# Unstable minimal surfaces in $\mathbb{R}^{n}$ and in products of hyperbolic
surfaces

Vladimir Marković  Vladimir Marković: University of Oxford, All Souls College,
Oxford, OX1 4AL, UK<EMAIL_ADDRESS>,  Nathaniel Sagman  Nathaniel
Sagman: Caltech, 1200 E California Blvd, Pasadena, CA, 91125, USA.
<EMAIL_ADDRESS>and  Peter Smillie  Peter Smillie: Caltech, 1200 E
California Blvd, Pasadena, CA, 91125, USA<EMAIL_ADDRESS>
###### Abstract.

We prove that every unstable equivariant minimal surface in $\mathbb{R}^{n}$
produces a maximal representation of a surface group into
$\prod_{i=1}^{n}\textrm{PSL}(2,\mathbb{R})$ together with an unstable minimal
surface in the corresponding product of closed hyperbolic surfaces. To do so,
we lift the surface in $\mathbb{R}^{n}$ to a surface in a product of
$\mathbb{R}$-trees, then deform to a surface in a product of closed hyperbolic
surfaces. We show that instability in one context implies instability in the
other two.

##  1\. Introduction

###  1.1. Minimal surfaces in products of hyperbolic surfaces

Let $\Sigma_{g}$ denote a closed surface of genus $g\geq 2$ and let
$\mathbf{T}_{g}$ be the Teichmüller space of marked complex structures on
$\Sigma_{g}$. Let $(X,d)$ be the hyperbolic plane, an $\mathbb{R}$-tree, or
product thereof with an action
$\sigma:\pi_{1}(\Sigma_{g})\to\mathrm{Isom}(X,d)$. For every Riemann surface
structure $S$ on $\Sigma_{g}$, with universal cover $\tilde{S}$, and
$\sigma$-equivariant Lipschitz map $f:\tilde{S}\to(X,d),$ there is a well-
defined notion of Dirichlet energy $\mathcal{E}(S,f)$ (see Section 2 for
details). For admissible $\sigma$, there is an essentially unique
$\sigma$-equivariant harmonic map $h:\tilde{S}\to(X,d)$, which satisfies

$\mathcal{E}(S,h)=\inf_{f}\mathcal{E}(S,f).$

This gives a function $\mathbf{E}_{\sigma}:\mathbf{T}_{g}\to\mathbb{R}$, by
$\mathbf{E}_{\sigma}(S)=\mathcal{E}(S,h)$. When $S$ is a critical point of
$\mathbf{E}_{\sigma}$, we say that $h$ is minimal; if $X$ is a manifold and
$h$ is an immersion, this is equivalent to $h(S)$ being a minimal surface.

One case of interest is when $\sigma$ is a product of Fuchsian representations
into $\textrm{PSL}(2,\mathbb{R})^{n}$ (also called a maximal representation),
in which case each component of the harmonic map is a diffeomorphism, and
critical points correspond to genuine minimal surfaces in a product of
hyperbolic surfaces. The work of Schoen-Yau [22] implies that in this case,
$\mathbf{E}_{\sigma}$ is proper, and therefore admits a global minimum, which
is a stable critical point. For $n=2$, Schoen proved that this is the unique
critical point of $\mathbf{E}_{\sigma}$ [23].

However, the first author proved in [16] that uniqueness fails when $n\geq 3$,
assuming the genus $g$ is large enough. See also the paper [15], which
provides a strengthening of Schoen’s result for $n=2$. The main goal of this
paper is to show that unstable equivariant minimal surfaces in
$\mathbb{R}^{n}$ yield unstable minimal surfaces in products of hyperbolic
surfaces. In particular, this strengthens the result from [16], while
providing a simpler and more revealing proof. When $n\geq 3$, there are many
unstable equivariant minimal surfaces in $\mathbb{R}^{n};$ most notably,
unstable minimal surfaces in tori, which Meeks [18], Hass-Pitts-Rubenstein
[7], and Traizet [24] have shown to be abundant, lift to unstable equivariant
minimal surfaces in $\mathbb{R}^{n}$.

We say that a critical point of $\mathbf{E}_{\sigma}$ is unstable if there
exists a $C^{2}$ path in $\mathbf{T}_{g}$ starting at the point and at which
the second derivative of $\mathbf{E}_{\sigma}$ along the path is negative.

######  Theorem A.

Let $n\geq 3.$ For every genus $g\geq 2$, there exists a maximal
representation
$\sigma:\pi_{1}(\Sigma_{g})\to\prod_{i=1}^{n}\textrm{PSL}(2,\mathbb{R})$ such
that $\mathbf{E}_{\sigma}:\mathbf{T}_{g}\to(0,\infty)$ admits an unstable
critical point. In particular, there are at least two minimal surfaces in the
product of hyperbolic surfaces determined by $\sigma.$

Labourie conjectured that for a Hitchin representation into a simple split
real Lie group $G$ of non-compact type, there exists a unique equivariant
minimal surface in the corresponding symmetric space. Labourie proves
existence in general [12], and that uniqueness holds when the rank of $G$ is
$2$ [13] (see also [1], where Collier-Tholozan-Toulisse prove the analogous
statement for maximal representations into Hermitian Lie groups of rank $2$).
The conjecture remains open in rank at least $3$, and [16] suggests that this
is the critical case.

The key idea of the proof of Theorem A is to reduce it to finding unstable
minimal surfaces in products of $\mathbb{R}$-trees (Theorem B2 below). The
unstable minimal surfaces are provided by Theorems C and D. We explain in the
forthcoming subsections.

###  1.2. Minimal surfaces in products of $\mathbb{R}$-trees

We give the definitions about harmonic maps to $\mathbb{R}$-trees in Section
2.2. Throughout the paper, let $S$ be a Riemann surface structure on
$\Sigma_{g}$ and $\operatorname{\textrm{QD}}(S)$ the space of holomorphic
quadratic differentials on $S$. The Riemann surface structure $S$ lifts to a
Riemann surface structure on the universal cover of $\Sigma_{g}$, which we
denote $\tilde{S}$. Given a non-zero $\phi\in\operatorname{\textrm{QD}}(S)$,
there are two natural ways of producing an equivariant harmonic map. First,
the leaf space of the vertical singular foliation of the lift $\tilde{\phi}$
to $\tilde{S}$ is an $\mathbb{R}$-tree $(T_{\phi},d)$. The action of
$\pi_{1}(\Sigma_{g})$ on $\tilde{S}$ descends to an action
$\rho:\pi_{1}(\Sigma_{g})\to\mathrm{Isom}(T_{\phi},d)$ by isometries. The
quotient map $\pi:\tilde{S}\to(T_{\phi},d)$ is harmonic and
$\rho$-equivariant, with Hopf differential $\phi/4$.

On the other hand, it is proved independently by Hitchin [8], Wan [25], and
Wolf [27] that there is a unique hyperbolic structure $M_{\phi}$ on
$\Sigma_{g}$ such that the identity map from $S$ to $M_{\phi}$ is harmonic
with Hopf differential $\phi$. Moreover Wolf shows that as $t\to\infty$,
$M_{t\phi}$ converges in a certain sense to the rescaled tree $(T_{\phi},2d)$
(see [28] for the precise statement).

Now let $\phi_{1},\ldots,\phi_{n}$ be $n$ nonzero holomorphic quadratic
differentials on the same surface $S$, and let $X$ be the product of the
$\mathbb{R}$-trees $(T_{\phi_{i}},2d_{i})$ arising from the construction
above. Let $\rho:\pi_{1}(\Sigma_{g})\to\mathrm{Isom}(X)$ be the product of the
actions $\rho_{i}$ on each factor. The energy function $\mathbf{E}_{\rho}$ on
$\mathbf{T}_{g}$ associated to $\rho$ is then the sum of the energy functions
$E_{\rho_{i}}$ associated to each component. Also for each positive $t>0$, let
$M_{i}^{t}$ be the hyperbolic structures associated to $t\phi_{i}$. We set
$\mathbf{E}_{\rho}^{t}$ to be the energy functional for the product of
Fuchsian representations associated to the $M_{i}^{t}$.

$S$ is a critical point for $\mathbf{E}_{\rho}$ if and only if it is a
critical point for $\mathbf{E}_{\rho}^{t}$ for every $t>0$. In other words,
minimality of the harmonic map into the product of surfaces is equivalent to
the minimality of the equivariant harmonic map into the product of
$\mathbb{R}$-trees. The condition occurs precisely when
$\sum_{i=1}^{n}\phi_{i}=0.$

Let $n\geq 2$. For $i=1,\dots,n$, let $\phi_{i}$ be nonzero holomorphic
quadratic differentials on the Riemann surface $S$ such that
$\sum_{i=1}^{n}\phi_{i}=0.$

######  Theorem B1.

$S$ is not a (local) minimum for $\mathbf{E}_{\rho}$ if and only if there
exists $t>0$ such that $S$ is not a (local) minimum for
$\mathbf{E}_{\rho}^{t}$. In this case, for all $s>t$, $S$ is not a (local)
minimum for $\mathbf{E}_{\rho}^{s}$.

######  Remark 1.1.

If $n=2$, Schoen’s result shows that the only critical point of
$\mathbf{E}^{t}_{\rho}$ is a minimum, and so by (1), the same is true of
$\mathbf{E}_{\rho}$. This was first proved by Wentworth who showed that,
provided existence, the equivariant minimal surface in a product of two
$\mathbb{R}$-trees is unique [26, Theorem 1.6].

######  Remark 1.2.

It appears to be unknown whether the energy functional on Teichmüller space
for harmonic maps to $\mathbb{R}$-trees is $C^{2}$. It is always $C^{1},$ and
real analytic near a Riemann surface such that the Hopf differential of the
harmonic map has only simple zeros (this is a generic condition) [17].

Theorem B1 can give critical points of $\mathbf{E}_{\rho}^{t}$ that are not
minima, but this is not quite strong enough to prove Theorem A, which is about
unstable critical points. To that end, we give a notion of instability in
products of $\mathbb{R}$-trees that will be suitable for our purposes. Let $S$
be a critical point for $\mathbf{E}_{\rho}$ with harmonic map
$\pi=(\pi_{1},\dots,\pi_{n})$. Given $C^{\infty}$ vector fields
$V_{1},\dots,V_{n}$ on $S$, let $r\mapsto f_{1}^{r},\dots,f_{n}^{r}:S\to S$ be
their flows, and construct the map $\pi_{r}=(\pi_{1}\circ
f_{1}^{r},\dots,\pi_{n}\circ f_{n}^{r})$. For any $C^{\infty}$ path of Riemann
surfaces $r\mapsto S_{r},$ there is a Beltrami form $\mu$ representing a point
$T_{S}\mathbf{T}_{g}$ that is tangent to our path at $r=0$.

######  Definition 1.3.

We define a quadratic form $\mathbf{L}:T_{S}\mathbf{T}_{g}\times
H^{0}(S,TS)^{n}\to\mathbb{R}$ by

$\mathbf{L}(\mu,V_{1},\dots,V_{n})=\frac{d^{2}}{dr^{2}}|_{r=0}\mathcal{E}(S_{r},\pi_{r}),$

where $r\mapsto S_{r}$ is any path tangent to $\mu$ at $r=0$.

The self-maps index of $S$ for $\mathbf{E}_{\rho}$ is the maximal dimension of
$T_{S}\mathbf{T}_{g}\times H^{0}(S,TS)^{n}$ on which $L$ is negative
definitef. If the index is positive, we say that $S$ is unstable.

We explain that $\mathbf{L}$ is well-defined in Section 2.2. $\mathbf{L}$ is
positive semi-definite on $\\{0\\}\times H^{0}(S,TS)^{n}$, and hence if
$\mathbf{L}$ is negative definite on a subspace $U\subset
T_{S}\mathbf{T}_{g}\times H^{0}(S,TS)^{n}$, then $U$ projects injectively to
$T_{S}\mathbf{T}_{g}\times\\{0\\}$. Moreover, for any variations $r\mapsto
S_{r}$ and $r\mapsto\pi_{r},$ we have
$\mathbf{E}_{\rho}(S_{r})\leq\mathcal{E}(S_{r},\pi_{r}),$ and hence if
$\mathbf{L}(\mu,V_{1},\dots,V_{n})<0,$ then
$\mathbf{E}_{\rho}(S_{r})<\mathbf{E}_{\rho}(S)$ for small $r$. See Remark 3.4
below for more motivation for the definition of $\mathbf{L}$.

######  Theorem B2.

The index of $\mathbf{E}_{\rho}^{t}$ at $S$ is non-decreasing with $t$, and
converges to the self-maps index of $S$ for $\mathbf{E}_{\rho}$ as
$t\to\infty.$ Consequently, $S$ is unstable for $\mathbf{E}_{\rho}$ if and
only if it is unstable for $\mathbf{E}_{\rho}^{t}$, for $t$ sufficiently
large.

Toward the proof of Theorem A, we only need the “only if” direction of Theorem
B2. We include the “if” direction and Theorem B1 because they show that
$\mathbb{R}$-trees are really at the heart of the result. A conjecture in
Higher Teichmüller theory is that high energy minimal maps into symmetric
spaces converge in an appropriate sense to minimal maps into buildings (see
[10]). This is the higher rank generalization of [28]. If our results extend
to this setting, then this would suggest that any counterexample to the
Labourie conjecture would have to come from an unstable minimal map into a
building.

###  1.3. Equivariant minimal surfaces in $\mathbb{R}^{n}$

In order to use Theorem B2 to prove Theorem A, we construct unstable surfaces
in products of $\mathbb{R}$-trees. We start by looking in a more familiar
place: Euclidean space $\mathbb{R}^{n}$.

For $i=1,\dots,n$, let $\alpha_{i}$ be a non-zero holomorphic $1$-form on the
Riemann surface $S$. Lifting to $1$-forms $\tilde{\alpha}_{i}$ on a universal
cover $\tilde{S}$ gives the data of a harmonic map to $\mathbb{R}^{n}$ via
integrating the real parts:

$h=(h_{1},\dots,h_{n}),\hskip
2.84526pth_{i}(z)=\textrm{Re}\int_{z_{0}}^{z}\tilde{\alpha_{i}},$

unique up to translation. The map $h$ intertwines the action of
$\pi_{1}(\Sigma_{g})$ on $\tilde{S}$ with some non-trivial homomorphism
$\chi:\pi_{1}(\Sigma_{g})\to\mathbb{R}^{n}$. The Hopf differential of $h_{i}$
is the square $\phi_{i}=\alpha_{i}^{2}$, which descends to a holomorphic
quadratic differential on $S$, by the equivariance property. $h$ is weakly
conformal if and only if $\sum_{i=1}^{n}\phi_{i}=0,$ which is equivalent to
$h$ being minimal.

By the construction of Section 1.2, the Hopf differentials $\phi_{i}$ also
define an action $\rho$ of $\pi_{1}(\Sigma_{g})$ on a product $X$ of
$\mathbb{R}$-trees and a $\rho$-equivariant minimal map $\pi$. The map $h$
naturally factors through $\pi$. Let $\mathbf{E}_{\chi}$ and
$\mathbf{E}_{\rho}$ be the corresponding energy functionals on Teichmüller
space. In the end we prove the following near-equivalence.

######  Theorem C.

For $n\geq 2$ and $i=1,\dots,n$, let $\alpha_{i}$ be nonzero holomorphic
$1$-forms on $S$ such that $\sum_{i=1}^{n}\alpha_{i}^{2}=0,$. Let $\rho$, and
$\chi$ be as above.

  1. (1)

If $S$ is not a (local) minimum for $\mathbf{E}_{\rho}$, then it is not a
(local) minimum for $\mathbf{E}_{\chi}$.

  2. (2)

The index of $\mathbf{E}_{\chi}$ at $S$ is equal to the self-maps index of
$\mathbf{E}_{\rho}$ for $S$. In particular, if $S$ is unstable for
$\mathbf{E}_{\chi}$, then $S$ is unstable for $\mathbf{E}_{\rho}.$

######  Remark 1.4.

As in Theorem B, the statement is not so interesting when $n=2$ since every
critical point is a stable minimum.

######  Remark 1.5.

Instability for $\mathbf{E}_{\chi}$ at $S$ is equivalent to instability for
the (equivariant) area functional on the space of all equivariant maps. The
second variations for both functionals have the same index (see [3, Theorem
3.4]).

The final ingredient needed to prove Theorem A is an example of an unstable
equivariant minimal surface in $\mathbb{R}^{n}$. Fortunately, these aren’t so
hard to find: when $n=3,$ every non-planar equivariant minimal surface in
$\mathbb{R}^{3}$ is unstable, since a constant section of the normal bundle is
destabilizing. Consequently, for any three $1$-forms on $S$ whose squares sum
to zero, as long as they span a 2-dimensional space, $S$ will be an unstable
point of $\mathbf{E}_{\chi}$ (we explain the details in Section 5.3).

The most natural example is the lift of a minimal surface in a flat $3$-torus;
there a many classical examples, such as the Schwarz P-surface of genus $3$
(see [18]). In fact, for every $g\geq 3$, every flat $3$-torus contains
infinitely many distinct unstable minimal surfaces of genus $g$ in the same
homotopy class (see [18], [7], and [24]).

By inclusion, this gives examples for every $n\geq 3$, as long as $g\geq 3$.
We can also perturb these examples to give even more. Unfortunately, if $g=2$,
then the only triples of 1-forms whose squares sum to zero are scalar
multiples of one another, so we cannot use Theorem C to prove Theorem A in the
case $g=2$.

###  1.4. A generalization of Theorem C, and the case $g=2$

In the general setting where we start with $n$ quadratic differentials summing
to zero that are not necessarily squares of abelian differentials, we may have
to lift to a branched covering of $S$ in order to get a harmonic map to
$\mathbb{R}^{n}$. Replacing $\pi_{1}(\Sigma)$ with the Deck group of this
branched covering, and $\chi$ with the corresponding representation of this
group, we prove that an analog of statement (2) from Theorem C still holds.

When $n=3$, the minimal surface in $\mathbb{R}^{3}$ arising from a branched
cover is no longer automatically equivariantly unstable for its energy
functional, because the normal bundle is not necessarily equivariantly trivial
with respect to the action of the Deck group of the branched covering.
Instead, we find a condition on the quadratic differentials that guarantees
that the bundle is equivariantly trivial, which then yields the following
theorem.

######  Theorem D.

Let $\phi_{1},\phi_{2},\phi_{3}$ be holomorphic quadratic differentials on $S$
that are not colinear and such that $\phi_{1}\phi_{2}\phi_{3}$ is the square
of a cubic differential. Then the corresponding equivariant minimal surface in
the product of $\mathbb{R}$-trees is unstable via self-maps for its energy
functional.

We show that the moduli space of solutions to this problem has dimension at
least $3g-3$ for every genus (Proposition 5.8). In particular, this gives us
unstable minimal surfaces in genus 2, and hence allows us to complete the
proof of Theorem A.

###  1.5. Acknowledgments

We would like to thank Phillip Engel and Mike Wolf for helpful discussions.
Vladimir Marković is supported by the Simons Investigator Award 409745 from
the Simons Foundation.

##  2\. Preliminaries

###  2.1. Harmonic maps to manifolds

Let $\nu$ be a smooth metric on $S$ compatible with the complex structure. Let
$(M,\sigma)$ be a closed Riemannian manifold, and $h:S\to M$ a $C^{2}$ map.
The energy density is the function

(1) $e(h)=\frac{1}{2}\textrm{trace}_{\nu}h^{*}\sigma,$

and the total energy is

(2) $\mathcal{E}(S,h)=\int_{S}e(h)dA,$

where $dA$ is the area form of $\nu$. We comment here that the energy density
$2$-form $e(h)dA$ does not depend on the choice of compatible metric $\nu$,
but only on the complex structure. $h$ is said to be harmonic if it is a
critical point for the energy $h\mapsto\mathcal{E}(S,h).$ The Hopf
differential is the quadratic differential on $S$ defined in a local
holomorphic coordinate $z$ by

(3) $\phi(h)(z)=h^{*}\sigma\Big{(}\frac{\partial}{\partial
z},\frac{\partial}{\partial z}\Big{)}(z)dz^{2},$

and it is holomorphic provided that $h$ is harmonic. We also consider maps
from $\tilde{S}\to(M,\sigma)$ that are equivariant with respect to a
representation $\rho:\tilde{S}\to\textrm{Isom}(M,\nu).$ Since $\rho$ is acting
by isometries, the energy density is invariant under the $\pi_{1}(\Sigma_{g})$
action on $\tilde{S}$ by deck transformations, and hence descends to a
function on $S$. In this way, we can define a total energy exactly as in (2),
and discuss harmonic maps. Similarly, the Hopf differential descends to a
holomorphic quadratic differential on $S$.

We give special attention to surfaces and the real line. When the target
$(M,\sigma)$ is a surface with conformal metric $\sigma$, then in holomorphic
coordinates $z$ on $S$ and $w$ on $M$, we write $\nu=\nu(z)|dz|^{2}$,
$\sigma=\sigma(w)|dw|^{2}$, and $h$ as a complex-valued function $h(z)$. The
energy density takes the form

$e(h)(z)=\frac{\sigma(h(z))}{\nu(z)}(|h_{z}|^{2}+|h_{\overline{z}}|^{2})(z),\hskip
2.84526pt\phi(h)(z)=\sigma(h(z))h_{z}\overline{(h_{\overline{z}})}dz^{2}.$

Considering equivariant maps to the real line, again in local coordinates,

$e(h)(z)=2\nu(z)^{-1}|h_{z}|^{2},\hskip 2.84526pt\phi(h)=h_{z}^{2}dz^{2}.$

For a harmonic map to a product space, the definitions (1) and (3) shows that
the energy density and the Hopf differential are the sum of the energy
densities and the Hopf differentials respectively of the component maps. So
for a mapping $h=(h_{1},\dots,h_{n})$ into a product of Riemann surfaces, or
an equivariant mapping into $\mathbb{R}^{n},$ the Hopf differential is the sum

(4) $\phi(h)=\sum_{i=1}^{n}\phi(h_{i}).$

$h$ is minimal if $\phi\equiv 0.$

If $(M,\sigma)$ is a negatively curved surface, it is well-known that there is
a unique harmonic map $h:S\to(M,\sigma)$ in the homotopy class of the identity
(see [2] for existence, and [6, Theorem H] for uniqueness). If we work on a
different Riemann surface structure $S^{\prime}$ on $\Sigma_{g}$, we get a
harmonic map from $S^{\prime}\to(M,\sigma)$ in the class of the identity, and
the total energy depends only on the class of $S^{\prime}$ in Teichmüller
space. Thus, we get a functional $\mathbf{E}:\mathbf{T}_{g}\to(0,\infty)$,
where $\mathbf{E}(S^{\prime})$ is the total energy of the harmonic map from
$S^{\prime}\to(M,\sigma).$ For a map into a product of surfaces, the energy
functional is the sum of the energy functionals of the component mappings.

###  2.2. Harmonic maps to $\mathbb{R}$-trees

######  Definition 2.1.

An $\mathbb{R}$-tree is a length space $(T,d)$ such that any two points are
connected by a unique arc, and every arc is a geodesic, isometric to a segment
in $\mathbb{R}$.

A point $x\in T$ is a vertex if the complement $T\backslash\\{x\\}$ has
greater than two components. Otherwise it is said to lie on an edge.

Let $S$ be a closed Riemann surface of genus $g\geq 2$. The horizontal (resp.
vertical) foliation of a holomorphic quadratic differential $\phi$ on $S$ is
the singular foliation whose non-singular leaves are the integral curves of
the line field on $S\backslash\phi^{-1}(0)$ on which $\phi$ is a negative real
number. The singularities are standard prongs at the zeros. Both foliations
come equipped with transverse measures $|\textrm{Im}\sqrt{\phi}|$ and
$|\textrm{Re}\sqrt{\phi}|$ respectively (see [5, Exposé 5] for precise
definitions).

In this paper, we work with the vertical foliation, unless specified
otherwise. Lifting to a singular measured foliation on a universal cover
$\tilde{S}$, we define an equivalence relation under which two points
$x,y\in\tilde{S}$ are equivalent if they lie on the same leaf. The quotient
space is denoted $T$, and we can push the transverse measure down via the
projection $\pi:\tilde{S}\to T$ to form a distance function $d$ such that
$(T,d)$ is an $\mathbb{R}$-tree, with an induced action
$\rho:\pi_{1}(S)\to\textrm{Isom}(T,d).$

According to Korevaar-Schoen, for Lipschitz maps $f$ from $\tilde{S}$ to
complete and non-positively curved (NPC) length spaces such as $(T,d)$, there
is a well-defined $L^{1}$ directional energy tensor $g_{ij}=g_{ij}(f)$ that
generalizes the pullback metric (see [11, Theorem 2.3.2]). In this way, one
can define a measurable energy density function by

(5) $e(f)=\frac{1}{2}\textrm{trace}_{\nu}g_{ij}(f).$

For an equivariant Lipschitz map $h$, the energy density $e(h)$ is invariant
under the group, and we define a total energy as in the smooth setting by

$\mathcal{E}(S,h)=\int_{S}e(h)dA.$

######  Definition 2.2.

We say that a $\rho$-equivariant map $h:\tilde{S}\to(T,d)$ is harmonic if,
among other $\rho$-equivariant maps, it is a critical point for the energy
$h\mapsto\mathcal{E}(S,h).$

For the projection map $\pi$, we can describe the energy density explicitly.
At a point on $p\in\tilde{S}$ on which $\phi(p)\neq 0$, the map locally
isometrically factors through a segment in $\mathbb{R}$. In a small enough
neighbourhood around that point, $e(\pi)$ is equal to the energy density of
the locally defined map to $\mathbb{R}$, which is computed as usual via the
formula (1). From this, we see that the energy density has a continuous
representative that is equal to $\nu^{-1}|\phi|/2$ everywhere.

The Hopf differential is well-defined for maps $f$ from $\tilde{S}$ to NPC
spaces as above: in local coordinates, it is given by

(6) $\frac{1}{4}(g_{11}(f)(z)-g_{22}(f)(z)-2ig_{12}(f)(z))dz^{2}.$

The projection map $\pi:\tilde{S}\to(T,d)$ is $\rho$-equivariant and harmonic,
with Hopf differential $\phi/4$. Instead of the equation (6), one can also see
this by using the local isometric factoring. As in the case of maps to
surfaces, a harmonic mapping into a product of trees is called minimal if the
Hopf differential–which splits as a sum as in (4)–vanishes.

Given $(T,d)$ as above, we always rescale the metric to $2d$, which makes it
so that the Hopf differential of $\pi:\tilde{S}\to(T,2d)$ is $\phi$. For any
other Riemann surface $S^{\prime}$ representing a point in $\mathbf{T}_{g}$,
there is a unique $\rho$-equivariant harmonic map from
$\tilde{S}^{\prime}\to(T,2d)$ (see [29]). Again like the surface case, the
representation $\rho$ then defines an energy functional on Teichmüller space.
The same holds for products of $\mathbb{R}$-trees with admissible actions.

Let’s now address the quadratic form $\mathbf{L}:T_{S}\mathbf{T}_{g}\times
H^{0}(S,TS)^{n}\to\mathbb{R}$. Given a $C^{\infty}$ path of Riemann surfaces
and a flow $r\mapsto f_{r},$ we consider

(7) $r\mapsto\mathcal{E}(S_{r},\pi\circ\tilde{f_{r}}).$

In [19, Section 5], Moore computes the derivative of the two-variable energy
for maps from a surface to a Riemannian manifold. Using the characterization
(5), one can word-for-word redo that computation, but with the measurable
density $e(\pi\circ\tilde{f}_{r})$, to see that (7) is $C^{\infty}$ in $r$.
Working with a minimal map into a product of trees and $n$ vector fields, one
does the computation $n$ times to get $\mathbf{L}$. We note that it only
depends on the tangent vectors and not the specific path of Riemann surfaces
and flow of maps, because a minimal map is a critical point for the two-
variable energy.

##  3\. Minimal surfaces in products of $\mathbb{R}$-trees

We first prove Theorem B2, and then Theorem B1.

###  3.1. The Reich-Strebel energy formula

Reich-Strebel computed a formula for the difference of energies of
quasiconformal maps (equation 1.1 in [20]). Let $h:S\to M$ and $f:S\to
S^{\prime}$ be quasiconformal maps between Riemann surfaces, with a conformal
metric on $M$. Let $\mu$ be the Beltrami form of $f$, and $\phi$ the Hopf
differential of $h$, which need not be holomorphic. Then,

(8) $\mathcal{E}(S^{\prime},h\circ
f^{-1})-\mathcal{E}(S,h)=-4\textrm{Re}\int_{S}\phi\cdot\frac{\mu}{1-|\mu|^{2}}+2\int_{S}e(h)\cdot\frac{|\mu|^{2}}{1-|\mu|^{2}}dA.$

The computation goes through just the same when $h$ maps $\tilde{S}$
equivariantly into an $\mathbb{R}$-tree. For a map $h$ to a tree $(T,d)$ with
Hopf differential $\psi$, the energy density satisfies $e(h)=2\nu^{-1}|\psi|.$
We obtain the Proposition below.

######  Proposition 3.1.

Let $h:\tilde{S}\to(T,d)$ be an equivariant harmonic map to an
$\mathbb{R}$-tree with Hopf differential $\psi$, and $f:S^{\prime}\to S$ a
quasiconformal map. Then the following formula holds:

(9)
$\mathcal{E}(S^{\prime},h\circ\tilde{f}^{-1})-\mathcal{E}(S,h)=-4\textrm{Re}\int_{S}\psi\cdot\frac{\mu}{1-|\mu|^{2}}+4\int_{S}|\psi|\cdot\frac{|\mu|^{2}}{1-|\mu|^{2}}.$

In the formula above, $\tilde{f}$ is the lift to $\tilde{S}$ and $\mu$ is the
Beltrami form.

###  3.2. Proof of Theorem B2

Let $\phi\in\operatorname{\textrm{QD}}(S)-\\{0\\}$. For $t>0,$ let $M_{t}$ be
the hyperbolic structure with hyperbolic metric $\sigma_{t}$ such that the
identity map $h_{t}:S\to M_{t}$ is harmonic with Hopf differential $t\phi$,
let $\mathcal{E}_{\rho}^{t}$ be the two-variable energy functional for
$M_{t}$, and $E_{\rho}^{t}$ the corresponding energy functional on Teichmüller
space. Similarly, let $\mathcal{E}_{\rho}$ and $E_{\rho}$ be the energies for
the $\mathbb{R}$-tree $(T,2d)$ determined by $\phi$ (with a rescaled metric).

The main step in the proof of Theorem B2 is Lemma 3.2. If we rescale
$\sigma_{t}$ by $t^{-1}$, then for any Riemann surface structure $S^{\prime}$
and $C^{2}$ map $f:S^{\prime}\to M_{t},$ the energy with respect to the target
metric $t^{-1}\sigma_{t}$ is $t^{-1}\mathcal{E}_{\rho}^{t}(S^{\prime},f)$. Let
$r\mapsto S_{r}$ be a path of Riemann surfaces and $r\mapsto f_{r}$ a flow
starting at the identity map. Lemma 3.2 shows that the second derivative in
$r$ of the energy of $h_{t}\circ f_{r}$ on $S_{r}$ with respect to the target
metric $t^{-1}\sigma_{t}$ converges as $t\to\infty$ to the second derivative
of the energy of $\pi\circ f_{r}:\tilde{S}_{r}\to(T,2d).$

######  Lemma 3.2.

For $s>t$,

$\frac{1}{t}\frac{d^{2}}{dr^{2}}|_{r=0}\mathcal{E}_{\rho}^{t}(S_{r},h_{t}\circ
f_{r})>\frac{1}{s}\frac{d^{2}}{dr^{2}}|_{r=0}\mathcal{E}_{\rho}^{s}(S_{r},h_{s}\circ
f_{r})>\frac{d^{2}}{dr^{2}}|_{r=0}\mathcal{E}_{\rho}(S_{r},\pi\circ\tilde{f}_{r}),$

and

$\lim_{t\to\infty}\frac{1}{t}\frac{d^{2}}{dr^{2}}|_{r=0}\mathcal{E}_{\rho}^{t}(S_{r},h_{t}\circ
f_{r})=\frac{d^{2}}{dr^{2}}|_{r=0}\mathcal{E}_{\rho}(S_{r},\pi\circ\tilde{f}_{r}).$

Toward the proof, we first record the lemma below about the growth of the
energy density.

######  Lemma 3.3.

Let $e(h_{t})$ be the energy density of $h_{t}$ with respect to the target
metric $\sigma_{t}$. Then for $s\geq t$,

(10) $\frac{e(h_{t})}{t}\geq\frac{e(h_{s})}{s},$

and the inequality is strict away from the zeros of $\phi$. Moreover,

(11) $\lim_{t\to\infty}\frac{e(h_{t})}{t}=2\nu^{-1}|\phi|.$

###### Proof.

Let $\mu_{t}$ and $\mu_{s}$ be the Beltrami forms of $h_{t}$ and $h_{s}$
respectively. It is proved in [27, Proposition 4.3] that away from the zeros
of $\phi$ (at which $|\mu_{t}|=0$ for every $t$), $|\mu_{t}|$ monotonically
increases to $1$ as $t\to\infty.$ A simple computation gives

$e(h)=\frac{2t|\phi|}{\nu}\cosh\log|\mu_{t}|^{-1},$

and likewise for $s.$ Therefore, (10) is equivalent to the inequality

(12) $\cosh\log|\mu_{t_{0}}|^{-1}\geq\cosh\log|\mu_{t}|^{-1}.$

Since $|\mu_{t}|<1$ everywhere, the inequality (12) follows. Using the
limiting behaviour of $|\mu_{t}|,$ we take $t\to\infty$ to obtain (11). ∎

###### Proof of Lemma 3.2.

Let $(\mu,V)\in T_{S}\mathbf{T}_{g}\times H^{0}(S,TS)$, and let $r\mapsto
f_{r}$ be the flow of $V$. Let $\mu_{r}$ be the Beltrami form of $f_{r}^{-1}$,
and $\alpha$ the $C^{\infty}$ $(1,-1)$-form and $\beta$ the $C^{\infty}$
function on $S$ described by

$\alpha(z)=\frac{d^{2}}{dr^{2}}\bigg{|}_{r=0}\frac{\mu_{r}(z)}{1-|\mu_{r}(z)|^{2}},\hskip
2.84526pt\beta(z)=\frac{d^{2}}{dr^{2}}\bigg{|}_{r=0}\frac{|\mu_{r}(z)|^{2}}{1-|\mu_{r}(z)|^{2}}.$

We use the Reich-Strebel fomula (8). For each $t>0$,

$\displaystyle\frac{1}{t}\frac{d^{2}}{dr^{2}}\bigg{|}_{r=0}\mathcal{E}_{\rho}^{t}(S_{r},h^{t}\circ
f_{r})$
$\displaystyle=\frac{1}{t}\frac{d^{2}}{dr^{2}}\bigg{|}_{r=0}\Big{(}\mathcal{E}_{\rho}^{t}(S_{r},h^{t}\circ(f_{r}^{-1})^{-1})-\mathcal{E}_{\rho}^{t}(S,h^{t})\Big{)}$
$\displaystyle=\frac{d^{2}}{dr^{2}}\bigg{|}_{r=0}\Big{(}-4\textrm{Re}\int_{S}\phi\cdot\frac{\mu_{r}}{1-|\mu_{r}|^{2}}+2\int_{S}\frac{e(h^{t})}{t}\cdot\frac{|\mu_{r}|^{2}}{1-|\mu_{r}|^{2}}dA\Big{)}$
$\displaystyle=-4\textrm{Re}\int_{S}\phi\cdot\alpha+2\int_{S}\frac{e(h^{t})}{t}\cdot\beta
dA.$

On the other hand, by the same computation, but using (9),

$\frac{d^{2}}{dr^{2}}\bigg{|}_{r=0}\mathcal{E}_{\rho}(S_{r},\pi\circ\tilde{f}_{r})=-4\textrm{Re}\int_{S}\phi\cdot\alpha+4\int_{S}|\phi(h)|\cdot\beta.$

By Lemma 3.3, for $s>t$,

$\int_{S}\frac{e(h^{t})}{t}\cdot\beta dA>\int_{S}\frac{e(h^{s})}{s}\cdot\beta
dA,$

so that $\frac{L_{t}}{t}>\frac{L_{s}}{s}$. By Lemma 3.3 again and the
dominated convergence theorem,

$\int_{S}\frac{e(h^{t})}{t}\cdot\beta dA\to 2\int|\phi|\cdot\beta$

in a strictly decreasing fashion as $t\to\infty$. The result follows.

∎

Moving onto the proof of Theorem B2, we resume the notation from the
introduction: for $i=1,\dots,n,$ $\phi_{i}$ are nonzero holomorphic quadratic
differentials on $S$ summing to $0$. The product of $\mathbb{R}$-trees
$(T_{i},2d_{i})$, which we denote by $X$, comes equipped with the action
$\rho=(\rho_{1}\times\dots\times\rho_{n}).$ For each positive $t>0$,
$M_{i}^{t}$ is the hyperbolic structure such that the identity map
$h_{i}^{t}:S\to M_{i}^{t}$ is harmonic and has Hopf differential $t\phi_{i}$.
The energy functionals are denoted $\mathcal{E}_{\rho}$ and
$\mathbf{E}_{\rho}$ for the trees and $\mathcal{E}_{\rho}^{t}$ and
$\mathbf{E}_{\rho}^{t}$ for the surfaces.

We define $\mathbf{L}_{t}:T_{s}\mathbf{T}_{g}\times
H^{0}(S,TS)^{n}\to\mathbb{R}$ for the harmonic map
$h_{t}=(h_{1}^{t},\dots,h_{n}^{t})$ in the same way as $\mathbf{L}$: if
$r\mapsto S_{r}$ is a path of Riemann surfaces tangent to a Beltrami form
$\mu$ at $r=0$, and $V_{1},\dots,V_{n}$ are vector fields giving rise to flows
$\mapsto f_{1}^{r},\dots,f_{n}^{r},$ then we set

$\mathbf{L}_{t}(\mu,V_{1},\dots,V_{n})=\frac{d^{2}}{dr^{2}}|_{r=0}\mathcal{E}_{\rho}^{t}(S_{r},h_{r}^{t}),$

where $h_{t}^{r}=(h_{1}^{t}\circ f_{1}^{r},\dots,h_{n}^{t}\circ f_{n}^{r}).$

######  Remark 3.4.

The two-variable energy for $\prod_{i=1}^{n}M_{i}^{t}$ is defined on
$\mathbf{T}_{g}\times\prod_{i=1}^{n}\textrm{Maps}(S,M_{i}^{t})$. Since any
small perturbation of the identity map is a diffeomorphism, the space on which
$\mathbf{L}_{t}$ acts is canonically isomorphic to the tangent space of
$\mathbf{T}_{g}\times\prod_{i=1}^{n}\textrm{Maps}(S,M_{i}^{t})$ at
$S\times\prod_{i=1}^{n}\textrm{id}$.

######  Remark 3.5.

Since $S$ is a critical point for the two-variables energies
$\mathcal{E}_{\rho}^{t}$ and $\mathcal{E}_{\rho}$, the second order
derivatives $\mathbf{L}_{t}$ and $\mathbf{L}$ depend only on the first order
data $\mu,V_{1},\dots,V_{n}$ (this is not true for the second variations of
each component harmonic map).

######  Proposition 3.6.

For $s>t$,

$\frac{\mathbf{L}_{t}}{t}>\frac{\mathbf{L}_{s}}{s}>\mathbf{L},$

and $\lim_{t\to\infty}\frac{\mathbf{L}_{t}}{t}=\mathbf{L}.$

###### Proof.

We invoke Lemma 3.2 $n$ times. ∎

######  Lemma 3.7.

The index of $\mathbf{L}_{t}$ is equal to the index of
$\mathbf{E}_{\rho}^{t}.$

###### Proof.

Let $r\mapsto S_{r}$ be a path of Riemann surfaces, tangent to the Beltrami
form $\mu$ at $r=0$, and suppose there exists $(V_{1},\dots,V_{n})\in
H^{0}(S,TS)^{n}$ such that $\mathbf{L}_{t}(\mu,V_{1},\dots,V_{n})<0$. For each
fixed $t>0,$ the maps $r\mapsto\mathbf{E}_{\rho}^{t}(S_{r})$ and
$r\mapsto\mathcal{E}_{\rho}^{t}(S_{r},h_{t}^{r})$ have zero first variation at
$r=0$. By the minimizing property for harmonic maps,
$\mathbf{E}_{\rho}^{t}(S_{r})\leq\mathcal{E}_{\rho}^{t}(S_{r},h_{t}^{r})$ for
every $r,$ and it follows that

$\frac{1}{t}\frac{d^{2}}{dr^{2}}|_{r=0}\mathbf{E}_{\rho}^{t}(S_{r})\leq\frac{1}{t}\frac{d^{2}}{dr^{2}}|_{r=0}\mathcal{E}_{\rho}^{t}(S_{r},h_{t}^{r})=\frac{1}{t}\mathbf{L}_{t}(\mu,V_{1},\dots,V_{n})<0.$

So, the index of $\mathbf{E}_{\rho}^{t}$ is at least that of $\mathbf{L}_{t}.$

For the other direction, assume $r\mapsto S_{r}$ lowers
$\mathbf{E}_{\rho}^{t}$ to second order, and for each $r>0,$ let
$k=(k_{1}^{r},\dots,k_{n}^{r}):S_{r}\to\prod_{i=1}^{n}M_{i}^{t}$ be the
harmonic map in the class of the identity. All $h_{i}^{t}$’s and $k_{i}^{r}$’s
are orientation-preserving diffeomorphisms. Set
$f_{i}^{r}=(h_{i}^{t})^{-1}\circ k_{i}^{r}$ and let $V_{i}$ be the
infinitesimal generator of the flow $r\mapsto f_{i}^{r}$. Then

$\frac{1}{t}\mathbf{L}_{t}(\mu,V_{1},\dots,V_{n})=\frac{1}{t}\frac{d^{2}}{dr^{2}}|_{r=0}\mathbf{E}_{\rho}^{t}(S_{r})<0,$

which gives the result. ∎

We now deduce Theorem B2.

###### Proof of Theorem B2.

Proposition 3.6 implies that the index of $\mathbf{L}_{t}$ is non-decreasing
with $t$, and converges to the self-maps index of $S$ for $\mathbf{E}_{\rho}$.
We then apply Lemma 3.7 to obtain the same statement for the index of
$\mathbf{E}_{\rho}^{t}$ at $S$. ∎

###  3.3. Proof of Theorem B1

The proof of Theorem B1 is similar to that of Theorem B2, so we don’t go
through every detail. The main difference is that we replace Lemma 3.2 with
Lemma 3.8 below.

As above, let $M_{t}$ be the hyperbolic structure on $\Sigma_{g}$ with
hyperbolic metric $\sigma_{t}$ such that the identity map has Hopf
differential $t\phi$, with energy functional $E_{\rho}^{t}$, and let
$E_{\rho}$ be the energy functional for the $\mathbb{R}$-tree $(T,2d)$ for
$\phi$.

######  Lemma 3.8.

For all Riemann surfaces $S^{\prime}$,

$\lim_{t\to\infty}\frac{E_{\rho}^{t}(S^{\prime})}{t}=E_{\rho}(S^{\prime}).$

In order to prove the lemma, we recall some facts about the Thurston
compactification of Teichmüller space. Let $\mathcal{S}$ be the set of
homotopically non-trivial simple closed curves on $\Sigma_{g}$ and
$\mathbb{R}^{\mathcal{S}}$ the product space with the weak topology. There is
an embedding

$\ell:\mathbf{T}_{g}\times\mathbb{R}^{+}\to\mathbb{R}^{\mathcal{S}}$

that associates the data of a hyperbolic metric $\sigma$ and
$s\in\mathbb{R}^{+}$ to the set of lengths of geodesic representatives of
curves in $\mathcal{S}$ with respect to the metric $s\sigma$. Every singular
measured foliation $(\mathcal{F},\mu)$ on $S$ also defines a point in
$\mathbb{R}^{\mathcal{S}},$ by taking $\mu$-transverse measures of simple
closed curves. Furthermore, there is an injective map

$\beta:\textrm{QD}(S)\to\mathbb{R}^{\mathcal{S}}$

that takes a quadratic differential to its vertical foliation, and then to
$\mathbb{R}^{\mathcal{S}}.$ Note that both $\ell$ and $\beta$ are homogeneous
with exponent $\tfrac{1}{2}$.

According to Thurston and Hubbard-Masur (see [5] and [9]), both $\ell$ and
$\beta$ are homeomorphisms onto their images, and
$\ell(\mathbf{T}_{g}\times\mathbb{R}^{+})\sqcup\beta(QD(S))$ is homeomorphic
to a cone over a closed ball, which we call $\mathcal{C}$ (the cone over the
Thurston compactification of Teichmuller space). The following result can be
gleaned from the results of [27].

######  Theorem 3.9.

For any Riemann surface $S$, let
$E_{S}:\mathbf{T}_{g}\times\mathbb{R}^{+}\sqcup QD(S)\to\mathbb{R}^{+}$ be the
function that associates to each point in $\mathbf{T}_{g}\times\mathbb{R}^{+}$
the energy of the unique harmonic map isotopic to the identity from $S$, and
to each point of $QD(S)$ the energy of the unique equivariant harmonic map to
the corresponding $\mathbb{R}$-tree. Then $E_{S}$ is continuous with respect
to the topology on $\mathcal{C}$.

We now explain how to deduce this theorem from the paper [27]. The first
ingredient is a de-projectivized version of Lemma 4.7 of that paper, whose
proof is identical to the proof of the lemma in the paper.

######  Lemma 3.10.

Suppose $(\lambda_{n})_{n=1}^{\infty}\subset\mathbf{T}_{g}$ leaves all compact
subsets of the Teichmüller space, and let $\phi_{n}$ be the Hopf differential
of the harmonic map from $S$ to $(S,\lambda_{n})$. Suppose
$(a_{n})_{n=1}^{\infty}\subset(\mathbb{R}^{+})^{\mathcal{S}}$ is a chosen
sequence. Then $\ell(\lambda_{n},a_{n})$ converges in
$\mathbb{R}^{\mathcal{S}}$ if and only if $\beta(a_{n}\phi_{n})$ does, and in
the case of convergence, the two sequences have the same limit.

The second ingredient is the following computation (in which each term is
linear in the scalars $a_{n}$, so the factors of $a_{n}$ are superfluous).

######  Lemma 3.11 (Lemma 3.2 in [27]).

In the notation of the previous lemma,

$||a_{n}\phi_{n}||_{L^{1}(S)}\leq
E_{S}(a_{n}\lambda_{n})\leq||a_{n}\phi_{n}||_{L^{1}(S)}+a_{n}|\chi(\Sigma_{g})|.$

###### Proof of Theorem 3.9.

For brevity, write $E=E_{S}$. First, $E$ is continuous on
$\mathbf{T}_{g}\times\mathbb{R}^{+}$ and $E(\phi)=||\phi||_{L^{1}(S)}$, which
is certainly continuous on $QD(S)$. To show that $E$ is continuous on all of
$\mathcal{C}$, we just need to show that if
$\ell(\lambda_{n},a_{n})\to\beta(\phi)$, then $E(a_{n}\lambda_{n})\to
E(\phi)$. By Lemma 3.10, $\beta(a_{n}\phi_{n})\to\beta(\phi)$ (where as above
$\phi_{n}$ is the Hopf differential of the harmonic map to $\lambda_{n}$), and
since $\beta$ is a homeomorphism onto its image, $a_{n}\phi_{n}\to\phi$ as
well, so $E(a_{n}\phi_{n})\to E(\phi)$. Finally, since $a_{n}$ must tend to
zero in order for the sequence $a_{n}\lambda_{n}$ to converge in
$\mathbb{R}^{\mathcal{S}}$, Lemma 3.11 shows that $E(a_{n}\phi_{n})$ and
$E(a_{n}\lambda_{n})$ have the same limit. ∎

Now the proof of Lemma 3.8 is easy.

###### Proof of Lemma 3.8.

By definition, $E^{t}_{\rho}(S^{\prime})=E_{S^{\prime}}(\sigma_{t})$, and
$E_{\rho}(S^{\prime})=E_{S^{\prime}}(\phi)$, so by the continuity of
$E_{S^{\prime}}$ and its homogeneity, we just need to show that
$\ell(\sigma_{t}/t)\to\beta(\phi)$ in $\mathcal{C}$. To prove this, we use
Lemma 3.10 applied to the surface $S$. Indeed, the Hopf differential of the
harmonic map from $S$ to $\sigma_{t}/t$ is $\phi$ by construction, and since
the constant sequence at $\phi$ trivially converges to $\phi$, Lemma 3.10
implies that $\ell(\sigma_{t}/t)$ does as well. ∎

Preparations aside, we prove Theorem B1. We return to all of the notation from
the introduction and the proof of Theorem B2. We don’t recall it in full, but
just record here that the energy functionals are $\mathbf{E}_{\rho}$ for the
product of $\mathbb{R}$-trees and $\mathbf{E}_{\rho}^{t}$ for the product of
surfaces. The proof is quite similar to that of Theorem B2, so we leave the
details of the computations to the reader.

###### Proof of Theorem B1.

Beginning with a Riemann surface $S^{\prime}$ such that
$\mathbf{E}_{\rho}(S^{\prime})<\mathbf{E}_{\rho}(S)$, applying Lemma 3.8 $n$
times yields that $\mathbf{E}_{\rho}^{t}(S^{\prime})<\mathbf{E}_{\rho}^{t}(S)$
for sufficiently large $t$.

Conversely, suppose that there exists $t>0$ such that
$\mathbf{E}_{\rho}^{t}(S^{\prime})<\mathbf{E}_{\rho}^{t}(S)$, and let
$k=(k_{1}^{t},\dots,k_{n}^{t}):S^{\prime}\to\prod_{i=1}^{n}M_{i}^{t}$ be the
$n$-tuple of harmonic diffeomorphisms with lower energy. Let $h_{i}^{t}$ be
the $i^{th}$ component of the harmonic map $h_{t}$, and set
$f_{i}^{t}=(h_{i}^{t})^{-1}\circ k_{i}^{t}$. Arguing similarly to the proof of
Lemma 3.2, Reich-Strebel formulas (8) and (9) and the monotonicity on the
level of energy densities from Lemma 3.3 show that for $s>t$,

$\displaystyle\frac{\mathbf{E}_{\rho}^{t}(S^{\prime})-\mathbf{E}_{\rho}^{t}(S)}{t}$
$\displaystyle=\frac{\sum_{i=1}^{n}\mathcal{E}_{\rho}^{t}(S^{\prime},h_{i}^{t}\circ
f_{i}^{t})-\mathbf{E}_{\rho}^{t}(S)}{t}$
$\displaystyle>\frac{\sum_{i=1}^{n}\mathcal{E}_{\rho}^{s}(S^{\prime},h_{i}^{s}\circ
f_{i}^{t})-\mathbf{E}_{\rho}^{s}(S)}{s}$
$\displaystyle>\sum_{i=1}^{n}\mathcal{E}_{\rho}(S^{\prime},\pi\circ\tilde{f}_{i}^{t})-\mathbf{E}_{\rho}(S).$

It follows from the minimizing property that

$\frac{\mathbf{E}_{\rho}^{t}(S^{\prime})-\mathbf{E}_{\rho}^{t}(S)}{t}>\frac{\mathbf{E}_{\rho}^{s}(S^{\prime})-\mathbf{E}_{\rho}^{s}(S)}{s}$

and

$\frac{\mathbf{E}_{\rho}^{t}(S^{\prime})-\mathbf{E}_{\rho}^{t}(S)}{t}>\mathbf{E}_{\rho}(S^{\prime})-\mathbf{E}_{\rho}(S),$

and hence the result follows. ∎

##  4\. Unstable equivariant minimal surfaces in $\mathbb{R}^{n}$

We recall the setup of Theorem C. For $n\geq 2$ and $i=1,\dots,n$, let
$\alpha_{i}$ be nonzero holomorphic $1$-forms on $S$ such that
$\sum_{i=1}^{n}\alpha_{i}^{2}=0$. Let $\chi$ be the action of $\pi_{1}(S)$ on
$\mathbb{R}^{n}$ corresponding to the 1-forms $\alpha_{i}$, and let $\rho$ be
the action of $\pi_{1}(S)$ on a product $X=\prod_{i}(T_{i},2d_{i})$ of trees
corresponding to the quadratic differentials $\phi_{i}=\alpha_{i}^{2}$. We
write $\mathcal{E}_{\chi}$ and $\mathcal{E}_{\rho}$ for the associated two-
variable energies, and $\mathbf{E}_{\chi}$ and $\mathbf{E}_{\rho}$ for the
energy functionals on Teichmüller space. Let $h=(h_{1},\ldots,h_{n})$ and
$\pi=(\pi_{1},\ldots,\pi_{n})$ be the $\chi$\- and $\rho$\- equivariant
minimal maps respectively.

###  4.1. Isometric folding

We begin with the statement (1) from Theorem C. The result is a consequence of
the proposition below.

######  Proposition 4.1.

$\mathbf{E}_{\rho}\geq\mathbf{E}_{\chi},$ with equality at $S$.

The key is that there is a natural map $F:X\to\mathbb{R}^{n}$ intertwining
$\rho$ and $\chi$. To see why, let’s focus on a single tree $T_{i}$. Along a
curve parametrizing a non-singular leaf for the vertical singular foliation of
$\phi_{i}$, $\alpha_{i}$ evaluates the tangent vectors to purely imaginary
numbers. Since $dh_{i}=\mathrm{Re}(\tilde{\alpha}_{i})$, we deduce that
$h_{i}$ is constant along the singular vertical foliation of
$\tilde{\phi}_{i}$. Hence, $h_{i}$ descends to a map
$F_{i}:T_{i}\to\mathbb{R}$, which we call the folding map of the tree. The map
$F=(F_{1},\ldots,F_{n})$ has the required equivariance.

######  Lemma 4.2.

If $S^{\prime}$ is any point of $\mathbf{T}_{g}$, and $\pi_{i}^{\prime}$ the
unique $\rho_{i}$-equivariant harmonic map from $\tilde{S}^{\prime}$ to
$(T_{i},2d_{i})$, then the energy density of $\pi^{\prime}_{i}$ is pointwise
equal to the energy density of $F_{i}\circ\pi_{i}^{\prime}$.

###### Proof.

Let $\psi_{i}$ be the Hopf differential of $\pi_{i}^{\prime}$. As discussed in
Section 2, for any point $p$ at which $\psi_{i}(p)\neq 0$, there exists a
neighbourhood $\Omega$ of $p$, an open interval $I\subset\mathbb{R}$, a map
$\hat{\pi}_{i}^{\prime}:\Omega\to I$, and an isometric inclusion
$\iota:I\to(T_{i},2d_{i})$ such that in $\Omega,$

$\pi_{i}^{\prime}=\iota\circ\hat{\pi}_{i}^{\prime}.$

By construction, the restriction of $F_{i}|_{\iota(I)}:\iota(I)\to\mathbb{R}$
is an isometric embedding. It follows by continuity that the energy densities
are equal everywhere. ∎

###### Proof of Proposition 4.1.

For any $S^{\prime}\in\mathbf{T}_{g}$, let $\pi^{\prime}$ be the
$\rho$-equivariant harmonic map to the product of trees. The map
$F\circ\pi^{\prime}$ is a $\chi$-equivariant Lipschitz map to
$\mathbb{R}^{n}$. By the minimizing property for harmonic maps,

$\mathbf{E}_{\chi}(S^{\prime})\leq\mathcal{E}_{\chi}(S^{\prime},F\circ\pi^{\prime}).$

By the lemma above,
$\mathcal{E}(S^{\prime},F\circ\pi^{\prime})=\mathbf{E}_{\rho}(S^{\prime}),$ so
we have

$\mathbf{E}_{\rho}\geq\mathbf{E}_{\chi}.$

Working on the Riemann surface $S$, $h_{i}=F_{i}\circ\pi_{i}$ for every $i$,
so $\mathbf{E}_{\chi}(S)=\mathcal{E}_{\chi}(S^{\prime},F\circ\pi),$ and we
have equality. ∎

######  Remark 4.3.

Maps of the form $F_{i}\circ\pi_{i}^{\prime}$ above are subtle. They are
harmonic apart from some preimages under $\pi_{i}^{\prime}$ of the vertices in
$(T_{i},2d_{i})$, which are typically disjoint arcs or connected sums of
disjoint arcs. Even though they have finite total energy, a Weyl lemma cannot
be applied because they fail to be twice weakly differentiable on these lines.
The map $x\mapsto|x|$ on $\mathbb{R}$ exhibits this type of behaviour.

We see immediately from Proposition 4.1 that if $S$ is not a global (resp.
local) minimum of $\mathbf{E}_{\rho}$, then it is not a global (resp. local)
minimum of $\mathbf{E}_{\chi}$. So (1) is proved. Furthermore, we are very
close to proving one direction of (2), once we recall the definition of the
self-maps index, and its basic properties. We do this after collecting some
standard facts about minimal surfaces in $\mathbb{R}^{n}$.

###  4.2. Energy and area

Let $f$ be any smooth $\chi$-equivariant map from $\tilde{\Sigma}_{g}$ to
$\mathbb{R}^{n}$. The differential of $f$ descends to a closed
$\mathbb{R}^{n}$-valued 1-form $\theta$ on $\Sigma_{g}$, and the cohomology
class of $\theta$ is prescribed by the representation $\chi$. The map $f$ also
defines a $\pi_{1}(\Sigma_{g})$-invariant area form
$dA_{f}=\sqrt{\mathrm{det}(\theta^{T}\theta)}$, and the area of $f$, which we
write $A(f)$, is defined to be the integral of this form over $\Sigma_{g}$. If
$S$ is a Riemann surface structure on $\Sigma_{g}$, then
$\mathcal{E}_{\chi}(S,f)\geq A(f)$, with equality if and only if $f$ is
minimal (in fact, the integrands are equal pointwise).

Now suppose we are in the setting of Theorem C, so that $h$ is a minimal
$\chi$-equivariant map from $\tilde{S}$ to $\mathbb{R}^{n}$. Let $B$ be the
branch locus of $h$ on $S$.

######  Lemma 4.4.

Let $h_{r}$ be a smooth $\chi$-equivariant variation of $h$ such that
$h_{r}=h$ in a neighborhood of $B$. Then for $r$ small enough, there is a
smooth variation of Riemann surface structures $S_{r}$ such that $h_{r}$ is
minimal with respect to $S_{r}$.

###### Proof.

For $r$ sufficiently small, the map $h_{r}$ is still an immersion away from
$B$, and hence uniquely defines a new conformal structure on $S-B$. Since $X$
is compactly supported away from $B$, this conformal structure patches to the
conformal structure of $S$ near $B$, and defines a new conformal structure
$S_{r}$ on $S$, with respect to which $h$ is minimal. ∎

We say that a smooth $\mathbb{R}^{n}$-valued vector field $W$ on $S$ supported
on $S-B$ is a normal variation of $h$ if it is perpendicular to the image of
$dh$ at each point of $S-B$. For any such $W$, let $\tilde{W}$ be the pullback
to $\tilde{S}$; then the family $h_{r}=h+r\tilde{W}$ is a $\chi$-equivariant
deformation of $h$ equal to $h$ on a neighborhood of $B$. Taking the
derivative of the corresponding $S_{r}$ at $r=0$ defines a linear map from the
space of normal variations supported on $S-B$ to the tangent space of
Teichmüller space at $S$. Let $V$ the graph of this map, viewed as a subspace
of $T_{S}\mathbf{T}_{g}\times
T_{h}\mathrm{Map}_{\chi}(\tilde{\Sigma}_{g},\mathbb{R}^{n})$. We have shown
that restricted to $V$, the Hessian of $\mathcal{E}_{\chi}$ at the critical
point $(S,h)$ is equal to the Hessian of $A$ at the critical point $h$. The
latter has the following formula:

######  Proposition 4.5 (Theorem 32 in [14]).

If $W$ is a normal variation supported in $S-B$, the second derivative of the
area of any equivariant variation $h_{r}$ with derivative $W$ at $r=0$ is
given by the quadratic form

(13) $Q(W)=\int_{S}|(dW)^{N}|^{2}-|\langle k,W\rangle|^{2}$

where $(dW)^{N}$ is the component of $dW$ normal to the image of $dh$, $k$ is
the vector-valued second fundamental form of $h(S)$, and the second term is
interpreted as the square norm of the scalar-valued 2-tensor $\langle
k,W\rangle$.

###  4.3. Lifting to $\mathbb{R}$-trees via self-maps

In this section, we study energy and area in the context of the
$\rho$-equivariant harmonic maps to products of $\mathbb{R}$-trees.
Specifically, we relate $Q$ to the quadratic form
$\mathbf{L}:T_{S}\mathbf{T}_{g}\times H^{0}(S;TS)^{n}\to\mathbb{R}$ defined in
the introduction, which defines the self-maps index for $\mathbf{E}_{\rho}$.
Let $H_{c}^{0}(S-B,TS)$ be the subspace of $H^{0}(S,TS)$ of smooth vector
fields supported on $S-B$.

The key to the proof of the second part of Theorem C is the result below.

######  Lemma 4.6.

Suppose that $W$ is a normal variation of $S$ with support in $S-B$, and such
that $Q(W)<0.$ Then there exists a harmonic Beltrami form $\mu$ on $S$ and
vector fields $V_{1},\dots,V_{n}\in H^{0}_{c}(S-B,TS)^{n}$ such that

$\mathbf{L}(\mu,V_{1},\dots,V_{n})<0.$

###### Proof.

Denote the coefficients of $W$ by $W^{i}$. For each $i=1,\ldots,n$, let
$V_{i}$ be the vector field which vanishes on $B$ and is equal to $W^{i}\nabla
x^{i}/|\nabla x_{i}|^{2}$ on $S-B$, where $\nabla x^{i}$ is the gradient of
the coordinate function $x^{i}$ on $S$, which is nonvanishing on $S-B$. We
point out that $V_{i}$ has compact support on $S-B$.

Let $f_{i}^{W}:\mathbb{R}\times S\to S$ be flow of $V_{i}$, so that
$f_{i}^{W}(r,\cdot)=f_{i}^{r}(\cdot)$. Then the family
$H:\mathbb{R}\times\tilde{S}\to\mathbb{R}^{n}$ defined by
$H_{i}(r,p)=h_{i}\circ f_{i}^{r}(p)$ has derivative $W$ at time zero.
Moreover, the family $\Pi:\mathbb{R}\times\tilde{S}\to\prod_{i}(T_{i},2d_{i})$
defined by $\Pi_{i}(r,p)=\pi_{i}\circ f_{i}^{r}(p)$ satisfies
$F_{i}\circ\Pi_{i}=H_{i}$, where $F_{i}$ is the folding map from $T_{i}$ to
$\mathbb{R}$. Let $\pi_{r}$ be the map $(\pi_{1}\circ
f_{1}^{r},\dots,\pi_{n}\circ f^{r}).$ By Lemma 4.4, there exists a
$C^{\infty}$ variation of conformal structures $r\mapsto S_{r}$ along which
$\mathcal{E}_{\rho}(S_{r},\pi_{r})=\mathcal{E}_{\chi}(S_{r},h_{r})=A(h_{r}),$
and we set $\mu$ to be the Beltrami form in $T_{S}\mathbf{T}_{g}$ tangent to
this path at time zero. If $Q(W)<0$, then taking the second variation of
$r\mapsto\mathcal{E}_{\rho}(S_{r},\pi_{r})$ yields
$\mathbf{L}(\mu,V_{1},\dots,V_{n})=Q(W)<0.$ ∎

###  4.4. Log cutoff

In order to construct destabilizing variations for $Q$, it is helpful to do
away with the condition that $W$ is supported on $S-B$. First, we need to say
what it means for $W$ to be a normal variation over all of $S$. The map
$S-B\to\mathbb{C}\mathbb{P}^{n-1}$, which sends $p$ to the (one-dimensional)
image of $(\alpha_{1},\ldots,\alpha_{n})$ at $p$, extends holomorphically to
all of $S$ by clearing denominators. Thus, the normal bundle also extends
analytically to all of $S$. The quadratic form $Q$ is still finite for normal
variations that are not necessarily supported on $S-B$.

For normal variations $W$, which are not necessarily supported on $S-B,$ we
will need to show that one can replace them with variations that are supported
on $S-B$ without changing the value of $Q$ too much. This is the log cut-off
trick. If $r$ is the radial coordinate in $\mathbb{C}$ then the function
$\log(r)/\log(\delta^{-1})+2$, defined between $r=\delta$ and $r=\delta^{2}$,
is equal to $1$ for $r=\delta$ and 0 for $r=\delta^{2}$ and has Dirichlet
energy

(14)
$\begin{split}\frac{1}{2}\int_{\delta^{2}}^{\delta}\frac{2\pi}{r\log(\delta^{-1})^{2}}=\frac{\pi}{\log(\delta^{-1})}.\end{split}$

The point is this this tends to zero as $\delta$ goes to zero. A good picture
is that $\log r$ is the height coordinate on a cylinder conformal to the
punctured disk, so our function is an affine function of the height of the
cylinder, and its derivative is small. The extension of this function by 0 and
1 is Lipschitz. For very minor reasons, it will be convenient to use a smooth
cutoff function, so we let $l_{\delta}(r)$ be a perturbation of
$\log(r)/\log(\delta^{-1})+2$ which extends smoothly by 0 and 1 and has
Dirichlet energy no more than $2\pi/\log(\delta^{-1})$.

We use this model to define a cut-off function as follows. For each point
$p_{i}$ of $B$, fix a holomorphic coordinate $z_{i}$ with $z_{i}(p_{i})=0$.
Then, for any value of $\delta$ small enough that each $z_{i}$ is defined on
the ball of radius $\delta$ around $p_{i}$ and these balls do not overlap, let
$\eta_{\delta}$ be the function on $S$ defined by

  * •

$\eta_{\delta}(p)=l_{\delta}(|z_{i}|)$ if $\delta^{2}\leq|z_{i}(p)|\leq\delta$
for some $i$

  * •

$\eta_{\delta}(p)=0$ if $|z_{i}(p)|\leq\delta^{2}$ for some $i$

  * •

$\eta_{\delta}(p)=1$ otherwise.

We now use the log cut-off trick to prove the following lemma.

######  Lemma 4.7.

Suppose that $W$ is normal variation of $h$ on $S$. Then given any
$\epsilon>0$, there is a constant $d(\epsilon,Q(W),\sup|W|)$ such that for all
$\delta<d$,

(15) $|Q(\eta_{\delta}W)-Q(W)|<\epsilon.$

###### Proof.

For $\delta$ to be determined, we compute $Q(\eta_{\delta}W)$. We first treat
the normal term in the formula (13) applied to the variation $\eta_{\delta}W$:

$\displaystyle\int_{\Sigma}|(d(\eta_{\delta}W))^{N}|^{2}$
$\displaystyle=\int_{\Sigma}|\eta_{\delta}(dW)^{N}+(Wd\eta_{\delta})^{N}|^{2}$
$\displaystyle=\int_{\delta^{2}\leq|z|\leq\delta}|\eta_{\delta}(dW)^{N}+(Wd\eta_{\delta})^{N}|^{2}+\int_{|z|\geq\delta}|(dW)^{N}|^{2},$

where $Wd\eta_{\delta}$ is the $\mathbb{R}^{n}$-valued $1$-form $W\otimes
d\eta_{\delta}.$ Hence,

$\displaystyle|Q(\eta_{\delta}W)-Q(W)|$
$\displaystyle\leq\int_{\delta^{2}\leq|z|\leq\delta}|\eta_{\delta}(dW)^{N}+(Wd\eta_{\delta})^{N}|^{2}+\int_{|z|\leq\delta}|(dW)^{N}|^{2}+\int_{S}(1-\eta_{\delta}^{2})|W|^{2}|k|^{2}$
$\displaystyle=\int_{\delta^{2}\leq|z|\leq\delta}|\eta_{\delta}(dW)^{N}+(Wd\eta_{\delta})^{N}|^{2}+O(\delta^{2}),$

since $1-\eta_{\delta}^{2}$ is supported in $|z|\leq\delta.$ By Cauchy-Schwarz
and (14),

$\displaystyle|Q(\eta_{\delta}W)-Q(W)|$
$\displaystyle\leq\int_{\delta^{2}\leq|z|\leq\delta}|\eta_{\delta}(dW)^{N}|^{2}+|(Wd\eta_{\delta})^{N}|^{2}+2|\eta_{\delta}(dW)^{N}|^{2}|(Wd\eta_{\delta})^{N}|^{2}+O(\delta^{2})$
$\displaystyle=O(\delta^{2})+O\Big{(}\frac{1}{\log\delta^{-1}}\Big{)}=O\Big{(}\frac{1}{\log\delta^{-1}}\Big{)}.$

Thus, we can choose $\delta>0$ so that the difference of second variations is
at most $\epsilon.$ ∎

A consequence is that we can speak without ambiguity of the index of $Q$.

######  Proposition 4.8.

The index of $Q$ on the space of all normal variations is equal to the index
of $Q$ on the subspace of normal variations supported in $S-B$.

###### Proof.

We just need to show that if there is a $k$-dimensional space of normal
variations on which $Q$ is negative definite, then there is another
$k$-dimensional space of normal variations supported in $S-B$ on which $Q$ is
still negative definite. Let $V$ be a $k$-dimensional space of normal
variations on which $Q$ is negative definite. Let $S(V)$ be the unit sphere in
$V$ with respect to any metric on $V$. Then for $\delta$ small enough,
$Q(\eta_{\delta}W)<0$ for every $W\in S(V)$. Since this implies
$\eta_{\delta}W\neq 0$, the space $\\{\eta_{\delta}W|W\in V\\}$ is a
$k$-dimensional subspace of normal variations supported in $S-B$ on which $Q$
is negative definite. ∎

We may now finish the proof of Theorem C.

###### Proof of Theorem C (2).

Let $k$ be the index of $\mathbf{E}_{\chi}$, and let $W\subset
T_{S}\mathbf{T}_{g}$ be a $k$-dimensional subspace on which the second
variation is negative definite. By the implicit function theorem, the unique
harmonic $1$-form in a given cohomology class varies smoothly with the
conformal structure of $S$. We can integrate this smoothly-varying $1$-form to
give a smooth equivariant variation of the harmonic map $h$. Projecting the
variation onto the normal bundle, we get from $W$ a vector space of normal
variations of $h$ on which the second derivative of $\mathbf{E}_{\chi}$ is
equal to $Q$. Since it is assumed to be positive definite, this space is still
$k$ dimensional.

By Proposition 4.8, we can replace this with a $k$-dimensional subspace of
normal variations supported on $S-B$ on which $Q$ is still negative definite.
Then by Lemma 4.6, there is a $k$-dimensional subspace of
$T_{S}\mathbf{T}_{g}\times H_{c}^{0}(S-B,TS)^{n}$ on which $\mathbf{L}$ is
negative definite, and so the index of $\mathbf{E}_{\rho}$ by self-maps is at
least $k$.

In the other direction, suppose $W^{\prime}$ is a $k$-dimensional subspace of
$T_{S}\mathbf{T}_{g}\times H^{0}(S,TS)^{n}$ on which $\mathbf{L}$ is negative
definite. Since $\mathbf{L}$ is positive semidefinite on $\\{0\\}\times
H^{0}(S,TS)^{n}$, the projection of $W^{\prime}$ to $T_{S}\mathbf{T}_{g}$ is
still $k$-dimensional. For maps to manifolds, the positive semidefinite
property follows from the computation [6, Theorem H], and we get the same
result in our setting by repeating the computation but using the measurable
energy density with the characterization (5). Since $\mathbf{E}_{\rho}$ is an
infimum over all maps, we get an upper bound for $\mathbf{E}_{\rho}$ near $S$
by a smooth function with negative definite Hessian at $S$. Recall that
Proposition 4.1 says that $\mathbf{E}_{\chi}\leq\mathbf{E}_{\rho}$, and so the
index of $\mathbf{E}_{\chi}$ at $S$ is at least $k$. ∎

##  5\. The general case

In this section, we generalize Theorem C to the situation in which the
quadratic differentials are not necessarily squares of abelian differentials.
We then specialize to dimension $3$ and give the proof of Theorem D.

###  5.1. The spectral curve

Let $S_{0}$ be a point of $\mathbf{T}_{g}$, and let $\phi_{1},\ldots,\phi_{n}$
be nonzero holomorphic quadratic differentials on $S_{0}$ summing to zero. To
this data, there is an associated spectral curve. By this, we mean a branched
covering $S$ of $S_{0}$ and abelian differentials $\alpha_{i}$ on $S$ that
square to the pullback of $\phi_{i}$, which is terminal in the sense that if
the $\alpha_{i}$ lift to squares on some other branched cover $R\to S_{0}$,
then this factors through a branched covering from $R\to S$. It is always a
$2^{n}$-fold branched covering of $S_{0}$, but may be disconnected, for
instance if any $\phi_{i}$ is already a square. By universality, $S$ has $n$
holomorphic involutions $\tau_{i}$, each of which negates $\alpha_{i}$ and
fixes $\alpha_{j}$ for $j\neq i$.

We let $\rho$ be the action of $\pi_{1}(S_{0})$ on the product $X$ of the
$\mathbb{R}$-trees $(T_{i},2d_{i})$ corresponding to the quadratic
differentials $\phi_{i}$, and $\pi$ the canonical equivariant map from
$\tilde{S}_{0}$ to $X$.

Since $S$ has $n$ abelian differentials whose squares sum to zero, the theory
of the previous section applies. For instance, we can integrate
$\mathrm{Re}(\tilde{\alpha}_{i})$ on a simply connected covering space to get
a harmonic map $h$ to $\mathbb{R}^{n}$, equivariant under a representation
$\chi$ of the Deck group, and well defined up to a constant on each component
of $S$. The energy density of this map descends not only to $S$, but all the
way to $S_{0}$, where it is equal to the energy density of $\pi$.

In the spirit of Proposition 4.8, we want to compare the index of
$\mathbf{E}_{\rho}$ through self-maps at $S_{0}$ to the index of the quadratic
form $Q$ associated to $h$. But to get the right comparison, we need to
restrict $Q$ to a subspace of the space of normal variations. Let
$G\cong(\mathbb{Z}/2\mathbb{Z})^{n}$ be the group generated by the $\tau_{i}$.
Let $\sigma$ be the action of $G$ on $\mathbb{R}^{n}$ such that each
$\tau_{i}$ acts by reflection in the $i$th coordinate hyperplane. Let
$NV^{\sigma}$ be the space of normal variations of $h$ that are
$\sigma$-equivariant.

######  Proposition 5.1.

The index of $\mathbf{E}_{\rho}$ by self-maps is equal to the index of $Q$ on
$NV^{\sigma}$.

###### Proof.

Let $k$ be the index of $Q$ on $NV^{\sigma}$. The first thing we want to do is
use Proposition 4.8 to find a $k$-dimensional space of $\sigma$-equivariant
normal variations on $S-B$ on which $Q$ is still negative definite. This works
fine if we choose our cutoff function $\eta_{\delta}$ to be
$\tau_{i}$-invariant. For instance, we can define $\eta_{\delta}$ to be the
pull-back to $S$ of the similarly-defined function on $S_{0}$; then the
dependence of the energy of $\eta_{\delta}$ with $\delta$ is the same up to a
factor of 2 coming from the relation $\log(|\sqrt{z}|)=\log(|z|)/2$.

Next, for every $W$ in this space, we get $n$ tangential vector fields
$V_{i}=W^{i}\nabla x^{i}/|\nabla x^{i}|^{2}$ on $S$, as in Proposition 4.6.
Since both $W^{i}$ and $dx^{i}$ transform the same way under each $\tau_{j}$,
we have $\tau_{j}(V_{i})=\pm W^{i}(\pm\nabla x^{i})/|\nabla x^{i}|^{2}$, where
each sign is $+$ if $i\neq j$ and $-$ if $i=j$. Hence each $V_{i}$ descends to
a vector field on $S_{0}-B$, which we still call $V_{i}$.

For each $i$, let $f_{i}^{W}:\mathbb{R}\times S\to S$,
$f_{i}^{W}(r,\cdot)=f_{i}^{r}(\cdot)$ be the flow of $V_{i}$. Let $h_{i}$ be
the component functions of $h$, $h_{i}^{r}=h_{i}\circ f_{i}^{r}$, and
$h_{r}=(h_{1}^{r},\dots,h_{n}^{r}).$ The conformal structures $S_{r}$ for
which each $h_{r}$ is conformal are still $G$-invariant, hence descend to
conformal structures $(S_{0})_{r}$ on $\Sigma_{g}$. Let $\pi^{r}_{i}=\pi\circ
f_{i}^{r}$. Even though the tree $T_{i}$ no longer folds to $\mathbb{R}$, the
energy density of $\pi^{r}_{i}$ on $(S_{0})_{r}$ is still pointwise equal to
the energy density of $h^{r}_{i}$ on $S^{t}$; indeed, both are equal to
$|(f_{i}^{r})^{*}\mathrm{Re}(\alpha_{i})|^{2}$. Therefore, the second
derivative of $\mathcal{E}_{\rho}$ is equal to $Q$ on this $k$-dimensional
space so the index of $\mathbf{E}_{\rho}$ by self-maps is at least $k$.

The other inequality is easier. If the index of $\mathbf{E}_{\rho}$ by self-
maps is $k$, then we can use the log-cutoff trick to find a $k$-dimensional
space of vector fields $V_{i}$ supported on $S_{0}-B$ and variations $\mu_{i}$
of conformal structure on which $L$ is negative definite. Lifting everything
to $S$ and differentiating the coordinate functions, we get a $k$-dimensional
space of equivariant variations of $h$ for which the second derivative of
energy is negative definite. Taking the normal components of these variations,
and using that energy dominates area, we get a $k$-dimensional subspace of
$NV^{\sigma}$ on which $Q$ is negative definite. ∎

###  5.2. Unstable minimal surfaces in $\mathbb{R}^{n}$

In order to finish the proof of Theorem A, we need to construct for each
$g\geq 2$ and $n\geq 3$, either an unstable equivariant minimal surface $S$ of
genus $g$ in $\mathbb{R}^{n}$, or a surface $S_{0}$ of genus $g$ whose
spectral curve is a $(\mathbb{Z}/2\mathbb{Z})^{n}$-equivariantly unstable
minimal surface in $\mathbb{R}^{n}$.

If $g\geq 3$, then as we discuss in the next section, there are plenty of
equivariant minimal surfaces of genus $g$ in $\mathbb{R}^{n}$. They are not
always unstable; for instance, if the minimal map is holomorphic with respect
to some complex structure on a linear subspace of $\mathbb{R}^{n}$, then it is
calibrated by the Kähler form, and hence stable. In general, it is not
straightforward to decide if a minimal surface in a flat space is unstable.

A special case is when the equivariant minimal surface is contained in a real
2-plane, and hence is stable. We call such a minimal surface flat. These at
least are easy to identify.

######  Proposition 5.2.

Let $\phi_{1},\dots,\phi_{n}\in QD(S_{0})$ sum to $0$, giving a
$\chi$-equivariant map $h:\tilde{S}\to\mathbb{R}^{n}$ as before. The vector
valued second fundamental form $k$ of $h(\tilde{S})$ vanishes identically if
and only if the quadratic differentials $\phi_{i}$ are all complex multiples
of one another.

###### Proof.

Let $h_{1},\dots,h_{n}$ denote the coordinate functions of $h$. Since
$\phi_{i}=((h_{i})_{z})^{2}dz^{2}$, the quadratic differentials are all
complex multiples of one another if and only if the functions $(h_{i})_{z}$
are. In one direction, assume $(h_{i})_{z}=a_{i}f(z)$ for some function $f(z)$
and some complex constants $a_{i}$. Then the image of the
$\mathbb{R}^{n}$-valued $1$-form $dh$ is contained in a two-dimensional
subspace, and by integrating, we see that image of $h$ is contained in an
affine subspace of $\mathbb{R}^{n}$. In particular, it is totally geodesic, so
the second fundamental form is zero. Conversely, if the second fundamental
form is zero, then the image of $dh$ is contained in some two-dimensional
linear subspace, and so the image of $h_{z}$ is contained in the
complexification of that subspace, which is two-dimensional. As $h$ is weakly
conformal, $\langle h_{z},h_{z}\rangle=0$; since the inner product is
nondegenerate on the complexification of any real two-dimensional subspace,
this shows that $h_{z}$ is contained in a complex line (we use analyticity to
deduce this as well at the branch points), and so the functions $(h_{i})_{z}$
are all complex multiples of one another. ∎

For the remaining section, we restrict to $n=3$. For $n\geq 3$, any isometric
inclusion of $\mathbb{R}^{3}$ into $\mathbb{R}^{n}$ gives examples in
$\mathbb{R}^{n}$. Let $\mathbf{M}_{g}$ be the moduli space of Riemann surfaces
of genus $g$, and let $E^{n}$ be the total space of the bundle over
$\mathbf{M}_{g}$ consisting of $n$-tuples of quadratic differentials that sum
to $0$. Instability of the corresponding equivariant minimal surfaces in
$\mathbb{R}^{n}$ is an open condition on $E^{n}$, so by perturbing the
$3$-dimensional examples we get many more.

###  5.3. Equivariant minimal surfaces in $\mathbb{R}^{3}$

Every non-flat equivariant minimal surface in $\mathbb{R}^{3}$ is unstable.
Indeed, in dimension 3, the expression $|\langle k,W\rangle|^{2}$ in the
formula for $Q(W)$ is equal to $2|K||W|^{2}$, where $K$ is the Gauss curvature
of the equivariant minimal surface. The normal bundle to the minimal surface
$h(\tilde{S})$ is a real line bundle on $S$. Since $S$ is always orientable,
the normal bundle is as well, and hence it is equivariantly trivial. If $N$ is
a unit normal section, and $\eta$ is any function on $S$, then the second
variation formula (13) takes the form

$Q_{0}(\eta N)=\int_{\Sigma}|\nabla\eta|^{2}-|K|^{2}|\eta|^{2}.$

As long as the curvature $K$ is anywhere nonzero, a constant section of $N$
will therefore be destabilizing: for $\eta=1,$

$Q_{0}(N)=\int_{\Sigma}-|K|^{2}<0.$

When $g\geq 3$, the moduli space of $(S,\alpha_{1},\alpha_{2},\alpha_{3})$,
where $S$ is a Riemann surface of genus $g$ and $\alpha_{i}$ are abelian
differentials on $S$ whose squares sum to zero, but are not all mutliples of
one another, is nonempty and has complex dimension $3g$ (in [4, Section 6] it
is shown that the quotient by the natural free actions of $\mathbb{C}^{*}$ and
$\textrm{SO}(3,\mathbb{C})$ has dimension $3g-4$). This proves Theorem A for
$g\geq 3$.

######  Remark 5.3.

In fact, in [21, Theorem 16], Ros proves that every non-flat minimally
immersed surface of genus $g$ in a 3-torus has index at least $2g/3-1$. The
result easily generalizes to any non-flat equivariant minimal immersion for
any representation, but we emphasize that it applies only to immersed
surfaces.

Unfortunately, there are no non-flat equivariant minimal surfaces of genus 2
in $\mathbb{R}^{3}$, stable or not. This is because the canonical map lands in
$\mathbb{P}^{1}$, so the canonical curve cannot be contained in a rank 3
quadric (or see the comment after Proposition 5.6). Hence, we are forced to
study $\sigma$-equivariant deformations of the spectral curve. The key that
makes this work is that the normal bundle of the spectral curve $S$ of
$(S_{0},\phi_{1},\phi_{2},\phi_{3})$ can be equivariantly trivial even if the
$\phi_{i}$ are not squares (in which case $S$ is just $8$ copies of $S_{0}$).

######  Proposition 5.4.

Suppose that the sextic differential $\phi_{1}\phi_{2}\phi_{3}$ is the square
of a cubic differential $c$. Then there is a $\sigma$-equivariant deformation
of $S$ of constant length 1.

###### Proof.

The cubic differential $c$ distinguishes two components of $S$; one on which
$\alpha_{1}\alpha_{2}\alpha_{3}=c$, and one on which it is equal to $-c$. Each
$\tau_{i}$ interchanges the two components of $S$. The subgroup
$\Gamma<(\mathbb{Z}/2\mathbb{Z})^{3}$ preserving the components acts on
$\mathbb{R}^{3}$ in an orientation-preserving way. Indeed, for each element
$\gamma\in\Gamma$, the determinant of the matrix describing the product of
hyperplane reflections is equal to the product of the monodromies of the
$\alpha_{i}$ under the action of $\gamma$. We can use the orientation of
$\mathbb{R}^{3}$, together with the orientation of the component of $S$, to
equivariantly orient the normal bundle. Since the normal bundle is a line
bundle, it therefore has an equivariant section of constant length. ∎

######  Remark 5.5.

If each $\phi_{i}$ is the square of an abelian differential $\alpha_{i}$, then
clearly $\phi_{1}\phi_{2}\phi_{3}=c^{2}$ with
$c=\alpha_{1}\alpha_{2}\alpha_{3}$.

If the quadratic differentials $\phi_{i}$ are not complex multiples of one
another, then neither are their lifts $\alpha_{i}$ to the spectral curve.
Hence, the minimal map from the lift of the spectral curve is non-flat, so any
$\sigma$-equivariant deformation of constant length will be destabilizing.

The final step is to show that there are non-flat solutions even in genus 2 to
the equations $\phi_{1}\phi_{2}\phi_{3}=c$ and $\phi_{1}+\phi_{2}+\phi_{3}=0$.
For any $g\geq 2$, let $\mathcal{P}_{g}$ be the moduli space of genus $g$
Riemann surfaces $S$ together with a triple of quadratic differentials
$\phi_{i}$ summing to zero whose product is a square and which are not all
complex multiples of one another.

######  Proposition 5.6.

The moduli space $\mathcal{P}_{2}$ has dimension $3$.

###### Proof.

Consider the three dimensional family of algebraic curves
$w^{2}=z(z-1)(z-a)(z-b)(z-c)$ for $(0,1,a,b,c)$ distinct complex numbers. This
is a finite covering of the moduli space of genus 2 Riemann surfaces. Every
holomorphic quadratic differential on a curve in this family is of the form
$p(z)(dz)^{2}/w^{2}$ for $p(z)$ a polynomial of degree at most 2. If the roots
of $p(z)$ are branch points of the curve, then the quadratic differential
vanishes to order two at the corresponding point of the curve. For arbitrary
$a$ and $b$, and $c$ to be determined, let

$\displaystyle\phi_{1}$ $\displaystyle=z(z-1)\frac{dz^{2}}{w^{2}}$
$\displaystyle\phi_{2}$ $\displaystyle=\mu(a,b)(z-a)\frac{dz^{2}}{w^{2}},$

where $\mu(a,b)=-b(b-1)/(b-a)$ is chosen so that $\phi_{1}+\phi_{2}$ vanishes
at $b$ (equivalently, that the corresponding quadratic polynomial vanishes at
$b$). A short computation shows that the other root of the polynomial for
$\phi_{1}+\phi_{2}$ is at $a(b-1)/(b-a)$, so if this happens to be the value
of $c$, then the sextic differential $\phi_{1}\phi_{2}\phi_{3}$ vanishes to
order two at each of the six branch points of the curve (including $\infty$).
Hence it is the square of the cubic differential $dz^{3}/w^{2}$, which
vanishes to order one at each of these points. Including a parameter rescaling
$\phi_{1}$, $\phi_{2}$, and $\phi_{3}$, this shows that $\mathcal{P}_{2}$ has
dimension 3. ∎

For example we could take $a=-1,b=i,$ and $c=-i$ to get a solution on the
hyperelliptic curve $w^{2}=z^{5}-z$. This suffices for the proof of Theorem A.

######  Remark 5.7.

Note that the triples of quadratic differentials in genus 2 are squares of
abelian differentials since the polynomials $z(z-1)$, etc., are not squares.
However, they still have even order zeros.

Together with [4, Section 6] and the Remark 5.5, this shows:

######  Proposition 5.8.

For every genus $g\geq 2$, $\mathcal{P}_{g}$ is nonempty and every component
has complex dimension at least $3g-3$.

We give a self-contained proof of this proposition, since it is very brief in
the reference.

###### Proof.

We have already proved this for genus 2. The canonical map of a hyperelliptic
curve of genus 3 is the vanishing locus of a nondegenerate quadric on
$\mathbb{CP}^{2}$; diagonalizing this quadric gives three abelian
differentials whose squares sum to zero on the curve. By Remark 5.5, these
give points in $\mathcal{P}_{3}$. Since the hyperelliptic locus has dimension
5, we get a sixth dimension from rescaling the abelian differentials. This
proves the result for $g=3$.

In general, taking unramified coverings of a point in $\mathcal{P}_{2}$ shows
that $\mathcal{P}_{g}$ is nonempty for every $g$. To get the bound on
dimension, we observe that $\mathcal{P}_{g}$ is, up to a double cover, the
intersection in the total space of the bundle $H^{0}(K^{3})$ over
$\mathbf{M}_{g}$ (dimension $14(g-1)$) of the sextic differentials that are
squares of cubic differenitals (dimension $8(g-1)$) and those that are the
product of three independent quadratic differentials summing to zero
(dimension $9(g-1)$). This gives a lower bound on the dimension of
$8(g-1)+9(g-1)-14(g-1)=3(g-1)$. ∎

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