# On the connected components of the moduli space of equivariant minimal
surfaces in $\CH^{2}$.

Ian McIntosh  Department of Mathematics
University of York
York YO10 5DD, UK<EMAIL_ADDRESS>
(Date: July 8, 2021)

###### Abstract.

An equivariant minimal surface in $\CH^{n}$ is a minimal map of the Poincaré
disc into $\CH^{n}$ which intertwines two actions of the fundamental group of
a closed surface $\Sigma$: a Fuchsian representation on the disc and an
irreducible action by isometries on $\CH^{n}$. The moduli space of these can
been studied by relating it to the nilpotent cone in each moduli space of
$PU(n,1)$-Higgs bundles over the conformal surface corresponding to the map.
By providing a necessary condition for points on this nilpotent cone to be
smooth this article shows that away from the points corresponding to branched
minimal immersions or $\pm$-holomorphic immersions the moduli space is smooth.
The argument is easily adapted to show that for $\mathbb{RH}^{n}$ the full
space of (unbranched) immersions is smooth. For $\CH^{2}$ we show the
connected components of the moduli space of minimal immersions are indexed by
the Toledo invariant and the Euler number of the normal bundle of the
immersion. This is achieved by studying the limit points of the
$\mathbb{C}^{\times}$-action on the nilpotent cone. It is shown that the limit
points as $t\to 0$ lead only to branched minimal immersions or
$\pm$-holomorphic immersions. In particular, the Euler number of the normal
bundle can only jump by passing through branched minimal maps.

###### Key words and phrases:

Minimal surface, Higgs bundle, complex hyperbolic plane, nilpotent cone

###### 1991 Mathematics Subject Classification:

20H10,53C43,58E20

##  1\. Introduction.

Let $\Sigma$ be a closed oriented surface of genus $g\geq 2$ and let $N$ be a
noncompact irreducible symmetric space. Denote by $G$ the identity component
of the isometry group of $N$. By an equivariant minimal surface we mean a
minimal immersion $f:\mathcal{D}\to N$ of the Poincaré disc $\mathcal{D}$
which intertwines the action of a Fuchsian representation
$c:\pi_{1}\Sigma\to\operatorname{Aut}(\mathcal{D})$ with an irreducible
representation $\rho:\pi_{1}\Sigma\to G$. By a theorem of Corlette [4]
(generalising a theorem of Donaldson for $N=\mathbb{RH}^{3}$ [5]), the triple
$(f,c,\rho)$ is uniquely determined by the pair $(c,\rho)$, and if one
considers that $f$ is essentially unchanged by pre-composition or post-
composition by isometries, then one only needs the conjugacy classes of $c$
and $\rho$. Hence there is a natural way to assign a topology to the set
$\mathcal{M}(\Sigma,N)$ of equivalence classes $[f,c,\rho]$ of equivariant
minimal surfaces by embedding it in
$\mathcal{T}_{g}\times\mathcal{R}(\pi_{1}\Sigma,G)$ where $\mathcal{T}_{g}$ is
the Teichmúller space of $\Sigma$ and $\mathcal{R}(\pi_{1}\Sigma,G)$ is the
character variety of $G$., i.e., the moduli space of reductive representations
of $\pi_{1}\Sigma$ into $G$ up to conjugacy.

This topology for $\mathcal{M}(\Sigma,N)$ was first proposed by Loftin and the
author in [12] but prior to this, in [11], we had studied the set
$\mathcal{M}(\Sigma,\CH^{2})$ and provided parametrisations for certain
subsets which we called “components”, simply to mean “parts whose union is the
whole”. We showed that each parametrisation equipped the component with the
structure of a connected smooth complex manifold, most of which had the same
dimension. These parametrisations exploited the nonabelian Hodge
correspondence, through which one can identify an equivariant minimal surface
with a $PU(2,1)$-Higgs bundle having certain properties. However, since in
[11] we did not equip the total space $\mathcal{M}(\Sigma,\CH^{2})$ with a
topology a priori, the question of its connected components had no framework
in which to be addressed. One of the aims of this article is to answer this
question now, given the topology described above (in a nutshell, the
components identified in [11] are, with a few exceptions, the connected
components). A key to understanding the connected components lies in having a
criterion for points of the moduli space to be smooth, and this is provided
below for $\mathcal{M}(\Sigma,N)$ when $N$ is either $\CH^{n}$ or
$\mathbb{RH}^{n}$. The second aim of this article is to show how these
components fit together inside the larger set
$\overline{\mathcal{M}}(\Sigma,\CH^{2})$ obtained by including all branched
minimal surfaces. We show that this is not smooth, since singularities occur
where the closures of the connected components of
$\mathcal{M}(\Sigma,\CH^{2})$ meet. Note that
$\overline{\mathcal{M}}(\Sigma,\CH^{2})$ was the space studied in [11], and
the distinctive role of branched immersions explained here was not at all
apparent during the writing of [11].

To summarise the results below, first recall that a $PU(n,1)$-Higgs bundle can
be represented by a pair $(E,\Phi)$ where $E\to\Sigma_{c}$ is a holomorphic
vector bundle of rank $n+1$ over the Riemann surface
$\Sigma_{c}\simeq\mathcal{D}/c$. This has a decomposition $E=V\oplus 1$, where
$V$ is rank $n$ and $1$ denotes the trivial line bundle. Then $\Phi$ is a
holomorphic one-form with values in
$\operatorname{Hom}(1,V)\oplus\operatorname{Hom}(V,1)$. We write
$\Phi=(\Phi_{1},\Phi_{2})$ to denote the two components. In general $(E,\Phi)$
corresponds to an equivariant, but possibly branched, minimal surface when
$\operatorname{tr}(\Phi^{2})=0$. Zeroes of $\Phi$ are branch points of the
minimal surface and therefore we require $\Phi$ to have no zeroes (i.e.,
$\Phi_{1},\Phi_{2}$ have no common zeroes) to obtain a point in
$\mathcal{M}(\Sigma,\CH^{n})$. From [11] we know that one of
$\Phi_{1},\Phi_{2}$ is identically zero precisely when the minimal immersion
is either holomorphic or anti-holomorphic. It is convenient to write
$\mathcal{M}(\Sigma,\CH^{n})$ as a disjoint union $\mathcal{V}\cup\mathcal{W}$
where $\mathcal{W}$ is the subvariety of $\pm$-holomorphic maps and
$\mathcal{V}$ is the complement. Our first main result concerns $\mathcal{V}$,
and in fact applies also to
$\mathcal{M}(\Sigma,\mathbb{RH}^{n})\subset\mathcal{V}$ using the totally
geodesic embedding of $\mathbb{RH}^{n}$ into $\CH^{n}$.

######  Theorem 1.1.

Both $\mathcal{V}\subset\mathcal{M}(\Sigma,\CH^{n})$ and
$\mathcal{M}(\Sigma,\mathbb{RH}^{n})$ are smooth manifolds. They have complex
dimension $(g-1)\dim(G)$ for $G=PU(n,1)$ and $SO_{0}(n,1)$ respectively.

In fact what we prove is that in the nilpotent cone $\mathcal{N}^{c}$ (i.e,
the locus of $\operatorname{tr}(\Phi^{2})=0$ in the Higgs bundle moduli space
$\mathcal{H}(\Sigma_{c},G)$ for $G=PU(n,1)$ or $SO_{o}(n,1)$) the regular
points of the Hitchin function $\operatorname{tr}(\Phi^{2})$ are precisely the
points for which neither $\Phi_{j}$ has a zero.

Next we restrict our attention to $n=2$. For $G=PU(2,1)$ the connected
components of $\mathcal{R}(\pi_{1}\Sigma,G)$ are indexed by the Toledo
invariant, $\tau$. Using Toledo’s convention, as in [11], this satisfies
$\tau\in\tfrac{2}{3}\mathbb{Z}$ and $|\tau|\leq 2(g-1)$. We denote the
subspace of triples $[f,c,\rho]$ for which $\rho$ has Toledo invariant $\tau$
by $\mathcal{M}(\Sigma,\CH^{2})_{\tau}$ and it is clear that these are
disconnected from each other. Minimal immersions which are not
$\pm$-holomorphic have complex and anti-complex points (points $p\in\Sigma$
for which $f_{*}T_{p}^{1,0}\Sigma$ is a subspace of either $T^{\prime}\CH^{2}$
or $T^{\prime\prime}\CH^{2}$). These give effective divisors, $D_{2}$ and
$D_{1}$ respectively. From [11] their degrees $d_{j}=\deg(D_{j})$ satisfy

$\displaystyle\tau=\tfrac{2}{3}(d_{2}-d_{1}),$
$\displaystyle\chi(T\Sigma^{\perp})=2(g-1)-d_{1}-d_{2},$ (1.1) $\displaystyle
0\leq 2d_{1}+d_{2}<6(g-1),$ $\displaystyle 0\leq d_{1}+2d_{2}<6(g-1),$ (1.2)

where $T\Sigma^{\perp}$ is the normal bundle of $f$, which means the quotient
of $T\mathcal{D}^{\perp}\subset f^{-1}T\CH^{2}$ by the natural action of
$\pi_{1}\Sigma$. The inequalities are necessary and sufficient conditions for
the existence of minimal immersions which are not $\pm$-holomorphic. In
particular, the degrees $d_{1},d_{2}$ are equivalent information to the pair
$\tau$ and $\chi(T\Sigma^{\perp})$. For holomorphic immersions the only
topological invariant is the Toledo invariant, which determines
$\chi(T\Sigma^{\perp})$ through

$\chi(T\Sigma^{\perp})=-\tfrac{3}{2}\tau+2(g-1).$ (1.3)

This follows directly by taking the quotient of the direct sum of holomorphic
bundles $f^{-1}T^{\prime}\CH^{2}=T^{1,0}\mathcal{D}\oplus
T\mathcal{D}^{\perp}$ by the $\pi_{1}\Sigma$-action.

Let $\mathcal{V}(d_{1},d_{2})\subset\mathcal{V}$ denote the subset of those
immersions whose divisors have these degrees fixed. It was shown in [11] that
each $\mathcal{V}(d_{1},d_{2})$ can be parametrised by a smooth connected
complex manifold of dimension $8(g-1)$ (actually in [11] we included the
branched minimal immersions, which occur when $D_{1}\cap D_{2}\neq\emptyset$,
but as these are described by a complex analytic subvariety removing these
does not affect the connectedness of $\mathcal{V}(d_{1},d_{2})$). In Lemma 5.1
below we show that the parametrisation in [11] is smoothly compatible with the
structure it inherits from Theorem 1.1. Since $\mathcal{V}(d_{1},d_{2})$ all
have the same dimension they are therefore the connected components of
$\mathcal{V}$. To establish the connected components of
$\mathcal{M}(\Sigma,\CH^{2})$ it remains to see how these fit together with
points of $\mathcal{W}$. Set
$\mathcal{W}_{\tau}=\mathcal{W}\cap\mathcal{M}(\Sigma,\CH^{2})_{\tau}$ for
each $\tau$. We need only consider $\tau\geq 0$ since
$\mathcal{M}(\Sigma,\CH^{2})$ carries a natural real involution
$[f,c,\rho]\mapsto[\bar{f},c,\bar{\rho}]$ for which
$\tau(\bar{\rho})=-\tau(\rho)$.

######  Theorem 1.2.

Let $\mathcal{M}(\Sigma,\CH^{2})_{\tau}$ denote the space of equivariant
minimal immersions with Toledo invariant $\tau$. Then
$\mathcal{M}(\Sigma,\CH^{2})_{0}$ has connected components
$\mathcal{V}(d_{1},d_{2})$ for $d_{2}=d_{1}$. For $0<\tau<2(g-1)$ the
connected components of $\mathcal{M}(\Sigma,\CH^{2})_{\tau}$ are
$\mathcal{V}(d_{1},d_{2})$ with $d_{1}\neq 0$ and
$\mathcal{V}(0,d_{2})\cup\mathcal{W}_{\tau}$, while
$\mathcal{M}(\Sigma,\CH^{2})_{\tau}=\mathcal{W}_{\tau}$ for $\tau=2(g-1)$.

In particular, the connected components are indexed by $\tau$ and
$\chi(T\Sigma^{\perp})$: when $d_{1}=0$ the two equations (1.1) become the
single equation (1.3).

This theorem is proved through a complete description of the limit points of
the $\mathbb{C}^{\times}$-action $t\cdot(E,\Phi)=(E,t\Phi)$ on $PU(2,1)$-Higgs
bundles, for both $t\to\infty$ and $t\to 0$. As well as proving the theorem,
it gives us information about how the closures of the connected components fit
together inside the space $\overline{\mathcal{M}}(\Sigma,\CH^{2})$ which
includes all branched minimal immersions. On the Higgs bundle side this
corresponds to the space of all $(c,E,\Phi)$ for which $(E,\Phi)$ lies in the
nilpotent cone $\mathcal{N}^{c}$ excluding the locus $\Phi=0$ (since this
corresponds to constant maps). Each $\mathcal{N}^{c}$ is known to be
stratified by the unstable manifolds of the downwards Morse flow for the Higgs
field energy $\|\Phi\|^{2}_{L^{2}}$: for each connected component $C$ of
critical points these agree with

$\mathcal{U}_{C}=\\{(E,\Phi):\lim_{t\to\infty}(E,t\Phi)\in C\\}.$

It was conjectured in [11], and is proven below, that except when $C$ consists
of local minima (there is precisely one such $C$ for each $\tau$ and it equals
$\mathcal{W}_{\tau}$) these unstable manifolds correspond to the fibres of the
analytic family $\mathcal{V}(d_{1},d_{2})$ over $\mathcal{T}_{g}$. We
understand how these fit together by examining the limit in the opposite
direction, as $t\to 0$. It is well-known that when $E$ is semi-stable (as a
vector bundle) the limit $\lim_{t\to 0}(E,t\Phi)$ must have $\Phi=0$ and
therefore this can only occur when $\tau=0$. When $E$ is strictly unstable
(and assuming $\tau\geq 0$ without loss of generality) we show that either
$(E,\Phi_{1})$ is a Hodge bundle, in which case this is limit, or the limit is
determined by the maximal destabilizing line subbundle of $V$ and always
corresponds to a branched minimal immersion.

Thus we have the the following geometric picture of how these pieces fit
together. Each fibre over $c$ of the space
$\overline{\mathcal{M}}(\Sigma,\CH^{2})_{\tau}$ is a union of these unstable
manifolds, and their closures intersect either at branched minimal surfaces or
at $\pm$-holomorphic surfaces. Removing these leaves a disjoint union of
components $\mathcal{V}(d_{1},d_{2})$, one for each critical manifold of non-
minima. In particular, one cannot pass from one component
$\mathcal{V}(d_{1},d_{2})$ to another in
$\overline{\mathcal{M}}(\Sigma,\CH^{2})$ without forcing the minimal immersion
to branch somewhere. Another way of saying this is that
$\chi(T\Sigma^{\perp})$ can jump but only by passing through branched maps.

Acknowledgements. The initial stages of this research were carried out while
the author was a visitor at the Simons Center for Geometry and Physics, Stony
Brook University, in April 2019 during the programme “The Geometry and Physics
of Hitchin Systems”, and then subsequently at the MFO Oberwolfach workshop
“The Geometry and Physics of Hitchin Bundles” in May 2019\. I gratefully
acknowledge the support of these institutions, and thank the organisers for
the opportunity to participate. I also thank Andy Sanders and John Loftin for
the stimulating discussions during the SCGP programme which raised some of the
questions answered here.

##  2\. Definitions.

Let $\mathbb{C}^{n,1}$ denote $\mathbb{C}^{n+1}$ equipped with the pseudo-
Hermitian inner product characterised by

$\langle u,u\rangle=|u_{1}|^{2}+\ldots|u_{n}|^{2}-|u_{n+1}|^{2}.$

Let $e_{1},\ldots,e_{n+1}$ denote the standard basis vectors for
$\mathbb{C}^{n+1}$. We model $\CH^{n}$ on the space of lines in
$\mathbb{CP}^{n}$ generated by a negative definite vector and fix a base point
$o=[e_{n+1}]$. Throughout this article we set $G=PU(n,1)$, the group of
orientation preserving isometries of $\CH^{n}$, and let $H\simeq U(n)$ denote
the maximal compact subgroup which fixes the base point $o$. The Lie algebra
$\mathfrak{g}=\mathfrak{su}(n,1)$ of $G$ has Cartan decomposition
$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$ corresponding to the
decomposition of elements

$\begin{pmatrix}A&u\\\ u^{\dagger}&a\end{pmatrix}=\begin{pmatrix}A&0\\\
0&a\end{pmatrix}+\begin{pmatrix}0&u\\\ u^{\dagger}&0\end{pmatrix}$ (2.1)

where $A\in\mathfrak{u}(n)$, $a=-\operatorname{tr}(A)$, $u\in\mathbb{C}^{n}$
and $u^{\dagger}$ is the Hermitian transpose.

Let $\Sigma$ be a closed oriented surface of genus $g\geq 2$ and let
$\mathcal{D}$ denote the Poincaré disc thought of as the universal cover of
$\Sigma$ (note that $\mathcal{D}\simeq\CH^{1}$). A Fuchsian representation
$c:\pi_{1}\Sigma\to PU(1,1)$ equips $\Sigma$ with a complex structure which
will be denoted $\Sigma_{c}$. We will assume this complex structure is
compatible with the orientation of $\Sigma$ and hence the set conjugacy
classes of such Fuchsian representations can be identified with the
Teichmüller space $\mathcal{T}_{g}$ of $\Sigma$. An equivariant minimal
surface in $\CH^{n}$ is the equivalence class $[f,c,\rho]$ of a triple
consisting of a Fuchsian representation $c$, an indecomposable representation
$\rho:\pi_{1}\Sigma\to G$ and a minimal immersion $f:\mathcal{D}\to\CH^{n}$
which intertwines the actions of $c$ and $\rho$, i.e.,

$f\circ c(\delta)=\rho(\delta)\circ
f,\quad\forall\delta\in\pi_{1}\mathcal{D}.$

Equivalence is with respect to the action of $PU(1,1)\times G$ by conjugation
on $(c,\rho)$ and the corresponding natural action on $f$. Indecomposability
of $\rho$ means exactly that its image does not lie in a proper subgroup of
$G$. By the same argument as [12, Lemma 2.3] it is equivalent to the condition
that $f$ is linearly full (i.e., its image does not lie in a totally geodesic
copy of $\CH^{k}$ for $k<n$).

######  Remark 2.1.

We could also say $\rho$ is irreducible in its standard real representation on
$\mathbb{C}^{n,1}$, although there is the potential for confusion with other
meanings of irreducibility (say, for the adjoint representation of the
complexification $G^{\mathbb{C}}$).

We use $\mathcal{M}(\Sigma,\CH^{n})$ to denote the set of equivariant minimal
surfaces and give it the topology of its natural embedding

$\mathcal{F}:\mathcal{M}(\Sigma,\CH^{n})\to\mathcal{T}_{g}\times\mathcal{R}(\pi_{1}\Sigma,G);\quad[f,c,\rho]\to[c,\rho],$
(2.2)

into the product of Teichmüller space with the character variety
$\mathcal{R}(\pi_{1}\Sigma,G)$ of reductive representations of $\pi_{1}\Sigma$
into $G$ modulo conjugacy. This map is injective by Corlette’s uniqueness
theorem for equivariant harmonic maps under these conditions [4]. We will
denote by $\mathcal{M}(\Sigma_{c},\CH^{n})$ the fibre over fixed
$c\in\mathcal{T}_{g}$.

Recall that, for each choice of $c$, non-abelian Hodge theory provides a
homeomorphism between $\mathcal{R}(\pi_{1}\Sigma,G)$ and the moduli space
$\mathcal{H}(\Sigma_{c},G)$ of polystable $G$-Higgs bundles over $\Sigma_{c}$
(and this homeomorphism is analytic away from singularities). Indeed, for any
simple noncompact $G$ there is a homeomorphism between
$\mathcal{T}_{g}\times\mathcal{R}(\pi_{1}\Sigma,G)$ and the universal Higgs
bundle moduli space $\mathcal{H}(\mathcal{C}_{g},G)$ over $\mathcal{T}_{g}$
[1, Thm 7.5] (here $\mathcal{C}_{g}$ denotes the universal Teichmüller curve
over $\mathcal{T}_{g}$). This is a complex analytic space for which the fibre
over $c\in\mathcal{T}_{g}$ is biholomorphic to $\mathcal{H}(\Sigma_{c},G)$.
This homeomorphism is smooth about smooth points and therefore we obtain an
embedding

$\mathcal{F}^{\prime}:\mathcal{M}(\Sigma,\CH^{n})\to\mathcal{H}(\mathcal{C}_{g},G),$
(2.3)

when $G=PU(n,1)$. This equips $\mathcal{M}(\Sigma,\CH^{n})$ with a complex
analytic structure.

We can easily characterise the image of $\mathcal{F}^{\prime}$. Recall that in
the projective equivalence class of the $G$-Higgs bundle $(E,\Phi)$, $E$ can
be taken to have the form $V\oplus 1$ where $V$ is a holomorphic rank $n$
bundle over $\Sigma_{c}$ and $1$ denotes the trivial line bundle. The Higgs
field $\Phi$ satisfies

$\Phi\in(\operatorname{Hom}(1,V)\oplus\operatorname{Hom}(V,1))\otimes K,$

and we will write $\Phi=(\Phi_{1},\Phi_{2})$ to indicate the two components
for this decomposition. It is well-known that the harmonic map corresponding
to $(E,\Phi)$ is weakly conformal (therefore branched minimal) when
$\operatorname{tr}(\Phi^{2})=0$. The branch points correspond exactly to the
zeroes of $\Phi$. Consequently we have the following elementary observation.

######  Lemma 2.1.

The pair $(E,\Phi)$ lies in the image of $\mathcal{F}^{\prime}$ if and only if
it is stable (hence indecomposable) with $\operatorname{tr}(\Phi^{2})=0$ and
$\Phi$ having no zeroes.

Each equivariant minimal surface $(f,c,\rho)$ has associated to it the Toledo
invariant $\tau$ of $\rho$ and we denote the subset of triples with fixed
$\tau$ by $\mathcal{M}(\Sigma,\CH^{n})_{\tau}$. Our convention will be that
$\tau=-\frac{2}{n+1}\deg(V)$ so that $|\tau|\leq 2(g-1)$: this matches
Toledo’s original definition and is the one used in [11].

We will also write $\mathcal{M}(\Sigma,\CH^{n})$ as a disjoint union
$\mathcal{V}\cup\mathcal{W}$ where $\mathcal{W}$ consists of all
$\pm$-holomorphic immersions and $\mathcal{V}$ is the complement. Each of
these has its subsets $\mathcal{V}_{\tau}$, $\mathcal{W}_{\tau}$ containing
those elements with fixed Toledo invariant. We note that under the non-abelian
Hodge correspondence elements of $\mathcal{W}$ correspond to Higgs bundles
with $\Phi_{2}=0$ for $\tau>0$ or $\Phi_{1}=0$ for $\tau<0$. Since we always
have $\Phi\neq 0$ we know $\mathcal{W}_{0}=\emptyset$. Note that
$\mathcal{W}_{\tau}$ corresponds to the length two Hodge bundles [11], which
are the minima of the Higgs field energy $\|\Phi\|^{2}_{L^{2}}$ [3].

From [11] we know that when $n=2$ the space $\mathcal{V}$ is a disjoint union
of subsets $\mathcal{V}(d_{1},d_{2})$, corresponding to $PU(2,1)$-Higgs
bundles for which $V$ and $\Phi$ are determined by an exact sequence [11]

$0\to
K^{-1}(D_{1})\stackrel{{\scriptstyle\Phi_{1}}}{{\to}}V\stackrel{{\scriptstyle\Phi_{2}}}{{\to}}K(-D_{2})\to
0,$ (2.4)

where $d_{j}=\deg(D_{j})$ satisfy the inequalities (1.2). There is a one-to-
one correspondence between Higgs bundles of this type and data
$(c,D_{1},D_{2},\xi)$ where $\xi$ is the extension class of this extension:
this was used in [11] to give each $\mathcal{V}(d_{1},d_{2})$ the structure of
a complex manifold of dimension $5(g-1)$.

Finally, we will want to consider the strictly larger space
$\overline{\mathcal{M}}(\Sigma,\CH^{2})$ which includes additionally the
branched minimal immersions. Under $\mathcal{F}^{\prime}$ it maps to the locus
$\operatorname{tr}(\Phi^{2})=0$ excluding the Higgs bundles with $\Phi=0$. We
will write $\bar{\mathcal{V}}$ and $\bar{\mathcal{W}}$ when we extend
$\mathcal{V}$ and $\mathcal{W}$ to include branched minimal immersions of the
respective types. The total space is disconnected into Toledo invariant pieces
$\overline{\mathcal{M}}(\Sigma,\CH^{2})_{\tau}$ and for $\tau\neq 0$ each of
these is connected. This follows from the fact that for $\tau\neq 0$ each
fibre over $c\in\mathcal{T}_{g}$ agrees with the component
$\mathcal{N}^{c}_{\tau}$ of the nilpotent cone in Toledo invariant $\tau$, and
these are connected by the downwards Morse flow. For $\tau=0$ this fibre is
$\mathcal{N}^{c}_{0}$ without the locus $\Phi=0$. Since this locus is the
critical manifold of minima we can no longer use the Morse flow argument, and
an understanding of the connected components would require a different
approach.

##  3\. Smooth points of $\mathcal{M}(\Sigma,N)$.

For $N=\CH^{n}$ or $\mathbb{RH}^{n}$ let
$\mathcal{V}\subset\mathcal{M}(\Sigma,N)$ denote the subvariety of points
where neither $\Phi_{j}$ is identically zero. For $\mathbb{RH}^{n}$ this is
the whole space while for $\CH^{n}$ it consists of the equivariant minimal
surfaces which are not $\pm$-holomorphic. Because we assume $\rho$ is
indecomposable the there are no singular points of
$\mathcal{T}_{g}\times\mathcal{R}(\pi_{1},G)$ in $\mathcal{V}$.

Recall that $\mathcal{C}_{g}$ denotes the universal Teichmüller curve over
$\mathcal{T}_{g}$. If we denote by $H^{0}(\mathcal{C}_{g},K^{2})$ the bundle
over $\mathcal{T}_{g}$ with fibre $H^{0}(\Sigma_{c},K^{2})$ then the map

$\mathcal{I}:\mathcal{H}(\mathcal{C}_{g},G)\to
H^{0}(\mathcal{C}_{g},K^{2});\quad(c,E,\Phi)\mapsto(c,\operatorname{tr}(\Phi^{2})),$

is complex analytic. We will denote its zero locus (i.e., the preimage of the
zero section of $H^{0}(\mathcal{C}_{g},K^{2})$) by $\mathcal{N}$: for real
rank one groups like $PU(n,1)$ and $SO_{0}(n,1)$ this is the union
$\cup_{c\in\mathcal{T}_{g}}\mathcal{N}^{c}$ of all nilpotent cones.

Let $\mathcal{N}^{o}\subset\mathcal{N}$ denote the open subvariety for which
$\Phi$ vanishes nowhere and neither $\Phi_{j}$ is identically zero. Then
$\mathcal{N}^{o}$ is the image of $\mathcal{V}$ in
$\mathcal{H}(\mathcal{C}_{g},G)$ and Theorem 1.1 is the statement that
$\mathcal{N}^{o}$ is smooth. We will prove this by showing that points of
$\mathcal{N}^{o}$ are regular points of $\mathcal{I}$. Since $\mathcal{I}$
maps $\mathcal{H}(\Sigma_{c},G)$ to $H^{0}(\Sigma_{c},K^{2})$ it suffices the
show that each point of
$(\mathcal{N}^{c})^{o}=\mathcal{N}^{o}\cap\mathcal{N}^{c}$ is a regular point
of $\mathcal{I}_{c}$, the restriction of $\mathcal{I}$ to the fibre over $c$.
We do this by generalising an argument used by Hitchin for
$SL(2,\mathbb{C})$-Higgs bundles [8] to compute the rank of
$d\mathcal{I}_{c}$. For this we must recall the hypercohomology description of
the tangent space to $\mathcal{H}(\Sigma_{c},G)$ [2].

Let $P\to\Sigma_{c}$ denote the principal $H^{\mathbb{C}}$-bundle whose
associated bundle is $E$, and let $P(\mathfrak{h}^{\mathbb{C}})$,
$\mathcal{P}(\mathfrak{m}^{\mathbb{C}})$ denote the holomorphic
$\mathfrak{h}^{\mathbb{C}}$ and $\mathfrak{m}^{C}$ bundles associated by the
adjoint action of $H^{\mathbb{C}}$ on these subspaces of
$\mathfrak{g}^{\mathbb{C}}$. One knows from [2] that when $(E,\Phi)$ is stable
the tangent space at this point is isomorphic to the first hypercohomology
$\mathbb{H}^{1}(\mathcal{A}^{*})$ of the complex

$\mathcal{A}^{0}\stackrel{{\scriptstyle\operatorname{ad}\Phi}}{{\to}}\mathcal{A}^{1}\to
0\to\ldots$ (3.1)

where $\mathcal{A}^{0}$ is the sheaf of local sections of
$P(\mathfrak{h}^{\mathbb{C}})$ and $\mathcal{A}^{1}$ is the sheaf of local
sections of $P(\mathfrak{m}^{\mathbb{C}})\otimes K$. As Hitchin [8] points
out, by considering the “second” spectral sequence for the hypercohomology of
$\mathcal{A}^{*}$ (i.e., the spectral sequence for the filtration by the
degree of Cech cochains) one obtains an isomorphism for
$\mathbb{H}^{1}(\mathcal{A}^{*})$ involving the kernel and cokernel sheaves

$0\to\mathit{Ker}\to\mathcal{A}^{0}\stackrel{{\scriptstyle\operatorname{ad}\Phi}}{{\to}}\mathcal{A}^{1}\to\mathit{Coker}\to
0,$ (3.2)

namely,

$\mathbb{H}^{1}(\mathcal{A}^{*})\simeq H^{1}(\mathit{Ker})\oplus
H^{0}(\mathit{Coker}).$ (3.3)

Now let $B$ denote the Killing form on $\mathfrak{g}^{\mathbb{C}}$,
specifically
$B(\eta,\xi)=\operatorname{tr}(\operatorname{ad}\eta\operatorname{ad}\xi)$. By
adjoint invariance $B(\Phi,\operatorname{ad}\Phi(\eta))=0$ for all $\eta$, so
that we have a well defined map of sheaves

$B(\Phi,\cdot):\mathit{Coker}\to K^{2};\quad\eta\mapsto B(\Phi,\eta),$ (3.4)

and an induced map from $H^{0}(\mathit{Coker})$ to $H^{0}(K^{2})$. When pre-
composed with the projection of $\mathbb{H}^{1}(\mathcal{A}^{*})$ onto
$H^{0}(\mathit{Coker})$ from (3.3), at a stable point $(E,\Phi)$, this induced
map agrees with $d\mathcal{I}_{c}$ [2, Remark 2.8(iv)]. Hence the rank of
$d\mathcal{I}_{c}$ at $(E,\Phi)$ equals the rank of $B(\Phi,\cdot)$ on
$H^{0}(\mathit{Coker})$.

######  Lemma 3.1.

Suppose $\operatorname{ad}\Phi$ has corank $1$, i.e., $\mathit{Coker}$ is a
rank $1$ sheaf. Then $d\mathcal{I}_{c}$ has maximal rank if and only if $\Phi$
has no zeroes.

###### Proof.

When $\mathit{Coker}$ is rank one $B(\Phi,\cdot)$ vanishes precisely at the
zeroes of $\Phi$ with the same divisor of zeroes, $D$, and hence
$\mathit{Coker}\simeq K^{2}(-D)$. Thus the induced map
$H^{0}(\mathit{Coker})\to H^{0}(K^{2})$ is maximal rank, and hence
$d\mathcal{I}$ has maximal rank, if and only if $\Phi$ has no zeroes. ∎

Now our aim is to show that $\operatorname{ad}\Phi$ has corank $1$ for points
of $\mathcal{V}$. First we do the case $N=\CH^{n}$.

######  Lemma 3.2.

For $G=PU(n,1)$, when neither $\Phi_{j}$ is identically zero the map
$\operatorname{ad}\Phi$ has corank one.

###### Proof.

Following [11], since $\operatorname{tr}(\Phi^{2})=0$ we have
$\Phi_{2}\circ\Phi_{1}=0$, and so we have a sequence

$0\to
K^{-1}(D_{1})\stackrel{{\scriptstyle\Phi_{1}}}{{\to}}V\stackrel{{\scriptstyle\Phi_{2}}}{{\to}}K(-D_{2})\to
0,$ (3.5)

where $D_{j}$ is the divisor of zeroes of $\Phi_{j}$. About each point
$p\in\Sigma$ we can choose a local holomorphic chart $(U,z)$ and a local
holomorphic frame $\sigma_{1},\ldots,\sigma_{n}$ for $V$ over $U$ satisfying

$\Phi(\sigma_{0})=z^{k}\sigma_{1}dz,\quad\Phi(\sigma_{n})=z^{l}dz,\quad\Phi(\sigma_{j})=0\
\text{for}\ j\neq n.$

Here $k,l$ are, respectively, the degrees of $D_{1}$ and $D_{2}$ at the point
$p$ and $\sigma_{0}$ denotes a local trivialising section of the trivial
bundle. With respect to this frame $\Phi$ is represented as a local section of
$\operatorname{End}(E)\otimes K$ by

$\xi=\begin{pmatrix}0&\ldots&0&z^{k}\\\ 0&\ddots&&0\\\ \vdots&&&\vdots\\\
0&\ldots&z^{l}&0\end{pmatrix}dz=\begin{pmatrix}0&z^{k}e_{1}\\\
z^{l}e_{n}^{t}&0\end{pmatrix}dz,$ (3.6)

where in the second expression the standard basis vectors $e_{1},\ldots,e_{n}$
for $\mathbb{C}^{n}$ have been used to write the matrix in block form. Let
$\chi$ be a local holomorphic section of $P(\mathfrak{h}^{\mathbb{C}})$,
identified with a locally holomorphic $\mathfrak{h}^{\mathbb{C}}$-valued
function. We compute

$[\xi,\chi]=\left[\begin{pmatrix}0&z^{k}e_{1}\\\
z^{l}e_{n}^{t}&0\end{pmatrix}dz,\begin{pmatrix}A&0\\\
0&a\end{pmatrix}\right]=\begin{pmatrix}0&z^{k}(aI_{n}-A)e_{1}\\\
z^{l}e_{n}^{t}(A-aI_{n})&0\end{pmatrix}dz,$ (3.7)

$a=-\operatorname{tr}(A)$. Hence $[\xi,\chi]=0$ holds over $U$ when the $2n$
equations

$(A+\operatorname{tr}(A))e_{1}=0,\quad(A^{t}+\operatorname{tr}(A))e_{n}=0,$

hold. In components these equations are

$2A_{11}+\sum_{i=2}^{n}A_{ii}=0,\ A_{j1}=0,\ A_{nk}=0,\
2A_{nn}+\sum_{i=1}^{n-1}A_{ii}=0,$

for $2\leq j\leq n$ and $1\leq k\leq n-1$. We see that this gives $2n-1$
independent equations since $A_{n1}=0$ appears twice. Hence
$\operatorname{ad}\xi$ has rank $2n-1=\dim\mathfrak{m}^{\mathbb{C}}-1$. ∎

Now we deal with the case $N=\mathbb{RH}^{n}$ and follow the description of
the Higgs bundles for this case given in [12]. These have the form $E=V\oplus
1$ where $V$ has $\det(V)=1$ and carries a non-degenerate quadratic form
$Q_{V}$. In this case $\Phi_{2}=\Phi_{1}^{\top}$ where the transpose is with
respect to $Q_{V}$. It follows that for $f$ to be an immersion both $\Phi_{j}$
vanish nowhere so both divisors $D_{1},D_{2}$ are trivial. As a smooth bundle
$V$ splits into a direct sum $K^{-1}\oplus W\oplus K$ and with respect to this
splitting we can write

$Q_{V}=\begin{pmatrix}0&0&1\\\ 0&Q_{W}&0\\\ 1&0&0\end{pmatrix}.$ (3.8)

Then $\Phi$ is represented by $\xi$ in (3.6) with $k=l=0$. For the Lie algebra
we use

$\mathfrak{so}(Q,1)=\left\\{\begin{pmatrix}A&u\\\
v^{t}&0\end{pmatrix}:A^{t}Q+QA=0,\
Qu=v\right\\}\simeq\mathfrak{so}(n+1,\mathbb{C}),$

where $A\in\operatorname{End}(\mathbb{C}^{n})$, $u,v\in\mathbb{C}^{n}$, $Q$ is
(3.8) with $Q_{W}=I_{n-2}$, and the transpose is the usual matrix transpose.
The appropriate (complexified) symmetric space decomposition is obtained from
the Cartan decomposition (2.1) above:

$\mathfrak{so}(Q,1)=\mathfrak{h}_{Q}^{\mathbb{C}}\oplus\mathfrak{m}_{Q}^{\mathbb{C}},\quad\mathfrak{h}_{Q}^{\mathbb{C}}=\mathfrak{so}(Q,1)\cap\mathfrak{h}^{\mathbb{C}},\
\mathfrak{m}_{Q}^{\mathbb{C}}=\mathfrak{so}(Q,1)\cap\mathfrak{m}^{\mathbb{C}}.$

######  Lemma 3.3.

For $G=SO_{0}(n,1)$ the map $\operatorname{ad}\Phi$ has corank one.

###### Proof.

Following the same arguments as for $\CH^{n}$ above but applied to
$\mathfrak{so}(Q,1)$ it suffices to show that the map
$\operatorname{ad}\xi:\mathfrak{h}_{Q}^{\mathbb{C}}\to\mathfrak{m}_{Q}^{\mathbb{C}}$
has corank $1$. Using the equations (3.7) but with
$\chi\in\mathfrak{m}_{Q}^{\mathbb{C}}$ we we see that the image of
$\operatorname{ad}\xi$ is isomorphic to

$\\{Ae_{1}:A^{t}=-QAQ\\}\subset\mathbb{C}^{n}.$

The symmetry $A^{t}=-AQA$ implies that $A_{n1}=0$ and this is the only
condition $Ae_{1}$ must satisfy. Therefore
$\operatorname{im}(\operatorname{ad}\xi)$ has codimension one in
$\mathfrak{m}_{Q}^{\mathbb{C}}$. ∎

To complete the proof of Theorem 1.1 we note that at regular points the
dimension of $\mathcal{I}^{-1}(0)$ is

$\displaystyle\dim\mathcal{H}(\mathcal{C}_{g},G)-\dim H^{0}(\Sigma_{c},K^{2})$
$\displaystyle=$
$\displaystyle\dim(\mathcal{T}_{g})+\dim(\mathcal{R}(\pi_{1}\Sigma,G))-\dim
H^{0}(\Sigma_{c},K^{2})$ $\displaystyle=$
$\displaystyle\dim(\mathcal{R}(\pi_{1}\Sigma,G).$

Here $\dim(\mathcal{R}(\pi_{1}\Sigma,G))$ means the dimension of the smooth
open subvariety of indecomposable representations. This has real dimension
$(2g-2)\dim(G)$.

######  Remark 3.1.

Theorem 1.1 fills a small gap in the proof of Theorem 4.4 of [12], which gives
a smooth parametrisation of the connected components of
$\mathcal{M}(\Sigma,\mathbb{RH}^{4})$. The proof that the parametrisation is a
local diffeomorphism needs the fact the $\mathcal{M}(\Sigma,\mathbb{RH}^{4})$
is non-singular.

##  4\. Limit points of the $\mathbb{C}^{\times}$-action for $n=2$.

It is well-known that $\mathbb{C}^{\times}$ acts on $G$-Higgs bundles by
$t\cdot(E,\Phi)=(E,t\Phi)$, $t\in\mathbb{C}^{\times}$, and that the fixed
points of this action are Hodge bundles, i.e., one can write
$E=\oplus_{j=1}^{m}E_{j}$ so that $\Phi:E_{j}\to E_{j+1}\otimes K$ with
$E_{m+1}=0$. The integer $m$ is called the _length_ of the Hodge bundle. It is
also well-known that these fixed points are exactly the critical points of the
Higgs field energy $\mathfrak{E}(E,\Phi)=\|\Phi\|^{2}_{L^{2}}$. Given one of
the connected components $C$ of $\mathbb{C}^{\times}$-fixed points define

$\mathcal{S}_{C}=\\{(E,\Phi):\lim_{t\to 0}(E,t\Phi)\in
C\\},\quad\mathcal{U}_{C}=\\{(E,\Phi):\lim_{t\to\infty}(E,t\Phi)\in C\\}.$

By a theorem of Kirwan these agree with, respectively, the stable and unstable
manifolds of $C$ along the downwards Morse flow (gradient flow of
$-\operatorname{grad}\mathfrak{E}$). One knows that $\cup_{C}\mathcal{S}_{C}$
equals $\mathcal{H}(\Sigma_{c},G)$ while $\cup_{C}\mathcal{U}_{C}$ equals the
nilpotent cone $\mathcal{N}^{c}$.

Our aim in this section is to identify the limit points as $t\to\infty$ and
$t\to 0$ inside $\overline{\mathcal{M}}(\Sigma,\CH^{2})$.

First we note that if $\Phi_{j}=0$ for either $j=1$ or $j=2$, the pair
$(E,\Phi)$ is plainly a Hodge bundle of length $2$, hence a
$\mathbb{C}^{\times}$-fixed point. So our interest is when neither $\Phi_{j}$
vanishes identically. In this case, using (2.4), as a $C^{\infty}$-bundle

$V\simeq_{C^{\infty}}K^{-1}(D_{1})\oplus K(-D_{2}).$ (4.1)

The $\bar{\partial}$-operator for the holomorphic structure of $V$ can be
written with respect to this smooth splitting as

$\bar{\partial}_{V}=\begin{pmatrix}\bar{\partial}_{1}&\beta\\\
0&\bar{\partial}_{2}\end{pmatrix},$ (4.2)

where $\beta$ is a smooth $(0,1)$-form taking values in
$\operatorname{Hom}(K(-D_{2}),K^{-1}(D_{1}))$): its Dolbeault cohomology class
$[\beta]$ corresponds to the extension class in $H^{1}(K^{-2}(D_{1}+D_{2}))$
which determines the extension (2.4). When $[\beta]=0$ it is easy to check
that this gives a Hodge bundle [11], which we will denote by
$(E^{\infty},\Phi^{\infty})$. Here $E^{\infty}=V^{\infty}\oplus 1$ where
$V^{\infty}$ is the holomorphically split bundle (4.1) and

$\Phi^{\infty}_{1}:K^{-1}(D_{1})\to
V^{\infty},\quad\Phi^{\infty}_{2}:V^{\infty}\to K(-D_{2}),$

are the natural inclusion and projection respectively. Notice that
$\Phi^{\infty}=\Phi$ as a map on smooth bundles.

######  Lemma 4.1.

For $(E,\Phi)\in\mathcal{V}$,
$\lim_{t\to\infty}(E,t\Phi)=(E^{\infty},\Phi^{\infty})$.

The proof is essentially the same as the proof of Prop. 4.9 in [12].

###### Proof.

We perform a gauge transformation. Set

$g_{t}=\begin{pmatrix}t&0&0\\\ 0&t^{-1}&0\\\ 0&0&1\end{pmatrix}.$ (4.3)

Then $g_{t}\Phi g_{t}^{-1}=t\Phi$ while

$g_{t}^{-1}\bar{\partial}_{E}g_{t}=\begin{pmatrix}\bar{\partial}_{1}&t^{-2}\beta&0\\\
0&\bar{\partial}_{2}&0\\\ 0&0&0&\bar{\partial}\end{pmatrix}.$ (4.4)

Let $E^{t}$ denote the holomorphic bundle for this $\bar{\partial}$-operator.
Then $(E,t\Phi)\simeq(E^{t},\Phi)$ hence

$\lim_{t\to\infty}(E,t\Phi)=\lim_{t\to\infty}(E^{t},\Phi)=(E^{\infty},\Phi^{\infty}).$

∎

Recall that in [11] the data $(c,D_{1},D_{2},[\beta])$ is used to parametrise
the component $\mathcal{V}(d_{1},d_{2})$ by fixing the degrees of the
divisors. Let $\overline{\mathcal{V}}(d_{1},d_{2})$ denote the closure of this
in $\overline{\mathcal{V}}$, i.e., where $D_{1},D_{2}$ can have points in
common. By [11] this is a complex analytic family over Teichmüller space: let
$\overline{\mathcal{V}}_{c}(d_{1},d_{2})$ denote the fibre over
$c\in\mathcal{T}_{g}$. The previous lemma shows that, under the map from
$\overline{\mathcal{V}}_{c}(d_{1},d_{2})$ to Higgs bundles over $\Sigma_{c}$,
$(c,D_{1},D_{2},[t^{-2}\beta])$ maps to $(E,t\Phi)$ and therefore the limit as
$t\to\infty$ is given by the trivial extension for $V$. Given that this
parametrisation of Higgs bundles is smooth (which is proved in Lemma 5.1
below) we have the following corollary, which was conjectured in [11].

######  Corollary 4.2.

$\overline{\mathcal{V}}_{c}(d_{1},d_{2})$ is diffeomorphic to the unstable
manifold $\,\mathcal{U}_{C}$ where $C$ is the critical manifold of Hodge
bundles for which $V=K^{-1}(D_{1})\oplus K(-D_{2})$ (i.e., the trivial
extension).

Now we turn to limits as $t\to 0$. For a stable Higgs bundle $(E,\Phi)$ with
Toledo invariant $\tau$ with $E=V\oplus 1$ we note that
$\tau=-\tfrac{2}{3}\deg(V)$ in our convention. Since the dual Higgs bundle has
opposite Toledo invariant we may assume that $\tau\geq 0$.

######  Proposition 4.3.

Let $(E,\Phi)$ be a stable Higgs bundle with $\tau\geq 0$ and let
$(E^{0},\Phi^{0})=\lim_{t\to 0}(E,t\Phi)$. Suppose $(E,\Phi)$ is not a Hodge
bundle and therefore is given by a non-trivial extension of the form (2.4) for
effective divisors $D_{1},D_{2}$. Then exactly one of the following holds:

  1. (i)

$E$ is a semistable bundle, in which case $\tau=0$ and
$(E^{0},\Phi^{0})=(Gr(E),0)$, where $Gr(E)$ is the associated graded bundle
arising from a Jordan-Hölder filtration of $E$.

  2. (ii)

$\tau>0$ and every line subbundle $L$ of $V$ satisfies
$\deg(L)<\tfrac{1}{3}\deg(V)$, in which case $(E^{0},\Phi^{0})=(E,\Phi_{1})$
is a length two Hodge bundle.

  3. (iii)

$\tau>0$, and the maximal destabilizing line subbundle $L$ of $V$ has
$\deg(L)=\tfrac{1}{3}\deg(V)$. In this case there is a positive divisor $D$
for which $L\simeq K(-D_{2}-D)$, $V/L\simeq K^{-1}(D_{1}+D)$ and
$(E^{0},\Phi^{0})$ is the polystable Higgs bundle

$(K^{-1}(D_{1}+D)\oplus 1,\varphi)\oplus(K(-D_{2}-D),0),$ (4.5)

where $\varphi$ is the projection of $\Phi_{1}:1\to V\otimes K$ to the
quotient.

  4. (iv)

$\tau\geq 0$, $V$ is unstable and its maximal destabilizing line subbundle $L$
has $\deg(L)>\tfrac{1}{3}\deg(V)$. In this case there is a positive divisor
$D$ so that $L,V/L$ have the isomorphisms in (iii) but now $(E^{0},\Phi^{0})$
is a length three Hodge bundle of the form

$(K^{-1}(D_{1}+D)\oplus K(-D_{2}-D)\oplus 1,\Phi^{0}),$ (4.6)

where $\Phi^{0}_{1}$ is the projection of $\Phi_{1}$ to $(V/L)\otimes K$, and
$\Phi^{0}_{2}$ is the restriction of $\Phi_{2}$ to $L$.

In case (i) the limit corresponds to a constant harmonic map. In all other
cases except (ii) with $d_{1}=0$ (i.e., $\Phi_{1}$ has no zeroes) the limit
corresponds to a branched minimal surface. In case (iii) these are branched
holomorphic maps to a totally geodesic $\CH^{1}$ in $\CH^{2}$.

###### Proof.

(i) If $E=V\oplus 1$ is a semistable bundle then
$0=\deg(1)\leq\tfrac{1}{3}\deg(V)$. By assumption $\tau\geq 0$, hence
$\tau=0$. Now $(E,0)$ is $S$-equivalent to $(Gr(E),0)$ [13], which is
polystable and hence the limit in moduli space. In particular, note that
either $V$ is polystable and $E=Gr(E)$, or $V$ is strictly semistable and a
non-trivial extension of its maximal degree line subbundle $L$ of degree $0$.
Then a Jordan-Hölder filtration for $E$ is $L\subset L\oplus 1\subset E$ and
$Gr(E)=L\oplus 1\oplus V/L$.

(ii) In this case write the holomorphic structure and Higgs field for
$(E,t\Phi)$ in block decomposition with respect to $E=V\oplus 1$:

$\bar{\partial}_{E}=\begin{pmatrix}\bar{\partial}_{V}&0\\\
0&\bar{\partial}\end{pmatrix},\quad\Phi=\begin{pmatrix}0&t\Phi_{1}\\\
t\Phi_{2}&0\end{pmatrix}.$

A simple computation shows that for

$g_{t}=\begin{pmatrix}tI_{V}&0\\\ 0&1\end{pmatrix}$

we have

$\lim_{t\to
0}g_{t}^{-1}\bar{\partial}_{E}g_{t}=\bar{\partial}_{E},\quad\lim_{t\to
0}g_{t}^{-1}t\Phi g_{t}=\begin{pmatrix}0&\Phi_{1}\\\ 0&0\end{pmatrix}.$

This represents the limit provided $(E,\Phi_{1})$ is stable as a Higgs bundle.
The $\Phi_{1}$-invariant subbundles of $E$ are
$1\oplus\operatorname{im}\Phi_{1}$, $V$ and any line subbundle of $V$. Since
$(E,\Phi)$ is stable as a Higgs bundle and $\deg(V)<0$ the first two already
satisfy the slope inequality, as does $\operatorname{im}\Phi_{1}$. So the
additional condition is that every other line subbundle $L$ of $V$ satisfies
$\deg(L)<\tfrac{1}{3}\deg(V)$. Note that the case $\tau=0$ is covered by part
(i).

(iii) First we note that $(V/L\oplus 1,\varphi)$ is a stable Higgs bundle,
since $V/L$ is the only $\varphi$-invariant proper subbundle and
$\deg(V/L)=\tfrac{2}{3}\deg(V)<0$ hence $\deg(V/L)<\tfrac{1}{2}\deg(V/L)$.
Also $\tfrac{1}{2}\deg(V/L)=\deg(L)$ hence (4.5) is a polystable Higgs bundle.
We also note that the restriction of $\Phi_{2}$ to $L$ is a holomorphic
section of $L^{-1}\otimes K(-D_{2})$ and this must have zeroes otherwise the
extension (2.4) splits. Let $D>0$ be the divisor of these zeroes, then
$L\simeq K(-D_{2}-D)$ and since $\det(V)\simeq\mathcal{O}(D_{1}-D_{2})$ it
follows that $V/L\simeq K^{-1}(D_{1}+D)$.

Now $V$ can be written as an extension of the form

$0\to L\to V\to V/L\to 0.$ (4.7)

This gives a $C^{\infty}$-isomorphism $E\simeq V/L\oplus L\oplus 1$. With
respect to such a decomposition we can write

$\bar{\partial}_{E}=\begin{pmatrix}\bar{\partial}_{V/L}&0&0\\\
\beta&\bar{\partial}_{L}&0\\\
0&0&\bar{\partial}\end{pmatrix},\quad\Phi=\begin{pmatrix}0&0&\phi_{13}\\\
0&0&\phi_{23}\\\ \phi_{31}&\phi_{32}&0\end{pmatrix}.$

Using the gauge transformation

$g_{t}=\begin{pmatrix}t&0&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix}.$

a straighforward calculation gives

$g_{t}^{-1}\bar{\partial}_{E}g_{t}=\begin{pmatrix}\bar{\partial}_{V/L}&0&0\\\
t\beta&\bar{\partial}_{L}&0\\\ 0&0&\bar{\partial}\end{pmatrix},\quad
g_{t}^{-1}(t\Phi)g_{t}=\begin{pmatrix}0&0&\phi_{13}\\\ 0&0&t\phi_{23}\\\
t^{2}\phi_{31}&t\phi_{32}&0\end{pmatrix}.$

Setting $t=0$ gives (4.5). Note that $\varphi=\varphi_{13}$ is the projection
of $\Phi_{1}:K^{-1}(D_{1})\to V$ onto $V/L$ and has divisor of zeroes
$D_{1}+D$ since it vanishes when either $\Phi_{1}=0$ or at the support of
$\operatorname{im}\Phi_{1}\cap L$. But $\Phi_{2}\circ\Phi_{1}=0$ so the
support of $D$ is all the points at which $\operatorname{im}\Phi_{1}$ is not
zero but lies in $L$.

(iv) As in the previous case we first show that the proposed limit (4.6) is a
stable Higgs bundle. Since the only proper $\Phi^{0}$ invariant subbundles are
$V/L$ and $V/L\oplus 1$, and since $\deg(V)\leq 0$, the stability condition is
the single inequality $\tfrac{1}{2}\deg(V/L)<\tfrac{1}{3}\deg(V)$, i.e.,
$\deg(L)>\tfrac{1}{3}\deg(V)$. Just as in part (iii) we note that $L\simeq
K(-D_{2}-D)$ where $D>0$ is the divisor of zeroes of $\Phi_{2}$ restricted to
$L$. Hence $V/L\simeq K^{-1}(D_{1}+D)$.

Now to show this gives the limit we use the same argument as part (iii) but
with

$g_{t}=\begin{pmatrix}t^{2}&0&0\\\ 0&1&0\\\ 0&0&t\end{pmatrix}.$

In this case

$g_{t}^{-1}\bar{\partial}_{E}g_{t}=\begin{pmatrix}\bar{\partial}_{V/L}&0&0\\\
t^{2}\beta&\bar{\partial}_{L}&0\\\ 0&0&\bar{\partial}\end{pmatrix},\quad
g_{t}^{-1}(t\Phi)g_{t}=\begin{pmatrix}0&0&\phi_{13}\\\ 0&0&t^{2}\phi_{23}\\\
t^{2}\phi_{31}&\phi_{32}&0\end{pmatrix}.$

Setting $t=0$ gives (4.6) with $\Phi^{0}_{1}=\varphi_{13}$, the projection of
$\Phi_{1}$ onto $(V/L)\otimes K$, and $\Phi^{0}_{2}=\varphi_{32}$, the
restriction of $\Phi_{2}$ to $L$. The latter has zeroes $D_{2}+D$ by
definition, and the former has zeroes $D_{1}+D$ for the same reason as in part
(iii). ∎

Note that this argument works perfectly well even when $D_{1},D_{2}$ already
have points in common (although $D$ will not be the full branch point divisor
in that case), and it tells us something about singularities of the nilpotent
cone. When $C$ consists of length three Hodge bundles $\mathcal{U}_{C}$ is
smooth and of dimension $5(g-1)$: this follows either from the Morse index
calculation of Gothen [6, Prop 3.2] or the dimension count in [11] for
$\overline{\mathcal{V}}_{c}(d_{1},d_{2})$, given Corollary 4.2. Any point on
$C$ which is a limit point as $t\to 0$ (i.e., from some
$\mathcal{U}_{C^{\prime}}$ for $C^{\prime}\neq C$) lies on both the stable and
unstable manifolds of $C$, which are transverse, and therefore the tangent
space at this point has dimension greater than $\mathcal{U}_{C}$. Thus such
limit points are singular points of the nilpotent cone. We can ask whether
every length three Hodge bundle of the form (4.6) (i.e., with common divisor
$D$) is such a singular point. The answer is no: by the following result there
are many such Hodge bundles which are not limit points. Recall from [11] that
the necessary and sufficient conditions for (4.6) to give a stable Higgs
bundle are the inequalities

$0\leq 2d_{1}^{\prime}+d_{2}^{\prime}<6(g-1),\quad 0\leq
d_{1}^{\prime}+2d_{2}^{\prime}<6(g-1),$ (4.8)

where $d_{j}^{\prime}=\deg(D_{j}+D)$.

######  Proposition 4.4.

Let $D_{1},D_{2},D$ be effective divisors, $D>0$, set $D_{j}^{\prime}=D_{j}+D$
and suppose their degrees $d_{1}^{\prime},d_{2}^{\prime}$ satisfy
$d_{1}^{\prime}\leq d_{2}^{\prime}$ and
$d_{1}^{\prime}+d_{2}^{\prime}<2(g-1)$. Then the Hodge bundle with $V$ given
by (4.6) does not lie at the limit as $t\to 0$ of a
$\mathbb{C}^{\times}$-orbit in a different unstable manifold.

###### Proof.

First, the two inequalities imply that (4.6) corresponds to a stable Higgs
bundle with $\tau\geq 0$. By Prop. 4.3 such a limit can only be obtained from
case (iv), where $V$ has unique maximal destabilizing line subbundle $L\simeq
K(-D_{2}^{\prime})$ and $V/L\simeq K^{-1}(D_{1}^{\prime})$. The existence of
this requires

$1\leq\dim H^{1}(K^{2}(-D_{1}^{\prime}-D_{2}^{\prime}))=\dim
H^{0}(K^{-1}(D_{1}^{\prime}+D_{2}^{\prime})),$

using Serre duality. But the right hand dimension is zero whenever
$\deg(K^{-1}(D_{1}^{\prime}+D_{2}^{\prime}))<0$, i.e., when
$d_{1}^{\prime}+d_{2}^{\prime}<2(g-1)$. ∎

##  5\. Connected components of $\mathcal{M}(\Sigma,\CH^{2})$.

Our aim is to prove Theorem 1.2. First we must recall some more facts from
[11]. From [11] $\mathcal{V}$ is a disjoint union
$\cup_{(d_{1},d_{2})}\mathcal{V}(d_{1},d_{2})$ where the non-negative integers
$d_{1},d_{2}$ satisfy the inequalities (4.8). Points of
$\mathcal{V}(d_{1},d_{2})$ are parametrised by the data $(c,D_{1},D_{2},\xi)$
where $D_{1},D_{2}$ are effective divisors of degree $d_{1},d_{2}$ and $\xi$
denotes the extension class for the extension (2.4). This class can be freely
chosen in $H^{1}(\Sigma_{c},K^{-2}(D_{1}+D_{2}))$ and this parametrisation
gives $\mathcal{V}(d_{1},d_{2})$ the structure of a complex analytic family
over $c\in\mathcal{T}_{g}$. The fibre over $c\in\mathcal{T}_{g}$, which we
will denote by $\mathcal{V}_{c}(d_{1},d_{2})$, is the holomorphic vector
bundle over

$\\{(D_{1},D_{2})\in S^{d_{1}}\Sigma_{c}\times S^{d_{2}}\Sigma_{c}:D_{1}\cap
D_{2}=\emptyset\\}$

with fibre $H^{1}(\Sigma_{c},K^{-2}(D_{1}+D_{2}))$.

We need to show first that this parametrisation is smooth with respect to the
smooth structure of $\mathcal{M}(\Sigma,\CH^{2})$ given above, which uses its
embedding in $\mathcal{T}_{g}\times\mathcal{R}(\pi_{1}\Sigma,G)$. Since we are
always dealing with points which are smooth in the latter we can use the
nonabelian Hodge correspondence to identify the smooth locus of
$\mathcal{R}(\pi_{1},G)$ with the smooth locus of Higgs bundle moduli space
$\mathcal{H}(\Sigma_{c},G)$. It therefore suffices to prove the following
lemma.

######  Lemma 5.1.

For each fixed $c\in\mathcal{T}_{g}$ we get a holomorphic embedding

$\mathcal{Z}:\overline{\mathcal{V}}_{c}(d_{1},d_{2})\to\mathcal{H}(\Sigma_{c},G),$

by assigning to $(D_{1},D_{2},\xi)$ the Higgs bundle $(E,\Phi)$ for which
$E=V\oplus 1$ where $V$ and $\Phi$ arise from the extension $\xi$ using (2.4).

We know from [11] that $\mathcal{Z}$ is one-to-one, so it suffices to show
that it is a holomorphic immersion.

Before we begin the proof we need to fix a convention for which way to
represent $1$-cocycles with values in a vector bundle. For a vector bundle $F$
with model fibre $A$ and local transition relations $\psi_{i}=a_{ij}\psi_{j}$
between local trivialisations $\psi_{i}$ over a Leray cover $\\{U_{i}\\}$, let
$(\xi_{ij},U_{i},U_{j})$ be a $1$-cocycle for a class in $H^{1}(F)$, i.e.,
$\xi_{ij}\in\Gamma(U_{i}\cap U_{j},F)$ satisfy $\xi_{ij}+\xi_{jk}=\xi_{ik}$.
Then we choose to represent this by the local functions

$c_{ij}=\psi_{i}(\xi_{ij}):U_{i}\cap U_{j}\to A,$

and the $1$-cocycle conditions are equivalent to

$c_{ij}+a_{ij}c_{jk}=c_{ik}.$

Note that the opposite convention, to use $\psi_{j}(\xi_{ij})$ instead, is
used in Gunning [7]. In what follows we need this for $H^{1}(\mathcal{A}^{0})$
and $H^{1}(K^{-2}(D_{1}+D_{2}))$.

###### Proof.

To describe the map $\mathcal{Z}$ concretely about a point
$(D_{1},D_{2},\xi)$, let $P_{1},\ldots,P_{d}$ be the points in the support of
$D_{1}+D_{2}$ and let $U_{0},\ldots,U_{d}$ be the Leray cover of $\Sigma_{c}$
for which each $U_{j}$ for $j\geq 1$ is an open disc about $P_{j}$, with
$U_{j}\cap U_{k}=\emptyset$ for $j\neq k$, and $U_{0}\cap U_{j}$ is an annulus
excluding an open disc about $P_{j}$. Since $V$ is determined by the extension
class $\xi$, $V$ has $1$-cocycle $g=\\{(g_{0j},U_{0},U_{j}):1\leq j\leq d\\}$
given by

$g_{0j}=\begin{pmatrix}\alpha_{0j}&\lambda_{0j}\\\ 0&\beta_{0j}\end{pmatrix}$
(5.1)

where

$\alpha_{0j}=z^{-n_{j}}\frac{dz_{0}}{dz_{j}},\quad\beta_{0j}=z_{j}^{m_{j}}\frac{dz_{j}}{dz_{0}},\quad\lambda_{0j}=\beta_{0j}\xi_{0j}dz_{0}^{2}.$

Here, for $j\geq 1$, $z_{j}$ is a local parameter in $U_{j}$ centred at
$P_{j}$ and $n_{j}$ is the degree of $D_{1}$ at $P_{j}$ while $m_{j}$ is the
degree of $D_{2}$ at $P_{j}$. By $dz_{0}$ we simply mean a non-vanishing
holomorphic $1$-form on $U_{0}$. By $\xi_{0j}$ we mean the local section of
$K^{-2}(D_{1}+D_{2})$ over $U_{0}\cap U_{j}$ which comprises a $1$-cocycle
representing the extension class $\xi$.

The $1$-cocycle $g$ determines a trivialisation $\chi_{j}$ of $V$, and dual
$\chi^{*}_{j}$ of $V^{*}$, over $U_{j}$. When we think of the Higgs field
components as $\Phi_{1}\in H^{0}(V\otimes K)$ and $\Phi_{2}\in
H^{0}(V^{*}\otimes K)$ these satisfy

$\chi_{0}(\Phi_{1})=\begin{pmatrix}dz_{0}\\\
0\end{pmatrix},\quad\chi^{*}_{0}(\Phi_{2})=\begin{pmatrix}0&dz_{0}\end{pmatrix},$

and for $j\geq 1$

$\chi_{j}(\Phi_{1})=\begin{pmatrix}z_{j}^{n_{j}}dz_{j}\\\
0\end{pmatrix},\quad\chi^{*}_{j}(\Phi_{2})=\begin{pmatrix}0&z_{j}^{m_{j}}dz_{j}\end{pmatrix}.$
(5.2)

An open neighbourhood around $D_{1}\in S^{d_{1}}\Sigma_{c}$ (respectively
$D_{2}\in S^{d_{2}}\Sigma_{c}$) is determined by collection of monic
polynomials $z_{j}^{n_{j}}+u_{j}(z_{j})$ (respectively
$z_{j}^{m_{j}}+v_{j}(z_{j})$) whose zeroes lie in $U_{j}$. Note that if
$P_{j}$ is not in the support of $D_{1}$ then $n_{j}=0$ and $u_{j}$ is the
zero polynomial (and likewise for $m_{j}$, $v_{j}$ when $P_{j}$ is not in the
support of $D_{2}$). The coefficients of the polynomials
$u_{j}(z_{j}),v_{j}(z_{j})$ provide a local chart about $(D_{1},D_{2})$. At a
pair $(D_{1}^{\prime},D_{2}^{\prime})$ in this neighbourhood of
$(D_{1},D_{2})$ any $\xi^{\prime}\in
H^{1}(K^{-2}(D_{1}^{\prime}+D_{2}^{\prime}))$ can be represented by a
$1$-cocycle $\\{(\xi^{\prime}_{0j},U_{0},U_{j})\\}$. The map $\mathcal{Z}$
maps $(D_{1}^{\prime},D_{2}^{\prime},\xi^{\prime})$ to the Higgs bundle
constructed as above but with $z_{j}^{n_{j}}$ replaced by
$z_{j}^{n_{j}}+u_{j}(z_{j})$, $z_{k}^{m_{k}}$ replaced by
$z_{k}^{m_{k}}+v_{k}(z_{k})$, and $\xi_{0j}$ replaced by $\xi^{\prime}_{0j}$.
It is easy to see that smooth variations of these parameters result in smooth
variations of the Higgs bundle.

Our aim now is to show that $d\mathcal{Z}$ has trivial kernel at each point.
First we describe the tangent space to
$\overline{\mathcal{V}}_{c}(d_{1},d_{2})$ at a given point. Since
$\overline{\mathcal{V}}_{c}(d_{1},d_{2})$ is a vector bundle over
$S^{d_{1}}\Sigma_{c}\times S^{d_{2}}\Sigma_{c}$ we can fix a local
trivialisation over a neighbourhood of $(D_{1},D_{2},\xi)$ and identify the
tangent space at that point with

$T_{D_{1}}S^{d_{1}}\Sigma_{c}\oplus T_{D_{2}}S^{d_{2}}\oplus
H^{1}(K^{-2}(D_{1}+D_{2})).$

We will write elements of this in the form $(w,y,\eta)$. Further, the vector
$w$ (and similarly $y$) can be described as follows. It is well-known that

$T_{D_{1}}S^{d_{1}}\Sigma_{c}=\oplus_{j}(\mathfrak{m}_{P_{j}}/\mathfrak{m}_{P_{j}}^{n_{j}+1}),$

where $\mathfrak{m}_{P_{j}}\subset\mathcal{O}_{P_{j}}$ is the maximal ideal of
locally holomorphic functions about $P_{j}$ which vanish at $P_{j}$. We will
identify

$\mathfrak{m}_{P_{j}}/\mathfrak{m}_{P_{j}}^{n_{j}+1}\simeq\\{z_{j}w_{j}(z_{j}):w_{j}\in\mathbb{C}[z_{j}],\
\deg(w_{j})\leq n_{j}-1\\},$

and therefore write $w\in T_{D_{1}}S^{d_{1}}\Sigma_{c}$ as a tuple
$w=(w_{1},\ldots,w_{k})$ of polynomials. This is tangent to the curve
$D_{1}(t)$ on $S^{d_{1}}\Sigma_{c}$ for which $D_{1}(t)$ is the divisor of
zeroes given by the locally defined polynomial $z_{j}^{n_{j}}+tw_{j}(z_{j})$
in $U_{j}$ (for $t$ sufficiently close to zero).

Given such a tangent vector $(w,y,\xi)$, let $(D_{1}(t),D_{2}(t),\xi(t))$ be
the curve it is tangent to, obtained as above using $u_{j}=tw_{j}$,
$v_{j}=ty_{j}$ and $\xi(t)=\xi+t\eta$ (using the local trivialisation of
$\overline{\mathcal{V}}_{c}(d_{1},d_{2})$). Let
$\mathcal{Z}(t)=(E(t),\Phi(t))$ denote the image curve, with $E(t)=V(t)\oplus
1$, and let $g_{0j}(t)$ denote the transition functions for $V(t)$, in the
form (5.1). These have entries

$\alpha_{0j}(t)=\frac{1}{z^{n_{j}}+tw_{j}}\frac{dz_{0}}{dz_{j}},\quad\beta_{0j}(t)=(z^{m_{j}}+ty_{j})\frac{dz_{j}}{dz_{0}},\quad\lambda_{0j}(t)=\beta_{0j}\xi(t)_{0j}dz_{0}^{2}.$

In particular, $\chi_{0}(\Phi_{1})$, $\chi^{*}_{0}(\Phi_{2})$ are time-
independent. The derivative $\mathcal{Z}^{\prime}(0)$ is represented by a
hypercohomology class in

$\mathbb{H}^{1}(\mathcal{A}^{*})\simeq\frac{\\{(A,B)\in\mathcal{C}^{1}(\mathcal{A}^{0})\oplus\mathcal{C}^{0}(\mathcal{A}^{1}):\delta
A=0,\ \delta B=[\Phi,A]\\}}{\\{(\delta
C,[\Phi,C]):C\in\mathcal{C}^{0}(\mathcal{A}^{0})\\}},$

following [2]. Specifically, it is the class of a pair $(A,B)$ for which $A$
is represented by the $1$-cocyle

$g^{\prime}(0)g^{-1}=\begin{pmatrix}\alpha^{\prime}(0)\alpha^{-1}&-\lambda\alpha^{\prime}(0)\alpha^{-1}\beta^{-1}+\lambda^{\prime}(0)\beta^{-1}\\\
0&\beta^{\prime}(0)\beta^{-1}\end{pmatrix}$ (5.3)

where we have dropped the Cech cocycle subscripts for notational simplicity.
If we write $B=(B_{1},B_{2})$ where $B_{j}=\Phi_{j}^{\prime}(0)$ then explicit
calculation gives

$\chi_{0}(B_{1})=\begin{pmatrix}0\\\
0\end{pmatrix},\quad\chi^{*}_{0}(B_{2})=\begin{pmatrix}0&0\end{pmatrix},$
$\chi_{j}(B_{1})=\begin{pmatrix}w_{j}dz_{j}\\\
0\end{pmatrix},\quad\chi^{*}_{j}(B_{2})=\begin{pmatrix}0&y_{j}dz_{j}\end{pmatrix}.$
(5.4)

If $\mathcal{Z}^{\prime}(0)=0$ there must exist
$C\in\mathcal{C}^{0}(\mathcal{A}^{0})$ for which $A=\delta C$ and
$B=[\Phi,C]$. We may assume $C$ is represented by an upper triangular
$0$-cocycle of the form

$\begin{pmatrix}a_{j}&b_{j}\\\ 0&c_{j}\end{pmatrix},$

where the entries are holomorphic functions in $U_{j}$. The Lie bracket
$[\Phi,C]$ corresponds to the pair $(-C\Phi_{1},\Phi_{2}C)$ and calculation
gives

$\chi_{j}(C\Phi_{1})=\begin{pmatrix}a_{j}z^{n_{j}}dz_{j}\\\
0\end{pmatrix},\quad\chi_{j}^{*}(\Phi_{2}C)=\begin{pmatrix}0&c_{j}z^{m_{j}}dz_{j}\end{pmatrix}.$

Comparing this with (5.4), since $\deg(w_{j})<n_{j}$ and $\deg(y_{j})<m_{j}$
it follows that $B=[\Phi,C]$ if and only if both sides are zero, i.e.,
$a_{j}=0=c_{j}$ and $w_{j},y_{j}$ are both identically zero. This in turn
means that the $1$-cocycle in (5.3), which represents $A$, is

$\begin{pmatrix}0&\eta_{0j}dz_{0}^{2}\\\ 0&0\end{pmatrix}.$

Now $A=\delta C$ is the condition that the $1$-cocycle $\eta\in
H^{1}(K^{-2}(D_{1}+D_{2}))$ is trivial. Thus the kernel of $d\mathcal{Z}$ is
trivial. ∎

It follows that each $\mathcal{V}(d_{1},d_{2})$ is smooth and connected as a
submanifold of $\mathcal{V}$, and each is open since they have the same
dimension. Thus we conclude:

######  Lemma 5.2.

The smooth submanifold $\mathcal{V}\subset\mathcal{M}(\Sigma,\CH^{2})$ is a
disjoint union of connected components $\mathcal{V}(d_{1},d_{2})$.

To complete the proof of Theorem 1.2 we need two more lemmas. Both concern
$\tau>0$. The first says that $\mathcal{V}(d_{1},d_{2})$ is disconnected from
$\mathcal{W}_{\tau}$ when $d_{1}\neq 0$. The second says
$\mathcal{V}(0,d_{2})$ is connected to $\mathcal{W}_{\tau}$.

######  Lemma 5.3.

For $\tau>0$,
$\overline{\mathcal{V}}(d_{1},d_{2})\cap\mathcal{W}_{\tau}=\emptyset$ when
$d_{1}\neq 0$.

###### Proof.

It suffices to prove that this is true for each fixed conformal structure $c$.
So fix $c$ and $\tau>0$, and let $C$ be the critical manifold of minima for
the Higgs field energy $\mathfrak{E}=\|\Phi\|_{L^{2}}^{2}$ for this value of
$\tau$. Let $C_{0}\subset C$ be the open subset of Higgs bundles for which
$\Phi$ has no zeroes. Let $C^{\prime}$ be the critical manifold of
$\mathfrak{E}$ for which $\overline{\mathcal{V}}_{c}(d_{1},d_{2})$ corresponds
to $\overline{\mathcal{U}}_{C^{\prime}}$, the closure of the unstable manifold
of the downward gradient flow of $\mathfrak{E}$ (equally, the unstable
manifold for the $\mathbb{C}^{\times}$-action). The assertion is that
$\mathcal{U}_{C^{\prime}}\cap C_{0}=\emptyset$ whenever $d_{1}\neq 0$. Suppose
$(E,\Phi)\in\overline{\mathcal{U}}_{C^{\prime}}\cap C_{0}$, then it is the
limit of a sequence $(E_{k},\Phi_{k})$ of Higgs bundles in
$\mathcal{U}_{C^{\prime}}$. We may assume without loss of generality that each
$\|\Phi_{k}\|_{L^{2}}^{2}$ lies below the penultimate critical value of
$\mathfrak{E}$, and therefore all $(E_{k},\Phi_{k})$ lie in the stable
manifold $\mathcal{S}_{C}$. Taking the limit as $t\to 0$ is a continuous map
(since $\pi:\mathcal{S}_{C}\to C$ is a vector bundle) and therefore

$(E,\Phi)=\lim_{k\to\infty}(E_{k}^{0},\Phi_{k}^{0}),$

where $(E_{k}^{0},\Phi_{k}^{0})=\lim_{t\to 0}(E_{k},t\Phi_{k})$. By Prop. 4.3
each $\Phi_{k}^{0}$ has zeroes unless $d_{1}=0$. Therefore $(E,\Phi)$ lies in
the closed subset $C\setminus C_{0}$. ∎

######  Lemma 5.4.

For every choice of $0<d_{2}<3(g-1)$ there is a Higgs bundle $(E,\Phi)$ in
$\mathcal{V}(0,d_{2})$ for which $\lim_{t\to 0}(E,t\Phi)$ lies in
$\mathcal{W}_{\tau}$ for $\tau=\tfrac{2}{3}d_{2}$.

###### Proof.

Since $d_{1}=0$ the Higgs bundle $(E,\Phi)$ must correspond to an extension of
the form

$0\to
K^{-1}\stackrel{{\scriptstyle\Phi_{1}}}{{\to}}V\stackrel{{\scriptstyle\Phi_{2}}}{{\to}}K(-D_{2})\to
0,$

where $\deg(D_{2})=d_{2}$. By Prop. 4.3(ii) it suffices to show we can find
such an extension for which $V$ is a stable bundle. This is equivalent to the
existence of an extension

$0\to 1\to\lambda\to K^{2}(-D_{2})\to 0,$

for which $s(\lambda)=\deg(\lambda)-2\max\deg(L)\geq 1$, where the maximum is
taken over all line subbundles of $\lambda$. Note that such extensions are
parametrised by the space $H^{1}(K^{-2}(D_{2}))\simeq
H^{0}(K^{3}(-D_{2}))^{*}$. Such a situation is covered by [10, Prop 1.1].
Specifically, define $d=\deg(K^{2}(-D_{2}))=4(g-1)-d_{2}$ and let $s=1$ when
$d$ is odd and $s=2$ when $d$ is even. This satisfies the conditions $4-d\leq
s\leq d$ required to apply [10, Prop 1.1] (in particular note that the
critical case $4-d=d$ can only occur for $d=2$, in which case $4-d=s=d$). Then
by [10, Prop 1.1] $s(\lambda)\geq s$ if and only if the secant variety

$\operatorname{Sec}_{\tfrac{1}{2}(d+s-2)}(\Sigma_{c})\subseteq\mathbb{P}H^{0}(K^{3}(-D_{2}))^{*},$

is a proper subvariety. Since $\deg(K^{3}(-D_{2}))>2(g-1)$ the Riemann-Roch
theorem gives the dimension of $\mathbb{P}H^{0}(K^{3}(-D_{2}))^{*}$ as
$5(g-1)-d_{2}-1=d+g-2$. By [9] this secant variety has dimension $d+s-3\leq
d-1$ is therefore a proper subvariety for $g\geq 2$. ∎

## References

  * [1] D Alessandrini & B Collier, The geometry of maximal components of the $PSp(4,\mathbb{R})$ character variety, Geom. Topol. 23 (2019), no. 3, 1251-1337.
  * [2] I Biswas & S Ramanan, An infinitesimal study of the moduli of Hitchin pairs, J. London Math. Soc., 49 (1994), 219-231.
  * [3] S B Bradlow, O Garcia-Prada & P B Gothen, Surface group representations and $U(p,q)$-Higgs bundles, J. Differential Geom. 64 (2003), 111-170.
  * [4] K Corlette, Flat $G$-bundles with canonical metrics, J. Diff. Geom. 28 (1988), 361-382.
  * [5] S K Donaldson, Twisted harmonic maps and the self-duality equations, Proc. London Math. Soc. (3) 55 (1987), 127-131.
  * [6] P B Gothen, Topology of $U(2,1)$ representation spaces, Bull. London Math. Soc. 34 (2002), 729-738.
  * [7] R Gunning, Lectures on vector bundles over Riemann surfaces, Princeton University Press, 1967\.
  * [8] N Hitchin, Critical loci for Higgs bundles, Commun. Math. Phys, 366 (2019), pp841-864.
  * [9] H Lange, Higher secant varieties of curves and the theorem of Nagata on ruled surfaces, Manuscripta Math. 47 (1984), 263-269.
  * [10] H Lange & M S Narasimhan, Maximal subbundles of rank two vector bundles on curves, Math. Ann. 266 (1983), 55-72.
  * [11] J Loftin & I McIntosh, Equivariant minimal surfaces in $\CH^{2}$ and their Higgs bundles, Asian J. Math., 23 (2019), 71-106.
  * [12] J Loftin & I McIntosh, The moduli spaces of equivariant minimal surfaces in $\mathbb{RH}^{3}$ and $\mathbb{RH}^{4}$ via Higgs bundles, Geom. Ded., 201 (2018), pp325-351.
  * [13] C S Seshadri, Space of unitary vector bundles on a compact Riemann surface, Ann. Math. 85 (1967), pp303-336.