2}}\bigg{|}_{t=0}\langle\partial
h_{t},\overline{\partial}h_{t}\rangle$
$\displaystyle=2\langle\nabla_{\frac{\partial}{\partial t}}\partial
h_{t},\nabla_{\frac{\partial}{\partial
t}}\operatorname{\overline{\partial}}h_{t}\rangle+\langle\nabla_{\frac{\partial}{\partial
t}}\nabla_{\frac{\partial}{\partial t}}\partial
h_{t},\operatorname{\overline{\partial}}h_{t}\rangle+\langle\partial
h_{t},\nabla_{\frac{\partial}{\partial t}}\nabla_{\frac{\partial}{\partial
t}}\operatorname{\overline{\partial}}h_{t}\rangle\bigg{|}_{t=0}$
$\displaystyle=2\langle\partial\dot{h},\overline{\partial}\dot{h}\rangle+\langle
R(\dot{h},\partial h)\dot{h},\overline{\partial}h\rangle+\langle\partial
h,R(\dot{h},\overline{\partial}h)\dot{h}\rangle+\langle\partial\ddot{h},\overline{\partial}h\rangle+\langle\partial
h,\overline{\partial}\ddot{h}\rangle$

By the symmetries of the Riemannian curvature tensor, the two curvature terms
are equal. For the rightmost two terms, we have

$\displaystyle\langle\partial\ddot{h},\overline{\partial}h\rangle+\langle\partial
h,\overline{\partial}\ddot{h}\rangle$
$\displaystyle=\partial\langle\ddot{h},\overline{\partial}h\rangle-\overline{\partial}\langle\partial
h,\ddot{h}\rangle$
$\displaystyle=d(\langle\ddot{h},\overline{\partial}h\rangle-\langle\partial
h,\ddot{h}\rangle)$

Finally, using that $h$ is conformal, the computation of the second term of
(9) simplifies to give

$\frac{d^{2}}{dt^{2}}\bigg{|}_{t=0}\frac{|\langle\partial h_{t},\partial
h_{t}\rangle|^{2}}{2\langle\partial
h,\overline{\partial}h\rangle^{2}}=\frac{4|\langle\partial\dot{h},\partial
h\rangle|^{2}}{\langle\partial h,\overline{\partial}h\rangle}.$

Putting this all together yields the proposition. ∎

If $h:S\to N$ is any minimal map, not necessarily an immersion, define
$\mathrm{Var}(h)$ to be the space of smooth sections of $h^{*}TN$ with compact
support on $S$. The stability form $Q_{h}$ of $h$ is the quadratic form on
$\mathrm{Var}(h)$ defined for $X\in\mathrm{Var}(h)$ by

(10) $Q_{h}(X)=2\int_{S}\langle\partial
X,\overline{\partial}X\rangle-\frac{2|\langle\partial X,\partial
h\rangle|^{2}}{\langle\partial h,\overline{\partial}h\rangle}+\langle
R(X,\partial h)X,\overline{\partial}h\rangle$

######  Proposition 3.2.

If $h$ is a minimal map and $h(t)$ is a variation with
$\dot{h}=X\in\mathrm{Var}(h)$, and $A(t)$ the total area of a compact
subsurface containing the support of $X$ and of $\ddot{h}$, then

$\frac{d^{2}}{dt^{2}}\bigg{|}_{t=0}A(t)=Q_{h}(X).$

###### Proof.

The area form is non-degenerate on the open set $U\subset S$ on which $h$ is
an immersion, and so we can apply the formula (8) on $U$. But it is easily
checked that both the area form and the right hand side of the formula (8)
smoothly extend to all of $S$. The result follows from dominated convergence.
The final term in formula (8) vanishes upon integration by the divergence
theorem. ∎

######  Definition 3.3.

We call $X$ a destabilizing variation of $h$ if $Q_{h}(X)<0$. $h$ is called
unstable if it has a destabilizing variation. The index $\mathrm{Ind}(h)$ of
$h$ is the maximal dimension of a space of destabilizing variations.

The log cutoff trick (see [29, section 4.4] for details and explanation)
implies the following proposition, which is essential to the proof of Theorem
B.

######  Proposition 3.4.

If $U=S-B$, for $B$ a discrete set, then
$\mathrm{Ind}(h|_{U})=\mathrm{Ind}(h)$.

Another elementary proposition of a similar flavor is:

######  Proposition 3.5.

If $N$ has nonpositive sectional curvature, $\iota:N^{\prime}\to N$ is the
inclusion of a totally geodesic subspace, and $h=\iota\circ h^{\prime}$, then
$\mathrm{Ind}(h)=\mathrm{Ind}(h^{\prime})$.

If $h$ is a minimal section of a flat Riemannian bundle $M$ over $S$, then the
second variation formula and Proposition 3.2 still hold, as one checks in
local flat charts, the only difference being that the bundle $h^{*}TM$ is
replaced everywhere by the bundle $h^{*}T^{\mathrm{vert}}M$, and the Levi-
Civita connection on a fiber $M_{z}$ is replaced by its natural extension to
the whole bundle $T^{\mathrm{vert}}M$. For instance, $\partial h$ is now a
$(1,0)$-form valued in $h^{*}T^{\mathrm{vert}}M$. We extend definitions of
variations, stability operator, and index to minimal sections in the obvious
way. Propositions 3.4 and 3.5 also generalize immediate to sections of flat
Riemannian bundles.

####  3.1.1. Minimal harmonic bundles

We record the stability form for a minimal harmonic bundle $(E,D,H)$. We use
the isomorphism $\varphi$ to identify $\mathrm{Var}(H)$ with the space of
smooth sections of $\mathrm{End}^{0}_{H}(E)$. One way of thinking about this
identification is that if $X$ is a smooth section of
$\mathrm{End}^{0}_{H}(E)$, then the one-parameter family of metrics

$H^{t}=e^{-t\overline{X}^{\vee}}He^{-tX}$

has derivative $X$ at time zero.

If $X$ is a variation, then $\partial X$ should takes values in the
complexified tangent bundle, so it becomes an $\mathrm{End}^{0}(E)$-valued
$1$-form. Recall that under this isomorphism, the (complexified) Riemannian
metric becomes $\langle X,Y\rangle=2n\operatorname{\textrm{tr}}(XY)$. We write
the corresponding Hermitian norm as
$|X|^{2}_{H}=2n\operatorname{\textrm{tr}}(XX^{*_{H}})$. The operator
$\partial$ on this bundle is the $(1,0)$-part of $\nabla^{H}$ defined in 2.3.

######  Proposition 3.6.

Let $(E,D,H)$ be a minimal harmonic bundle on a compact surface $S$ with
associated Higgs field $\phi$. If $X$ is a smooth section of
$\mathrm{End}_{H}^{0}(E)$, then the stability form of the section $[H]$ of the
associated flat Riemannian bundle $\mathbb{P}\mathrm{Met}(E)$ is

$Q_{H}(X)=2\int_{S}|\partial X|_{H}^{2}-\frac{2|\langle\partial
X,\phi\rangle|^{2}}{|\phi|_{H}^{2}}+|[X,\phi]|_{H}^{2}.$

###### Proof.

By definition, $\phi=\varphi(\partial H)$, and since the identification
$\varphi$ is implicit, we should replace “$\partial h$” in Equation (10) with
$\phi$. The curvature term becomes

$\langle
R(X,\phi)X,\phi^{*_{H}}\rangle=2n\operatorname{\textrm{tr}}(-[[X,\phi],X]\phi^{*_{H}})=2n\operatorname{\textrm{tr}}([X,\phi][\phi^{*_{H}},X])=|[X,\phi]|^{2}_{H}.$

∎

###  3.2. The limiting objects

Generically regular semisimple Higgs bundles are introduced in [30]. We
slightly relax the condition. Let
$(E,\operatorname{\overline{\partial}}_{E},\phi)$ be a Higgs bundle.

######  Definition 3.7.

$\phi$ is generically semisimple if it is semisimple away from a discrete
subset of $S$.

If $\phi$ is generically semisimple, then there exists a discrete subset
$B\subset S$ and an integer $m\leq n$ such that for every $p\in S-B$, there
exists a neighbourhood $U\subset S-B$ of $p$ over which
$(E,\overline{\partial}_{E},\phi)$ splits in the sense that we have $m$
holomorphic $1$-forms $\phi_{1},\dots,\phi_{m}$ such that
$\phi_{i}\neq\phi_{j}$ at every point in $U$ for $i\neq j$, and a
decomposition of the Higgs bundle $E|_{U}=\oplus_{i=1}^{m}E_{i}$, with respect
to which $\phi$ decomposes as $\phi=\sum_{i=1}^{m}\phi_{i}\pi_{i},$ where
$\pi_{i}$ is the projection onto $E_{i}$ with respect to the given
decomposition of $E|_{U}$. A generically semisimple Higgs bundle is
generically regular semisimple if $m=n$.

######  Definition 3.8.

The critical set $B$ of $\phi$ is the smallest set such that
$(E,\overline{\partial}_{E},\phi)$ splits into $m$ distinct eigenspaces in a
neighborhood of every point of $S-B$.

Let $(E,\overline{\partial}_{E},\phi)$ be a generically semisimple Higgs
bundle, and let $B$ be the critical set of $\phi$. For any neighborhood $U$,
define $\phi_{i}$ and $\pi_{i}$ as above. Over $S-B$, these patch together to
form an $m$-dimensional subbundle $F_{\phi}\subset\textrm{End}(E)|_{S-B}$
locally spanned by the projections $\pi_{i}$. Let $F_{\phi}^{0}$ be the
codimension one subbundle $F^{0}_{\phi}=F\cap\mathrm{End}^{0}(E)$.

######  Definition 3.9.

We call $F^{0}_{\phi}$ the toral bundle of $\phi$ and $F_{\phi}$ the extended
toral bundle.

######  Remark 3.10.

Equivalently, the fiber of $F^{0}_{\phi}$ over each point $p$ of $S-B$ is the
center of the centralizer of $\phi(p)$ in $\mathrm{End}^{0}(E)$. If $\phi$ is
regular semisimple, this centralizer is already abelian.

Fix $(E,\overline{\partial}_{E},\phi)$, and let $F$ be its extended toral
bundle. $F$ has a natural flat connection, flat metric, and real structure
defined by the condition that the $\pi_{i}$ are real and flat. If $X$ is a
section of $F$, let $dX=\partial X+\operatorname{\overline{\partial}}X$ be its
derivative with respect to the flat connection, and let $X^{\dagger}$ be its
conjugate with respect to the flat structure. If $X=\sum_{i}X_{i}\pi_{i}$,
then

$X^{\dagger}=\sum_{i}\overline{X_{i}}\pi_{i}$

and

$dX=\sum_{i}dX_{i}\pi_{i}.$

Since the complex metric on $F$ is just the restriction $\langle
X,Y\rangle=2n\operatorname{\textrm{tr}}(XY)$ from $\mathrm{End}^{0}(E)$, the
corresponding Hermitian metric is given by
$|X|^{2}=2n\operatorname{\textrm{tr}}(XX^{\dagger})$. If $r_{i}$ is the rank
of $E_{i}$, then

(11)
$\langle\pi_{i},\pi_{j}^{\dagger}\rangle=2n\operatorname{\textrm{tr}}(\pi_{i}\pi_{j}^{\dagger})=2nr_{i}\delta_{ij}.$

The toral bundle $F^{0}$ of $\phi$ inherits the flat, metric, and real
structures from $F$. Let $F^{0}_{\mathbb{R}}$ be its real subbundle, i.e., the
$\mathbb{R}$-span of the $\pi_{i}$. Since the $\pi_{i}$’s are holomorphic, the
flat connection is compatible with the holomorphic structure on
$\textrm{End}(E)$, and hence $\phi$ can be viewed as a holomorphic
$F^{0}$-valued $1$-form. Let $\psi=\phi+\phi^{\dagger}$. Since $\psi$ is a
closed $F^{0}_{\mathbb{R}}$-valued $1$-form, it defines an affine bundle
$M_{\phi}$ whose vertical tangent bundle is $F^{0}_{\mathbb{R}}$ together with
a section $f$ of $M_{\phi}$. The construction is as follows. Let
$(U_{\alpha},w_{\alpha\beta})$ be a presentation of $F^{0}_{\mathbb{R}}$ by an
open cover and locally constant transition functions, and $(\psi_{\alpha})$ a
representative of $\psi$, with $\psi_{\alpha}=w_{\alpha\beta}\psi_{\beta}$. On
each $U_{\alpha}$, choose $f_{\alpha}$ with $df_{\alpha}=\psi_{\alpha}$. Since
$dw_{\alpha}=0$, the functions
$t_{\alpha\beta}:=f_{\alpha}-w_{\alpha\beta}f_{\beta}$ are locally constant.
Rearranging this gives

$f_{\alpha}=w_{\alpha\beta}f_{\beta}+t_{\alpha\beta}.$

The affine transformations $x\mapsto w_{\alpha\beta}x+t_{\alpha\beta}$ define
the bundle $M_{\phi}$, and $(f_{\alpha})$ defines a section $f$ such that
$df=\psi$, and therefore $\partial f=\phi$.

######  Definition 3.11.

We call $M_{\phi}$ the appartment bundle associated to
$(E,\operatorname{\overline{\partial}}_{E},\phi)$.

By equation (11), the natural metric on $F^{0}_{\mathbb{R}}$ is parallel with
respect to the flat connection, and therefore the appartment bundle $M_{\phi}$
is a flat Riemannian bundle. The section $f$ is harmonic by construction.
Assume that $\operatorname{\textrm{tr}}(\phi^{2})=0$, so that the Higgs bundle
$(E,\operatorname{\overline{\partial}}_{E},\phi)$ is minimal. Then the section
$f$ is also minimal.

######  Definition 3.12.

The pair $(M_{\phi},f)$ is the limiting object associated to the stable
minimal generically semisimple Higgs bundle
$(E,\operatorname{\overline{\partial}}_{E},\phi)$. Its index is by definition
the index of $f$.

A variation $X\in\mathrm{Var}(f)$ is a smooth section of $F^{0}_{\mathbb{R}}$
with compact support on $S-B$. Since the curvature of the fibers of $M_{\phi}$
vanishes, the stability form of $f$ is given by

(12) $Q_{f}(X)=2\int_{S}|\partial X|^{2}-\frac{2|\langle\partial
X,\phi\rangle|^{2}}{|\phi|^{2}}$

####  3.2.1. Cameral covers

We now give an equivalent description of the limiting object, which is a
slight modification of the equivariant map associated to the section $f$ as in
section 1.4. The monodromy of $F^{0}$ only permutes the projections $\pi_{i}$
(and moreover can only permute any two projections $\pi_{i}$ of the same
rank), and hence it is contained in a (finite) product of symmetric groups.
Therefore there is a finite and minimal degree branched covering $\tau:C\to
S$, unbranched over $S-B$, which trivializes the monodromy of $F^{0}$. This is
the cameral cover of $(S,\phi)$ (see [21, Section 3] and [10]).

Over $C-\tau^{-1}(B)$, there is a global splitting
$\tau^{*}E|_{S-B}=\oplus_{i}E_{i}$ with projections $\pi_{i}$, and
$\tau^{*}\phi|_{S-B}$ acts on each $E_{i}$ by multiplication by a holomorphic
$1$-form $\phi_{i}$. We extend the (bounded) $1$-forms $\phi_{i}$ via the
removable singularities theorem to all of $C$. Each harmonic $1$-form
$\psi_{i}=\phi_{i}+\overline{\phi_{i}}$ determines a cohomology class, of
which it is the unique harmonic representative. Choosing a basepoint $z_{0}$
on $C$, the cohomology class defines a representation from $\pi_{1}(C)$ to
$\mathbb{R}$. Lifting to a harmonic $1$-form $\tilde{\psi}_{i}$ on the
universal cover $\tilde{C}$ and integrating from a lift of the basepoint gives
a harmonic function

$\tilde{f}_{i}(z)=\int_{z_{0}}^{z}\tilde{\psi}_{i}$

which is equivariant by this representation. Scaling each $\tilde{f}_{i}$ by
$\sqrt{r_{i}}$ and putting them together, we have a representation
$\chi:\pi_{1}(C)\to\mathbb{R}^{m}$ and a $\chi$-equivariant minimal map
$\tilde{f}$ from $\tilde{C}$ to $\mathbb{R}^{m}$.

Define the equivariant area of $\tilde{f}$ to be the integral of its area form
over a fundamental domain of the action of $\pi_{1}(C)$ on $\tilde{C}$. Note
that this is $\mathrm{deg}(\tau)$ times the area of the section $f$ of
$M_{\phi}$. Since the representation $\chi$ acts by translations, a variation
of the map $\tilde{f}$ that preserves $\chi$-equivariance is the same thing as
a $\chi$-invariant map from $\tilde{C}$ to $\mathbb{R}^{m}$. Of course, this
is the same thing as a map from $C$ to $\mathbb{R}^{m}$. The stability
operator for the equivariant area is

$Q_{\tilde{f}}(X)=2\mathrm{deg}(\tau)\int_{C}|\partial
X|^{2}-\frac{2|\langle\partial X,\partial\tilde{f}\rangle|^{2}}{|\partial
f|^{2}}.$

Call a variation $X$ $\tau$-equivariant if it is the pullback of a section of
$F^{0}_{\mathbb{R}}$ on $S$. Define the $(\tau,\chi)$ equivariant index of
$\tilde{f}$ to be the index of the stability operator restricted to
$\tau$-equivariant variations.

The log cut-off trick (Proposition 3.4) shows that we need only consider
variations to be supported on $C-\tau^{-1}(B)$. But a $\tau$-equivariant
varation of $\tilde{f}$ supported on $C-\tau^{-1}(B)$ is the same as a
variation of the section $f$, and the resulting area forms and stability
operators are the same up to a factor of $\mathrm{deg}(\tau)$. We conclude:

######  Proposition 3.13.

The index of $f$ is the same as the $(\tau,\chi)$-equivariant index of
$\tilde{f}$.

###  3.3. Convergence under the $\mathbb{R}^{+}$ action

There is a $\mathbb{C}^{*}$-action on the moduli space of Higgs bundles,

$\gamma\cdot(E,\overline{\partial}_{E},\phi)=(E,\overline{\partial}_{E},\gamma\phi).$

By Proposition 2.12, the action restricts to the space of stable Higgs
bundles. In this paper, we are only interested in the restricted
$\mathbb{R}^{+}$-action. Since stable Higgs bundles have unique harmonic
metrics, the non-abelian Hodge correspondence transfers the
$\mathbb{R}^{+}$-action to an $\mathbb{R}^{+}$-action on the moduli space of
projectively flat bundles with irreducible holonomy. Let
$(E,\overline{\partial}_{E},\phi)$ be a minimal stable Higgs bundle. By taking
$R\mapsto(E,\overline{\partial}_{E},R\phi),$ we obtain a family of minimal
harmonic bundles $R\mapsto(E,D_{R},H_{R})$ with holonomy representations
$\rho_{R}.$

######  Remark 3.14.

The apartment bundle $M_{R\phi}$, and the harmonic section $f_{R\phi}$ depend
on $R$ because the translational part is being rescaled by $R$. However, the
special toral bundle $F_{R\phi}^{0}$ of $(E,R\phi)$ is the same for every $R$,
and hence the space of variations of $f_{R}$ is well-defined independent of
$R$. The stability operator is independent of $R$ as well.

Recall that the statement of Theorem B is that the liminf as $R\to\infty$ of
the indices of the minimal harmonic bundles $(E,D_{R},H_{R})$ is at least the
index of the limiting object $(M_{\phi},f)$. Let $Q_{H_{R}}$ be the stability
operator of $(E,D_{R},H_{R})$, and let $Q_{f}$ be the stability operator of
$(M_{\phi},f)$. Recall that we can identify $\mathrm{Var}(H_{R})$ with the
space of smooth sections of $\mathrm{End}^{0}_{H_{R}}(E)$, whereas
$\mathrm{Var}(f)$ is the space of smooth sections of $F^{0}_{\mathbb{R}}$ with
compact support on $S-B$. In order to relate the operators $Q_{H_{R}}$ to $Q$,
we need to turn variations of $f$ into variations of $H_{R}$. Fortunately,
there is a completely natural way to do this. If $X$ is a variation of $f$, it
is in particular a section of $\mathrm{End}^{0}(E)$, so we can simply project
to $\mathrm{End}^{0}_{H_{R}}(E)$. Namely, let $X_{R}$ be the section of
$\mathrm{End}_{H_{R}}^{0}(E)$ defined by

(13) $X_{R}=\frac{X+X^{*_{R}}}{2}$

where $*_{R}$ is an abbreviation of $*_{H_{R}}$. We will prove

######  Proposition 3.15.

For any minimal stable Higgs bundle $(E,\overline{\partial}_{E},\phi)$ with
$\phi$ generically semisimple, and any $X\in\mathrm{Var}(f)$, we have

$\lim_{R\to\infty}Q_{H_{R}}(X_{R})=Q_{f}(X).$

Once we have proved this, the proof of Theorem B, at least in the case of
$\mathrm{PGL}_{n}(\mathbb{R})$, follows immediately by applying the following
elementary lemma to $Q_{R}=Q_{H_{R}}$ and $Q=Q_{f}$.

######  Lemma 3.16.

For $R\in\mathbb{R}^{+}$, let $V_{R}$ be a family of vector spaces with
quadratic forms $Q_{R}$. If $V$ is a vector space with quadratic form $Q$ of
finite index, and for each $R$, there is a linear map $X\mapsto X_{R}$ from
$V$ to $V_{R}$ such that $\lim_{R\to\infty}Q_{R}(X_{R})=Q(X)$ for each $R$,
then $\liminf_{R\to\infty}\mathrm{Ind}(Q_{R})\geq\mathrm{Ind}(Q)$.

###### Proof.

Let $W$ be a $k$-dimensional subspace of $V$ on which $Q$ is negative
definite, where $k=\mathrm{Ind}(Q)$. For each $R$, $X\mapsto Q_{R}(X_{R})$ is
a quadratic form on the finite dimensional space $W$, so the pointwise
convergence to $Q(X)$ implies locally uniform convergence on $W$. Hence, for
large enough $R$, $Q_{R}(X_{R})$ is also negative definite. For such $R$, the
dimension of $\\{X_{R}|X\in W\\}$ must also be $k$, since otherwise we would
have $Q_{R}(X_{R})=0$ for some vector $X$, but this is impossible as
$Q_{R}(X_{R})$ is negative definite. Hence, for such $R$, we have
$\mathrm{Ind}(Q_{R})\geq k$. ∎

Proving Proposition 3.15 requires an analysis of the behaviour of the minimal
harmonic bundles $(E,D_{R},H_{R})$ as $R$ tends to infinity, and for this we
turn to Mochizuki’s work [30]. In order to slightly simplify the statements of
results below, as well as the proof of Proposition 3.15, we fix a metric
$\sigma_{0}$ on $S-B$ to define the norm of $1$-forms.

######  Theorem 3.17 (Proposition 2.3 and 2.10 in [30]).

Suppose that over a domain $U$ in $S$, the bundle $E$ splits as
$E=\bigoplus_{i}E_{i}$ and $\phi$ acts on $E_{i}$ by multiplication by a
$1$-form $\phi_{i}$. Let $\pi_{i}$ be the projection to $E_{i}$ determined by
the splitting, and let $(\pi^{\prime}_{i})_{R}$ be the Hermitian orthogonal
projection to $E_{i}$. If $U^{\prime}$ is any relatively compact subdomain of
$U$, then there are constants $C_{1}$ and $\epsilon$ depending on $U$,
$U^{\prime}$, the rank of $E$, and the $1$-forms $\phi_{i}$ such that

(14) $|\pi_{i}-(\pi^{\prime}_{i})_{R}|_{H_{R}}<C_{1}e^{-\epsilon R}$

and

(15) $|\partial_{H_{R}}\pi_{i}|_{H_{R},\sigma_{0}}<C_{1}e^{-\epsilon R}$

on $U^{\prime}.$

Mochizuki gives a precise description of the constant $\epsilon R$, in terms
of the gaps between the eigenvalues of $R\phi_{i}$. All we need is the linear
dependence on $R$. We remark that if $E$ is polystable, the constants $C_{1}$
and $\epsilon$ do not depend on the choice of harmonic metric $H_{R}$.

The second point is stated in Mochizuki with the additional assumption that
the $E_{i}$ are one-dimensional. But the only place he uses it is in the short
proof of his Theorem 2.9, and the only property he uses is that $\phi$
commutes with endomorphisms of $E_{i}$. This is true in our case because we
specified that $\phi$ acts on $E_{i}$ by a multiple of the identity. This
slight generalization will be useful for applying Mochizuki’s results to
groups other than $\textrm{PGL}(n,\mathbb{C})$ in section 3.4.

We recall that if $Z$ is a section of the toral bundle $F^{0}_{\phi}$, then
$Z^{\dagger}$ and $dZ=\partial Z+\operatorname{\overline{\partial}}Z$ are the
conjugate and derivative respectively of $Z$ with respect to the real
structure and flat connection on $F^{0}_{\phi}$.

In the following corollary, so as to reduce the number of subscripts, we
abbreviate $H_{R}$ by $R$ in the adjoint, the norm, and the connection.

######  Corollary 3.18.

Let $(E,\operatorname{\overline{\partial}}_{E},\phi)$ be a generically
semisimple stable Higgs bundle with critical set $B$ and toral bundle
$F^{0}\subset\mathrm{End}^{0}(E)|_{S-B}$. Let $Z$ be a smooth section of
$F^{0}$ on $S-B$ and let $U$ be a subdomain of $S-B$ with compact closure.
Then there are constants $C_{2},\epsilon>0$ depending on
$(E,\operatorname{\overline{\partial}}_{E},\phi),U,|Z|$, and
$|\operatorname{\overline{\partial}}Z|_{\sigma_{0}}$ such that

  1. (i)

$|Z^{*_{R}}-Z^{\dagger}|_{R}$

  2. (ii)

$|\partial_{R}Z^{*_{R}}-\partial Z^{\dagger}|_{R,\sigma_{0}}$

  3. (iii)

$|\partial_{R}Z-\partial Z|_{R,\sigma_{0}}$, and

  4. (iv)

$||Z|_{R}-|Z||$

are all bounded by $C_{2}e^{-\epsilon R}$ on $U$.

###### Proof.

First suppose that $(E,\overline{\partial}_{E},\phi)$ satisfies the splitting
condition of Theorem 3.17 on a neighborhood of $U$, and write
$Z=\sum_{i}Z_{i}\pi_{i}$.

  1. (i)

Let $(\rho_{i})_{R}=\pi-(\pi_{i}^{\prime})_{R}$. Since
$(\pi_{i}^{\prime})_{R}$ is self-adjoint with respect to $H_{R}$, we have
$(\rho_{i})_{R}^{*_{R}}=\pi_{i}^{*_{R}}-(\pi_{i}^{\prime})_{R}$ and so

$|\pi_{i}-\pi_{i}^{*_{R}}|_{R}=|(\rho_{i})_{R}-(\rho_{i})_{R}^{*_{R}}|_{R}\leq
2|(\rho_{i})_{R}|_{R}\leq 2C_{1}e^{-\epsilon R}$

by equation (14) from Theorem 3.17. So for some constant $C_{2}$,

$|Z^{*_{R}}-Z^{\dagger}|_{R}\leq\sum_{i}|Z_{i}||\pi_{i}^{*_{R}}-\pi_{i}|_{R}\leq
C_{2}e^{-\epsilon R}.$

  2. (ii)

Writing $Z=\sum_{i}Z_{i}\pi_{i}$ and using $\partial_{R}\pi_{i}^{*_{R}}=0,$ we
see

$|\partial_{R}Z^{*_{R}}-\partial
Z^{\dagger}|_{R,\sigma_{0}}\leq\sum_{i}|\partial\overline{Z}_{i}|_{\sigma_{0}}|\pi_{i}^{*_{R}}-\pi_{i}|_{R}\leq
C_{2}e^{-\epsilon R}.$

  3. (iii)

Writing $Z=\sum_{i}Z_{i}\pi_{i}$ yet again,

$|\partial_{R}Z-\partial
Z|_{R,\sigma_{0}}\leq\sum_{i}|Z_{i}||\partial_{R}\pi_{i}|_{R,\sigma_{0}}\leq
C_{2}e^{-\epsilon R}$

where the last inequality is equation (15) from Theorem 3.17.

  4. (iv)

We have
$|Z|_{R}^{2}-|Z|^{2}=\sum_{i,j}Z_{i}\overline{Z_{j}}(\operatorname{\textrm{tr}}(\pi_{i}\pi_{j}^{*_{R}})-\operatorname{\textrm{tr}}(\pi_{i}\pi_{j}))$,
and

$|(\operatorname{\textrm{tr}}(\pi_{i}\pi_{j}^{*_{R}})-\operatorname{\textrm{tr}}(\pi_{i}\pi_{j}))|_{R}\leq|\pi_{i}|_{R}|\pi_{j}^{*_{R}}-\pi_{j}|_{R}.$

Recall $r_{i}=\mathrm{rank}(E_{i}).$ Then expanding the inner product, we find

$|\pi_{i}-\pi_{i}^{*_{R}}|^{2}_{R}=2|\pi_{i}|^{2}_{R}-2r_{i},$

from which we conclude that $|\pi_{i}|_{R}$ is uniformly bounded in $R$. Thus

$|\pi_{i}|_{R}|\pi_{j}^{*_{R}}-\pi_{j}|_{R}\leq C_{2}e^{-\epsilon R}$

for some value of $C_{2}$.

In general, we can cover $U$ by finitely many sets $U^{\prime}_{\alpha}$ such
that the conditions of Theorem 3.17 are satisfied on a neighborhood of
$U^{\prime}_{\alpha}$, and take $\epsilon$ (resp. $C_{2}$) to be the minimum
(resp. maximum) of the value for each $U^{\prime}_{\alpha}$. ∎

We may now prove Proposition 3.15.

###### Proof of Proposition 3.15.

From Proposition 3.6,

$Q_{H_{R}}(X_{R})=2\int_{S}|\partial_{R}X_{R}|_{R}^{2}-\frac{2|\langle\partial_{R}X_{R},\phi\rangle|^{2}}{|\phi|_{R}^{2}}+R^{2}|[X_{R},\phi]|_{R}^{2}.$

On the other hand from (12) we have

$Q_{f}(X)=2\int_{S}|\partial X|^{2}-\frac{2|\langle\partial
X,\phi\rangle|^{2}}{|\phi|^{2}}.$

We prove uniform convergence of each of the first two terms, and uniform
convergence of the third term to zero. Starting with the first,

$\big{|}|\partial_{R}X_{R}|_{R}-|\partial
X|_{R}\big{|}\leq|\partial_{R}X_{R}-\partial
X|_{R}\leq\bigg{|}\frac{\partial_{R}X}{2}-\frac{\partial
X}{2}\bigg{|}_{R}+\bigg{|}\frac{\partial_{R}X^{*_{R}}}{2}-\frac{\partial
X}{2}\bigg{|}_{R}$

which is $O(e^{-\epsilon R})$ by (ii) and (iii) of Corollary 3.18, and so
combining with (iv) we see that $|\partial_{R}X_{R}|_{R}-|\partial X|$ is also
$O(e^{-\epsilon R}).$

For the second term we have exponential convergence of the denominators by
(iv) and for the numerators,

$\big{|}|\langle\partial_{R}X_{R},\phi\rangle|-|\langle\partial
X,\phi\rangle|\big{|}\leq|\langle\partial_{R}X_{R}-\partial
X,\phi\rangle|\leq|\partial_{R}X_{R}-\partial X|_{R}|\phi|_{R}.$

The term $|\phi|_{R}$ is uniformly bounded by (iv) and
$|\partial_{R}X_{R}-\partial X|_{R}$ is exponentially decreasing by the
previous equation.

Finally, since every section of $F_{0}$ commutes with $\phi$, we have
$[X,\phi]=0$ and so

$|[X_{R},\phi]|=|[\frac{X^{*_{R}}-X}{2},\phi]|\leq\frac{1}{2}|X^{*_{R}}-X|_{R}|\phi|_{R}$

Since $|\phi|_{R}$ is uniformly bounded by (iv) and $|X^{*_{R}}-X|_{R}$ is
exponentially small by (i), this term goes to zero (despite the factor of
$R^{2}$, which is dwarfed by the exponential decay).

∎

###### Proof of Theorem B for $\mathrm{PGL}(n,\mathbb{R})$.

Apply Proposition 3.15 and Lemma 3.16.

∎

###  3.4. Convergence for $G$-Higgs bundles

This subsection is not necessary for the proof of Theorem A for
$\mathrm{PSL}(n,\mathbb{R})$.

Fix an admissible triple $(G,K,\nu)$. Let $(P,A^{0,1},\phi)$ be a stable
$G$-Higgs bundle that is minimal with respect to $\nu$. Suppose that $\phi$ is
generically semisimple.

For each $R>0$, let $(M_{R},h_{R})$ be the associated minimal $G$-harmonic
bundle. The left hand side of Theorem B is about the index of $h_{R}$ with
respect to the space of smooth sections of $h_{R}^{*}T^{\textrm{vert}}M_{R}.$
Recall that this is identified with $P^{c}\times_{K}\mathfrak{p}$. In order to
apply the results of the previous section, we use Proposition 2.20 to fix an
essentially faithful representation
$\sigma=\prod_{i}\sigma_{i}:G\to\prod_{i}\mathrm{SL}_{n_{i}}(\mathbb{C})$ of
$G$ and constants $a_{i}$ such that the induced map of symmetric spaces is a
totally geodesic isometry. By Proposition 3.5, since the product of symmetric
spaces $\prod_{i}\mathbb{P}\mathrm{Met}(\mathbb{C}^{n_{i}})$ is nonpositively
curved, we should be able to express the index of $h_{R}$ using this isometry.
In terms of harmonic bundles, this expectation is realized as follows.

For each $i$, let
$(E_{i},\operatorname{\overline{\partial}}_{E_{i}},\phi_{i})$ be the Higgs
bundle associated to $(P,A^{0,1},\phi)$ by $\sigma_{i}$. It is possible that
$E_{i}$ not stable, but only polystable; if so, fix once and for all a
relative scaling of its stable factors as in Remark 2.14, which determines for
each $R\in\mathbb{R}^{+}$ a harmonic bundle $(E_{i},D^{R}_{i},H^{R}_{i})$. The
bundle $\mathbb{P}\mathrm{Met}(E_{i})$ inherits the metric
$a_{i}\operatorname{\textrm{tr}}$, and for each $R$ the induced map of flat
Riemannian bundles from $M_{R}$ to $\prod_{i}\mathbb{P}\mathrm{Met}(E_{i})$ is
a fiberwise totally geodesic isometry, sending $h_{R}$ to
$\prod_{i}[H^{i}_{R}]$. In particular the section
$[H_{R}]:=\prod_{i}[H^{i}_{R}]$ is minimal, and its index is the same as the
index of $h_{R}$.

Let $X_{R}$ be a variation of $[H_{R}]$, which we can write as
$X_{R}=\sum_{i}X^{R}_{i}$, where each $X^{R}_{i}$ is a section of
$\mathrm{End}^{0}_{H^{i}_{R}}(E_{i})$. The following two propositions are easy
generalizations of Propositions 3.6 and 3.15 respectively.

######  Proposition 3.19.

The stability operator of $H_{R}$ applied to $X_{R}$ is

$Q_{H_{R}}(X_{R})=\int_{S}\Big{(}\sum_{i}a_{i}|\partial_{H_{R}^{i}}X^{R}_{i}|_{H_{R}^{i}}^{2}\Big{)}-\frac{2\Big{|}\sum_{i}a_{i}\langle
X^{R}_{i},\phi_{i}\rangle\Big{|}^{2}}{\sum_{i}a_{i}|\phi_{i}|_{H_{R}^{i}}^{2}}+R^{2}\Big{(}\sum_{i}a_{i}|[X^{R}_{i},\phi_{i}]|_{H_{R}^{i}}^{2}\Big{)}.$

###### Proof.

This is a consequence of the formula (10), The curvature splits over the
direct sum, so we can apply the same manipulations from Proposition 3.6 to get
the last term. ∎

Now let $B^{\sigma}$ be the union of the critical sets of each $\phi_{i}$, and
define $F_{\phi}^{\sigma}$ to be the fiber product of the toral bundles
$F^{0}_{\phi_{i}}$ over $S-B^{\sigma}$ for each $\phi_{i}$. Each
$F^{0}_{\phi_{i}}$ inherits the rescaled complex metric
$a_{i}\operatorname{\textrm{tr}}$ from $\mathrm{End}^{0}(E_{i})$, and together
these give a parallel metric on $F_{\phi}^{\sigma}$. Let $M_{\phi}^{\sigma}$
be the fiber product of the apartment bundles $M_{\phi_{i}}$ for each
$\phi_{i}$, and $f^{\sigma}$ the product of the canonical sections $f_{i}$.
Then the minimality of $\phi$ with respect to $\nu$ implies that $f^{\sigma}$
is minimal with respect to the metric on $M_{\phi}^{\sigma}$. Let
$Q_{f^{\sigma}}$ be its stability operator.

A variation $X$ of $f^{\sigma}$ is a compactly supported section of
$F^{\sigma}_{\phi}$. Given $X$, for each $R$, let $X^{i}_{R}$ be the
$H^{i}_{R}$-self-adjoint part of $X^{i}$ as in (13), and set
$X_{R}=\sum_{i}X^{i}_{R}$. Then $X_{R}$ is a variation of $[H_{R}]$, and we
have:

######  Proposition 3.20.

For any section $X\in\mathrm{Var}(f^{\sigma})$,

$\lim_{R\to\infty}Q_{H_{R}}(X_{R})=Q_{f^{\sigma}}(X).$

###### Proof.

Exactly parallel to formula (12), the stability operator $Q_{f^{\sigma}}$ is
given by

$Q_{f^{\sigma}}(X)=\int_{S}\Big{(}\sum_{i}a_{i}|\partial
X_{i}|^{2}\Big{)}+\frac{\Big{|}\sum_{i}a_{i}\langle
X_{i},\phi_{i}\rangle\Big{|}^{2}}{\sum_{i}a_{i}|\phi_{i}|^{2}}.$

In the proof of Proposition 3.15, we showed that each term of
$Q_{H_{R}}(X_{R})$ converges to the corresponding term of $Q_{f}(X)$, and even
showed the separate convergence of the numerator and denominator of the second
term. Hence, the same argument gives the convergence of $Q_{H_{R}}(X_{R})$ to
$Q_{f^{\sigma}}(X)$. ∎

Proposition 3.20 together with the index Lemma 3.16 therefore implies that

$\liminf_{R\to\infty}\mathrm{Ind}(H_{R})\geq\mathrm{Ind}(f^{\sigma}).$

Since $\mathrm{Ind}(h_{R})=\mathrm{Ind}(H_{R})$, we have almost proved Theorem
B. In fact, we have already acheived the main point which is to give a useful
lower bound for the index of $h_{R}$. To complete the proof, it remains only
to define the $G$-apartment bundle $M^{G}_{\phi}$ and its minimal section $f$,
and show that $f$ has the same index as $f^{\sigma}$.

####  3.4.1. $G$-apartment bundles

Our definition of $M^{G}_{\phi}$ is a natural generalization of the apartment
bundle for a classical Higgs bundle, corresponding to the group
$\mathrm{PGL}(n,\mathbb{C})$. We first define the $G$-toral bundle
$F^{G}_{\phi}$. It is convenient to define it first on a contractible open set
$U\subset S$, on which $P$ has been trivialized as $U\times K^{\mathbb{C}}$.
This allows us to view the Higgs field $\phi$ as a
$\mathfrak{p}^{\mathbb{C}}$-valued $1$-form. Fix a maximal toral subspace
$\mathfrak{a}$ of $\mathfrak{p}$, with roots $\Delta$ and Weyl group $W$.

Let $\mathcal{P}$ be the set of linear subspaces $V$ of
$\mathfrak{a}^{\mathbb{C}}$ which can be defined by the vanishing of a subset
of $\Delta$. If $V\in\mathcal{P}$, let $V^{\prime}$ be the open subset which
is not contained in any smaller subspace in $\mathcal{P}$. Noting that $W$
acts on $\mathcal{P}$, there is a $W$-invariant stratification of
$\mathfrak{a}^{\mathbb{C}}$ indexed by the quotient $\mathcal{P}/W$ whose
strata are the $W$-orbits of $V^{\prime}$ for $V\in\mathcal{P}$. Since it is
$W$-invariant, it determines a stratification of the locus of semisimple
elements of $\mathfrak{p}^{\mathbb{C}}$. Suppose that $\phi$ is semisimple on
all of $U$. Since $\phi$ is holomorphic it will lie in some stratum away from
a finite set $B_{U}$ of $U$. The set $B_{U}$ does not depend on the
trivialization of $P$ over $U$.

######  Definition 3.21.

The critical set $B$ of $\phi$ is the union of the set on which $\phi$ is not
semisimple with each set $B_{U}$ defined above for a covering $U$.

Let $z$ be a point of $U-B_{U}$, and let $V\in\mathcal{P}$ be a subspace in
the stratum determined by $\phi$. That means that there is an element $k\in
K^{\mathbb{C}}$ conjugating $\phi$ at $z$ into $V^{\prime}$. Since
$V^{\prime}$ is open in its stratum of $\mathfrak{a}^{\mathbb{C}}$, we can
extend $k$ to a gauge transformation, still called $k$, in a neighborhood
$U_{z}$ of $z$ conjugating $\phi$ into $V^{\prime}$. Over $U_{z}$, we want to
define the $G$-toral bundle $F^{G}_{\phi}$ to be equal to
$\mathrm{Ad}_{k^{-1}}V$.

We claim this is independent of the choice of $k$. Indeed, if $k^{\prime}$
were any other choice of gauge transformation, then
$\mathrm{Ad}{k^{\prime}k^{-1}}$ would act on $\mathfrak{a}^{\mathbb{C}}$ by an
element of $W$. Because $W$ acts by reflections in the root hyperplanes, the
stabilizer in $W$ of $\mathrm{Ad}_{k}\phi$ stabilizes the whole subspace $V$.
Hence $\mathrm{Ad}_{(k^{\prime})^{-1}}V=\mathrm{Ad}_{k^{-1}}V$. It follows
that it is also independent of the trivialization of $P$, since any two
trivializations also differ by a gauge transformation.

Since the definition of $F^{G}_{\phi}$ over $U_{z}$ is independent of choices,
these patch together to form a bundle over $S-B$ where $B=\cup_{U}B_{U}$ is by
definition the critical set of $\phi$. This is the $G$-toral bundle
$F^{G}_{\phi}$.

######  Remark 3.22.

An equivalent definition of the fiber of $F^{G}_{\phi}$ over a point of $S-B$
is the intersection of $\mathrm{ad}\mathfrak{p}^{\mathbb{C}}$ with the center
of the centralizer of $\phi$ in $\mathrm{ad}\mathfrak{p}^{\mathbb{C}}$.

The $G$-toral bundle $F^{G}_{\phi}$ has a canonical flat connection and real
structure with respect to which the roots of $\mathfrak{a}^{\mathbb{C}}$ are
real and parallel in any of the local identifications of $F^{G}_{\phi}$ with
$V\subset\mathfrak{a}^{\mathbb{C}}$. Since it is a subbundle of
$\mathrm{ad}\mathfrak{p}^{\mathbb{C}}$, it inherits the metric $\nu$, which is
parallel with respect to the flat connection. As before, $\phi$ is a
holomorphic $F^{G}_{\phi}$-valued $1$-form, so by integrating its real part,
we obtain an affine flat Riemannian bundle $M^{G}_{\phi}$ with section $f$.
This is the $G$-apartment bundle. The proof of Theorem B is completed by

######  Proposition 3.23.

$\mathrm{Ind}(f)=\mathrm{Ind}(f^{\sigma}).$

###### Proof.

The representation $\sigma$ defines an embedding, which we also call $\sigma$,
of $\mathrm{ad}\mathfrak{p}^{\mathbb{C}}$ to
$\prod_{i}\mathrm{End}^{0}(E_{i})$ which is compatible with the complex linear
metrics on each. We need to understand the relationship between $F^{G}_{\phi}$
and $\sigma^{-1}(F^{\sigma}_{\phi})$.

Fix a maximal toral subalgebra $\mathfrak{a}$ of $\mathfrak{p}$. Say that an
element of $\mathfrak{a}^{\vee}$ is a root of $\sigma$ if it is the difference
of two weights of the same factor $\sigma_{i}$. The roots of $\sigma$ are
always invariant by the Weyl group $W$, and always contain the roots of
$\mathfrak{g}$. Hence they define a $W$-invariant refinement of the
stratification of $\mathfrak{a}^{\mathbb{C}}$ by $\mathcal{P}/W$, and in turn
a refinement of the stratification of semisimple elemnts of
$\mathfrak{p}^{\mathbb{C}}$. It is not hard to see from the definition of
toral bundles that $\sigma^{-1}(F^{\sigma}_{\phi})$ is built from this
stratification in exactly the same way that $F^{G}_{\phi}$ is built from the
original stratifcation; namely, the critical set $B^{\sigma}$ is the finite
set at which $\phi$ is in a smaller stratum, and with respect to a local
trivialization near $z\in S-B^{\sigma}$ such that $\phi$ is contained in
$\mathfrak{a}^{\mathbb{C}}$, the fiber of $\sigma^{-1}(F^{\sigma}_{\phi})$ at
$z$ is the smallest subspace of $\mathfrak{a}^{\mathbb{C}}$ in the
$W$-invariant stratification containing $\phi(z)$. This implies that $B\subset
B^{\sigma}$, and that over $S-B^{\sigma}$,
$\sigma^{-1}(F^{\sigma}_{\phi})\subset F^{G}_{\phi}$.

First suppose that $\sigma^{-1}(F^{\sigma}_{\phi})=F^{G}_{\phi}$. Since
$\sigma$ is injective on Lie algebras, it restricts to an isomorphism over
$S-B^{\sigma}$ from $F^{G}_{\phi}$ to $F^{\sigma}_{\phi}$. We have already
observed that this isomorphism is an isometry with respect to the complex
linear metric. Furthermore, since the weights of $\sigma$, which determine the
real and flat structures of $F^{\sigma}_{\phi}$, are constant real linear
combinations of the roots of $\mathfrak{g}$, we see that the isomorphism also
preserves the real and flat structures. Hence, $\sigma$ defines an isomorphism
of the apartment bundles $M^{G}_{\phi}$ and $M^{\sigma}_{\phi}$ intertwining
the sections $f$ and $f^{\sigma}$. The only difference is that variations of
$f^{\sigma}$ must vanish near $B^{\sigma}$, whereas variations of $f$ need
only vanish near $B$. By the log cutoff trick (Proposition 3.4), the index is
the same.

In the general case where $\sigma^{-1}(F^{\sigma}_{\phi})$ is only a subbundle
of $F^{G}_{\phi}$, it is still a locally constant subbundle since it is cut
out by roots of $\sigma$, which are locally constant. Therefore,
$M^{\sigma}_{\phi}$ becomes identified with a totally geodesic subbundle of
$M^{G}_{\phi}$. By Proposition 3.5, the index of $f$ and $f^{\sigma}$ is still
the same.

∎

######  Remark 3.24.

Even though we have used Proposition 3.5 to compare indices several times, it
was not really necessary. The destabilizing variations $X$ of $f^{\sigma}$ in
$M^{\sigma}_{\phi}$ induced from variations of the section $f$ of the
$G$-apartment bundle $M^{G}_{\phi}$ already give varations $X_{R}$ of $H_{R}$
which are tangential to the image of $M_{R}$ inside
$\prod_{j}\mathbb{P}\mathrm{Met}(E_{j})$. This is because if $X=\sigma(\xi)$
for a section $\xi$ of $\mathrm{ad}\mathfrak{p}^{\mathbb{C}}$ then
$X_{R}=\sigma(\frac{\xi+\xi^{*_{c}}}{2})$.

##  4\. Unstable minimal surfaces and the Labourie conjecture

After recalling the Hitchin section, we prove Theorem A for
$G=\textrm{PSL}(n,\mathbb{R})$, $n\geq 4$. We then prove Theorem A in general
and deduce Corollary A.

###  4.1. The Hitchin section

Let $(G,K)$ be an admissible pair of rank $l$. Fix homogeneous generators
$p_{1},\ldots,p_{l}$ of $\mathcal{O}(\mathfrak{p})^{K}$, and let
$m_{1},\ldots,m_{l}$ be their degrees. In our terminology, the Hitchin map
with respect to the basis $p_{1},\ldots,p_{l}$ sends a $G$-Higgs bundle
$(P,A^{0,1},\phi)$ to

$(p_{1}(\phi),\ldots,p_{l}(\phi))\in\bigoplus_{i=1}^{l}H^{0}(S,\mathcal{K}^{m_{i}}).$

The space $\bigoplus_{i=1}^{l}H^{0}(S,\mathcal{K}^{m_{i}})$ is called the
Hitchin base.

If $G$ is the adjoint form of a connected split real simple group, Hitchin
constructs for each point
$(\alpha_{1},\ldots,\alpha_{l})\in\bigoplus_{i=1}^{l}H^{0}(S,\mathcal{K}^{m_{i}})$
a $G$-Higgs bundle $s(\alpha_{1},\ldots,\alpha_{l})$. He proves:

######  Theorem 4.1 (Theorem 7.5 in [20]).

With respect to the natural complex structure on the moduli space
$\mathcal{M}_{G}(S)$ of $G$-Higgs bundles over $S$, $s$ is a holomorphic
section of the Hitchin map. Furthermore, $s(\alpha_{1},\ldots\alpha_{l})$ is
always stable and $s$ is an isomorphism onto a connected component of
$\mathcal{M}_{G}(S)$.

By $G$-NAH II, the image of the Hitchin section $s$ corresponds to a connected
component of the representation variety $\mathrm{Rep}(\Sigma_{g},G)$. This
component of the representation variety is called the Hitchin component, which
we write as $\mathrm{Hit}(G)$.

If $G$ is a product of adjoint forms of split real simple groups, we can take
the generators $p_{i}$ to be the union of a basis of generators for each
factor. Hitchin’s construction gives a $G$-Higgs bundle for each factor. We
extend the Hitchin section to products by simply taking the product of
$G_{i}$-Higgs bundles for each factor $G_{i}$. Theorem 4.1 remains true for
such groups $G$.

######  Remark 4.2.

A metric $\nu$ on $(G,K)$ determines a degree two invariant polynomial on
$\mathfrak{p}^{\mathbb{C}}$ by $X\mapsto\nu(X^{2})$. Since it should be clear
in context, we write $\nu$ for the polynomial.

######  Remark 4.3.

If $(E,\overline{\partial}_{E},\phi)$ lies in the image of the Hitchin
section, then $\phi$ never vanishes [20, Section 5]. It follows that minimal
harmonic maps associated to Hitchin representations are immersions.

###  4.2. Unstable minimal surfaces for Hitchin representations

Let $(G,K,\nu)$ be an admissible triple with $G$ the adjoint form of a split
real semisimple group, and let $N=G/K$. Let $\mathbf{T}_{g}$ be the
Teichmüller space of a surface $\Sigma_{g}$ of genus $g$. Recall that if $M$
is a flat Riemannian $N$-bundle with irreducible holonomy
$\rho:\pi_{1}(\Sigma_{g})\to G$, $G$-NAH I produces a harmonic section $h$.
Let $\mathbf{E}_{\rho}(S)$ be the total energy of $h$ with respect to $\nu$,
which we view as a function on $\mathbf{T}_{g}$. The function
$\mathbf{E}_{\rho}$ is smooth [36]. If $S$ is a critical point in
$\mathbf{T}_{g}$, then the section $h$ is minimal with respect to $\nu$, and
its index as a minimal map is equal to the index of $\mathbf{E}_{\rho}$ at $S$
[14, Theorem 3.4]. In the case that $\rho$ is in the Hitchin component, the
result below follows from work of Labourie [24, Theorem 1.0.3].

######  Theorem 4.4 (Labourie).

If $\rho$ is Hitchin, then $\mathbf{E}_{\rho}$ is proper on $\mathbf{T}_{g}$.

Consequently, $\mathbf{E}_{\rho}$ admits a global minimum, which must in
particular be a stable critical point. Therefore, every Hitchin representation
has at least one stable equivariant minimal surface. Hence, to prove that
there exists a Hitchin representation with more than one equivariant minimal
surface, it suffices to produce a Hitchin representation with an unstable
minimal surface. This is what we do below. But first, we state a well-known
fact which is the key to the construction. Note that a $\mathbb{R}^{n}$-valued
cohomology class of $\Sigma_{g}$ defines an action of the Deck group of any
covering of $\Sigma_{g}$ on $R^{n}$ by translations.

######  Proposition 4.5.

Let $\Sigma_{g}$ be a closed surface of genus $g$ with $g\geq 3$. For any
$n\geq 3$, there is a cohomology class $\beta\in
H^{1}(\Sigma_{g},\mathbb{R}^{n})$, a conformal structure $S$ on $\Sigma_{g}$,
and an unstable minimal map $\tilde{f}$ from a covering space
$\tilde{\Sigma}_{g}$ of $\Sigma_{g}$ to $\mathbb{R}^{n}$, equivariant by the
action of the Deck group determined by $\beta$.

###### Proof.

Here is a very direct proof. In the case $g=3$ and $n=3$, we take
$\tilde{\Sigma}_{g}$ to be the triply periodic Schwarz P-surface, $\tilde{f}$
the inclusion, $S$ the conformal structure of the quotient by
$\mathbb{Z}^{3}$, and $\beta$ the coresponding cohomology class. This surface
has equivariant index one [34]; to show that the index is at least one, it
suffices to consider a normal variation of constant length. If $g>3$, we can
take a branched covering of the (quotient of the) Schwarz P-surface, and if
$n>0$, we can linearly isometrically map $\mathbb{R}^{3}$ into
$\mathbb{R}^{n}$. ∎

In fact, there are many such examples, and, at least for large genus, many of
very large index. For a discussion in the case $n=3$, see [29, section 5.3].

By the usual correspondence between equivariant maps and flat bundles, we can
also interpret this as a flat Riemannian bundle $M_{\beta}$, whose fibers are
$\mathbb{R}^{n}$ and transition functions are translations given by the
cohomology class $\beta$, together with a minimal section $f$. The vertical
tangent bundle of $M_{\beta}$ is just the trivial $\mathbb{R}^{n}$ bundle. The
equivariant index of $\tilde{f}$ is the same as the index of $f$. This
proposition indicates that we should be looking for instability in Higgs
bundles for which the $G$-toral bundle $F^{G}_{\phi}$ is trivial.

We first prove Theorem A for the main case of interest,
$G=\textrm{PSL}(n,\mathbb{R})$, $n\geq 4$. The proof is simple and distills
the main ideas.

Recall from section 2.4 that
$\mathfrak{p}^{\mathbb{C}}\subset\mathfrak{g}^{\mathbb{C}}=\mathfrak{sl}(n,\mathbb{C})$
is the subset of complex symmetric matrices. We choose the elementary
symmetric polynomials $p_{2},\dots,p_{n}$ as our basis for
$\mathcal{O}(\mathfrak{p})^{K}$ defining the Hitchin base. For
$a=\textrm{diag}(a_{1},\dots,a_{n})$ in the subalgebra
$\mathfrak{a}^{C}\subset\mathfrak{p}^{C}$ of diagonal matrices, the $p_{i}$
are defined

$p_{2}(a)=\sum_{1\leq i<j\leq n}a_{i}a_{j},\hskip
2.84526ptp_{3}(a)=\sum_{1\leq i<j<k\leq n}a_{i}a_{j}a_{k},\hskip
2.84526pt\dots\hskip 2.84526ptp_{n}(a)=\prod_{i=1}^{n}a_{i}.$

By the Chevalley restriction theorem, this defines $p_{i}$ on all of
$\mathcal{O}(\mathfrak{p})^{K}.$ Note that if $q$ is a polynomial in
$\mathbb{C}$ with roots $\lambda_{1},\dots,\lambda_{n},$ then

(16) $q(z)=\sum_{i=0}^{n}(-1)^{n-i}p_{n-i}(\lambda)z^{i},$

where $\lambda=\textrm{diag}(\lambda_{1},\dots,\lambda_{n})$ and
$p_{1}(\lambda)=\sum_{i=1}^{n}\lambda_{i}$ (which vanishes on our Lie
algebra).

###### Proof of Theorem A for $\textrm{PSL}(n,\mathbb{R})$.

As above, let
$\mathfrak{a}^{\mathbb{C}}\subset\mathfrak{p}^{\mathbb{C}}\subset\mathfrak{g}^{\mathbb{C}}$
be the subalgebra of diagonal matrices,

$\mathfrak{a}^{\mathbb{C}}=\\{\textrm{diag}(a_{1},\dots,a_{n}):a_{i}\in\mathbb{C},\sum_{i=1}^{n}a_{i}=0\\}.$

Let $(\tilde{S},\beta,\tilde{f})$ be an unstable minimal map to
$\mathbb{R}^{n-1}$ as in Proposition 4.5. There is a
$\textrm{SO}(n-1,\mathbb{R})$-torsor of isometries from $\mathbb{R}^{n-1}$ to
the subalgebra of real points $\mathfrak{a}$, and we have the freedom to
choose one, $\iota$, such that no two components of $\iota(\partial\tilde{f})$
agree at every point of $\tilde{S}.$ Let
$\textrm{diag}(\phi_{1},\dots,\phi_{n})=\iota(\partial\tilde{f}).$ This is the
holomorphic derivative of the section $f$ of $M_{\beta}$ in a particular
trivialization, so we denote it by $\partial f$. From the choice of
hyperplane, $p_{1}(\partial f)=0$, and since $f$ is minimal and the mapping
from $\mathbb{R}^{n-1}$ to $\mathfrak{a}$ is an isometry, $p_{2}(\partial
f)=0$. By our choice of isometry $\iota$, $\partial f$ is generically regular
semisimple.

For each $i=2,\dots,n$, set $\alpha_{i}=p_{i}(\phi_{i})$, defining a point
$(\alpha_{2},\dots,\alpha_{n})$ in the Hitchin base, and let
$(E,\overline{\partial}_{E},\phi)$ be the Higgs bundle associated to
$s((\alpha_{2},\dots,\alpha_{n}))$ via the standard representation. Since
$\alpha_{2}=0$, $p_{2}(\phi)=0,$ so the Higgs bundle
$(E,\overline{\partial}_{E},\phi)$ is minimal. By (16), the characteristic
polynomials of $\phi$ and $\partial f$ agree. Hence, $\phi$ is not just
generically regular semisimple, but it is globally diagonalizable on the
complement of its critical set $B$, with eigen-$1$-forms
$\phi_{1},\dots,\phi_{n}$.

Since $\phi$ is globally diagonalizable on $S-B$, the extended toral bundle
$F_{\phi}$—spanned by globally defined projections to the eigenspaces of
$\phi$—is trivial. The toral bundle $F_{\phi}^{0}$ is then trivial as well.
Equivalently, $S$ is equal to its cameral cover. After trivializing the
vertical tangent bundle of the apartment bundle $M_{\phi}$ to
$\mathbb{R}^{n}$, the holomorphic derivative of the minimal section agrees
with $\partial f$ up to an isometry. By Theorem B, for $R$ large enough the
index of the minimal map $h_{R}$ associated to
$(E,\overline{\partial}_{E},\phi)$ is bounded below by the index of the
limiting object, which is equal to the equivariant index of $\tilde{f}$, hence
positive. This completes the proof. ∎

The proof of the full Theorem A follows the $\textrm{PSL}(n,\mathbb{R})$ case
in an abstract setting.

###### Proof of Theorem A.

Let $G$ be the adjoint form of a split real group of real rank $l\geq 3$, with
a maximal compact $K$ and decomposition
$\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$. Let $\mathfrak{a}$ be a maximal
toral subalgebra of $\mathfrak{p}$ and let $\nu$ be an invariant metric on
$G/K$. We set $(\tilde{S},\beta,\tilde{f})$ to be an unstable minimal map to
$\mathbb{R}^{l}$ from Proposition 4.5. Choose an isometry between
$\mathbb{R}^{l}$ and $\mathfrak{a}$ (with the metric $\nu$) such that no root
of $\mathfrak{a}$ vanishes on the derivative $\partial\tilde{f}$ at every
point of $\tilde{\Sigma}_{g}$.

Let $M_{\beta}$ be the flat Riemannian bundle with translational holonomy
given by $\beta$, with section $f$ descended from $\tilde{f}$. Using the
chosen isometry, we can identify the vertical tangent bundle of $M_{\beta}$
with the trivial $\mathfrak{a}$ bundle, so that $\partial f$ is a $(1,0)$-form
valued in $\mathfrak{a}^{\mathbb{C}}$. Since $f$ is minimal, $\nu((\partial
f)^{2})=0$.

Let $(p_{1},\ldots,p_{n})$ be a homogeneous generating set for the invariant
polynomials on $\mathfrak{p}^{\mathbb{C}}$, and for each $i=1,\ldots,n$, set
$\alpha_{i}=p_{i}(\partial f)$. Then $(\alpha_{1},\ldots,\alpha_{n})$ is a
point in the Hitchin base. Let
$(P,A^{0,1},\phi)=s((\alpha_{1},\ldots,\alpha_{n}))$, where $s$ is the Hitchin
section. Since $s$ is a section of the Hitchin map, every
$K^{\mathbb{C}}$-invariant polynomial on $\mathfrak{p}^{\mathbb{C}}$ takes the
same values on $\phi$ as on $\partial f$. In particular, since $\nu$ is an
invariant polynomial, the $G$-Higgs bundle $(P,A^{0,1},\phi)$ is minimal with
respect to $\nu$. Moreover, since $\partial f$ is generically regular
semisimple by the choice of isometry, and $s$ is a section of the Hitchin map,
Lemma 2.21 shows that $\phi$ is conjugate to $\partial f$ in any local
trivialization of $P$, and hence that $\phi$ is also generically regular
semisimple. Let $B\subset S$ be the critical set of $\phi$, which is to say
the set at which it is not regular semisimple. Let $F^{G}_{\phi}$ be the toral
bundle of $\phi$ over $S-B$ and $M^{G}_{\phi}$ be the $G$-apartment bundle of
$\phi$ with minimal section $f^{G}$.

From Theorem $B$, for $R$ sufficiently large, the minimal harmonic map
associated to $(P,A^{0,1},R\phi)$ has index bounded below by the index of the
minimal section $f^{G}$. Since the index of $f$ is positive, we are done if we
can produce an isomorphism of flat Riemannian bundles between $M_{\beta}$ and
$M_{\phi}^{G}$ that intertwines $f$ and $f^{G}$. It is even enough to produce
an isomorphism between $\mathfrak{a}^{\mathbb{C}}$ and $F^{G}_{\phi}$ which
intertwines $\partial f$ and $\phi$, since the associated apartment bundles
are then constructed by integration in the same way.

Fix a local trivialization of $P$ on $U\subset S-B$. By the second part of
Lemma 2.21, in this trivialization every gauge transformation conjugating
$\phi$ to $\partial f$ defines the same isomorphism from $F^{G}_{\phi}$ to
$\mathfrak{a}^{\mathbb{C}}$. Since the isomorphism is unique, these local
isomorphisms patch together to give a global flat real isometric isomorphism
between the bundle $F^{G}_{\phi}$ and the trivial bundle
$\mathfrak{a}^{\mathbb{C}}$ identifying $\phi$ with $\partial f$. This
completes the proof of Theorem A.

∎

###  4.3. Non-uniqueness of area minimizers

To conclude the paper, we prove Corollary A, which states that area minimizers
for Hitchin representations need not be unique. First we recall the Labourie
map.

Fix a connected split real group $G$ in adjoint form with Cartan decomposition
and metric $\nu$ on $G/K$. Given a conformal structure $S$, let

$\mathbf{H}_{G}(S)=\bigoplus_{i=1}^{l}H^{0}(S,\mathcal{K}^{m_{i}})$

be the Hitchin base for $G$ with respect to a homogeneous basis $\\{p_{i}\\}$
of invariant polynomials on $\mathfrak{p}$. Let $\mathbf{H}_{G}$ be the bundle
over $\mathbf{T}_{g}$ whose fiber over $S$ is $\mathbf{H}_{G}(S)$. Since $\nu$
defines a polynomial in $\mathcal{O}(\mathfrak{p})^{K}$, $G$-Higgs bundles
over $S$ with Higgs field $\phi$ satisfying $\nu(\phi^{2})=0$ are taken by the
Hitchin map onto a subspace $\mathbf{M}_{G}(S)\subset\mathbf{H}_{G}(S)$ of
complex codimension $3g-3=\mathrm{dim}(H^{0}(S,\mathcal{K}^{2}))$. Varying
over $\mathbf{T}_{g}$, we obtain a codimension $3g-3$ subbundle
$\mathbf{M}_{G}\subset\mathbf{H}_{G}$ that parametrizes the space of minimal
maps associated to Hitchin representations. The Labourie map is the map

$L_{G}:\mathbf{M}_{G}\to\mathrm{Hit}(G)$

to the Hitchin component of the representation variety, defined by applying
the Hitchin section for $S$ on the fiber $\mathbf{M}_{G}(S)$. Labourie’s
existence result [24, Theorem 1.0.3] shows that $L_{G}$ is surjective. Theorem
A is equivalent to the following

######  Theorem 4.6.

For every split real semisimple $G$ of rank at least $3$ with maximal compact
subgroup $K$ and invariant metric $\nu$, the Labourie map $L_{G}$ is not
injective.

The theorem implies the following lemma.

######  Lemma 4.7.

There is no continuous section of the Labourie map.

###### Proof.

Suppose that $s:\textrm{Hit}(G)\to\textbf{M}_{G}$ is a continuous section of
$L_{G}$. The spaces $\mathbf{M}_{G}$ and $\mathrm{Hit}(G)$ are both
topologically open balls of the same real dimension $(2g-2)\dim G$. In the
case of $\mathrm{Hit}(G)$, this is because the Hitchin section from a fixed
Hitchin base is a homoeomorphism onto $\mathrm{Hit}(G)$, and since
$\mathrm{dim}\mathbf{T}_{g}=\mathrm{dim}H^{0}(\mathcal{K}^{2})$ this also
shows the dimensions of $\textbf{M}_{G}$ and $\textrm{Hit}(G)$ are the same.
Since $s$ is a section, it is both injective and proper. By Brouwer’s
invariance of domain, $s$ is homeomorphism. But then the Labourie map is also
a homeomorphism, which contradicts Theorem A. ∎

To prove Corollary A, we argue by contradiction, and suppose that for every
Hitchin representation $\rho,$ there exists a unique equivariant minimizing
minimal surface. If this is the case, we can define a section

$s_{\mathrm{min}}:\textrm{Hit}(G)\to\mathbf{M}_{G}$

that associates each representation to the conformal structure of the minimal
surface and the point in the Hitchin base for the Higgs bundle data. By Lemma
4.7, the proof of Corollary A will follow from the lemma below.

######  Lemma 4.8.

Assuming its existence, the map $s_{\textrm{min}}$ is continuous.

The auxiliary result we use is contained in a paper of Tholozan [38].

######  Definition 4.9.

Let $X$ and $Y$ be two metric spaces, and $(F_{y})_{y\in Y}$ a family of maps
$F_{y}:X\to\mathbb{R}$ depending continuously on $Y$ in the compact-open
topology. We say that $(F_{y})_{y\in Y}$ is locally uniformly proper if for
every $y_{0}\in Y,$ there exists a neighbourhood $U$ of $y_{0}$ such that for
any $C\in\mathbb{R},$ there exists a compact set $K\subset X$ such that for
all $y\in U$ and $x\in X\backslash K$, $F_{y}(x)>C.$

######  Lemma 4.10 (Proposition 2.6 of [38]).

Let $X$ and $Y$ be two metric spaces, and $(F_{y})_{y\in Y}$ a locally
uniformly proper family of maps $F_{y}:X\to\mathbb{R}$ depending continuously
on $Y$. Assume that each $F_{y}$ achieves its minimum at a unique point
$x_{m}(y)\in X$. Then the map from $Y\to X$ defined by

$y\mapsto x_{m}(y)$

is continuous.

Fix a hyperbolic metric $g_{0}$ on $\Sigma_{g}$. For
$\gamma\in\pi_{1}(\Sigma_{g})$ let

$\ell(\rho(\gamma))=\textrm{inf}_{x\in G/K}d(\rho(\gamma)x,x)$

be the translation length for $\rho(\gamma)$ and $\ell_{g_{0}}(\gamma)$ the
$g_{0}$-length of the geodesic representative of $\gamma$. Hitchin
representations are well-displacing, which in [24] means that there exists
constants $A,B>0$ such that

$\ell(\rho(\gamma))\geq A\ell_{g_{0}}(\gamma)-B.$

More generally, it is proved by Guichard-Wienhard that an Anosov
representation defines a quasi-isometric embedding from the Cayley graph of
$\pi_{1}(\Sigma_{g})$ to $G$ (with a left-invariant metric), and are hence
well-displacing [19].

###### Proof of Lemma 4.8.

We take $X=\mathbf{T}_{g}$, $Y=\textrm{Hit}(G),$ and
$F_{\rho}=\mathbf{E}_{\rho}:\mathbf{T}_{g}\to\mathbb{R}.$
$\rho\mapsto\mathbf{E}_{\rho}$ is smooth in $\rho$; this is a consequence of
the implicit function theorem [36]. Assuming
$(\mathbf{E}_{\rho})_{\rho\in\textrm{Hit}(G)}$ is locally uniformly proper,
Lemma 4.10 asserts that the map $m:\textrm{Hit}(G)\to\mathbf{T}_{g}$ taking
$\rho$ to the conformal structure of the area minimizing surface is
continuous.

Toward continuity of $s_{\textrm{min}}$, it suffices to work locally in
$\textbf{M}_{G}$, where we can write
$\textrm{s}_{min}(\rho)=(m(\rho),\alpha(\rho)),$ with
$\alpha(\rho)\in\mathbf{M}_{G}(m(\rho))$. Fixing a basepoint on $\Sigma_{g}$,
every Hitchin representation $\rho:\pi_{1}(\Sigma_{g})\to G$ defines a section
$\delta_{\rho}:\mathbf{T}_{g}\to\mathbf{H}_{G}$ as follows. For each Riemann
surface, we obtain a $G$-Higgs bundle from $\rho$ using $G$-NAH I, and then we
apply the Hitchin map to get a point in $\mathbf{H}_{G}$. Assembling these
sections into a map
$\delta:\textrm{Hit}(G)\times\mathbf{T}_{g}\to\mathbf{H}_{G},$
$\delta(\rho,\cdot)=\delta_{\rho}(\cdot),$ it is well-understood that $\delta$
is continuous (essentially a consequence of the fact that harmonic maps vary
smoothly [36]). We express

$\alpha(\rho)=\delta(\rho,m(\rho))$

and deduce that $s_{\textrm{min}}$ is continuous.

It remains to justify that $(\mathbf{E}_{\rho})_{\rho\in\textrm{Hit}(G)}$ is
locally uniformly proper, which amounts to going through Labourie’s proof that
each $\mathbf{E}_{\rho}$ is proper and observing that the estimates can be
made locally uniform in $\rho$. This requires no new insight, so we only give
a brief explanation and point to the relevant references.

Let $S_{0}$ be a fixed marked Riemann surface structure on $\Sigma_{g}$. In
his proof of [24, Theorem 1.0.3], Labourie shows

$\mathbf{E}_{\rho}(S)\geq A_{\rho}(\textrm{inter}(S,S_{0}))^{2},$

where $\textrm{inter}(\cdot,\cdot):\mathbf{T}_{g}^{2}\to\mathbb{R}$ is the
intersection function (see [2]), and $S\mapsto\textrm{inter}(S,S_{0})$ is
known to be proper on $\mathbf{T}_{g}$ (see [2, Proposition 4] or [24,
Proposition 6.2.4]). Stepping into the proof, $A_{\rho}$ depends only on the
minimal well-displacing constant $A$ for $\rho$. It can be gleaned from the
proofs of Propositions 3.3.5 and 3.3.7 of [24] that this constant is uniformly
locally controlled with $\rho$. This establishes the result. ∎

###### Proof of Corollary A.

Assuming uniqueness of area minimizers, we construct the section
$s_{\mathrm{min}}:\textrm{Hit}(G)\to\mathbf{M}_{G}$. We apply Lemma 4.7 and
Lemma 4.8 to find a contradiction.

∎

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