On cyclic Higgs bundles

Abstract

In this paper, we derive a maximum principle for a type of elliptic systems and apply it to analyze the Hitchin equation for cyclic Higgs bundles. We show several domination results on the pullback metric of the (possibly branched) minimal immersion f associated to cyclic Higgs bundles. Also, we obtain a lower and upper bound of the extrinsic curvature of the image of f. As an application, we give a complete picture for maximal $Sp(4,{\mathbb {R}})$-representations in the $2g-3$ Gothen components and the Hitchin components.

1 Introduction

Let S be a closed, oriented surface of genus $g\ge 2$ and G be a reductive Lie group. Let $\Sigma$ be a Riemann surface over S and denote its canonical line bundle by $K_{\Sigma }$. A G-Higgs bundle over $\Sigma$ is a pair $(E,\phi )$ where E is a holomorphic vector bundle and $\phi$ is a holomorphic section of $End(E)\otimes K_{\Sigma }$ plus extra conditions depending on G. The non-abelian Hodge theory developed by Corlette [11], Donaldson [14], Hitchin [16] and Simpson [24], provides a one-to-one correspondence between the moduli space of representations from $\pi _1(S)$ to G with the moduli space of G-Higgs bundles over $\Sigma$. The correspondence is through looking for an equivariant harmonic map from ${\widetilde{\Sigma }}$ to the symmetric space G / K, where K is the maximal compact subgroup of G, for a given representation $\rho$.

In this paper, we are interested in the direction of the non-abelian Hodge correspondence from the moduli space of Higgs bundles to the space of equivariant harmonic maps. More explicitly, given a stable G-Higgs bundle $(E,\phi )$ on $\Sigma$, there exists a unique Hermitian metric h compatible with G-structure satisfying the Hitchin equation

$$\begin{aligned} F^{\nabla ^{h}}+[\phi ,\phi ^{*_{h}}]=0, \end{aligned}$$

called the harmonic metric, which gives the equivariant harmonic map from ${\widetilde{\Sigma }}$ to G / K. So for a given Higgs bundle $(E,\phi )$, we would like to deduce geometric properties of the corresponding equivariant harmonic map: ${\widetilde{\Sigma }}\rightarrow G/K$.

We are particularly interested in cyclic Higgs bundles. They are $SL(n,{\mathbb {C}})$-Higgs bundles of the following form

$$\begin{aligned} E=L_1\oplus L_2\oplus \cdots \oplus L_n, \quad \phi = \left( \begin{array}{cccccc} 0 &{}\quad &{}\quad &{}\quad \gamma _n\\ \gamma _1 &{}\quad 0 &{}\quad &{}\quad \\ &{}\quad \ddots &{}\quad \ddots &{}\quad \\ &{}\quad &{}\quad \gamma _{n-1} &{}\quad 0 \end{array} \right) :E\rightarrow E\otimes K_{\Sigma }, \end{aligned}$$

(1)

where $L_k$ is a holomorphic line bundle and $\gamma _k$ is a holomorphic section of $L_{k}^{-1}L_{k+1}K_{\Sigma }$, $k=1,\ldots , n$$(L_{n+1}=L_1$). Here $\gamma _k\ne 0$, $k=1,\ldots ,n-1$. Call such a Higgs bundle $(E,\phi )$ a cyclic Higgs bundle parameterized by $\{\gamma _1,\ldots ,\gamma _n\}$. For G a subgroup of $SL(n,{\mathbb {C}})$, we call $(E,\phi )$ a cyclic G-Higgs bundle if it is a G-Higgs bundle and it is cyclic as an $SL(n,{\mathbb {C}})$-Higgs bundle.

The terminology “cyclic Higgs bundles” first appeared in [2]. The notion here is slightly different from the one in [2], where the notion “cyclic” there is referred to the group G. One may also view cyclic Higgs bundles as a special type of quiver bundles in [1]. Cyclic Higgs bundles are special in G-Higgs bundles for G of higher rank. The harmonic metric for a cyclic Higgs bundle is diagonal, making it possible to analyze the solution to the Hitchin equation and hence the corresponding harmonic map. So studying cyclic Higgs bundles could give us a hint on predicting what may happen to general Higgs bundles.

If a representation $\rho :\pi _1(S)\rightarrow SL(n,{\mathbb {C}})$ corresponds to a cyclic Higgs bundle over one Riemann surface $\Sigma$, it might be possible that $\rho$ also corresponds to a cyclic Higgs bundle over another Riemann surface $\Sigma '$. By Labourie [19], this cannot happen for Hitchin representations into split real rank 2 Lie groups. Later Collier [7] generalizes this result to maximal $Sp(4,{\mathbb {R}})$-representations.

If $n\ge 3$, the associated harmonic map for a stable cyclic Higgs bundle is conformal and hence is a (possibly branched) minimal immersion. In [12], the authors studied the pullback metric and curvature of the minimal immersion for cyclic Higgs bundles in the Hitchin component (the $q_n$ case). In this paper, we derive a maximum principle for the elliptic systems. The maximum principle is very useful for the Toda-type equation with variable coefficients, which appears in the Hitchin equation for cyclic Higgs bundles. With this powerful tool, we generalize and improve the results in [12] and discover some new phenomena.

1.1 Monotonicity of pullback metrics

Let $(E,\phi )$ be a stable cyclic Higgs bundle and $n\ge 3$. Let f be the corresponding harmonic map and it is in fact a branched minimal surface. The Riemannian metric on $SL(n,{\mathbb {C}})/SU(n)$ is induced by the Killing form on $sl(n,{\mathbb {C}})$. Then the pullback metric of f is given by

$$\begin{aligned} g=2n\cdot \text {tr}(\phi \phi ^{*_{h}})dz\otimes d\bar{z}, \end{aligned}$$

where h is the harmonic metric. Though at branch points $g=0$, we still call g a “metric”.

There is a natural ${\mathbb {C}}^*$-action on the moduli space $M_{Higgs}$ of $SL(n,{\mathbb {C}})$-Higgs bundles:

$$\begin{aligned} {\mathbb {C}}^*\times {\mathcal {M}}_{Higgs}\longrightarrow & {} {\mathcal {M}}_{Higgs}\\ t\cdot [(E,\phi )]= & {} [(E,t\phi )] \end{aligned}$$

Theorem 1.1

Let $(E,\phi )$ be a stable cyclic Higgs bundle. Then along the ${\mathbb {C}}^*$-orbit of $(E,\phi )$, outside the branched points, as |t| increases, the pullback metric $g^t$ of the corresponding branched minimal immersions strictly increases.

We will discuss the stability of cyclic Higgs bundles in Sect. 2.5.

Remark 1.2

If we integrate the pullback metric, it gives the Morse function f (up to a constant scalar) on the moduli space of Higgs bundles as the $L^2$-norm of $\phi$:

$$\begin{aligned} f(E,\phi )=\int _{\Sigma }\text {tr}(\phi \phi ^{*_h}) \sqrt{-1}dz\wedge d{\bar{z}}. \end{aligned}$$

(2)

As a corollary, along the ${\mathbb {C}}^*$-orbit of a stable cyclic Higgs bundle $(E,\phi )$, the Morse function $f(E,t\phi )$ strictly increases as |t| increases. The Morse function is the main tool to determine the topology of the moduli space of Higgs bundles, for example, in Hitchin [16, 17], Gothen [15]. In fact, Hitchin in [16] showed that with respect to the Kähler metric on the moduli space, the gradient flow of the Morse function is exactly the ${\mathbb {R}}^*$-part of ${\mathbb {C}}^*$-action. Hence, along the ${\mathbb {C}}^*$-orbit of any Higgs bundles $(E,\phi )$, the Morse function $f(E,t\phi )$ strictly increases as |t| increases.

Here we improve the integral monotonicity to pointwise monotonicity along the ${\mathbb {C}}^*$-orbit of cyclic Higgs bundles.

Consider a family of stable cyclic Higgs bundles $(E,\phi ^t)$ parameterized by $\{\gamma _1,\ldots ,t\gamma _n\}$. For $t\in {\mathbb {C}}^*$, the family $(E,\phi ^t)$ is gauge equivalent to $(E,t^{\frac{1}{n}}\phi )$. If the cyclic Higgs bundle parameterized by $\{\gamma _1,\ldots ,\gamma _{n-1},0\}$ is also stable, we extend the monotonicity of ${\mathbb {C}}^*$-family to the ${\mathbb {C}}$-family.

Theorem 1.3

Let $(E,\phi ^t)$ be a family of stable cyclic Higgs bundle parameterized by $\{\gamma _1,\ldots ,t\gamma _n\}$ for $t\in {\mathbb {C}}$. Then outside the branched points, as |t| increases, the pullback metric $g^t$ of corresponding branched minimal immersions strictly increases.

Remark 1.4

If the cyclic Higgs bundle parameterized by $\{\gamma _1,\ldots ,\gamma _{n-1},0\}$ is stable, it lies in the moduli space of Higgs bundles and is fixed by the ${\mathbb {C}}^*$-action. Note that the fixed points of ${\mathbb {C}}^*$-action are exactly the critical points of the Morse function as shown in Hitchin [16].

1.2 Curvature of cyclic Higgs bundles in the Hitchin component

The cyclic Higgs bundles in the Hitchin component are of the following form

$$\begin{aligned} E=K^{\frac{n-1}{2}}\oplus K^{\frac{n-3}{2}}\oplus \cdots \oplus K^{\frac{3-n}{2}}\oplus K^{\frac{1-n}{2}},\quad \phi = \left( \begin{array}{cccccc} 0 &{}\quad &{}\quad &{}\quad q_n\\ 1 &{}\quad 0 &{}\quad &{}\quad \\ &{}\quad \ddots &{}\quad \ddots &{}\quad \\ &{}\quad &{}\quad 1 &{}\quad 0 \end{array} \right) , \end{aligned}$$

(3)

where $q_n$ is a holomorphic n-differential. If $q_n=0$, the Higgs bundle is called Fuchsian.

We then study the geometry of the corresponding equivariant harmonic maps into the symmetric space $SL(n,{\mathbb {R}})/SO(n)$.

Theorem 1.5

Let $(E,\phi )$ be a cyclic Higgs bundle in the Hitchin component. Let $\sigma$ be the tangent plane of the image of f, then the curvature $K_{\sigma }$ in $SL(n,{\mathbb {R}})/SO(n)$ satisfies

$$\begin{aligned} -\frac{1}{n(n-1)^2}\le K_{\sigma }<0. \end{aligned}$$

Remark 1.6

The sectional curvature K of $SL(n,{\mathbb {R}})/SO(n)$ and $SL(n,{\mathbb {C}})/SU(n)$ satisfies $-\frac{1}{n}\le K\le 0$ (see Proposition 5.1). For general Higgs bundles, one should not expect there is such a nontrivial lower bound at immersed points. For example, in the case of cyclic Higgs bundles parameterized by $\{\gamma _1,\ldots ,\gamma _n\}$ where $n-1$ terms of $\gamma _i$’s vanish at point p, $K_{\sigma }(p)=-\frac{1}{n}$.

Remark 1.7

• (1) The upper bound is shown in [12]. Here we give a new proof. As shown in [9], along the family of cyclic Higgs bundles parameterized by $\{1,\ldots ,1,tq_n\}$, $K_{\sigma }^t$ approaches to 0 away from the zeros of $q_n$ as $|t|\rightarrow \infty$. So the upper bound $K_{\sigma }<0$ is sharp.

• (2) The lower bound $-\frac{1}{n(n-1)^2}$ can only be achieved at some point in the case $n=2,3$.

• (3) In the Fuchsian case, i.e. $q_n=0$, the sectional curvature $K_{\sigma }$ is $-\frac{6}{n^2(n^2-1)}$. However, it is strictly larger than the lower bound of $K_{\sigma }$ for $q_n\ne 0$ case when $n>3$ (see Proposition 5.9). Hence, one cannot expect the curvature in the Fuchsian case could serve as a lower bound of $K_{\sigma }$ for general Hitchin representations.

1.3 Comparison inside the real Hitchin fibers

The Hitchin fibration is a map from the moduli space of $SL(n,{\mathbb {C}})$-Higgs bundles to the direct sum of the holomorphic differentials

$$\begin{aligned} h:M_{Higgs}\longrightarrow & {} \bigoplus \limits _{j=2}^nH^0(\Sigma , K^j)\ni (q_2,q_3,\ldots ,q_n)\nonumber \\ (E,\phi )\longmapsto & {} (\text {tr}(\phi ^2),\text {tr}(\phi ^3),\ldots ,\text {tr}(\phi ^n)). \end{aligned}$$

(4)

Each Hitchin fiber contains a special point: the intersection with the Hitchin component. Note that cyclic Higgs bundles $(E,\phi )$ lie in the Hitchin fiber at $(0,\ldots ,0,n\cdot q_n)$, where $q_n=(-1)^{n-1}\det \phi$.

In Proposition 6.1, we show that the harmonic metric in the cyclic Higgs bundle in the Hitchin component dominates the ones for other cyclic $SL(n,{\mathbb {R}})$-Higgs bundles in the same Hitchin fiber in a certain sense. As the applications in lower rank $n=2,3,4$, we compare the pullback metrics of the corresponding harmonic maps.

Theorem 1.8

Let $(\tilde{E},{\tilde{\phi }})$ be a cyclic Higgs bundle in the Hitchin component and $(E,\phi )$ be a distinct stable cyclic $SL(n,{\mathbb {R}})$-Higgs bundle in Sect. 2.3 such that $\det \phi =\det {{\tilde{\phi }}}$. For $n=2,3,4$, the pullback metrics $g,\tilde{g}$ of the corresponding harmonic maps satisfy $g<\tilde{g}.$

Under the assumptions above, the Morse function satisfies $f(E,\phi )<f(\tilde{E},\tilde{\phi })$.

By Hitchin’s work in [16], all stable $SL(2,{\mathbb {R}})$-Higgs bundles are cyclic. Apply Theorem 1.8 to $SL(2,{\mathbb {R}})$-representations, we recover the following result shown in [13].

Corollary 1.9

For any non-Fuchsian irreducible $SL(2,{\mathbb {R}})$-representation $\rho$ and any Riemann surface $\Sigma$, there exists a Fuchsian representation j such that the pullback metric of the corresponding j-equivariant harmonic map $f_j:{\widetilde{\Sigma }}\rightarrow {\mathbb {H}}^2$ dominates the one for $f_{\rho }$.

Remark 1.10

Deroin and Tholozan in [13] show a stronger result by comparing Fuchsian representations with all $SL(2,{\mathbb {C}})$-representations. Inspired by this result, they conjecture that in the Hitchin fiber, the Hitchin section maximizes the translation length. Our Theorem 1.8 here is exactly in the same spirit, but using the pullback metric rather than the translation length.

We expect that Theorem 1.8 holds for general Higgs bundles rather than just cyclic Higgs bundles.

Conjecture 1.11

Let $(\tilde{E},{\tilde{\phi }})$ be a Higgs bundle in the Hitchin component and $(E,\phi )$ be a distinct polystable $SL(n,{\mathbb {C}})$-Higgs bundle in the same Hitchin fiber. Then the pullback metrics $g,\tilde{g}$ of corresponding harmonic maps satisfy $g<\tilde{g}.$

As a result, the Morse function satisfies $f(E,\phi )<f(\tilde{E},\tilde{\phi })$.

1.4 Maximal $Sp(4,{\mathbb {R}})$-representations

For each reductive representation $\rho$ into a noncompact Hermitian Lie group G, we can define the Toledo invariant $\tau (\rho )$ satisfying the Milnor-Wood inequality $|\tau (\rho )|\le \text {rank}(G)(g-1)$ shown in [6]. One can also check a general purely Higgs bundle definition of the Toledo invariant in [3]. The representation $\rho$ with $|\tau (\rho )|=\text {rank}(G)(g-1)$ is called maximal. Maximal representations are Anosov [5] and hence discrete and faithful.

In the case for $Sp(4,{\mathbb {R}})$, there are $3\cdot 2^{2g}+2g-4$ connected components of maximal representations containing $2^{2g}$ isomorphic components of Hitchin representations [17] and $2g-3$ exceptional components called Gothen components [15]. Labourie in [19] shows that any Hitchin representation for $Sp(4,{\mathbb {R}})$ corresponds to a cyclic Higgs bundle over a unique Riemann surface. Together with the description in [4, 15] and Collier’s work [7], any maximal $Sp(4,{\mathbb {R}})$-representation in the Gothen components corresponds to a cyclic Higgs bundle over a unique Riemann surface of the form

$$\begin{aligned} E=N\oplus NK^{-1}\oplus N^{-1}K\oplus N^{-1}, \quad \phi = \left( \begin{array}{cccccc} 0 &{}\quad &{}\quad &{}\quad \nu \\ 1 &{}\quad 0 &{}\quad &{}\quad \\ &{}\quad \mu &{}\quad 0 &{}\quad \\ &{}\quad &{}\quad 1 &{}\quad 0 \end{array} \right) , \end{aligned}$$

(5)

where $g-1<\deg N<3g-3$. Note that if $N=K^{\frac{3}{2}}$, the above Higgs bundle corresponds to a Hitchin representation. As a result, for any $Sp(4,{\mathbb {R}})$-representation in the Hitchin components or the Gothen components, there is a unique $\rho$-equivariant minimal immersion of $\widetilde{S}$ in $Sp(4,{\mathbb {R}})/U(2)$. Recently, this result is reproved and generalized to maximal SO(2, n)-representations in Collier–Tholozan–Toulisse [10].

The above cyclic Higgs bundles with $\nu =0$ play a similar role as the Fuchsian case: they are the fixed points of the ${\mathbb {C}}^*$-action. We call the corresponding representations $\mu$-Fuchsian representations. The only difference with the Fuchsian case is that they form a subset inside each component rather than one single point. As a corollary of Theorem 1.3, the space of $\mu$-Fuchsian representations serves as the minimum set in its component of maximal $Sp(4,{\mathbb {R}})$-representations in the following sense.

Corollary 1.12

For any maximal $Sp(4,{\mathbb {R}})$-representation $\rho$ in the Gothen components (or the Hitchin components), there exists a $\mu$-Fuchsian (or Fuchsian) representation j in the same component of $\rho$ such that the pullback metric of the unique j-equivariant minimal immersion $f_j:\widetilde{S}\rightarrow Sp(4,{\mathbb {R}})/U(2)$ is dominated by the one for $f_{\rho }$.

To consider the curvature, as a corollary of Theorem 1.5 and Proposition 5.9, we have

Corollary 1.13

For any Hitchin representation $\rho$ for $Sp(4,{\mathbb {R}})$, the sectional curvature $K_{\sigma }$ in $Sp(4,{\mathbb {R}})/U(2)$ of the tangent plane $\sigma$ of the unique $\rho$-equivariant minimal immersion satisfies

• (1) $K_{\sigma }=-\frac{1}{40}$, if $\rho$ is Fuchsian;

• (2) $-\frac{1}{36}<K_{\sigma }<0$ and $\exists ~ p$ such that $K_{\sigma }(p)<-\frac{1}{40},$ if $\rho$ is not Fuchsian.

Similarly, we obtain an upper and lower bound on the curvature of minimal immersions for maximal representations.

Theorem 1.14

For any maximal $Sp(4,{\mathbb {R}})$-representation $\rho$ in the Gothen components, the sectional curvature $K_{\sigma }$ in $Sp(4,{\mathbb {R}})/U(2)$ of tangent plane $\sigma$ of the unique $\rho$-equivariant minimal immersion satisfies

• (1) $-\frac{1}{8}\le K_{\sigma }<-\frac{1}{40}$ and the lower bound is sharp, if $\rho$ is $\mu$-Fuchsian;

• (2) $-\frac{1}{8}\le K_{\sigma }<0$, if $\rho$ is not $\mu$-Fuchsian.

Remark 1.15

As shown in [21], along the family of $(E,t\phi )$, away from zeros of $\det \phi \ne 0$, the sectional curvature goes to zero as $|t|\rightarrow \infty$. So the upper bounds in Part (2) in both Corollary 1.13 and Theorem 1.14 are sharp.The sectional curvature K in $Sp(4,{\mathbb {R}})/U(2)$ satisfies $-\frac{1}{4}\le K\le 0$. So the lower bounds in Corollary 1.13 and Theorem 1.14 are nontrivial.

As an immediate corollary of Theorem 1.8 for $n=4$, comparing maximal representations in the Gothen components with Hitchin representations, we obtain

Corollary 1.16

For any maximal $Sp(4,{\mathbb {R}})$-representation $\rho$ in the Gothen components, there exists a Hitchin representation $\delta$ such that the pullback metric of the unique $\delta$-equivariant minimal immersion $f_{\delta }:\widetilde{S}\rightarrow Sp(4,{\mathbb {R}})/U(2)$ dominates the one for $f_{\rho }$.

1.5 Maximum principle

We derive a maximum principle for the elliptic systems. It is the main tool we use throughout this paper.

Basically, we consider the following linear elliptic system

$$\begin{aligned} \triangle _{g} u_i+\langle X,\nabla u_i \rangle +\sum _{j=1}^{n}c_{ij}u_j=f_i, \quad 1\le i\le n. \end{aligned}$$

Roughly speaking, suppose the functions $c_{ij}$’s satisfy the following assumptions:

• (a) cooperative: $c_{ij}\ge 0,~ i\ne j$,

• (b) column diagonally dominant: $\sum _{i=1}^{n}c_{ij}\le 0,~ 1\le j\le n$,

• (c) fully coupled: the index set $\{1,\ldots ,n\}$ cannot be split up in two disjoint nonempty sets $\alpha ,\beta$ such that $c_{ij}\equiv 0$ for $i\in \alpha ,j\in \beta .$

Then the maximum principle holds, that is, if $f_i\le 0$ for $1\le i\le n$, then $u_i\ge 0$ for $1\le i\le n$. The precise statement is Lemma 3.1.

In the literature, it is common to require the existence of a positive supersolution, which is equivalent to the maximum principle, see [20]. So for variable coefficients, people usually suppose $c_{ij}$ satisfy the row sum condition $\sum _{j=1}^{n}c_{ij}\le 0,~ 1\le i\le n$, say [23]. The column sum condition $\sum _{i=1}^{n}c_{ij}\le 0,~ 1\le j\le n$, or in other words column diagonally dominant condition, rarely appeared in the literature. The similar column sum condition first appeared in [20, Theorem 3.3].

To the knowledge of the authors, the maximum principles in the literature do not directly imply Lemma 3.1.

Structure of the article The article is organized as follows. In Sect. 2, we recall some fundamental results about the Higgs bundle and introduce the cyclic Higgs bundles. In Sect. 3, we show a maximum principle for the elliptic systems, the main tool of this article. In Sect. 4, we show the monotonicity of the pullback metrics of the branched minimal immersions. In Sect. 5, we obtain a lower and upper bound for the extrinsic curvature of the minimal immersions for cyclic Higgs bundles in the Hitchin component. In Sect. 6, we compare the harmonic metrics of cyclic Higgs bundle in the Hitchin component with other cyclic $SL(n,{\mathbb {R}})$-Higgs bundles in the same Hitchin fiber. In Sect. 7, we apply our results to maximal $Sp(4,{\mathbb {R}})$-representations.

2 Preliminaries

In this section, we recall some facts in the theory of the Higgs bundles. One may refer to [2, 12, 19]. Let $\Sigma$ be a closed Riemann surface of genus $\ge 2$ and $K=K_{\Sigma }$ be the canonical line bundle over $\Sigma$. For $p\in \Sigma$, let $\pi _1=\pi _1(\Sigma ,p)$ be the fundamental group of $\Sigma$. Let ${\tilde{\Sigma }}$ be the universal cover of $\Sigma$.

An $SL(n,{\mathbb {C}})$-Higgs bundle over $\Sigma$ is a pair $(E,\phi )$, where E is a holomorphic vector bundle with $\det E={\mathcal {O}}$ and $\phi$ is a trace-free holomorphic section of $End(E)\bigotimes K$. We call $(E,\phi )$ is stable if for any proper $\phi$-invariant holomorphic subbundle F, $\frac{\text {deg}F}{\text {rank}F}<\frac{\text {deg}E}{\text {rank}E}$. We call $(E, \phi )$ is polystable if $(E,\phi )$ is a direct sum of stable Higgs bundles of degree 0.

2.1 Higgs bundles and harmonic maps

Theorem 2.1

(Hitchin [16] and Simpson [24]) Let $(E,\phi )$ be a stable $SL(n,{\mathbb {C}})$-Higgs bundle. Then there exists a unique Hermitian metric h on E compatible with $SL(n,{\mathbb {C}})$ structure, called the harmonic metric, solving the Hitchin equation

$$\begin{aligned} F^{\nabla ^h}+[\phi ,\phi ^{*_h}]=0, \end{aligned}$$

(6)

where ${\nabla ^h}$ is the Chern connection of h, in local holomorphic trivialization,

$$\begin{aligned} F^{\nabla ^h}={\overline{\partial }}(h^{-1}\partial h), \end{aligned}$$

and $\phi ^{*_{h}}$ is the adjoint of $\phi$ with respect to h, in the sense that

$$\begin{aligned} h(\phi (u),v)=h(u,\phi ^{*_{h}}(v)),\quad u,v\in E, \end{aligned}$$

in local frame, $\phi ^{*_{h}}=\bar{h}^{-1}{\bar{\phi }}^{\top }\bar{h}$.

Denote

$$\begin{aligned}&G=SL(n,{\mathbb {C}}),\quad K=SU(n)\\&\mathfrak {g}=sl(n,{\mathbb {C}}),\quad \mathfrak {k}=su(n),\quad \mathfrak {p}=\{X\in sl(n,{\mathbb {C}}):{\bar{X}}^t=X\},\quad \mathfrak {g}=\mathfrak {k}\oplus \mathfrak {p}.\\&\text {The Killing form } B(X,Y)=2n\cdot \text {tr}(XY) \end{aligned}$$

The harmonic metric h gives rise to a flat $SL(n,{\mathbb {C}})$ connection $D=\nabla ^h+\Phi =\nabla ^{h}+\phi +\phi ^{*_{h}}$. The holonomy of D gives a representation $\rho :\pi _1\rightarrow SL(n,{\mathbb {C}})$ and the bundle (E, D) is isomorphic to ${\widetilde{\Sigma }}\times _{\rho }{\mathbb {C}}^n$ with the associated flat connection. A Hermitian metric h on E is equivalent to a reduction $i:P_K\rightarrow P_G$ from the frame bundle $P_G={\widetilde{\Sigma }}\times _{\rho }G$ of $E={\widetilde{\Sigma }}\times _{\rho }{\mathbb {C}}^n$ to the unitary frame bundle $P_K$ of E with respect to h. Then the metric descends to be a section of $P_G/K={\widetilde{\Sigma }}\times _{\rho }G/K$ over $\Sigma$. Equivalently, it gives a $\rho$-equivariant map $f: {\tilde{\Sigma }}\rightarrow G/K$. Denote by $\tilde{P}_K$ the pullback of the principle K-bundle $G\rightarrow G/K$ by f. Note that $\pi ^*P_K=\tilde{P}_K$, where $\pi$ is the covering map $\pi : {\tilde{\Sigma }}\rightarrow \Sigma$. The Maurer–Cartan form $\omega$ of G gives a flat connection on $P_G$, we still use $\omega$ to denote the connection. It coincides with the flat connection D. Consider $i^{*}\omega$, which is a $\mathfrak {g}$-valued one form on $P_K$. Decomposing $i^{*}\omega =A+\Phi$ from $\mathfrak {g}=\mathfrak {k}\oplus \mathfrak {p}$, where A is $\mathfrak {k}$-valued and $\Phi$ is $\mathfrak {p}$-valued. Then $A\in \Omega ^1(P_K,\mathfrak {k})$ is a connection on $P_K$ and $\Phi$ is a section of $T^*\Sigma \otimes (P_{K}\times _{Ad_{K}}\mathfrak {p})$. By complexification, $\Phi$ is also a section of

$$\begin{aligned}&(T^*{\Sigma }\otimes {\mathbb {C}})\otimes ({P}_{K}\times _{Ad_{K}}\mathfrak {p}\otimes {\mathbb {C}})\\&\quad =(T^*{\Sigma }\otimes {\mathbb {C}})\otimes ({P}_{K^{{\mathbb {C}}}}\times _{Ad_{K^{{\mathbb {C}}}}}\mathfrak {p}^{{\mathbb {C}}}) \\&\quad =(K\oplus \bar{K})\otimes ({P}_G\times _{Ad_G}\mathfrak {g})=(K\oplus \bar{K})\otimes End_0(E)\end{aligned}$$

where $End_0(E)$ the trace-free endomorphism bundle of E. With respect to the decomposition $(K\oplus \bar{K})\otimes End_0(E)$, $\Phi =\phi +\phi ^*$.

With respect to the decomposition $\mathfrak {g}=\mathfrak {k}+\mathfrak {p}$, we can decompose $\omega =\omega ^{\mathfrak {k}}+\omega ^{\mathfrak {p}}$, where $\omega ^{\mathfrak {k}}\in \Omega ^1(G,\mathfrak {k}), \omega ^{\mathfrak {p}}\in \Omega ^1(G,\mathfrak {p})$. Moreover, $\omega ^{\mathfrak {p}}$ descends to be an element in $\Omega ^1(G/K,G\times _{Ad_K}\mathfrak {p})$. In fact, using the Maurer–Cartan form $\omega ^{\mathfrak {p}}\in \Omega ^1(G/K,G\times _{Ad_K}\mathfrak {p})$ over G / K: $T(G/K)\cong G\times _{Ad_K}\mathfrak {p}$. Then $\mathfrak {g}=\mathfrak {k}\oplus \mathfrak {p}$ gives an $Ad_K$-invariant orthogonal decomposition and the Killing form B on $\mathfrak {g}$ is positive on $\mathfrak {p}$. The Killing form B induces a Riemannian metric $\tilde{B}$ on G / K: for two vectors $Y_1,Y_2\in T_p(G/K)$,

$$\begin{aligned} \tilde{B}(Y_1,Y_2)=B(\omega ^{\mathfrak {p}}(Y_1),\omega ^{\mathfrak {p}}(Y_2)). \end{aligned}$$

Then $f^*\omega ^{\mathfrak {p}}$ is a section of $T^*{\tilde{\Sigma }}\otimes (\tilde{P}_K\times _{Ad_K}\mathfrak {p})$ over ${\tilde{\Sigma }}$.

By comparing the two decompositions of the Maurer–Cartan form $\omega$, we obtain:

$$\begin{aligned} f^*\omega ^{\mathfrak {p}}=\pi ^*\Phi . \end{aligned}$$

So for every tangent vector $X\in T{{\widetilde{\Sigma }}}$, under the isomorphism by the Maurer–Cartan form

$$\begin{aligned} \omega ^{\mathfrak {p}}:T(G/K)\cong G\times _{Ad_K}\mathfrak {p}, \end{aligned}$$

we have

$$\begin{aligned} \omega ^{\mathfrak {p}}(f_*(X))=f^*\omega ^{\mathfrak {p}}(X)=\pi ^*\Phi (X)=\Phi (\pi _*(X)). \end{aligned}$$

(7)

We consider the pullback metric g on $\Sigma$, $g=\pi _{*}f^{*}\tilde{B}$. Since f is $\rho$-equivariant and $\tilde{B}$ is G-invariant, g is well defined. Then $\forall X,Y\in T\Sigma$, locally choose any lift $\tilde{X},\tilde{Y}\in T{\tilde{\Sigma }}$,

$$\begin{aligned} g(X,Y)=\tilde{B}(f_*(\tilde{X}),f_*(\tilde{Y}))=B(\omega ^{\mathfrak {p}}(f_*(\tilde{X})),\omega ^{\mathfrak {p}}(f_*(\tilde{Y}))=B(\Phi (X),\Phi (Y)). \end{aligned}$$

Later in the paper, we may ignore this covering map $\pi$ for short. Then we have

$$\begin{aligned} \text {Hopf}(f)=g^{2,0}=2n\text {tr}(\phi \phi ),\quad g^{1,1}=2n\text {tr}(\phi \phi ^{*_{h}})dz\otimes d\bar{z}. \end{aligned}$$

If $\text {Hopf}(f)=0$, then as a section of $K\otimes \bar{K}$, the Hermitian metric is

$$\begin{aligned}g=g^{1,1}=2n\text {tr}(\phi \phi ^{*_{h}})dz\otimes d{\bar{z}}.\end{aligned}$$

The associated Riemannian metric of g is $g+\bar{g}$ on $\Sigma$, i.e., $2n\text {tr}(\phi \phi ^{*_{h}})dz\cdot d\bar{z}$, where

$$\begin{aligned} dz\cdot d\bar{z}=dz\otimes d\bar{z}+d\bar{z}\otimes dz=2|dz|^2=2(dx^2+dy^2). \end{aligned}$$

We focus on the cyclic Higgs bundles introduced below.

2.2 Cyclic Higgs bundles

A cyclic Higgs bundle is an $SL(n,{\mathbb {C}})$-Higgs bundle $(E,\phi )$ of the form (1), where $L_k$ is a holomorphic line bundle over $\Sigma$ and $\gamma _k$ is a holomorphic section of $L_{k}^{-1}L_{k+1}K$, $k=1,\ldots , n$. The subscript is counted modulo n, i.e., $n+1\equiv 1$. Here $\det {E}={\mathcal {O}}$ and $\gamma _k\ne 0$, $k=1,\ldots ,n-1$.

We will discuss the stability of the cyclic Higgs bundles in Sect. 2.5.

Following the proof in Baraglia [2], Collier [7, 8], we have

Proposition 2.2

For a stable cyclic Higgs bundle $(E,\phi )$, the harmonic metric h is diagonal, i.e.

$$\begin{aligned} h=\text {diag}(h_1,h_2,\ldots , h_n) \end{aligned}$$

where each $h_k$ is a Hermitian metric on $L_{k}$.

Denote $L\otimes \bar{L}=|L|^2$, then $h_k$ is a smooth section of $|L_k|^{-2}$. Given a local holomorphic frame, we use $\gamma _k$ also to denote the local coefficient function of the section $\gamma _k$. Then locally the Hitchin equation is

$$\begin{aligned} \triangle \log h_k+|\gamma _k|^2h_k^{-1}h_{k+1}-|\gamma _{k-1}|^2h_{k-1}^{-1}h_k=0,\quad k=1,\ldots , n, \end{aligned}$$

where $\triangle =\partial _z\partial _{{\bar{z}}}$, $|\gamma _k|^2=\gamma _k{\bar{\gamma }}_k$ as a local function.

When $n\ge 3$, the Hopf differential of the harmonic map $\text {Hopf}(f)=\text {tr}(\phi ^2)=0$. And f is immersed at p if and only if $\phi (p)\ne 0$. At a point p where $\phi (p)=0$, f is branched at p. Then outside the branch points, the harmonic map is conformal. The pullback metric is given by

$$\begin{aligned} g=2n\text {tr}(\phi \phi ^{*_{h}})=2n\left( \sum _{k=1}^{n}|\gamma _k|^2h_k^{-1}h_{k+1}\right) dz\otimes d\bar{z}. \end{aligned}$$

Remark 2.3

For $n=2$, we consider the (1, 1) part of the pullback metric g instead.

2.3 Cyclic $SL(n,{\mathbb {R}})$-Higgs bundles

An $SL(n,{\mathbb {R}})$-Higgs bundle over $\Sigma$ is a triple $(E,\phi ,Q)$, where $(E,\phi )$ is an $SL(n,{\mathbb {C}})$-Higgs bundle and Q is a non-degenerate holomorphic quadratic form on E such that $Q(\phi u,v)=Q(u,\phi v)$ for $u,v\in E$. Such a $(E,\phi ,Q)$ corresponds to a representation

$$\begin{aligned} \rho :\pi _1\rightarrow SL(n,{\mathbb {R}})\hookrightarrow SL(n,{\mathbb {C}}). \end{aligned}$$

For $n=2m$, the cyclic $SL(n,{\mathbb {R}})$-Higgs bundle is of the following form

$$\begin{aligned} E= & {} L_1\oplus \cdots \oplus L_m\oplus L_m^{-1}\oplus \cdots \oplus L_1^{-1},\quad \\ \phi= & {} \left( \begin{array}{cccccccc} 0 &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \nu \\ \gamma _1 &{}\quad \ddots &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \\ &{}\quad \ddots &{}\quad 0 &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \\ &{}\quad &{}\quad \gamma _{m-1} &{}\quad 0 &{}\quad &{}\quad &{}\quad &{}\quad \\ &{}\quad &{}\quad &{}\quad \mu &{}\quad 0 &{}\quad &{}\quad &{}\quad \\ &{}\quad &{}\quad &{}\quad &{}\quad \gamma _{m-1} &{}\quad 0 &{}\quad &{}\quad \\ &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \ddots &{}\quad \ddots &{}\quad \\ &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \gamma _{1} &{}\quad 0 \end{array} \right) , \end{aligned}$$

and the holomorphic quadratic form

(8)

For a stable cyclic $SL(n,{\mathbb {R}})$-Higgs bundle, by Proposition 2.2, $h=\text {diag}(h_1,\ldots ,h_m,h_{m}^{-1},\ldots ,h_1^{-1})$. Locally, the Hitchin equation is

$$\begin{aligned} \triangle \log h_1+|\gamma _1|^2h_1^{-1}h_{2}-|\nu |^2h_{1}^{2}= & {} 0,\\ \triangle \log h_k+|\gamma _k|^2h_k^{-1}h_{k+1}-|\gamma _{k-1}|^2h_{k-1}^{-1}h_k= & {} 0,\quad k=2,\ldots , m-1,\\ \triangle \log h_m+|\mu |^2h_m^{-2}-|\gamma _{m-1}|^2h_{m-1}^{-1}h_m= & {} 0. \end{aligned}$$

The pullback metric is $g=2n(|\nu |^2h_{1}^{2}+|\mu |^2h_m^{-2}+2\sum _{k=1}^{m-1}|\gamma _k|^2h_k^{-1}h_{k+1})dz\otimes d\bar{z}.$

For $n=2m+1$, the cyclic $SL(n,{\mathbb {R}})$-Higgs bundle is of the following form

$$\begin{aligned} E= & {} L_1\oplus \cdots \oplus L_m\oplus {\mathcal {O}}\oplus L_m^{-1}\oplus \cdots \oplus L_1^{-1},\\ \phi= & {} \left( \begin{array}{ccccccccc} 0 &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \nu \\ \gamma _1 &{}\quad \ddots &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \\ &{}\quad \ddots &{}\quad 0 &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \\ &{}\quad &{}\quad \gamma _{m-1} &{}\quad 0 &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \\ &{}\quad &{}\quad &{}\quad \mu &{}\quad 0 &{}\quad &{}\quad &{}\quad &{}\quad \\ &{}\quad &{}\quad &{}\quad &{}\quad \mu &{}\quad 0 &{}\quad &{}\quad &{}\quad \\ &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \gamma _{m-1} &{}\quad 0 &{}\quad &{}\quad \\ &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \ddots &{}\quad \ddots &{}\quad \\ &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \gamma _{1} &{}\quad 0 \end{array} \right) \end{aligned}$$

and Q is of the form (8). In this case, $h=\text {diag}(h_1,\ldots ,h_m,1,h_{m}^{-1},\ldots ,h_1^{-1})$. Locally, the Hitchin equation is

$$\begin{aligned} \triangle \log h_1+|\gamma _1|^2h_1^{-1}h_{2}-|\nu |^2h_{1}^{2}= & {} 0,\\ \triangle \log h_k+|\gamma _k|^2h_k^{-1}h_{k+1}-|\gamma _{k-1}|^2h_{k-1}^{-1}h_k= & {} 0,\quad k=2,\ldots , m-1,\\ \triangle \log h_m+|\mu |^2h_m^{-1}-|\gamma _{m-1}|^2h_{m-1}^{-1}h_m= & {} 0. \end{aligned}$$

The pullback metric is $g=2n(|\nu |^2h_{1}^{2}+2|\mu |^2h_m^{-1}+2\sum _{k=1}^{m-1}|\gamma _k|^2h_k^{-1}h_{k+1})dz\otimes d\bar{z}.$

2.4 Hitchin fibration and cyclic Higgs bundles in the Hitchin component

The Hitchin fibration is a map from $M_{Higgs}(SL(n,{\mathbb {C}}))$ to $\bigoplus _{j=2}^nH^0(\Sigma , K^j)$ given by (4). In [17], Hitchin defines a section $s_h$ of this fibration whose image consists of stable Higgs bundles with the corresponding flat connections having holonomy in $SL(n,{\mathbb {R}}).$ Furthermore, the section $s_h$ maps surjectively to the connected component (called Hitchin component) of the $SL(n,{\mathbb {R}})$-Higgs bundle moduli space which naturally contains an embedded copy of Teichmüller space. The Teichmüller locus corresponds to the image of $(q_2,0,\ldots ,0)$. Such a $(E,\phi )$ corresponds to a representation $\rho$ which can be factored through $SL(2,{\mathbb {R}})$,

$$\begin{aligned} \rho :\pi _1\rightarrow SL(2,{\mathbb {R}})\xrightarrow {\iota } SL(n,{\mathbb {R}})\hookrightarrow SL(n,{\mathbb {C}}), \end{aligned}$$

where $\iota$ is the canonical irreducible representation. The cyclic Higgs bundles in the Hitchin component correspond to the image of $s_h$ at $(0,\ldots ,0,n\cdot q_n)$ and they are of the form (3).

In the Fuchsian case, i.e., $q_n=0$, the solution to the Hitchin equation is explicit: for $n=2m$,

$$\begin{aligned} h^{-1}_{k}h_{k+1}=\frac{1}{2}k(n-k)g_{0},\quad 1\le k\le m-1,\qquad h^{-2}_{m}=\frac{1}{2}m^2g_0; \end{aligned}$$

for $n=2m+1$,

$$\begin{aligned} h^{-1}_{k}h_{k+1}=\frac{1}{2}k(n-k)g_{0},\quad 1\le k\le m-1,\qquad h^{-1}_m=\frac{1}{2}m(m+1)g_0. \end{aligned}$$

Here $g_0$ is the hyperbolic metric such that $\triangle \log g_0=g_0$.

2.5 Stability for cyclic Higgs bundles

In this subsection, let $(E,\phi )$ be a cyclic Higgs bundle parameterized by $\{\gamma _1,\ldots ,\gamma _n\}$ of the form (1) and we discuss its stability.

Proposition 2.4

For $\gamma _n=0$, $(E,\phi )$ is stable if and only if $\sum _{i=1}^k\deg (L_{n+1-i})<0$ for all $1\le k\le n-1$.

Proof

Let Z be the union of the zeros of $\gamma _k$, $1\le k\le n-1$. At $p\in \Sigma {\setminus } Z$, it is easy to see $\phi$-invariant subspaces are of the form $\bigoplus _{i=1}^k L_{n+1-i}$ for $1\le k\le n$. Since Z is discrete, a proper $\phi$-invariant subbundle must be of the form $\bigoplus _{i=1}^k L_{n+1-i}$ for $1\le k\le n-1$. We obtain the desired result by definition. $\square$

Proposition 2.5

For $\gamma _n\ne 0$, if $\sum _{i=1}^k\deg (L_{n+1-i})<0$ for all $1\le k\le n-1$, then $(E,\phi )$ is stable.

Proof

We use the same argument in [17, p. 453]. Since stability is an open condition, we have $(E,\phi ^{t})$ is stable for small t, where $\phi ^t$ is parameterized by $\{\gamma _1,\ldots ,\gamma _{n-1},t\gamma _n\}$. Choose $\mu =t^{-\frac{1}{n}}$. Since stability holds under ${\mathbb {C}}^*$-action, $(E, \mu \phi ^t)$ is stable. But $(E,\phi )$ is gauge equivalent to $(E,\mu \phi ^t)$ because the holomorphic structure on E does not change under $\beta$ and $\phi =\beta ^{-1}(\mu \phi ^t)\beta$, where $\beta =diag(1,\mu ,\mu ^2,\ldots ,\mu ^{n-1})$ is a gauge transformation on E. Then $(E,\phi )$ is stable. $\square$

We also show the stability of $(E,\phi )$ for the case $\gamma _n\ne 0$ under a different assumption.

Proposition 2.6

For $\gamma _n\ne 0$, if there does not exist a holomorphic differential $\omega$ such that $\gamma _1\gamma _2\cdots \gamma _n=\omega ^{l}$ for some positive integer $l>1$, l divides n, then $(E,\phi )$ is stable.

Proof

We will prove that there is no holomorphic proper $\phi$-invariant subbundle V. Suppose there is such a subbundle V of rank k, let W be a smooth complement of V and has the induced holomorphic structure as the quotient bundle E / V. With respect to the smooth splitting $E=V+W$, the holomorphic structure on E and Higgs field are decomposed as

$$\begin{aligned} {\bar{\partial }}_E=\left( \begin{array}{cc} {\bar{\partial }}_V &{}\quad A\\ &{}\quad {\bar{\partial }}_W \end{array} \right) , \quad \phi =\left( \begin{array}{cc} \phi _V &{}\quad \beta \\ &{}\quad \phi _W \end{array} \right) . \end{aligned}$$

The holomorphicity of $\phi$ implies ${\bar{\partial }}_V\phi _V=0$ and ${\bar{\partial }}_W\phi _W=0$. Consider the characteristic polynomial of $\phi$, we have

$$\begin{aligned} \det (\phi -\lambda I)=\det (\phi _V-\lambda I)\det (\phi _W-\lambda I) \end{aligned}$$

and

$$\begin{aligned} \det (\phi -\lambda I)=(-1)^n\lambda ^n+\det \phi . \end{aligned}$$

For each point $p\in \Sigma$, choose a local holomorphic coordinate chart, then $\det \phi$ is a local holomorphic function and at p it is a complex number c. Let $x_1,x_2,\ldots , x_n$ be the roots of $(-1)^n\lambda ^n+c=0$.

Then at p, $\det (\phi _V-\lambda I)$ is of the form $(x_{i_1}-\lambda )(x_{i_2}-\lambda )\cdots (x_{i_k}-\lambda )$ and $\det \phi _V=x_{i_1}x_{i_2}\cdots x_{i_k}$. So at p, we obtain

$$\begin{aligned} (\det \phi _V)^n=(\det \phi )^k. \end{aligned}$$

(9)

Since $\det \phi _V,\det \phi$ are both globally defined holomorphic differentials, Eq. (9) holds globally.

Let $(n,k)=d$, $n=n_1d,k=k_1d$. We have $(\frac{(\det \phi _V)^{n_1}}{(\det \phi )^{k_1}})^d=1,$ implying that $\frac{(\det \phi _V)^{n_1}}{(\det \phi )^{k_1}}$ is a holomorphic function and must be a constant. Therefore there exists a holomorphic k-differential $\omega _1$ such that $(\det \phi )^{k_1}=\omega _1^{n_1}.$ Since $(n_1,k_1)=1$, there exists integer a, b such that $an_1+bk_1=1$. Then we have

$$\begin{aligned} \det \phi =(\det \phi )^{an_1+bk_1}=((\det \phi )^a\omega _1^{b})^{n_1}. \end{aligned}$$

So we finish the proof since $\det \phi =(-1)^{n-1}\gamma _1\cdots \gamma _n$. $\square$

Remark 2.7

It may happen that the cyclic Higgs bundle $(E,\phi )$ is not stable if there is a holomorphic differential $\omega$ such that $\gamma _1\gamma _2\cdots \gamma _n=\omega ^{l}$ for some positive integer $l>1$, l divides n. For instance, for $n=4,k=2$, let $E=K^{\frac{1}{2}}\oplus K^{-\frac{1}{2}}\oplus K^{\frac{1}{2}}\oplus K^{-\frac{1}{2}}$, $\phi$ be parameterized by $\{1,q_2,1,q_2\}$ for $q_2$ being a holomorphic quadratic differential. Consider the holomorphic bundle map $s:V=K^{\frac{1}{2}}\oplus K^{-\frac{1}{2}}\rightarrow E$ by $s=\begin{pmatrix}1&{}\quad 0\\ 0&{}\quad 1\\ 1&{}\quad 0\\ 0&{}\quad 1\end{pmatrix}$. Since the map s is of rank 2 everywhere, the image s(V) is a holomorphic subbundle of E of rank 2, which is biholomorphic to $K^{\frac{1}{2}}\oplus K^{-\frac{1}{2}}$. Clearly s(V) is $\phi$-invariant.

3 Maximum principle for systems

The main tool we use in this paper is the following maximum principle for systems on the surface $\Sigma$ equipped with a Riemannian metric g. We abuse the same notation g to denote both the metric $g(z)dz\otimes d{\bar{z}}$ and the local function g(z) on the surface. Define $\triangle _{g}=g^{-1}\triangle$, which is globally defined, called the Laplacian with respect to the metric $gdz\otimes d{\bar{z}}$.

Lemma 3.1

For each $1\le i\le n$, let $u_i$ be a $C^2$ function on $\Sigma {\setminus } P_i$, where $P_i$ is an isolated subset of $\Sigma$ ($P_i$ can be empty). Suppose $u_i$ approaches to $+\infty$ around $P_i$. Let $P=\bigcup _{i=1}^{n} P_i$. Let $c_{ij}$’s be continuous and bounded functions on $\Sigma {\setminus } P$, $1\le i,j\le n$. Suppose $c_{ij}$’s satisfy the following assumptions: in $\Sigma {\setminus } P$,

• (a) cooperative: $c_{ij}\ge 0,~ i\ne j$,

• (b) column diagonally dominant: $\sum _{i=1}^{n}c_{ij}\le 0,~ 1\le j\le n$,

• (c) fully coupled: the index set $\{1,\ldots ,n\}$ cannot be split up in two disjoint nonempty sets $\alpha ,\beta$ such that $c_{ij}\equiv 0$ for $i\in \alpha ,j\in \beta .$

Let $f_i$ be non-positive continuous functions on $\Sigma {\setminus } P$, $1\le i\le n$ and X be a continuous vector field on $\Sigma$. Suppose $u_i$ satisfies

$$\begin{aligned} \triangle _{g} u_i+\langle X,\nabla u_i \rangle +\sum _{j=1}^{n}c_{ij}u_j=f_i \text { in } \Sigma {\setminus } P, \quad 1\le i\le n. \end{aligned}$$

Consider the following conditions:

• Condition (1)  $(f_1,\ldots ,f_n)\ne (0,\dots ,0)$, i.e., there exists $i_{0}\in \{1,\ldots , n\}$, $p_0\in \Sigma {\setminus } P$, such that $f_{i_0}(p_0)\ne 0$;

• Condition (2)  P is nonempty;

• Condition (3)  $\sum _{i=1}^n u_i\ge 0$.

Then either condition (1) or (2) imply $u_{i}> 0$, $1\le i\le n$. And condition (3) implies either $u_{i}> 0$, $1\le i\le n$ or $u_{i}\equiv 0$, $1\le i\le n$.

Proof

Let $A=\{1,\ldots ,n\}$. For $S\subseteq A$, set $u_S=\Sigma _{i\in S}u_i$. If $S=\phi$, set $u_S=0$. Let $P_S=\bigcup _{i\in S}P_i$. Then

$$\begin{aligned} \triangle _{g} u_{S}+\langle X,\nabla u_S \rangle +\sum _{i\in A}\sum _{l\in S}c_{li}u_i\le 0\quad \text {in } \Sigma {\setminus } P. \end{aligned}$$

Then

$$\begin{aligned} \triangle _{g} u_{S}+\langle X,\nabla u_S \rangle +\sum _{j\notin S}\sum _{l\in S}c_{lj}u_j+\sum _{k\in S}\sum _{l\in S}c_{lk}u_{k}\le 0\quad \text {in } \Sigma {\setminus } P. \end{aligned}$$

Then for $S\ne \phi$,

$$\begin{aligned}&\triangle _{g} u_{S}+\langle X,\nabla u_S \rangle +\sum _{j\notin S}\sum _{l\in S}c_{lj}(u_{\{j\}\cup S}-u_{S})\\&\quad +\sum _{k\in S}\left( \sum _{l\in S}c_{lk}\right) (u_{S}-u_{S{\setminus } \{k\}})\le 0\text { in } \Sigma {\setminus } P. \end{aligned}$$

Set

$$\begin{aligned} b_{S}=\min _{\Sigma } u_{S}, \quad \check{b}_{S}=\min _{j\notin S,k\in S}\{b_{\{j\}\cup S},b_{S{\setminus } \{k\}}\},\quad b=\min _{S\subseteq A} b_{S}, \end{aligned}$$

Notice that all these constants are finite. By the assumptions (a)(b), in $\Sigma {\setminus } P$, $c_{lj}\ge 0$ for $l\in S,j\notin S$, and $\sum _{l\in S}c_{lk}\le 0$ for $k\in S$, then

$$\begin{aligned}&\triangle _{g} \check{b}_{S}+\langle X,\nabla \check{b}_S \rangle +\sum _{j\notin S}\sum _{l\in S}c_{lj}(u_{\{j\}\cup S}-\check{b}_{S})\\&\qquad +\sum _{k\in S}\left( \sum _{l\in S}c_{lk}\right) (\check{b}_{S}-u_{S{\setminus } \{k\}})\ge 0 \text { in } \Sigma {\setminus } P. \end{aligned}$$

Then

$$\begin{aligned}&\triangle _{g} \left( u_{S}-\check{b}_{S}\right) +\left\langle X,\nabla \left( u_{S}-\check{b}_{S}\right) \right\rangle \\&\qquad +\left( -\sum _{j\notin S}\sum _{l\in S}c_{lj}+\sum _{k\in S}\left( \sum _{l\in S}c_{lk}\right) \right) \left( u_{S}-\check{b}_{S}\right) \le 0\text { in } \Sigma {\setminus } P. \end{aligned}$$

Step 1: We show that under either condition (1) or (2), $u_{S}\ge \check{b}_{S}$ for any $S\subset A$; under condition (3), $u_{S}\ge \check{b}_{S}$ for $S\subsetneq A$. In particular, $b_{S}\ge \check{b}_{S}$ for $S\subset A$ under condition (1) and (2) and for $S\subsetneq A$ under condition (3).

If not, since $u_{S}-\check{b}_{S}$ approaches to $+\infty$ around $P_S$ and is continuous outside $P_S$, $u_{S}-\check{b}_{S}$ must attain a negative minimum in $\Sigma {\setminus } P_S$. First, we suppose $u_{S}-\check{b}_{S}$ is not a constant. By the assumptions (a)(b), in $\Sigma {\setminus } P$,

$$\begin{aligned} -\sum _{j\notin S}\sum _{l\in S}c_{lj}+\sum _{k\in S}\left( \sum _{l\in S}c_{lk}\right) \le 0. \end{aligned}$$

Then by the strong maximum principle for a single equation, the minimal point $p\notin \Sigma {\setminus } P$. So $p\in P{\setminus } P_S$. Since P is isolated, we consider $p_n\in \Sigma {\setminus } P$, $p_n\rightarrow p$. Then

$$\begin{aligned}&\limsup _{p_narrow p}\left( \phantom {\sum _{j\notin S}}\triangle _{g} (u_{S}-\check{b}_{S})+\langle X,\nabla (u_{S}-\check{b}_{S})\rangle \right. \\&\quad \quad \left. +\left( -\sum _{j\notin S}\sum _{l\in S}c_{lj}+\sum _{k\in S}\left( \sum _{l\in S}c_{lk}\right) \right) (u_{S}-\check{b}_{S}) \right) (p_n)\le 0. \end{aligned}$$

By the continuity,

$$\begin{aligned}&\lim _{p_n\rightarrow p} \left( \triangle _{g} (u_{S}-\check{b}_{S})+\langle X,\nabla (u_{S}-\check{b}_{S}) \rangle \right) (p_n)\nonumber \\&\quad = \left( \triangle _{g} (u_{S}-\check{b}_{S})+\langle X,\nabla (u_{S}-\check{b}_{S}) \rangle \right) (p)\ge 0. \end{aligned}$$

So

$$\begin{aligned} \limsup _{p_n\rightarrow p} \left( \left( -\sum _{j\notin S}\sum _{l\in S}c_{lj}+\sum _{k\in S}\left( \sum _{l\in S}c_{lk}\right) \right) (u_{S}-\check{b}_{S}) \right) (p_n)\le 0. \end{aligned}$$

If there exists a subsequence $p_{n_k}$ such that $(-\sum _{j\notin S}\sum _{l\in S}c_{lj}+\sum _{k\in S}(\sum _{l\in S}c_{lk}))(p_{n_k})$ approaches to a negative number, then

$$\begin{aligned} \lim _{p_{n_k}\rightarrow p} \left( \left( -\sum _{j\notin S}\sum _{l\in S}c_{lj}+\sum _{k\in S}\left( \sum _{l\in S}c_{lk}\right) \right) (u_{S}-\check{b}_{S}) \right) (p_{n_k})>0. \end{aligned}$$

Contradiction. Since $P_S$ is isolated, we have $-\sum _{j\notin S}\sum _{l\in S}c_{lj}+\sum _{k\in S}(\sum _{l\in S}c_{lk})$ is continuous in $\Sigma {\setminus } P_S$. Then

$$\begin{aligned}&\triangle _{g} \left( u_{S}-\check{b}_{S}\right) +\left\langle X,\nabla \left( u_{S}-\check{b}_{S}\right) \right\rangle \nonumber \\&\quad \quad +\left( -\sum _{j\notin S}\sum _{l\in S}c_{lj}+\sum _{k\in S}\left( \sum _{l\in S}c_{lk}\right) \right) \left( u_{S}-\check{b}_{S}\right) \le 0\text { in } \Sigma {\setminus } P_S, \end{aligned}$$

Then by the strong maximum principle for a single equation, $u_{S}-\check{b}_{S}$ cannot achieve its negative minimum in $\Sigma {\setminus } P_S$ unless it is a constant. Contradiction. Second, if $u_{S}-\check{b}_{S}$ is a negative constant, then by the assumptions (a)(b), in $\Sigma {\setminus } P$,

$$\begin{aligned} -\sum _{j\notin S}\sum _{l\in S}c_{lj}+\sum _{k\in S}\left( \sum _{l\in S}c_{lk}\right) \equiv 0. \end{aligned}$$

Then in $\Sigma {\setminus } P$, $\sum _{l\in S}c_{lk}\equiv 0$ for $k\in S$. Then by the assumptions (a)(b), $c_{ij}\equiv 0$ in $\Sigma {\setminus } P$, for $j\in S, i\notin S$, which is a contradiction to the assumption (c) unless $S=A$. If $S=A$, for condition (2), we have $u_S$ cannot be a constant. And for condition (1), $u_{S}-\check{b}_{S}$ is a negative constant implies $\sum _{i\in A}f_i\equiv 0$, which also gives a contraction. So we obtain $u_{S}\ge \check{b}_{S}$ on the whole $\Sigma$. For condition (3), we obtain $u_{S}\ge \check{b}_{S}$ for $S\subsetneq A$. So we finish the claim.

Step 2: We show $b=0$.

Since $u_{S}=0$ for $S=\phi$, we have $b \le 0$. If $b<0$, suppose b is achieved by $S_0$, and $|S_0|$ is the smallest among all minimizers. Then $S_0\ne \phi$. Under condition (1) and (2), $u_{S_0}\ge \check{b}_{S_0}$ is automatically true. Under condition (3), we have $u_A\ge 0$ and hence $S_0\subsetneq A$, $u_{S_0}\ge \check{b}_{S_0}$.

Since $c_{ij}$ are bounded, suppose $-\sum _{j\notin S_0}\sum _{l\in S_0}c_{lj}+\sum _{k\in S_0}(\sum _{l\in S_0}c_{lk})\ge -M$, where M is a positive constant. Then in $\Sigma {\setminus } P$,

$$\begin{aligned}&\triangle _{g} \left( u_{S_0}-\check{b}_{S_0} \right) +\left\langle X,\nabla \left( u_{S_0}-\check{b}_{S_0} \right) \right\rangle -M(u_{S_0}-\check{b}_{S_0})\\&\quad \quad \le -\left( M+ \left( -\sum _{j\notin S_0}\sum _{l\in S_0}c_{lj}+\sum _{k\in S_0} \left( \sum _{l\in S_0}c_{lk} \right) \right) \right) \left( u_{S_0}-\check{b}_{S_0} \right) . \end{aligned}$$

We have proved $u_{S_0}-\check{b}_{S_0}\ge 0$. Then by the continuity,

$$\begin{aligned} \triangle _{g} (u_{S_0}-\check{b}_{S_0})+\langle X,\nabla (u_{S_0}-\check{b}_{S_0}) \rangle -M(u_{S_0}-\check{b}_{S_0})\le 0 \text { in } \Sigma {\setminus } P_{S_0}. \end{aligned}$$

Since $b\le \check{b}_{S_0}\le b_{S_0}$ and $u_{S_0}$ achieves b, we have $\check{b}_{S_0}=b$. Then by the strong maximum principle, $u_{S_{0}}\equiv \check{b}_{S_0}=b$. Then

$$\begin{aligned} \triangle _{g} b+\langle X,\nabla b \rangle +\sum _{j\notin S_0}\sum _{l\in S_0}c_{lj}(u_{\{j\}\cup S_0}-b)+\sum _{k\in S_0}\left( \sum _{l\in S_0}c_{lk}\right) (b-u_{S_0{\setminus } \{k\}})\le 0\text { in } \Sigma {\setminus } P. \end{aligned}$$

Then by the assumptions (a)(b),

$$\begin{aligned} \left( \sum _{l\in S_0}c_{lk}\right) (b-u_{S_0{\setminus } \{k\}})\equiv & {} 0\text { in } \Sigma {\setminus } P, \text { for } k\in S_0. \end{aligned}$$

If $b-u_{S_0{\setminus } \{k\}}=0$ at one point, then $\check{b}_{S_0{\setminus } \{k\}}=b$, which is a contradiction since $|S_0|$ is the smallest. So in $\Sigma {\setminus } P$, $\sum _{l\in S_0}c_{lk}\equiv 0$ for $k\in S_0$. As the argument above, it is a contradiction to the assumption (c). Then we obtain $b=0$, in particular, $u_{i}\ge 0$, $1\le i\le n$.

Step 3: We finish the proof.

Since $u_{i}\ge 0$, we have in $\Sigma {\setminus } P$,

$$\begin{aligned} \triangle _g u_i+\langle X,u_i \rangle +c_{ii}u_i\le 0, \quad 1\le i\le n. \end{aligned}$$

Then as the argument above, by the strong maximum principle, there exists a subset $Z\subseteq A$, such that $u_i\equiv 0$ for $i\in Z$ and $u_j>0$ for $j\notin Z$. Then for $i\in Z$, in $\Sigma {\setminus } P$, $0\le \sum _{j\notin Z}c_{ij}u_j=f_i\le 0$. Since $u_j>0$ for $j\notin Z$, $c_{ij}\equiv 0$ for $i\in Z,~j\notin Z$. Suppose condition (1) $(f_1,\ldots ,f_n)\not \equiv (0,\dots ,0)$ or condition (2) P is nonempty holds, we can rule out the possibility $Z=A$. Suppose condition (3) $\sum _{i=1}^n u_i\ge 0$ holds, Z must be empty or A. So either $u_{i}>0$, $1\le i\le n$ or $u_i\equiv 0$ for $1\le i\le n$ . $\square$

The assumption (c) is easy to check by the following procedure. If $1\in \alpha$, consider $\beta _1=\{j: c_{1j}\equiv 0\}$, $\alpha _1=\{1,\ldots ,n\}{\setminus } \beta _1$. Then $\alpha _1\cap \beta =\phi$. Then $\alpha _1\subseteq \alpha$. Denote $\alpha _0=\{1\}$. If $\alpha _{1}\subseteq \alpha _0$, then $\alpha =\alpha _0$ gives such a partition. If $\alpha _{1}\nsubseteq \alpha _0$, consider $\beta _2=\{j: c_{ij}\equiv 0, i\in \alpha _{0}\cup \alpha _1 \}$, $\alpha _2=\{1,\ldots ,n\}{\setminus } \beta _2$. Then $\alpha _2\subseteq \alpha$. If $\alpha _2\subseteq \alpha _0\cup \alpha _1$, then $\alpha =\alpha _0\cup \alpha _1$ gives such a partition. If $\alpha _{2}\nsubseteq \alpha _0\cup \alpha _1$, consider $\beta _3=\{j: c_{ij}\equiv 0, i\in \bigcup _{k=0}^{2}\alpha _k \}$, $\alpha _3=\{1,\ldots ,n\}{\setminus } \beta _3$. Repeat this procedure, then either we obtain a partition $\alpha ,\beta$ such that $c_{ij}\equiv 0$ for $i\in \alpha ,j\in \beta$ or we show that $1\notin \alpha$. If $1\notin \alpha$, repeat the procedure above for $2,3,\ldots , n$. Then we can show whether such a partition exists or not.

Later in the paper, the maximum principle above may be applied to the non-linear version under certain assumptions, by using the linearization

$$\begin{aligned}&F(u_1,\ldots ,u_n,x)-F(v_1,\ldots ,v_n,x)\\&\quad =\sum _{j=1}^{n}(u_j-v_j)\int _{0}^{1}\frac{\partial F}{\partial u_j}\,(tu_1+(1-t)v_1,\ldots ,tu_n+(1-t)v_n,x)dt. \end{aligned}$$

For the problems involving poles, we need to check whether the coefficient after linearization is bounded.

4 Monotonicity of pullback metrics

In this section, we first consider the family of the cyclic Higgs bundles $(E,\phi ^t)$ parameterized by $\{\gamma _1,\ldots ,\gamma _{n-1},t\gamma _n\},$$n\ge 3$ for $t\in {\mathbb {C}}$. We show the monotonicity of the pullback metrics of the corresponding branched minimal immersions along the family.

Proposition 4.1

Consider a family of stable cyclic Higgs bundles parameterized by $\{\gamma _1,\ldots ,t\gamma _n\}$, $q_n\ne 0$, $n\ge 3$, $t\in {\mathbb {C}}^*$ and $h^t$ be the corresponding harmonic metrics. Then as |t| increases, $h_k^{-1}h_{k+1}$, $k=1,\ldots ,n-1$ and $t^2h_n^{-1}h_1$ strictly increase. As a result, outside the branch points, the pullback metric $g^t$ of the corresponding branched minimal immersions strictly increases.

Proof

We show that for $0<|t'|<|t|$, all the terms for t dominate the corresponding terms for $t'$.

Let $u_k=h_k^{-1}h_{k+1}$, $k=1,\ldots , n-1$, $u_n=|t|^2h_n^{-1}h_1$. Then

$$\begin{aligned} \triangle \log u_k+|\gamma _{k+1}|^2u_{k+1}-2|\gamma _k|^2u_k+|\gamma _{k-1}|^2u_{k-1}=0,\quad k=1,\ldots , n. \end{aligned}$$

And $\tilde{u}_k$ is similarly defined for $t'$, satisfying

$$\begin{aligned} \triangle \log \tilde{u}_k+|\gamma _{k+1}|^2\tilde{u}_{k+1}-2|\gamma _k|^2\tilde{u}_k+|\gamma _{k-1}|^2\tilde{u}_{k-1}=0,\quad k=1,\ldots , n. \end{aligned}$$

Let $v_k=\log (u_k\tilde{u}^{-1}_{k})$, then

$$\begin{aligned}&\triangle v_k+|\gamma _{k+1}|^2\tilde{u}_{k+1}(e^{v_{k+1}}-1)-2|\gamma _k|^2\tilde{u}_{k}(e^{v_k}-1)\\&\quad \quad +|\gamma _{k-1}|^2\tilde{u}_{k-1}(e^{v_{k-1}}-1)=0,\quad k=1,\ldots , n. \end{aligned}$$

Let

$$\begin{aligned} c_k=g_0^{-1}|\gamma _k|^2\tilde{u}_k\int _{0}^{1}e^{(1-t)(v_k)}dt,\quad k=1,\ldots , n. \end{aligned}$$

Then $v_k$’s satisfy

$$\begin{aligned} \triangle _{g_0} v_k+c_{k-1}v_{k-1}-2c_kv_k+c_{k+1}v_{k+1}=0,\quad k=1,\ldots ,n. \end{aligned}$$

It is easy to check that the above system of equations satisfies the assumptions in Lemma 3.1 and condition (3), since $\sum _{k=1}^nv_k=2\log (\frac{|t|}{|t'|})>0$. One can apply Lemma 3.1, then $v_k>0$, $k=1,\ldots , n$. Then we obtain $u_k>\tilde{u}_k, ~k=1,\ldots ,n.$

Finally, the monotonicity of $g^t$ follows from $g^t=2n(\sum _{k=1}^{n-1}|\gamma _k|^2h_k^{-1}h_{k+1}+|\gamma _n|^2t^2h_n^{-1}h_1)dz\otimes d\bar{z}$. $\square$

For $t\in {\mathbb {C}}^*$, the family $(E,\phi ^t)$ is gauge equivalent to $(E,t^{\frac{1}{n}}\phi )$. Then we obtain the following.

Corollary 4.2

Let $(E,\phi )$ be a stable cyclic Higgs bundle parameterized by $\{\gamma _1,\ldots ,\gamma _n\}$, $n\ge 3$, $\gamma _n\ne 0$. Let $g^t$ be the pullback metric corresponding to $t\phi$ for $t\in {\mathbb {C}}^*$. Then outside the branch points, along the ${\mathbb {C}}^*$-orbit, $g^t$ strictly increases as |t| increases.

If the cyclic Higgs bundles parameterized by $\{\gamma _1,\ldots ,\gamma _{n-1},0\}$ is stable, we can extend the monotonicity of the pullback metric of ${\mathbb {C}}^*$-family to ${\mathbb {C}}$-family.

Proposition 4.3

Let $(E,\phi )$ be a stable cyclic Higgs bundles parameterized by $\{\gamma _1,\ldots ,\gamma _n\}$, $\gamma _n\ne 0$, $n\ge 3$, and h be the corresponding harmonic metrics on E. Let $(E,{{\tilde{\phi }}})$ be a stable cyclic Higgs bundle parameterized by $\{\gamma _1,\ldots ,\gamma _{n-1},0\}$. Then $h_k^{-1}h_{k+1}$, $k=1,\ldots ,n-1,$ for $(E,\phi )$ strictly dominate the ones for $(E,{{\tilde{\phi }}})$. As a result, outside the branch points, the pullback metric g of the corresponding branched minimal immersion for $(E,\phi )$ strictly dominates the one for $(E,{{\tilde{\phi }}})$.

Proof

The equation for $h_k$ is

$$\begin{aligned} \triangle \log h_k+|\gamma _k|^2h_k^{-1}h_{k+1}-|\gamma _{k-1}|^2h_{k-1}^{-1}h_k=0,\quad k=1,\ldots , n. \end{aligned}$$

Let $u_k=h_k^{-1}h_{k+1}$, $k=1,\ldots , n$. Then

$$\begin{aligned} \triangle \log u_1+|\gamma _2|^2u_2-2|\gamma _1|^2u_1= & {} -|\gamma _n|^2u_n \le 0,\\ \triangle \log u_k+|\gamma _{k+1}|^2u_{k+1}-2|\gamma _k|^2u_k+|\gamma _{k-1}|^2u_{k-1}= & {} 0,\quad k=2,\ldots , n-2,\\ \triangle \log u_{n-1}-2|\gamma _{n-1}|^2u_{n-1}+|\gamma _{n-2}|^2u_{n-2}= & {} -|\gamma _{n}|^2u_{n} \le 0. \end{aligned}$$

And $\tilde{h}_k,\tilde{u}_k$ are similarly defined for $t=0$.

Let $v_k=\log (u_k\tilde{u}^{-1}_{k})$, $k=1,\ldots , n-1$. Then

$$\begin{aligned} \triangle v_1+|\gamma _2|^2\tilde{u}_{2}(e^{v_2}-1)-2|\gamma _1|^2\tilde{u}_1(e^{v_1}-1)\le & {} 0,\\ \triangle v_k+|\gamma _{k+1}|^2\tilde{u}_{k+1}(e^{v_{k+1}}-1)-2|\gamma _k|^2\tilde{u}_{k}(e^{v_k}-1)+|\gamma _{k-1}|^2\tilde{u}_{k-1}(e^{v_{k-1}}-1)= & {} 0,\quad k=2,\ldots , n-2,\\ \triangle v_{n-1}-2|\gamma _{n-1}|^2\tilde{u}_{n-1}(e^{v_{n-1}}-1)+|\gamma _{n-2}|^2\tilde{u}_{n-2}(e^{v_{n-2}}-1)\le & {} 0. \end{aligned}$$

Let $c_k=g_0^{-1}|\gamma _k|^2\tilde{u}_k\int _{0}^{1}e^{(1-t)v_k}dt,~ k=1,\ldots , n-1.$ Then $v_k$’s satisfy

$$\begin{aligned} \triangle _{g_0} v_1-2c_1v_1+c_2v_2\le & {} 0,\\ \triangle _{g_0} v_k+c_{k-1}v_{k-1}-2c_kv_k+c_{k+1}v_{k+1}= & {} 0,\quad k=2,\ldots ,n-2,\\ \triangle _{g_0} v_{n-1}+c_{n-2}v_{n-2}-2c_{n-1}v_{n-1}\le & {} 0. \end{aligned}$$

It is easy to check that the above system of equations satisfies the assumptions in Lemma 3.1 and condition (1), since $\gamma _n\ne 0$. Applying Lemma 3.1, $v_k>0$, $k=1,\ldots , n-1$. Then we obtain $u_k>\tilde{u}_k, ~k=1,\ldots ,n-1.$$\square$

Remark 4.4

Propositions 4.1 and 4.3 generalize the metric domination theorem in [12] in two aspects: (1) from dominating the Fuchsian case to monotonicity along the ${\mathbb {C}}$-family; (2) from cyclic Higgs bundles in the Hitchin component to general cyclic Higgs bundles.

5 Curvature of cyclic Higgs bundles in the Hitchin component

In this section, we would like to obtain a lower and upper bound for the extrinsic curvature of the branched minimal immersion associated to cyclic Higgs bundles. Let’s first get to know how big the range of the sectional curvature of the symmetric space is.

Proposition 5.1

Let G be $SL(n,{\mathbb {C}}), SL(n,{\mathbb {R}}), Sp(n,{\mathbb {R}})(n=2m)$ and the maximal compact subgroup K be SU(n), SO(n), U(m) respectively. The sectional curvature K of the associated symmetric space G / K satisfies

$$\begin{aligned} -\frac{1}{n}\le K\le 0, \end{aligned}$$

where (1) for $SL(n,{\mathbb {C}}), SL(n,{\mathbb {R}})$, $-\frac{1}{n}$ can be achieved by the tangent plane at eK spanned by

$$\begin{aligned} E_{ij}+E_{ji}, E_{ii}-E_{jj} \quad \text { for any }1\le i<j\le n. \end{aligned}$$

(2) for $Sp(n,{\mathbb {R}})$ where $n=2m$, $-\frac{1}{n}$ can be achieved by the tangent plane at eK spanned by

$$\begin{aligned} E_{i,m+i}+E_{m+i,i},E_{ii}-E_{m+i,m+i}\quad \text {for any}~ 1\le i\le m. \end{aligned}$$

Proof

Suppose the Cartan decomposition of the Lie algebra is $\mathfrak {g}=\mathfrak {k}+\mathfrak {p}$. The sectional curvature of the plane spanned by the vectors $Y_1,Y_2\in T_p(G/K)$ is (see [18] for reference)

$$\begin{aligned} K(Y_1\wedge Y_2)=\frac{B([\omega ^{\mathfrak {p}}(Y_1),\omega ^{\mathfrak {p}}(Y_2)],[\omega ^{\mathfrak {p}}(Y_1),\omega ^{\mathfrak {p}}(Y_2)])}{B(\omega ^{\mathfrak {p}}(Y_1),\omega ^{\mathfrak {p}}(Y_1))\cdot B(\omega ^{\mathfrak {p}}(Y_2),\omega ^{\mathfrak {p}}(Y_2))-B(\omega ^{\mathfrak {p}}(Y_1),\omega ^{\mathfrak {p}}(Y_2))^2}. \end{aligned}$$

So it is enough by only checking $Y_1,Y_2\in T_{eK}(G/K)=\mathfrak {p}$. The upper bound is obvious since B is negative definite on $\mathfrak {k}$, where $[\omega ^{\mathfrak {p}}(Y_1),\omega ^{\mathfrak {p}}(Y_2)]$ lies in.

Now we show the lower bound. Let $\sigma$ be the plane $\text {span}\{Y,Z\}$ where $Y,Z\in \mathfrak {p}$ satisfy

$$\begin{aligned} \text {tr}(YZ)=0, \quad \text {tr}(Y^2)=\text {tr}(Z^2). \end{aligned}$$

The Killing form $B(Y,Z)=2n\cdot \text {tr}(YZ)$. Define $U=Y+iZ, V=Y-iZ$, then

$$\begin{aligned} K_{\sigma }= & {} -\frac{\text {tr}(UVUV)-\text {tr}(U^2V^2)}{n\cdot \text {tr}(UV)^2}\\\ge & {} -\frac{\text {tr}(UVUV)}{n\cdot \text {tr}(UV)^2}\quad \text {using } \text {tr}(U^2V^2)=\text {tr}(U^2\overline{U^2}^T)\ge 0\\\ge & {} -\frac{1}{n}\quad \text {using } \text {tr}(A^2)\le \text {tr}(A)^2. \end{aligned}$$

The equality holds if and only if $U^2=0$ and $UV=U{\overline{U}}^T$ has only one nonzero eigenvalue. In terms of Y, Z, the equality holds if and only if $Y^2=Z^2$, $YZ+ZY=0$, and $Y^2+Z^2+i(ZY-YZ)$ has only one nonzero eigenvalue. The rest is by direct calculation. $\square$

For general cyclic Higgs bundles, one should not expect a nontrivial lower bound of the extrinsic curvature at immersed points since it could achieve the plane of the most negative curvature in $SL(n,{\mathbb {C}})/SU(n)$.

Proposition 5.2

For stable cyclic Higgs bundles parameterized by $\{\gamma _1,\ldots ,\gamma _n\}$, if there exists a point such that $n-1$ terms of $\gamma _i$’s are equal to zero, the sectional curvature of the tangent plane of the associated harmonic map at this point is $-\frac{1}{n}$.

Proof

Firstly, $n=2$ case is obvious. Let $n\ge 3$. The associated harmonic map is a possibly branched minimal immersion. The tangent plane $\sigma$ of the minimal immersion at f(p) inside G / K is spanned by $Y_{f(p)}=f_*(\frac{\partial }{\partial x})$ and $Z_{f(p)}=f_*(\frac{\partial }{\partial y})$. Using the formula (7) in Sect. 2,

$$\begin{aligned}&\omega ^{\mathfrak {p}}(Y)=\Phi \left( \frac{\partial }{\partial z}\right) =(\phi +\phi ^*)\left( \frac{\partial }{\partial x}\right) =\phi \left( \frac{\partial }{\partial z}\right) +\phi ^*\left( \frac{\partial }{\partial \bar{z}}\right) ,\\&\omega ^{\mathfrak {p}}(Z)=\Phi \left( \frac{\partial }{\partial y}\right) =(\phi +\phi ^*)\left( \frac{\partial }{\partial y}\right) =\sqrt{-1}\phi \left( \frac{\partial }{\partial z}\right) -\sqrt{-1}\phi ^*\left( \frac{\partial }{\partial \bar{z}}\right) . \end{aligned}$$

One may refer the details in Section 2 in [12]. Hence

$$\begin{aligned}{}[\omega ^{\mathfrak {p}}(Y),\omega ^{\mathfrak {p}}(Z)]=-2\sqrt{-1}\left[ \phi \left( \frac{\partial }{\partial z}\right) ,\phi ^*\left( \frac{\partial }{\partial {\bar{z}}}\right) \right] =-2\sqrt{-1}[\phi ,\phi ^*]\left( \frac{\partial }{\partial z}, \frac{\partial }{\partial {\bar{z}}}\right) \end{aligned}$$

Since f is conformal, we have $Y\perp Z$. Then the sectional curvature of the plane $\sigma$ is

$$\begin{aligned} K_{\sigma }= & {} K(Y\wedge Z)=\frac{B([\omega ^{\mathfrak {p}}(Y),\omega ^{\mathfrak {p}}(Z)],[\omega ^{\mathfrak {p}}(Y),\omega ^{\mathfrak {p}}(Z)])}{B(\omega ^{\mathfrak {p}}(Y),\omega ^{\mathfrak {p}}(Y))B(\omega ^{\mathfrak {p}}(Z),\omega ^{\mathfrak {p}}(Z))}\nonumber \\= & {} -\frac{B([\phi ,\phi ^*],[\phi ,\phi ^*])}{B(\phi ,\phi ^*)B(\phi ,\phi ^*)}=-\frac{\text {tr}([\phi ,\phi ^*][\phi ,\phi ^*])}{2n\cdot \text {tr}(\phi \phi ^*)^2}\nonumber \\= & {} -\frac{(h_n^{-1}h_1|\gamma _n|^2-h_1^{-1}h_2|\gamma _1|^2)^2+(h_1^{-1}h_2|\gamma _1|^2-h_2^{-1}h_3|\gamma _2|^2)^2+\cdots +(h_{n-1}^{-1}h_n|\gamma _{n-1}|^2-h_n^{-1}h_1|\gamma _n|^2)^2}{2n(h_n^{-1}h_1|\gamma _n|^2+h_1^{-1}h_2|\gamma _1|^2 +\cdots +h_{n-1}^{-1}h_n|\gamma _{n-1}|^2)^2}.\nonumber \\ \end{aligned}$$

(10)

In particular, if at point p, there exists $k_0$ such that $\gamma _i=0$, for $i\ne k_0$, and $\gamma _{k_0}\ne 0$. Then

$$\begin{aligned} K_p=-\frac{2(h_{k_0-1}^{-1}h_{k_0}|\gamma _{k_0}|^2)^2}{2n\cdot (h_{k_0-1}^{-1}h_{k_0})^2|\gamma _{k_0}|^2)^2} =-\frac{1}{n}. \end{aligned}$$

$\square$

Remark 5.3

For example, consider the cyclic Higgs bundle $\left( L\oplus {\mathcal {O}}\oplus L^{-1},\begin{pmatrix}0&{}\quad 0&{}\quad \beta \\ \alpha &{}\quad 0&{}\quad 0\\ 0&{}\quad \alpha &{}\quad 0\end{pmatrix}\right)$, where $\deg L<\deg K, 0\ne \alpha \in H^0(L^{-1}K), 0\ne \beta \in H^0(L^2K)$. Suppose in addition that the zeros of $\beta$ do not contain all zeros of $\alpha$. Then at any point where $\alpha =0,\beta \ne 0$, the map is an immersion and the extrinsic curvature is $-\frac{1}{3}$.

So we restrict ourselves to cyclic Higgs bundles in the Hitchin components. In this case, we obtain a nontrivial lower and upper bound on the extrinsic curvature of the associated minimal immersion into G / K.

In the rest part of this section, let $q_n$ denote a nonzero holomorphic differential and Z denote the set of zeros of $q_n$. Let $(E,\phi )$ be a cyclic Higgs bundle in the Hitchin component with $\det \phi =(-1)^{n-1}q_n$ and $(E,\tilde{\phi })$ be the Fuchsian case. Let $h,\tilde{h}$ be the corresponding harmonic metrics.

For $n=2m$ even, define

$$\begin{aligned} \nu _1=\frac{h_1^2|q_n|^2}{h_1^{-1}h_2},\quad \nu _k=\frac{h_{k-1}^{-1}h_k}{h_{k}^{-1}h_{k+1}},\quad k=2,\ldots , m-1,\quad \nu _m=\frac{h_{m-1}^{-1}h_m}{h_m^{-2}}. \end{aligned}$$

Similarly, define

$$\begin{aligned} \tilde{\nu }_1=0,\quad \tilde{\nu }_k=\frac{\tilde{h}_{k-1}^{-1}\tilde{h}_k}{\tilde{h}_k^{-1}\tilde{h}_{k+1}},\quad k=2,\ldots ,m-1,\quad \tilde{\nu }_m=\frac{\tilde{h}_{m-1}^{-1}\tilde{h}_m}{\tilde{h}_m^{-2}}. \end{aligned}$$

By the explicit description of $\tilde{h}$, $\tilde{h}_k^{-1}\tilde{h}_{k+1}=\frac{1}{2}k(n-k)g_0$ for $k=1,\ldots ,m-1$ and $\tilde{h}^{-2}_{m}=\frac{m(n-m)}{2}g_0$. Here $g_0$ is the hyperbolic metric.

For $n=2m+1$ odd, $\nu _k,\tilde{\nu }_k, \tilde{h}_k$ are as above except $\nu _m=\frac{h_{m-1}^{-1}h_m}{h_m^{-1}}$, $\tilde{\nu }_m=\frac{\tilde{h}_{m-1}^{-1}\tilde{h}_m}{\tilde{h}_m^{-1}}$, $\tilde{h}^{-1}_m=\frac{m(n-m)}{2}g_0$.

Proposition 5.4

In the above settings,

$$\begin{aligned} \frac{(k-1)(n-k+1)}{k(n-k)}=\tilde{\nu }_k<\nu _k<1, \quad \quad k=1,\ldots , m. \end{aligned}$$

Remark 5.5

The inequality $\nu _k<1$ recovers Lemma 5.3, $q_n$ case in [12]. Here we give a new proof using Lemma 3.1 directly.

Proof

We only prove the case for $n=2m$. The proof is similar for $n=2m+1$.

The equation system for $h_k$ is

$$\begin{aligned} \triangle \log h_1+h_1^{-1}h_{2}-|q_n|^2h_{1}^{2}= & {} 0,\\ \triangle \log h_k+h_k^{-1}h_{k+1}-h_{k-1}^{-1}h_k= & {} 0,\quad k=2,\ldots , m-1,\\ \triangle \log h_m+h_m^{-2}-h_{m-1}^{-1}h_m= & {} 0. \end{aligned}$$

Let

$$\begin{aligned} u_0=\log (|q_n|^2h^2_1), \quad u_k=\log (h^{-1}_kh_{k+1}),\quad 1\le k\le m-1,\quad u_m=\log (h^{-2}_m). \end{aligned}$$

By the holomorphicity of $q_n$, $\triangle \log |q_n|=0$ on $X{\setminus } Z$. Then on $X{\setminus } Z$,

$$\begin{aligned} \triangle u_0+2e^{u_1}-2e^{u_0}= & {} 0,\\ \triangle u_k+e^{u_{k+1}}-2e^{u_k}+e^{u_{k-1}}= & {} 0, \quad k=1,\ldots , m-1,\\ \triangle u_m-2e^{u_m}+2e^{u_{m-1}}= & {} 0. \end{aligned}$$

To prove $\nu _k<1$, let $v_k=u_{k+1}-u_k$, $c_k=\int _0^1e^{tu_{k+1}+(1-t)u_k}dt$, $k=0,\ldots , m-1$. Then on $X{\setminus } Z$,

$$\begin{aligned} \triangle v_0-3c_0v_0+c_1v_1= & {} 0, \\ \triangle v_k+c_{k-1}v_{k-1}-2c_kv_k+c_{k+1}v_{k+1}= & {} 0, \quad k=1,\ldots , m-2,\\ \triangle v_{m-1}+c_{m-2}v_{m-2}-3c_{m-1}v_{m-1}= & {} 0. \end{aligned}$$

Note that only $v_0$ has poles at zeros of $q_n$. To apply Lemma 3.1, we check that $c_0$ is bounded. In fact, around the zeros of $q_n$,

$$\begin{aligned} c_0=\int _{0}^{1}e^{tu_1+(1-t)u_0}dt=\int _{0}^{1}(|q_n|^2h_1^2)^{1-t}e^{tu_1}dt\le C. \end{aligned}$$

It is then easy to check that the above system of equations satisfies the assumptions in Lemma 3.1 and condition (2), since the set of poles (i.e. the set Z of zeros of $q_n$) is nonempty. Applying Lemma 3.1, $v_k>0$, $k=0,\ldots , m-1$. Then we obtain $\nu _k<1, ~k=1,\ldots , m.$

To prove $\nu _k>\tilde{\nu }_k$, define

$$\begin{aligned} u_k=\log (h^{-1}_kh_{k+1}), \quad 1\le k\le m-1, \quad u_m=\log (h^{-2}_m). \end{aligned}$$

Then

$$\begin{aligned} \triangle (u_{2}-u_1)+e^{u_{3}}-3e^{u_{2}}+3e^{u_{1}}= & {} |q_n|^2h_1^2\ge 0,\\ \triangle (u_{k+1}-u_k)+e^{u_{k+2}}-3e^{u_{k+1}}+3e^{u_{k}}-e^{u_{k-1}}= & {} 0,\quad k=2,\ldots , m-2,\\ \triangle (u_{m}-u_{m-1})-3e^{u_{m}}+4e^{u_{m-1}}-e^{u_{m-2}}= & {} 0. \end{aligned}$$

And $\tilde{u}_k$’s are similarly defined for the Fuchsian case, satisfying

$$\begin{aligned} \triangle (\tilde{u}_{2}-\tilde{u}_1)+e^{\tilde{u}_{3}}-3e^{\tilde{u}_{2}}+3e^{\tilde{u}_{1}}= & {} 0,\\ \triangle (\tilde{u}_{k+1}-\tilde{u}_k)+e^{\tilde{u}_{k+2}}-3e^{\tilde{u}_{k+1}}+3e^{\tilde{u}_{k}}-e^{\tilde{u}_{k-1}}= & {} 0,\quad k=2,\ldots , m-2,\\ \triangle (\tilde{u}_{m}-\tilde{u}_{m-1})-3e^{\tilde{u}_{m}}+4e^{\tilde{u}_{m-1}}-e^{\tilde{u}_{m-2}}= & {} 0. \end{aligned}$$

To estimate $(u_{k+1}-u_k)-(\tilde{u}_{k+1}-\tilde{u}_k)$, we have for $k=2,\ldots , m-2$,

$$\begin{aligned}&(e^{u_{k+2}}-3e^{u_{k+1}}+3e^{u_{k}}-e^{u_{k-1}})-(e^{\tilde{u}_{k+2}}-3e^{\tilde{u}_{k+1}}+3e^{\tilde{u}_{k}}-e^{\tilde{u}_{k-1}})\\&\quad =e^{\tilde{u}_{k+2}}(e^{u_{k+2}-\tilde{u}_{k+2}}-e^{u_{k+1}-\tilde{u}_{k+1}})-2e^{\tilde{u}_{k+1}}(e^{u_{k+1}-\tilde{u}_{k+1}}-e^{u_{k}-\tilde{u}_{k}})\\&\qquad +e^{\tilde{u}_{k}}(e^{u_{k}-\tilde{u}_{k}}-e^{u_{k-1}-\tilde{u}_{k-1}})\\&\qquad +(e^{\tilde{u}_{k+2}}-e^{\tilde{u}_{k+1}})(e^{u_{k+1}-\tilde{u}_{k+1}}-1)-2(e^{\tilde{u}_{k+1}}-e^{\tilde{u}_{k}})(e^{u_{k}-\tilde{u}_{k}}-1)\\&\qquad +(e^{\tilde{u}_{k}}-e^{\tilde{u}_{k-1}})(e^{u_{k-1}-\tilde{u}_{k-1}}-1)\\&\quad =e^{\tilde{u}_{k+2}}(e^{u_{k+2}-\tilde{u}_{k+2}}-e^{u_{k+1}-\tilde{u}_{k+1}})-2e^{\tilde{u}_{k+1}}(e^{u_{k+1}-\tilde{u}_{k+1}}-e^{u_{k}-\tilde{u}_{k}})\\&\qquad +e^{\tilde{u}_{k}}(e^{u_{k}-\tilde{u}_{k}}-e^{u_{k-1}-\tilde{u}_{k-1}})\\&\qquad +(e^{u_{k+1}-\tilde{u}_{k+1}}-e^{u_{k}-\tilde{u}_{k}})(e^{\tilde{u}_{k+2}}-e^{\tilde{u}_{k+1}})-(e^{u_{k}-\tilde{u}_{k}}-e^{u_{k-1}-\tilde{u}_{k-1}})(e^{\tilde{u}_{k}}-e^{\tilde{u}_{k-1}})\\&\qquad +(e^{u_k-\tilde{u}_{k}}-1)(e^{\tilde{u}_{k+2}}-3e^{\tilde{u}_{k+1}}+3e^{\tilde{u}_{k}}-e^{\tilde{u}_{k-1}}). \end{aligned}$$

Since $\tilde{u}_{k+1}-\tilde{u}_k$ is a globally defined constant function, the equation of $\tilde{u}_{k+1}-\tilde{u}_k$ gives

$$\begin{aligned} e^{\tilde{u}_{k+2}}-3e^{\tilde{u}_{k+1}}+3e^{\tilde{u}_{k}}-e^{\tilde{u}_{k-1}}=0. \end{aligned}$$

Then

$$\begin{aligned}&(e^{u_{k+2}}-3e^{u_{k+1}}+3e^{u_{k}}-e^{u_{k-1}})-(e^{\tilde{u}_{k+2}}-3e^{\tilde{u}_{k+1}}+3e^{\tilde{u}_{k}}-e^{\tilde{u}_{k-1}})\\&\quad =e^{\tilde{u}_{k+2}}(e^{u_{k+2}-\tilde{u}_{k+2}}-e^{u_{k+1}-\tilde{u}_{k+1}})+(e^{\tilde{u}_{k+2}}-3e^{\tilde{u}_{k+1}})(e^{u_{k+1}-\tilde{u}_{k+1}}-e^{u_{k}-\tilde{u}_{k}})\\&\qquad +e^{\tilde{u}_{k-1}}(e^{u_{k}-\tilde{u}_{k}}-e^{u_{k-1}-\tilde{u}_{k-1}}). \end{aligned}$$

Similarly, for $k=1$,

$$\begin{aligned}&(e^{u_{3}}-3e^{u_{2}}+3e^{u_{1}})-(e^{\tilde{u}_{3}}-3e^{\tilde{u}_{2}}+3e^{\tilde{u}_{1}})\\&\quad =e^{\tilde{u}_{3}}(e^{u_{3}-\tilde{u}_{3}}-e^{u_{2}-\tilde{u}_{2}})+(e^{\tilde{u}_{3}}-3e^{\tilde{u}_{2}})(e^{u_{2}-\tilde{u}_{2}}-e^{u_{1}-\tilde{u}_{1}}), \end{aligned}$$

for $k=m-1$,

$$\begin{aligned}&(-3e^{u_{m}}+4e^{u_{m-1}}-e^{u_{m-2}})-(-3e^{\tilde{u}_{m}}+4e^{\tilde{u}_{m-1}}-e^{\tilde{u}_{m-2}})\\&\quad =-3e^{\tilde{u}_{m}}(e^{u_{m}-\tilde{u}_{m}}-e^{u_{m-1}-\tilde{u}_{m-1}})+(4e^{\tilde{u}_{m-1}}-3e^{\tilde{u}_{m}})(e^{u_{m-1}-\tilde{u}_{m-1}}-e^{u_{m-2}-\tilde{u}_{m-2}}). \end{aligned}$$

Let $v_k=(u_{k+1}-u_k)-(\tilde{u}_{k+1}-\tilde{u}_k)$, $k=1,\ldots , m-1$. Let $c_k=\int _{0}^{1}e^{t(u_{k+1}-\tilde{u}_{k+1})+(1-t)(u_{k}-\tilde{u}_{k})}dt$, $k=1,\ldots , m-1$. Then

$$\begin{aligned} \triangle v_1+e^{\tilde{u}_{3}}c_{2}v_{2}+(e^{\tilde{u}_{3}}-3e^{\tilde{u}_{2}})c_1v_1\ge & {} 0\\ \triangle v_k+e^{\tilde{u}_{k+2}}c_{k+1}v_{k+1}+(e^{\tilde{u}_{k+2}}-3e^{\tilde{u}_{k+1}})c_kv_k+e^{\tilde{u}_{k-1}}c_{k-1}v_{k-1}= & {} 0,\quad k=2,\ldots , m-2,\\ \triangle v_{m-1}-3e^{\tilde{u}_{m}}c_{m-1}v_{m-1}+(4e^{\tilde{u}_{m-1}}-3e^{\tilde{u}_{m}})c_{m-2}v_{m-2}= & {} 0. \end{aligned}$$

To apply the maximum principle, we need to check

$$\begin{aligned} e^{\tilde{u}_{k+2}}-2e^{\tilde{u}_{k+1}}+e^{\tilde{u}_{k}}\le 0,\quad k=1,\ldots , m-2. \end{aligned}$$

This is from the equation of $\tilde{u}_{k+1}$ and the fact $\tilde{u}_{k+1}=\text {const}+\log g_0$, $\triangle \log g_0=g_0$. Other conditions to apply the maximum principle hold clearly (for $e^{\tilde{u}_{m-1}}\le e^{\tilde{u}_m}$, it is from Lemma 5.4), so we obtain the desired result. $\square$

The Higgs bundles in the Hitchin component induce harmonic immersions $f:{\widetilde{\Sigma }}\rightarrow SL(n,{\mathbb {R}})/SO(n)$. We want to investigate that, as an immersed submanifold, how $f({\widetilde{\Sigma }})$ sits in the symmetric space.

Theorem 5.6

Let $f:{\widetilde{\Sigma }}\rightarrow SL(n,{\mathbb {R}})/SO(n)$ be the harmonic map associated to cyclic Higgs bundles in the Hitchin component. Then the sectional curvature $K_{\sigma }$ of the tangent plane $\sigma$ of the image of f in G / K satisfies

$$\begin{aligned} -\frac{1}{n(n-1)^2}\le K_{\sigma }<0. \end{aligned}$$

The equality can be achieved only if $n=2,3$.

Proof

In the case $n=2$, the extrinsic curvature is constantly $-\frac{1}{2}$. Now we consider $n\ge 3$ case. We only prove the case for $n=2m$. The proof is similar for $n=2m+1$. Using the curvature formula (10), the sectional curvature of the plane $\sigma$ is

$$\begin{aligned} K_{\sigma }= & {} -\frac{(h_1^2|q_n|^2-h_1^{-1}h_2)^2+(h_1^{-1}h_2-h_2^{-1}h_3)^2+\cdots +(h_{m-1}^{-1}h_m-h_m^{-2})^2}{n(h_1^2|q_n|^2+2h_1^{-1}h_2+2h_2^{-1}h_3 +\cdots +2h_{m-1}^{-1}h_m+h_m^{-2})^2}. \end{aligned}$$

Then $K_\sigma <0$ follows from Proposition 5.4.

To show $K_{\sigma }\ge -\frac{1}{n(n-1)^2}$, let $\mu _k=\nu _k^{-1}$, then

$$\begin{aligned} K_\sigma\ge & {} -\frac{(h_1^{-1}h_2)^2+(h_1^{-1}h_2-h_2^{-1}h_3)^2+\cdots +(h_{m-1}^{-1}h_m-h_m^{-2})^2}{n(2h_1^{-1}h_2+2h_2^{-1}h_3+\cdots +2h_{m-1}^{-1}h_m+h_m^{-2})^2}\\= & {} -\frac{1+(1-\mu _2)^2+(1-\mu _3)^2\mu _2^{2}+\cdots +(1-\mu _m)^2\mu _2^{2}\cdots \mu _{m-1}^{2}}{n(2+2\mu _2+2\mu _2\mu _3+\cdots +\mu _2\cdots \mu _{m})^2}. \end{aligned}$$

Define the functions $G_k, H_k$ for $3\le k\le m+1$ as follows. For $3\le k\le m-1$,

$$\begin{aligned}&G_k=(1-\mu _k)^2+(1-\mu _{k+1})^2\mu _k^2+\cdots +(1-\mu _m)^2\mu _k^2\cdots \mu _{m-1}^2\\&H_k=2+2\mu _k+2\mu _k\mu _{k+1}+\cdots +2\mu _k\cdots \mu _{m-1}+\mu _k\cdots \mu _m \end{aligned}$$

and

$$\begin{aligned} G_m=(1-\mu _m)^2,\quad H_m=2+\mu _m,\quad G_{m+1}=0, \quad H_{m+1}=1. \end{aligned}$$

The derivatives in $\mu _k$ for $3\le k\le m$ are,

$$\begin{aligned} (G_k)_{\mu _k}=2\mu _k(1+G_{k+1})-2,\quad (H_k)_{\mu _k}=H_{k+1} \end{aligned}$$

Define $F_k$ as a function of $\mu _k$, for $3\le k\le m+1$,

$$\begin{aligned} F_k(\mu _k)=\frac{1+G_k}{(2(k-2)+H_k)^2}. \end{aligned}$$

So $K_{\sigma }\ge -\frac{1}{n}F_2.$ For $3\le k\le m$,

$$\begin{aligned} F_k(\mu _k)=\frac{1+G_k}{(2(k-2)+H_k)^2}=\frac{1+(1-\mu _k)^2+\mu _k^2G_{k+1}}{(2(k-1) +\mu _kH_{k+1})^2}. \end{aligned}$$

We claim:

Lemma 5.7

$F_2<F_3$.

Lemma 5.8

$F_k<F_{k+1}$, for $3\le k\le m$.

Combining Lemmas 5.7 and 5.8, the sectional curvature

$$\begin{aligned} K_{\sigma }\ge -\frac{1}{n}F_2>-\frac{1}{n}F_{m+1}=\frac{-1}{n(n-1)^2},\quad \text {for }~ m>1. \end{aligned}$$

$\square$

Proof of Lemma 5.7

The derivative of $F_2$ in $\mu _2$ is

$$\begin{aligned} (F_2)_{\mu _2}= & {} \frac{2(2(\mu _2(1+G_{3})-1)-(2-\mu _2)H_{3})}{(1+(1-\mu _2)^2+\mu _2^2G_3)(2+\mu _2H_3)^3}\\= & {} \frac{2F}{(1+(1-\mu _2)^2+\mu _2^2G_3)(2+\mu _2H_3)^3}, \end{aligned}$$

where $F=2(\mu _2(1+G_{3})-1)-(2-\mu _2)H_{3}$. Then

$$\begin{aligned} F< & {} 2(\tilde{\mu }_2(1+G_{3})-1)-(2-\tilde{\mu }_2)H_{3}\\= & {} 2\left( \frac{2(n-2)}{n-1}(1+G_{3})-1\right) -\left( 2-\frac{2(n-2)}{n-1}\right) H_{3}\\= & {} \frac{2}{n-1}(n-3+2(n-2)G_{3}-H_{3})\\= & {} \frac{2}{n-1}(n-3+2(n-2)((1-\mu _3)^2+\cdots +(1-\mu _m)^2\mu _3^2\cdots \mu _{m-1}^2)\\&-(2+2\mu _3+\cdots +2\mu _3\cdots \mu _{m-1})-\mu _3\cdots \mu _m)\\= & {} \frac{2}{n-1}(n-3+P_m), \end{aligned}$$

where $P_k=2(n-2)((1-\mu _3)^2+\cdots +(1-\mu _k)^2\mu _3^2\cdots \mu _{k-1}^2) -(2+2\mu _3+\cdots +2\mu _3\cdots \mu _{k-1})-(n+1-2k)\mu _3\cdots \mu _k$, for $3\le k\le m$.

Claim: $P_{k+1}< P_{k}$, for $3\le k\le m-1$.

$$\begin{aligned} P_{k+1}= & {} 2(n-2)((1-\mu _3)^2+\cdots +(1-\mu _{k+1})^2\mu _3^2\cdots \mu _{k}^2)\\&-(2+2\mu _3+\cdots +2\mu _3\cdots \mu _{k})-(n-1-2k)\mu _3\cdots \mu _{k+1}\\= & {} 2(n-2)((1-\mu _3)^2+\cdots (1-\mu _{k})^2\mu _3^2\cdots \mu _{k-1}^2)-(2+2\mu _3+\cdots +2\mu _3\cdots \mu _{k})\\&+2(n-2)(1-\mu _{k+1})^2\mu _3^2\cdots \mu _{k}^2-(n-1-2k)\mu _3\cdots \mu _{k+1} \end{aligned}$$

The last term $2(n-2)(1-\mu _{k+1})^2\mu _3^2\cdots \mu _{k}^2-(n-1-2k)\mu _3\cdots \mu _{k+1}$ satisfies

$$\begin{aligned}&2(n-2)(1-\mu _{k+1})^2\mu _3^2\cdots \mu _{k}^2-(n-1-2k)\mu _3\cdots \mu _{k+1}\nonumber \\&\quad = \mu _3\cdots \mu _{k}(2(n-2)\mu _3\cdots \mu _{k}(1-\mu _{k+1})^2-(n-1-2k)\mu _{k+1})\nonumber \\&\quad< \mu _3\cdots \mu _{k}(2(n-2)\tilde{\mu }_3\cdots \tilde{\mu }_k(1-\mu _{k+1})^2-(n-1-2k)\mu _{k+1})\nonumber \\&\quad = \mu _3\cdots \mu _{k}(k(n-k)(1-\mu _{k+1})^2-(n-1-2k)\mu _{k+1})\nonumber \\&\quad = \mu _3\cdots \mu _{k} k(n-k)\left( \mu _{k+1}^2-\left( 2+\frac{n-1-2k}{k(n-k)}\right) \mu _{k+1}+1\right) \\&\qquad \text {Since }1<\mu _{k+1}< \tilde{\mu }_{k+1}=\frac{(k+1)(n-1-k)}{k(n-k)}=1+\frac{n-1-2k}{k(n-k)},\text { by Proposition }5.4.\\&\quad < -(n-1-2k)\mu _3\cdots \mu _k. \end{aligned}$$

Hence $P_{k+1}< P_k$. So

$$\begin{aligned} P_m< & {} P_3=2(n-2)(1-\mu _3)^2-2-(n-5)\mu _3\\\le & {} 2(n-2)\left( \mu _3^2-\left( 2+\frac{n-5}{2(n-2)}\right) \mu _3+1\right) < -(n-3). \end{aligned}$$

Hence $F<0$ and then $(F_2)_{\mu _2}<0$. Therefore $F_2(\mu _2)< F_2(1)=F_3$. $\square$

Proof of Lemma 5.8

The derivative of $F_k$ with respect to $\mu _k$ is

$$\begin{aligned} (F_k)_{\mu _k}= & {} \frac{2(2(k-1)(\mu _k+\mu _kG_{k+1}-1) -(2-\mu _k)H_{k+1})}{(2(k-1)+\mu _kH_{k+1})(1+(1-\mu _k)^2+\mu _k^2G_{k+1})^3}. \end{aligned}$$

By Proposition 5.4, $\mu _k< \tilde{\mu }_k=\frac{k(n-k)}{(k-1)(n+1-k)}$,

$$\begin{aligned} G_k< & {} (1-\tilde{\mu }_k)^2+(1-\tilde{\mu }_{k+1})^2\tilde{\mu }_k^2+\cdots +(1-\tilde{\mu }_m)^2\tilde{\mu }_k^2\cdots \tilde{\mu }_{m-1}^2\\= & {} \frac{1^2+3^2+\cdots +(n+1-2k)^2}{(k-1)^2(n+1-k)^2}=\frac{(n+1-2k)(n+2-2k)(n+3-2k)}{6(k-1)^2(n+1-k)^2}. \end{aligned}$$

By Proposition 5.4, $\mu _k>1$, then $H_k> n+3-2k.$

The term $2(k-1)(\mu _k-1+\mu _kG_{k+1})-(2-\mu _k)H_{k+1}$ satisfies

$$\begin{aligned}&2(k-1)(\mu _k-1+\mu _kG_{k+1})-(2-\mu _k)H_{k+1}\\&\quad< 2(k-1)((\tilde{\mu }_k-1)+\tilde{\mu }_kG_{k+1})-(2-\tilde{\mu }_k)H_{k+1}\\&\quad<\frac{2(k-1)}{(k-1)(n+1-k)}\left( (n+1-2k)+\frac{(n-1-2k)(n-2k)(n+1-2k)}{6k(n-k)}\right) \\&\qquad -\left( 2-\frac{k(n-k)}{(k-1)(n+1-k)}\right) (n+1-2k)\\&\quad = \frac{n+1-2k}{(k-1)(n+1-k)}\left( \frac{(n-1-2k)(n-2k)(k-1)}{3k(n-k)}-(k-2)(n-k)\right) \\&\quad = \frac{(n+1-2k)(n-k)}{(k-1)(n+1-k)}\left( \frac{(n-1-2k)(n-2k)(k-1)}{3(n-k)^2k}-(k-2)\right) \\&\quad<\frac{(n+1-2k)(n-k)}{(k-1)(n+1-k)}\left( \frac{1}{3}-(k-2)\right) <0,\quad \text {for } \ge 3. \end{aligned}$$

Hence $F_k$ decreases as $\mu _k$ increases. Then $F_k(\mu _k)< F_k(1)=F_{k+1}$. $\square$

In the Fuchsian case, the sectional curvature $K_{\sigma }$ is constantly $-\frac{6}{n^2(n^2-1)}$. Note that $-\frac{6}{n^2(n^2-1)}>-\frac{1}{n(n-1)^2}$ for $n>3$.

Proposition 5.9

Under the same assumptions in Theorem 5.6, if $n>3$, at each zero p of $q_n$, $K_{\sigma }< -\frac{6}{n^2(n^2-1)}$.

Proof

In the case $n=2m\ge 4$,

$$\begin{aligned} K_{\sigma }(p)= & {} -\frac{(h_1^{-1}h_2)^2+(h_1^{-1}h_2-h_2^{-1}h_3)^2+\cdots +(h_{m-1}^{-1}h_m-h_m^{-2})^2}{n(2h_1^{-1}h_2+2h_2^{-1}h_3+\cdots +2h_{m-1}^{-1}h_m+h_m^{-2})^2}\\= & {} -\frac{(h_1^{-1}h_2)^2+(h_1^{-1}h_2-h_2^{-1}h_3)^2+\cdots +(h_{m-1}^{-1}h_m-h_m^{-2})^2}{n((2m-1)(h_1^{-1}h_2) -(2m-3)(h_1^{-1}h_2-h_2^{-1}h_3)-\cdots -(h_{m-1}^{-1}h_m-h_m^{-2}))^2}\\&\text {by using the Cauchy--Schwarz inequality and }\nu _k>\tilde{\nu }_k\text { for }k=2,\ldots ,m\\< & {} -\frac{1}{n((2m-1)^2+(2m-3)^2+\cdots +1^2)}\\= & {} -\frac{6}{n^2(n^2-1)}. \end{aligned}$$

The proof for the case $n=2m+1$ is similar. $\square$

6 Comparison inside the real Hitchin fibers at $(0,\ldots ,0,q_n)$

In this section, we restrict to the $SL(n,{\mathbb {R}})$-Higgs bundles and would like to compare the data of cyclic Higgs bundles in the same Hitchin fiber. We first compare the harmonic metrics for cyclic $SL(n,{\mathbb {R}})$-Higgs bundles in the same Hitchin fiber at $(0,\ldots ,0,n\cdot q_n)$, that is, $\det \phi =(-1)^{n-1}q_n$.

Proposition 6.1

Let $(\tilde{E},{\tilde{\phi }})$ be a cyclic Higgs bundle in the Hitchin component and $(E,\phi )$ be a distinct stable cyclic $SL(n,{\mathbb {R}})$-Higgs bundle in Sect. 2.3 satisfying $\det \phi =\det {{\tilde{\phi }}}$. Let $h,\tilde{h}$ be the corresponding harmonic metrics.

• (1) For $n=2m$,

$$\begin{aligned} h_k>|\gamma _k|^2\cdots |\gamma _{m-1}|^2|\mu |\tilde{h}_k,~k=1,\ldots , m-1,&\qquad h_m>|\mu |\tilde{h}_m.\\ h^{-1}_{m+1-k}>|\nu ||\gamma _1|^2\cdots |\gamma _{m-k}|^2\tilde{h}_k,~k=1,\ldots , m-1,&\qquad h^{-1}_{1}>|\nu |\tilde{h}_m. \end{aligned}$$

• (2) For $n=2m+1$,

$$\begin{aligned} h_k>|\gamma _k|^2\cdots |\gamma _{m-1}|^2|\mu |^2\tilde{h}_k,~k=1,\ldots , m-1,&\qquad h_m>|\mu |^2\tilde{h}_m. \end{aligned}$$

Proof

We only prove the inequalities on the first line for $n=2m$. For other cases, the proofs are similar. Define a new Hermitian metric on each $L_k$,

$$\begin{aligned} {\hat{h}}_k=|\gamma _k|^2\cdots |\gamma _{m-1}|^2|\mu |\tilde{h}_k,~k=1,\ldots , m-1, \quad {\hat{h}}_m=|\mu |\tilde{h}_m. \end{aligned}$$

By the holomorphicity of $\gamma _k$, $\triangle \log |\gamma _k|=0$ outside the zeros of $\gamma _k$ (similar for $\mu ,\nu$). Then ${\hat{h}}$ satisfies, outside the zeros of $q_n$, locally

$$\begin{aligned} \triangle \log {\hat{h}}_1+|\gamma _1|^2{\hat{h}}_1^{-1}{\hat{h}}_{2}-|\nu |^2{\hat{h}}_{1}^{2}= & {} 0,\\ \triangle \log {\hat{h}}_k+|\gamma _k|^2{\hat{h}}_k^{-1}{\hat{h}}_{k+1}-|\gamma _{k-1}|^2{\hat{h}}_{k-1}^{-1}{\hat{h}}_k= & {} 0, \quad k=2,\ldots , m-1,\\ \triangle \log {\hat{h}}_m+|\mu |^2{\hat{h}}_m^{-2}-|\gamma _{m-1}|^2{\hat{h}}_{m-1}^{-1}{\hat{h}}_m= & {} 0. \end{aligned}$$

Notice that ${\hat{h}}$ satisfies the same equation system as h, but have zeros.

Define $u_i=\log (h_i/{\hat{h}}_i)$ and $u_i$ goes to $+\infty$ around the set $P_i$, the zeros of ${\hat{h}}_i$. Let

$$\begin{aligned} c_1= & {} g_0^{-1}|\nu |^2{\hat{h}}_{1}^{2}\int _{0}^{1}e^{(1-t)u_1}dt,\\ c_k= & {} g_0^{-1}|\gamma _k|^2{\hat{h}}_{k-1}^{-1}{\hat{h}}_{k}\int _{0}^{1}e^{(1-t)u_k}dt,\quad k=2,\ldots , m\\ c_{m+1}= & {} g_0^{-1}|\mu |^2{\hat{h}}_{m}^{-2}\int _{0}^{1}e^{(1-t)u_m}dt. \end{aligned}$$

Then $u_i$’s satisfy

$$\begin{aligned} \triangle _{g_0} u_1-(c_2+2c_1)u_1+c_2u_2= & {} 0,\\ \triangle _{g_0} u_k+c_{k+1}u_{k+1}-(c_k+c_{k+1})u_k+c_{k}u_{k-1}= & {} 0, \quad k=2,\ldots , m-1,\\ \triangle _{g_0} u_m-(2c_{m+1}+c_{m})u_m+c_{m}u_{m-1}= & {} 0. \end{aligned}$$

We need to check whether the coefficients are bounded. The $c_i$’s are indeed bounded from the fact $\int _{0}^{1}x^{1-t}dt\le C$ around $x=0$. It is then easy to check that the above system of equations satisfies the assumptions in Lemma 3.1 and condition (2), since the set $P=\bigcup _i P_i$ of poles is nonempty. Applying Lemma 3.1, we obtain $u_k>0$, $k=1,\ldots , m$. $\square$

Concerning the associated harmonic maps $f:{\widetilde{\Sigma }}\rightarrow G/K$, we show that the pullback metric of the harmonic map for the cyclic Higgs bundle in the Hitchin component dominates the ones for other cyclic $SL(n,{\mathbb {R}})$-Higgs bundles in the Hitchin fiber for $n=2,3,4$.

Theorem 6.2

Let $(\tilde{E},{\tilde{\phi }})$ be a cyclic Higgs bundle in the Hitchin component and $(E,\phi )$ be a distinct stable cyclic $SL(n,{\mathbb {R}})$-Higgs bundle in Sect. 2.3 such that $\det \phi =\det {{\tilde{\phi }}}$. In the case $n=2, 3, 4$, the pullback metrics $g,\tilde{g}$ of corresponding harmonic maps satisfy $g<\tilde{g}.$

Proof

For $n=2, \mu \nu =q_2$, locally

$$\begin{aligned} \frac{1}{4}g=q_2dz^2+(|\nu |^2h^2+|\mu |^2h^{-2})dz\cdot d{\bar{z}}+{\bar{q}}_2d{\bar{z}}^2\\ \frac{1}{4}\tilde{g}=q_2dz^2+(|\mu |^2|\nu |^2\tilde{h}^2+\tilde{h}^{-2})dz\cdot d{\bar{z}}+{\bar{q}}_2d{\bar{z}}^2 \end{aligned}$$

So

$$\begin{aligned} \frac{1}{4}g\left( \frac{\partial }{\partial z},\frac{\partial }{\partial \bar{z}}\right)= & {} (|\nu |h-|\mu |h^{-1})^2+2|\mu \nu |,\quad \frac{1}{4}\tilde{g}\left( \frac{\partial }{\partial z},\frac{\partial }{\partial \bar{z}}\right) \\= & {} (|\mu ||\nu |\tilde{h}-\tilde{h}^{-1})^2+2|\mu \nu |. \end{aligned}$$

From Proposition 6.1, $|\mu ||\nu |\tilde{h}\le |\mu |h^{-1}<\tilde{h}^{-1}$, $|\mu ||\nu |\tilde{h}\le |\nu |h<\tilde{h}^{-1}$. Then

$$\begin{aligned} (|\nu |h-|\mu |h^{-1})^2<(|\mu ||\nu |\tilde{h}-\tilde{h}^{-1})^2, \end{aligned}$$

which implies $g<\tilde{g}$.

For $n=3, \mu ^2\nu =q_3$, we claim $|\nu |^2\tilde{h}h^2<1$. The Hitchin equation is reduced to

$$\begin{aligned} \triangle \log (|\nu |^2\tilde{h}h^2)+\tilde{h}^{-1}-|\mu |^4|\nu |^2\tilde{h}^2+2|\mu |^2h^{-1}-2|\nu |^2h^2=0. \end{aligned}$$

Let $u=|\nu |^2\tilde{h}h^2$, $a=|\mu |^2\tilde{h}h^{-1}$. Then

$$\begin{aligned} \triangle \log u+\tilde{h}^{-1}(1+2a-(2+a^2)u)=0. \end{aligned}$$

Notice that $u\equiv 1$ is a supersolution, then by the maximum principle, $u<1$. For the pullback metric $g,\tilde{g}$, locally,

$$\begin{aligned} \frac{1}{6}g=|\nu |^2h^2+2|\mu |^2h^{-1},\quad \frac{1}{6}\tilde{g}=|\nu |^2|\mu |^4\tilde{h}^2+2\tilde{h}^{-1}. \end{aligned}$$

Let $x=|\nu |h$, $\tilde{x}=|\mu ||\nu |^2\tilde{h}$, $A=|q_3|=|\nu ||\mu |^2$. Outside the zeros of $\mu \nu$, from Proposition 6.1, $x<\tilde{x}$. Then

$$\begin{aligned} \frac{1}{6}(g-\tilde{g})= & {} \left( x^2+\frac{2A}{x}\right) -\left( \tilde{x}^2+\frac{2A}{\tilde{x}}\right) =\frac{(x-\tilde{x})}{x\tilde{x}}((x+\tilde{x})x\tilde{x}-2A)\\< & {} \frac{(x-\tilde{x})}{x\tilde{x}}(2x^2\tilde{x}-2A) =\frac{2|\nu ||\mu |^2(x-\tilde{x})}{x\tilde{x}}(|\nu |^2\tilde{h}h^2-1)<0. \end{aligned}$$

So outside the zeros of $q_3=\mu ^2\nu$, we obtain $g<\tilde{g}$. We can easily see it also holds at the zeros of $q_3$.

For $n=4, \gamma ^2\mu \nu =q_4$, locally

$$\begin{aligned} \frac{1}{8}g= & {} |\nu |^2h^2_1+2|\gamma |^2h_1^{-1}h_2+|\mu |^2h^{-2}_2 =(|\nu |h_1-|\mu |h^{-1}_2)^2\\&+2|\nu ||\mu |h_1h^{-1}_2+2|\gamma |^2h_1^{-1}h_2,\\ \frac{1}{8}\tilde{g}= & {} |\mu |^2|\nu |^2|\gamma |^4\tilde{h}^2_1+2\tilde{h}_1^{-1}\tilde{h}_2+\tilde{h}^{-2}_2 =(|\mu ||\nu ||\gamma |^2\tilde{h}_1-\tilde{h}^{-1}_2)^2\\&+2|\nu ||\mu ||\gamma |^2\tilde{h}_1\tilde{h}^{-1}_2 +2\tilde{h}_1^{-1}\tilde{h}_2. \end{aligned}$$

From Proposition 6.1, $|\mu ||\nu ||\gamma |^{2}\tilde{h}_1\le |\mu |h^{-1}_2<\tilde{h}^{-1}_2$, $|\mu ||\nu ||\gamma |^{2}\tilde{h}_1\le |\nu |h_1<\tilde{h}^{-1}_2$. Then

$$\begin{aligned} (|\mu ||\nu ||\gamma |^2\tilde{h}_1-\tilde{h}^{-1}_2)^2>(|\nu |h_1-|\mu |h^{-1}_2)^2. \end{aligned}$$

Let $x=|\gamma |^2h_1^{-1}h_2$, $\tilde{x}=\tilde{h}^{-1}_1\tilde{h}_2$, $A=|q_4|=|\mu ||\nu ||\gamma |^2$.

Claim: $x<\tilde{x}$ and $x\tilde{x}> A$, outside the zeros of $\gamma$. Then the desired result follows from the basic identity $x+\frac{A}{x}-\tilde{x}-\frac{A}{\tilde{x}}=(x-\tilde{x})(1-\frac{A}{x\tilde{x}})$.

To show $x<\tilde{x}$, let $u=\frac{x}{\tilde{x}}=|\gamma |^2h_1^{-1}h_2\tilde{h}_1\tilde{h}^{-1}_2$. Then u satisfies

$$\begin{aligned}&\triangle \log u-2\tilde{h}^{-1}_1\tilde{h}_{2}(u-1)+2|\mu ||\nu ||\gamma |^2\tilde{h}_1\tilde{h}^{-1}_{2}(u^{-1}-1)\\&\quad \quad +(|\nu |h_1-|\mu |h^{-1}_2)^2-(|\mu ||\nu ||\gamma |^2\tilde{h}_1-\tilde{h}^{-1}_2)^2=0. \end{aligned}$$

Then

$$\begin{aligned} \triangle \log u-2\tilde{h}^{-1}_1\tilde{h}_{2}(u-1)+2|\nu ||\mu ||\gamma |^2\tilde{h}_1\tilde{h}^{-1}_{2}(u^{-1}-1)>0. \end{aligned}$$

Notice that 1 is a subsolution, then by the strong maximum principle, $u<1$.

To show $x\tilde{x}> A$, let $u=\frac{A}{x\tilde{x}}=|\mu ||\nu |h_1h^{-1}_2\tilde{h}_1\tilde{h}^{-1}_2$. Then u satisfies

$$\begin{aligned}&\triangle \log u+(2\tilde{h}^{-1}_1\tilde{h}_2+2|\gamma |^2h^{-1}_1h_2)(1-u)\\&\quad -(|\nu |h_1-|\mu |h^{-1}_2)^2-(|\mu ||\nu ||\gamma |^2\tilde{h}_1-\tilde{h}^{-1}_2)^2=0. \end{aligned}$$

Then

$$\begin{aligned} \triangle \log u+(2\tilde{h}^{-1}_1\tilde{h}_2+2|\gamma |^2h^{-1}_1h_2)(1-u)>0. \end{aligned}$$

Notice that $u\equiv 1$ is a solution, then by the strong maximum principle, $u<1$. At the zeros of $\gamma$, we can also obtain $g<\tilde{g}$ from $|\mu ||\nu |h_1h^{-1}_2\tilde{h}_1\tilde{h}^{-1}_2<1$. So we finish the proof. $\square$

As an immediate corollary in terms of representations for $n=2$, we recover the following result shown in [13].

Corollary 6.3

For any non-Fuchsian irreducible $SL(2,{\mathbb {R}})$-representation $\rho$ and any Riemann surface $\Sigma$, there exists a Fuchsian representation j such that the pullback metric of the corresponding j-equivariant harmonic map $f_j:{\widetilde{\Sigma }}\rightarrow {\mathbb {H}}^2$ dominates the one for $f_{\rho }$.

Proof

Fix a Riemann surface $\Sigma$. For any irreducible $SL(2,{\mathbb {R}})$-representation $\rho$, it corresponds to a stable cyclic Higgs bundle parameterized by $\{\alpha ,\beta \}$ over $\Sigma$ by [16]. Then we choose the Fuchsian representation j corresponding to the cyclic Higgs bundle parameterized by $q_2=\alpha \beta$ over $\Sigma$. The statement follows from Theorem 6.2. $\square$

7 Maximal $Sp(4,{\mathbb {R}})$-representations

In this section, we consider maximal $Sp(4,{\mathbb {R}})$-representations as explained in Sect. 1.4. Firstly, we obtain the following corollary from Theorem 4.3.

Corollary 7.1

For any maximal $Sp(4,{\mathbb {R}})$-representation $\rho$ in the Gothen components (or the Hitchin components), there exists a $\mu$-Fuchsian (or Fuchsian) representation j of $\pi _1(S)$ such that the pullback metric of the unique j-equivariant minimal immersion $f_j:\widetilde{S}\rightarrow Sp(4,{\mathbb {R}})/U(2)$ is dominated by the one for $f_{\rho }$.

Proof

As in Sect. 1.4, any maximal representation in the Gothen components corresponds to cyclic Higgs bundle parameterized by $\{1,\mu ,1,\nu \}$ of the form (5) over a unique Riemann surface $\Sigma$. Then we choose the $\mu$-Fuchsian representation corresponding to cyclic Higgs bundle parameterized by $\{1,\mu ,1,0\}$ over $\Sigma$. Then the statement follows from Theorem 4.3. Similar for the Hitchin representations for $Sp(4,{\mathbb {R}})$. $\square$

We obtain bounds on the extrinsic curvature of minimal immersions for the Hitchin representations for $Sp(4,{\mathbb {R}})$ as an immediate corollary of Theorem 5.6. Similarly, we obtain estimates on the extrinsic curvature of minimal immersions for maximal representations in the Gothen components.

Theorem 7.2

For any maximal $Sp(4,{\mathbb {R}})$-representation $\rho$ in the Gothen components, the sectional curvature $K_{\sigma }$ in $Sp(4,{\mathbb {R}})/U(2)$ of the tangent plane $\sigma$ of the unique $\rho$-equivariant minimal immersion satisfies

• (1) $-\frac{1}{8}\le K_{\sigma }<-\frac{1}{40}$ and the lower bound is sharp, if $\rho$ is $\mu$-Fuchsian;

• (2) $-\frac{1}{8}\le K_{\sigma }<0$, if $\rho$ is not $\mu$-Fuchsian.

Proof

It is sufficient to work with cyclic Higgs bundle parameterized by $\{1,\mu ,1,\nu \}$ of the form (5). The Hitchin equation in this case is

$$\begin{aligned} \triangle \log {h}_1+{h}_1^{-1}h_{2}-|\nu |^2{h}_{1}^{2}= & {} 0,\\ \triangle \log {h}_2+|\mu |^2{h}_2^{-2}-h_1^{-1}h_2= & {} 0. \end{aligned}$$

Using the curvature formula (10), the sectional curvature of the tangent plane $\sigma$ of the minimal immersion is

$$\begin{aligned} K_{\sigma }= & {} -\frac{(h_1^{2}|\nu |^2-h_1^{-1}h_{2})^2+(h_1^{-1}h_2-h_2^{-2}|\,u|^2)^2}{4\cdot (h_1^{2}|\nu |^2+2h_1^{-1}h_{2}+h_{2}^{-2}|\mu |^2)^2}. \end{aligned}$$

Let $f_1=\frac{h_1^2|\nu |^2}{h_1^{-1}h_2}$ and $f_2=\frac{h_2^{-2}|\mu |^2}{h_1^{-1}h_2}$. So

$$\begin{aligned} K_{\sigma }=-\frac{(f_1-1)^2+(f_2-1)^2}{4(2+f_1+f_2)^2}. \end{aligned}$$

For the left inequality. Claim: $f_1,f_2<\frac{4}{3}$.

The equation for $f_1$ is, outside zeros of $\nu$,

$$\begin{aligned}&\triangle \log f_1+[3(1-f_1)-(f_2-1)]h_1^{-1}h_2=0,\\&\quad \Longrightarrow \triangle _{h_1^{-1}h_2}\log f_1+3(1-f_1)-(f_2-1)=0,\\&\quad \Longrightarrow \triangle _{h_1^{-1}h_2}\log f_1+3(1-f_1)+1\ge 0. \end{aligned}$$

So at the maximum of $f_1$, $3(1-f_1)+1\le 0$, hence $f_1\le \frac{4}{3}$. By the strong maximum principle, $f_1<\frac{4}{3}$. It is similar for $f_2$. The claim is proven.

Using the fact that $0\le f_1,f_2<\frac{4}{3}$,

$$\begin{aligned} K_{\sigma }=-\frac{(f_1-1)^2+(f_2-1)^2}{4(2+f_1+f_2)^2}\ge -\frac{1+1}{16}=-\frac{1}{8}. \end{aligned}$$

Note that $K_{\sigma }$ only achieves $-\frac{1}{8}$ if $f_1=f_2=0$. This only happens at common zeros of $\mu$ and $\nu$. Note that at zeros of $\mu$ in $\mu$-Fuchsian case, the curvature $K_{\sigma }=-\frac{1}{8}$.

For the right inequality. Claim: $f_1f_2<1$.

The equation for $f_1f_2$ is, outside zeros of $\mu \nu$,

$$\begin{aligned}&\triangle \log f_1f_2-2(|\mu |^2h_2^{-2}+|\nu |^2{h}_{1}^{2})+4{h}_1^{-1}h_{2}=0.\\&\quad \Longrightarrow \triangle \log f_1f_2-4|\mu \nu |h_1h_2^{-1}+4{h}_1^{-1}h_{2}\ge 0.\\&\quad \Longrightarrow \triangle _{g_0} f_1f_2-4((f_1f_2)^{\frac{1}{2}}-1){h}_1^{-1}h_{2}/g_0\ge 0. \end{aligned}$$

So at the maximum of $f_1f_2$, $f_1f_2\le 1$. Hence $f_1f_2\le 1$ on the whole surface. By the strong maximum principle, $f_1f_2<1$. Therefore $K_{\sigma }<0$.

In the $\mu$-Fuchsian case, $\nu =0$. So $f_1=0$ and again $f_2<\frac{4}{3}$. Then using $(f_2+2)^{-1}\in (\frac{3}{10},\frac{1}{2}]$,

$$\begin{aligned} K_{\sigma }= & {} -\frac{1+(f_2-1)^2}{4(f_2+2)^2}=-\frac{(f_2+2-3)^2+1}{4(f_2+2)^2}\\= & {} -\frac{10}{4}\left( (f_2+2)^{-1}-\frac{3}{10}\right) ^2-\frac{1}{40}<-\frac{1}{40}. \end{aligned}$$

$\square$

At last, we compare representations in the Gothen components with the Hitchin components in the following sense.

Corollary 7.3

For any maximal $Sp(4,{\mathbb {R}})$-representation $\rho$ in the Gothen components, there exists a Hitchin representation $\delta$ of $\pi _1(S)$ such that the pullback metric of the unique $\delta$-equivariant minimal immersion $f_{\delta }:\widetilde{S}\rightarrow Sp(4,{\mathbb {R}})/U(2)$ dominates the one for $f_{\rho }$.

Proof

For any maximal representation $\rho$ in the Gothen components, it corresponds to a cyclic Higgs bundle parameterized by $\{1,\mu ,1,\nu \}$ over a unique Riemann surface $\Sigma$. Then we choose the Hitchin representation $\delta$ corresponding to cyclic Higgs bundle in the Hitchin component with $\det \phi =-\mu \nu$ over $\Sigma$. The statement then follows from Theorem 6.2 for $n=4$. $\square$