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Equivariant harmonic maps depend real analytically on the representation

Abstract

We consider harmonic maps into symmetric spaces of non-compact type that are equivariant for representations that induce a free and proper action on the symmetric space. We show that under suitable non-degeneracy conditions such equivariant harmonic maps depend in a real analytic fashion on the representation they are associated to. The main tool in the proof is the construction of a family of deformation maps which are used to transform equivariant harmonic maps into maps mapping into a fixed target space so that a real analytic version of the results in [4] can be applied.

Introduction

In this article we consider equivariant harmonic maps and study the question whether these maps depend nicely (e.g. smoothly or real analytically) on the representation they are association to.

Let us first introduce some notation. Throughout this article we let M be a closed real analytic Riemannian manifold, \({\widetilde{M}}\) its universal cover and \(\Gamma = \pi _1(M)\) its fundamental group. Also we let G be a real semisimple Lie group without compact factors and \(X=G/K\) its associated symmetric space. If \(\rho :\Gamma \rightarrow G\) is a reductive representation of \(\Gamma \) in G, then by work of Corlette ([3]) and Labourie ([10]) there exists a \(\rho \)-equivariant harmonic map \(f :{\widetilde{M}} \rightarrow X\). A map f is called \(\rho \)-equivariant if

$$\begin{aligned} f(\gamma m) = \rho (\gamma ) f(m) \text { for all } m \in {\widetilde{M}} \text { and } \gamma \in \Gamma . \end{aligned}$$

These maps were used by Corlette to prove a version of super rigidity. They were also used by Hitchin and Simpson in the development of the Non-Abelian Hodge correspondence which gives an identification between representation varieties and moduli spaces of Higgs bundles over Kähler manifolds.

In [4] Eells and Lemaire prove that, under suitable non-degeneracy conditions, harmonic maps between closed Riemannian manifolds depend smoothly on the metrics on both the domain and the target (see also [14]). We extend this result in two directions. First, we show it holds also in the real analytic category. Namely, we show in Proposition 3.3 that, under the same non-degeneracy condition, harmonic maps depend in a real analytic fashion on the metrics on the domain and target. Secondly, we show that this result can be used to prove that certain equivariant harmonic maps depend in a real analytic fashion on the representation they are associated to. For this we restrict ourselves to families of representations that satisfy a condition that we will call uniformly free and proper (see Definition 2.3). This assumption allows us to consider equivariant harmonic maps \({\widetilde{M}} \rightarrow X\) as maps from M to the quotients of the symmetric space X by the actions induced by the representations. In this way we can transform the setting of equivariant harmonic maps into a setting that is closer to that of [4] and Proposition 3.3.

The main result (Theorem 2.8) of this article is as follows. If \((\rho _t)_t\) is a real analytic family of representations of \(\Gamma \) in G that is uniformly free and proper and is such that \(\rho _0\) is reductive and the centraliser of its image contains no hyperbolic elements, then for all t in a neighbourhood of zero there exist \(\rho _t\)-equivariant harmonic maps depending real analytically on t. Similarly, we also prove real analytic dependence on the metric on the domain M.

In general a representation \(\Gamma \rightarrow G\) need not induce a free and proper action on the symmetric space X. Hence, the assumption that the family \((\rho _t)_t\) is uniformly free and proper does constitute a restriction on the applicability of the theorem. In the approach we present here, where we let equivariant harmonic maps descend to a quotient, such an assumption seems, to the author, indispensable. Hence, for a more general statement, one without the uniformly free and proper assumption, it seems likely that a different approach will be necessary.

Despite this constraint, our result is well adapted to the study of the equivariant harmonic maps that arise in higher Teichmüller theory. Namely, most representations studied in higher Teichmüller theory satisfy the Anosov condition and as observed in Remark 2.4 families of such representations always satisfy the uniformly free and proper condition.

In Sect. 2.4.1 we illustrate some applications of the main theorem to higher Teichmüller theory. More specifically, we apply our results to families of Hitchin representations. Such families satisfy all assumptions of the main theorem (see Proposition 2.12). As a result we can characterise certain sets as real analytic subsets of Teichmüller space and the set of Hitchin representations. Namely, in Corollary 2.13 we prove that the set of points at which the equivariant harmonic maps are not immersions is a real analytic subvariety of the universal Teichmüller curve crossed with the set of Hitchin representations. Similarly, in Corollary 2.14 we prove that the set of points Y in Teichmüller space and representations \(\rho \) such that Y can be realised as a minimal surface in \(X/\rho (\Gamma )\) is a real analytic subvariety of Teichmüller space crossed with the set of Hitchin representations.

Statement of the results

We first collect some preliminary definitions and results needed to give a statement of the main theorem.

Harmonic maps

If (Mg) and (Nh) are Riemannian manifolds with M compact, then a \(C^1\) map \(f :(M,g) \rightarrow (N,h)\) is called harmonic if it is a critical point of the Dirichlet energy functional

$$\begin{aligned} E(f) = \frac{1}{2} \int _{M} |\!|df|\!|^2 {{\,\mathrm{vol}\,}}_g. \end{aligned}$$

Here we view df as a section of the bundle \(T^*M\otimes f^*TN\). A metric on this vector bundle is induced by the metrics g and h. A harmonic map satisfies the Euler–Lagrange equation \(\tau (f) = 0\) where \(\tau (f) = {{\,\mathrm{tr}\,}}_{g} \nabla df\) is the tension field of f. Here \(\nabla \) denotes the connection on \(T^*M\otimes f^*TN\) induced by the Levi-Civita connections of g and h. If the domain M is not compact, then a map is called harmonic if its tension field vanishes identically.

At a critical point the Hessian of the energy functional is given by

$$\begin{aligned} \nabla ^2 E(f)(X,Y) = \int _{M} [\langle {\nabla X}, {\nabla Y}\rangle - {{\,\mathrm {tr}\,}}_g \langle {R^N(X, df \cdot ) df \cdot }, {Y}\rangle ] {{\,\mathrm {vol}\,}}_g \end{aligned}$$

for \(X,Y \in \Gamma ^1(f^*TN)\). Here \(\nabla \) denotes the connection on \(f^*TN\) induced by the Levi-Civita connection on TN and \(R^N\) denotes the Riemannian curvature tensor of (Nh). The non-degeneracy condition imposed on harmonic maps in [4] is that a harmonic map f is a non-degenerate critical point of the energy, i.e. \(\nabla ^2 E(f)\) is a non-degenerate bilinear form. In [15] Sunada proved that if the target is a locally symmetric space of non-positive curvature this condition is satisfied if and only if the harmonic map is unique. We collect these non-degeneracy conditions in the following lemma. Recall that we denote by G a real semisimple Lie group without compact factors and by \(X = G/K\) the associated symmetric space. An element \(g \in G\) is called hyperbolic if it can be written as \(g = \exp _G(X)\), where \(X\in \mathfrak {p}, X\ne 0\) and \({{\,\mathrm{Lie}\,}}(G) = \mathfrak {k}+ \mathfrak {p}\) is a Cartan decomposition of the Lie algebra of G.

Lemma 2.1

(Sunada) Let \(\rho :\Gamma \rightarrow G\) be a representation and assume that the induced action of \(\Gamma \) on X is free and proper. Denote by N the locally symmetric space \(N = X /{{\,\mathrm{im}\,}}\rho \). Assume, furthermore, that there exists a \(\rho \)-equivariant harmonic map \({\widetilde{f}} :{\widetilde{M}} \rightarrow X\). The map \({\widetilde{f}}\) descends to a harmonic map \(M\rightarrow N\) which we will denote by f. The following are equivalent:

  1. i

    f is a non-degenerate critical point of E.

  2. ii

    f is the unique harmonic map in its homotopy class.

  3. iii

    \({\widetilde{f}}\) is the unique \(\rho \)-equivariant harmonic map.

  4. iv

    The centraliser of \({{\,\mathrm{im}\,}}\rho \) in G contains no hyperbolic elements.

Proof

We can write \(\nabla ^2E(f) =- \int _M \langle {J_f \cdot },{\cdot }\rangle {{\,\mathrm {vol}\,}}_g\) where

$$\begin{aligned} J_f(X) = {{\,\mathrm{tr}\,}}_g \nabla ^2 X + {{\,\mathrm{tr}\,}}_g R^N(X, df\cdot )df\cdot \end{aligned}$$

is the Jacobi operator at f. As discussed in [4, p.35] the Hessian \(\nabla ^2 E(f)\) is non-degenerate precisely when \(\ker J_f = 0\). It follows from [15, Proposition 3.2] that the set

$$\begin{aligned} {{\,\mathrm{Harm}\,}}(M,N, f) = \{h :M \rightarrow N \mid h \text { is harmonic and homotopic to } f\} \end{aligned}$$

is a submanifold of \(W^k(M,N)\) (the space of maps from M to N equipped with the \(W^{k,2}\) Sobolev topology) with tangent space at h equal to \(\ker J_h\). Because N has non-positive curvature the space \({{\,\mathrm{Harm}\,}}(M,N,f)\) is connected ([8, Theorem 8.7.2]). We see that f is a non-degenerate critical point of the energy if and only if \({{\,\mathrm{Harm}\,}}(M,N,f)\) contains only f. This established the equivalence of (i) and (ii).

We will demonstrate the equivalence of statements (ii), (iii) and (iv) using proofs by contraposition. We begin by proving the equivalence of (ii) and (iii). First, if a harmonic map \(h :M \rightarrow N\) exists that is homotopic to f but also distinct from f, then h can be lifted to a \(\rho \)-equivariant harmonic map \({\widetilde{h}} :{\widetilde{M}} \rightarrow X\) that is different from \({\widetilde{f}}\). It follows that (iii) implies (ii). Now assume that (iii) does not hold and so there exists a \(\rho \)-equivariant harmonic map \({\widetilde{h}} :{\widetilde{M}} \rightarrow X\) that is distinct from \({\widetilde{f}}\). This map descends to a harmonic map \(h :M \rightarrow N\). If this map is different from f, then we see immediately that (ii) does also not hold. If, on the other hand, \(f = h\) (which happens for example if \({\widetilde{h}}\) is a translate of \({\widetilde{f}}\) by an element of \({{\,\mathrm{im}\,}}\rho \)), then we argue as follows. We consider the homotopy \({\widetilde{F}} :{\widetilde{M}} \times [0,1] \rightarrow X\) between \({\widetilde{f}}\) and \({\widetilde{h}}\) that is given by \({\widetilde{F}}(x,t) = \gamma _x(t)\) where \(\gamma _x :[0,1] \rightarrow X\) is the unique geodesic in X that connects \({\widetilde{f}}(x)\) to \({\widetilde{h}}(x)\). For each fixed \(t\in [0,1]\) the map \({\widetilde{F}}(\cdot , t) :{\widetilde{M}} \rightarrow X\) is also \(\rho \)-equivariant hence the homotopy descends to a map \(F :M \times [0,1] \rightarrow N\). We have \(F(\cdot ,0) = F(\cdot , 1) = f\) and for each \(x\in M\) the path \(t\mapsto F(x,t)\) is a non-constant geodesic loop. The arguments in the proof of [8, Theorem 8.7.2] show that each of the maps \(F(\cdot , t) :M \rightarrow N\) is also harmonic. So we see there exists a family of distinct harmonic maps in the homotopy class of f and hence also in this case (ii) does not hold. We conclude that indeed (ii) and (iii) are equivalent.

Finally, we prove the equivalence of (iii) and (iv). First, suppose there exists a hyperbolic isometry \(g\in G\) that centralises \({{\,\mathrm{im}\,}}\rho \). Consider the map \({\widetilde{h}} = g \cdot {\widetilde{f}} :{\widetilde{M}} \rightarrow X\). Because g is an isometry the map \({\widetilde{h}}\) is also harmonic and because g commutes with all elements \(\rho (\gamma ), \gamma \in \Gamma \) it follows that \({\widetilde{h}}\) is also \(\rho \)-equivariant. Furthermore, \({\widetilde{h}}\) is distinct from \({\widetilde{f}}\) because the isometry g has no fixed points in X. It follows that (iii) implies (iv). The other direction follows from [15, Lemma 3.4]. Namely, there it is proved that if \({\widetilde{h}} :{\widetilde{M}} \rightarrow X\) is a \(\rho \)-equivariant harmonic map different from \({\widetilde{f}}\), then a hyperbolic isometry \(g\in G\) that centralises \({{\,\mathrm{im}\,}}\rho \) exists such that \({\widetilde{h}} = g\cdot {\widetilde{f}}\). This finishes the proof. \(\square \)

The main existence result in the theory of equivariant harmonic maps is the following theorem of Corlette.

Proposition 2.2

([3]) Assume G is a real semisimple algebraic Lie group. A representation \(\rho :\Gamma \rightarrow G\) is reductive if and only if there exists a \(\rho \)-equivariant harmonic map \(f:{\widetilde{M}} \rightarrow X\).

When G is an algebraic Lie group a representation \(\rho :\Gamma \rightarrow G\) is called reductive if the Zariski closure of its image in G is a reductive subgroup. A generalisation of the definition of a reductive representation and a corresponding existence result for equivariant harmonic maps if G is not an algebraic group can be found in [10].

Families of representations and metrics

We will index families of representations or metrics by open balls in \(\mathbb {R}^n\). For \(\epsilon >0\) we denote by \(D_\epsilon \) the open \(\epsilon \) ball in \(\mathbb {R}^n\) centred at 0. Let \((\rho _t)_{t\in D_\epsilon }\) be a family of representations \(\rho _t :\Gamma \rightarrow G\). Such a family induces a natural action of \(\Gamma \) on \(X\times D_\epsilon \) given by \(\gamma \cdot (x,t) = (\rho _t(\gamma )x, t)\). We make the following properness and freeness assumption on the families of representations we will consider.

Definition 2.3

We call a family of representations uniformly free and proper if the induced action on \(X\times D_\epsilon \) is free and proper.

In particular, each representation in such a family acts freely and properly on X.

Remark 2.4

Anosov representations constitute a large class of representations for which any deformation is automatically a uniformly free and proper family. These representations where introduced by Labourie in [11] and their definition was extended by Guichard and Wienhard in [5] to representations of hyperbolic groups. In our setting we can consider Anosov representations whenever M admits a metric of strictly negative curvature because then \(\Gamma = \pi _1(M)\) is a hyperbolic group. The set of Anosov representations is open in \({{\,\mathrm{Hom}\,}}(\Gamma , G)\) (see [5, Theorem 5.13] or [9, Theorem 7.33]). Moreover, it follows from [9, Theorem 7.33] that continuous families of Anosov representations satisfy the uniformly free and proper assumption. So if \((\rho _t)_{t\in D_\epsilon }\) is a continuous family of representations with \(\rho _0\) an Anosov representation, then, after possibly shrinking \(\epsilon \), the family of representations \((\rho _t)_{t\in D_\epsilon }\) is uniformly free and proper.

On families of representations we will make the following regularity assumption.

Definition 2.5

A family of representations \((\rho _t)_{t\in D_\epsilon }\) of \(\Gamma \) in G is called real analytic if for every \(\gamma \in \Gamma \) the map \(D_\epsilon \rightarrow G :t \mapsto \rho _t(\gamma )\) is real analytic.

Remark 2.6

A family of representations can be seen as a map from \(D_\epsilon \) into \({{\,\mathrm{Hom}\,}}(\Gamma , G)\), the set of representations of \(\Gamma \) into G. If G is an algebraic subgroup of \({{\,\mathrm{GL}\,}}(n,\mathbb {R})\) and if S is a generating set of \(\Gamma \) with relations R, then \({{\,\mathrm{Hom}\,}}(\Gamma ,G)\) can be realised as the closed subset of \({{\,\mathrm{GL}\,}}(n,\mathbb {R})^S\) consisting of tuples \((g_1,...,g_n)\) satisfying the relations \(r(g_1,...,g_n)=1\) for \(r\in R\). In this way we realise \({{\,\mathrm{Hom}\,}}(\Gamma ,G)\) as a real algebraic variety. We note that in this case a family of representations is real analytic if and only if the map \(D_\epsilon \rightarrow {{\,\mathrm{Hom}\,}}(\Gamma , G)\) is real analytic.

Finally, for families of metrics we make the following regularity assumption.

Definition 2.7

We call a family \((g_t)_{t\in D_\epsilon }\) of Riemannian metrics on M a real analytic family of metrics if \((x,t) \mapsto g_t(x)\) induces a real analytic map \(M\times D_\epsilon \rightarrow S^2 T^*M\).

Mapping spaces

If M and N are real analytic manifolds we denote by \(C^{k,\alpha }(M,N)\) (\(k\in \mathbb {N}, 0<\alpha <1\)) the space of k-times differentiable maps from M to N such that the k-th derivatives are \(\alpha \)-Hölder continuous. We equip these spaces with the topology of uniform \(C^{k,\alpha }\) convergence on compact sets. To be more explicit, for a compact set \(K\subset \mathbb {R}^m\) we define the \(C^{k,\alpha }\) Hölder norm for functions in \(C^{k,\alpha }(K, \mathbb {R}^n)\) as

$$\begin{aligned} |\!|f|\!|_{K, k, \alpha } = \max _{\overset{x\in K}{i=1,\ldots k}} |\!|D^i f(x)|\!| + \max _{x\ne y \in K} \frac{|\!|f(x)-f(y)|\!|}{|\!|x-y|\!|^\alpha }. \end{aligned}$$

Then we define that a sequence of maps \((f_j)_{j\in \mathbb {N}}\) in \(C^{k,\alpha }(M,N)\) converges to a map \(f\in C^{k,\alpha }(M,N)\) if and only if for all charts \((U,\phi )\) of M, \((V,\psi )\) of N and compact sets \(K'\subset U\) we have \(|\!|\psi \circ f_j \circ \phi ^{-1} - \psi \circ f \circ \phi ^{-1}|\!|_{\phi (K'), k, \alpha } \rightarrow 0\) as \(j\rightarrow \infty \). If the domain manifold M is compact, then the spaces \(C^{k,\alpha }(M,N)\) can be equipped with a natural real analytic Banach manifold structure.

In the statement of our results we will formulate the condition that a family of maps \((f_t)_{t\in D_\epsilon }\) is a real analytic family as the requirement that \(t\mapsto f_t\) induces a real analytic map into a mapping space that is a Banach manifold. Here we follow the approach that Eells and Lemaire take in [4] for the smooth case. However, the mapping space \(C^{k,\alpha }({\widetilde{M}}, X)\) which naturally contains the equivariant maps \({\widetilde{M}} \rightarrow X\) can not be given a suitable Banach manifold structure because \({\widetilde{M}}\) is not compact. In order to avoid this issue we make use of the observation that equivariant maps are determined by their values on a fundamental domain. This will allow us to interpret these maps as an element of a mapping space that is a Banach manifold.

When M is a closed manifold we let \(\Omega \subset {\widetilde{M}}\) be a bounded domain containing a fundamental domain for the action of \(\Gamma \) on \({\widetilde{M}}\). We note that a map \({\widetilde{M}} \rightarrow X\) that is equivariant with respect to any representation is completely determined by its restriction to \(\Omega \). Furthermore, \(\rho _n\)-equivariant maps \(f_n\) converge to a \(\rho \)-equivariant map f uniformly on compacts if and only if the restrictions \(f_n\vert _\Omega \) converge uniformly to \(f\vert _\Omega \).

We will consider the space of bounded functions from \(\Omega \) to X. For this we equip M with a background metric and for simplicity we identify X with \(\mathbb {R}^n\) via the exponential map \(\exp _o :T_o X \rightarrow X\) based at some basepoint \(o\in X\). The metric on M induces a \(C^{k,\alpha }\) norm on the space of functions \(\Omega \rightarrow \mathbb {R}^n \cong X\). We denote by \(C^{k,\alpha }_b(\Omega , X)\) the space of functions for which this norm is bounded. This space can be equipped with the structure of a real analytic Banach manifold. For this we observe that equipped with the \(C^{k,\alpha }\) norm the space \(C^{k,\alpha }_b(\Omega , X)\) becomes a Banach space (note that the linear structure comes from the identification \(X \cong \mathbb {R}^n\) and carries no direct geometric meaning). The Banach manifold structure is obtained by declaring this to be a global chart. One can check that the Banach manifold structure is independent of the choice of background metric on M and basepoint in X (see Lemma 3.4). We would like to note that, although the use of the identification \(X \cong \mathbb {R}^n\) is somewhat ad hoc, if we replace the domain \(\Omega \) by a closed manifold M, then the above construction yields the usual Banach manifold structure on the space \(C^{k,\alpha }(M,X)\).

Main result

The main result of this article can be stated as follows.

Theorem 2.8

Let \((g_t)_{t\in D_\epsilon }\) be a real analytic family of metrics on M and let \((\rho _t)_{t\in D_\epsilon }\) be a real analytic family of representations of \(\Gamma \) in G. We assume that the family \((\rho _t)_{t\in D_\epsilon }\) is uniformly free and proper. Suppose \(\rho _0\) is reductive and the centraliser \(Z_G({{\,\mathrm{im}\,}}\rho _0)\) contains no hyperbolic elements. Then for every \(k \in \mathbb {N}, 0< \alpha <1\) there exists a \(\delta >0\) smaller then \(\epsilon \) and a unique continuous map \(F :D_\delta \rightarrow C^{k,\alpha }({\widetilde{M}}, X)\) such that each F(t) is a \(\rho _t\)-equivariant harmonic map \(({\widetilde{M}}, g_t)\rightarrow X\) and the restricted map \(F(\cdot )\vert _{\Omega } :D_\delta \rightarrow C^{k,\alpha }_b(\Omega , X)\) is real analytic.

Remark 2.9

Let us briefly clarify the statement of the theorem by spelling out what it means for the map \(F(\cdot )\vert _{\Omega } :D_\delta \rightarrow C^{k,\alpha }_b(\Omega , X)\) to be real analytic. The situation can be simplified by again using the fact that \(C^{k,\alpha }_b(\Omega , X)\) admits a global chart. So we pick an identification \(X \cong \mathbb {R}^n\) and use this to identify \(C^{k,\alpha }_b(\Omega , X) = C^{k,\alpha }_b(\Omega , \mathbb {R}^n)\). The mapping \(F(\cdot )\vert _{\Omega }\) is real analytic if for every \(s\in D_\delta \) there exist maps \(f_{k_1, \ldots , k_n} \in C^{k,\alpha }_b(\Omega ,\mathbb {R}^n)\) such that for all t in a neighbourhood of s the power series

$$\begin{aligned} P(t) = \sum _{k_1,\ldots , k_n=0}^\infty f_{k_1, \ldots , k_n} (t_1 - s_1)^{k_1} \cdots (t_n - s_n)^{k_n} \end{aligned}$$

converges uniformly and \(F(t)\vert _{\Omega } = P(t)\). This implies in particular that for every fixed \(x\in \Omega \) the value of F(t)(x) depends real analytically on \(t\in D_\delta \).

Remark 2.10

The above result is also true in the smooth category. More precisely we can define, analogous to Definitions 2.5 and 2.7, the notion of smooth families of metrics and representations. Then Theorem 2.8 also holds when we replace ‘real analytic’ by ‘smooth’. For brevity we will not prove the smooth case here but the reader can easily check that the proof goes through also in this case.

If each \(\rho _t\) is reductive and has trivial centraliser, then by applying Theorem 2.8 at each \(t\in D_\epsilon \) we obtain immediately the following corollary.

Corollary 2.11

Let \((g_t)_{t\in D_\epsilon }\), \((\rho _t)_{t\in D_\epsilon }\) be as in Theorem 2.8. Suppose that for every \(t\in D_\epsilon \) the representation \(\rho _t\) is reductive and \(Z_G({{\,\mathrm{im}\,}}\rho _t) = 0\). Then for all \(k\in \mathbb {N}, 0< \alpha < 1\) there exists a unique continuous map \(F :D_\epsilon \rightarrow C^{k,\alpha }({\widetilde{M}}, X)\) such that each \(F(t) :({\widetilde{M}},g_t)\rightarrow X\) is a \(\rho _t\)-equivariant harmonic map and the restricted map \(F(\cdot )\vert _\Omega :D_\epsilon \rightarrow C^{k,\alpha }_b(\Omega ,X)\) is real analytic.

Hitchin representations

We briefly mention how the above results can be applied when we consider Hitchin representations. In this section we let \(M = S\) be a closed surface of genus \(g \ge 2\) and as before \(\Gamma = \pi _1(S)\). In this case the harmonicity of a map \(f:S \rightarrow N\) depends only on the conformal class of the metric on S. Also, in this section we let G be a split real Lie group. A split real Lie group is a Lie group whose Lie algebra is a split real form of a complex Lie algebra. For example, the non-exceptional split real Lie groups are \({{\,\mathrm{SL}\,}}(n,\mathbb {R})\), \({{\,\mathrm{Sp}\,}}(n,\mathbb {R})\), \({{\,\mathrm{SO}\,}}(n,n)\) and \({{\,\mathrm{SO}\,}}(n,n+1)\).

Associated to \(\Gamma \) and G is the representation variety

$$\begin{aligned} {{\,\mathrm{Rep}\,}}(\Gamma , G) = {{\,\mathrm{Hom}\,}}_{\text {red}}(\Gamma ,G)/G \end{aligned}$$

which consists of reductive representations \(\Gamma \rightarrow G\) considered up to conjugation by elements in G. In [6] Hitchin proved that \({{\,\mathrm{Rep}\,}}(\Gamma , G)\) contains a connected component, now called the Hitchin component, which contains \(\mathcal {T}(S)\), the Teichmüller space of S, in a natural way. A representation \(\Gamma \rightarrow G\) whose conjugacy class lies in the Hitchin component is called a Hitchin representation. We denote by \({{\,\mathrm{Hom_{Hit}}\,}}(\Gamma ,G)\) the component of \({{\,\mathrm{Hom}\,}}(\Gamma , G)\) consisting of Hitchin representations.

Hitchin representations enjoy many special properties. They are reductive because, by definition, their conjugacy class lies in the Hitchin component which is a subset of \({{\,\mathrm{Rep}\,}}(\Gamma ,G)\). The centraliser of the image of a Hitchin representation is trivial (see [2, Lemma 2.9]). Furthermore, in [11] Labourie showed that Hitchin representations are Anosov representations. It follows, as discussed in Remark 2.4, that continuous families of Hitchin representations satisfy the uniformly free and proper assumption as in Definition 2.3.

It follows that there exists a map \(F :\mathcal {T}(S) \times {{\,\mathrm{Hom_{Hit}}\,}}(\Gamma ,G) \rightarrow C^{k,\alpha }({\widetilde{S}},X)\) assigning to each \((J, \rho )\) the unique \(\rho \)-equivariant harmonic map \(({\widetilde{S}},J) \rightarrow X\). A chart of \({{\,\mathrm{Hom_{Hit}}\,}}(\Gamma ,G)\) modelled on \(D_\epsilon \) can be seen as real analytic family of representations that is uniformly free and proper. Furthermore, it follows from [18] that it is possible to choose metrics \(g_J\) on S representing points in Teichmüller space \(J\in \mathcal {T}(S)\) depending on J in a real analytic fashion. By applying Corollary 2.11 to charts around points \((J,\rho ) \in \mathcal {T}(S) \times {{\,\mathrm{Hom_{Hit}}\,}}(\Gamma ,G)\) we obtain the following proposition.

Proposition 2.12

For all \(k\in \mathbb {N}, 0< \alpha < 1\) the map

$$\begin{aligned} F :\mathcal {T}(S) \times {{\,\mathrm{Hom_{Hit}}\,}}(\Gamma ,G) \rightarrow C^{k,\alpha }({\widetilde{S}}, X) \end{aligned}$$

assigning to each \((J,\rho )\) the unique \(\rho \)-equivariant harmonic map \(({\widetilde{S}},J) \rightarrow X\) is continuous and the restricted map \(F(\cdot ,\cdot )\vert _\Omega :\mathcal {T}(S) \times {{\,\mathrm{Hom_{Hit}}\,}}(\Gamma ,G) \rightarrow C_b^{k,\alpha }(\Omega ,X)\) is real analytic.

We discuss three corollaries to this result.

First we observe that the family of harmonic maps given by F can also be interpreted as a single map with the universal Teichmüller curve as domain. Namely, let \(\Xi (S)\) be the universal Teichmüller curve of S. It is a trivial fibre bundle over \(\mathcal {T}(S)\) with fibres homeomorphic to S. It is equipped with a complex structure such that the fibre \(\Xi (S)_J\) over \(J \in \mathcal {T}(S)\) together with the marking provided by the trivialization \(\Xi (S)_J \cong \mathcal {T}(S)\times S\) determines the point J in Teichmüller space (see [7, Sect. 6.8]). The universal cover \({\widetilde{\Xi }}(S)\) is a trivial fibre bundle over \(\mathcal {T}(S)\) with fibres homeomorphic to \({\widetilde{S}}\). Let \(F' :{\widetilde{\Xi }}(S)\times {{\,\mathrm{Hom_{Hit}}\,}}(\Gamma ,G) \rightarrow X\) be the map which on each fibre \({\widetilde{\Xi }}(S)_J \times \{\rho \} \cong {\widetilde{S}}\) is given by the \(\rho \)-equivariant harmonic map \(({\widetilde{\Xi }}(S)_J, J) \rightarrow X\). It follows from Proposition 2.12 that this map is real analytic.

Corollary 2.13

The set

$$\begin{aligned} I&= \{((J,x),\rho ) \in {\widetilde{\Xi }}(S)\times {{\,\mathrm{Hom_{Hit}}\,}}(\Gamma ,G) \mid \\&F'(J,\cdot , \rho ) :{\widetilde{\Xi }}(S)_J \rightarrow X \text { is not an immersion at } x\} \end{aligned}$$

is a real analytic subvariety of \({\widetilde{\Xi }}(S)\times {{\,\mathrm{Hom_{Hit}}\,}}(\Gamma ,G)\).

It is conjectured (see for example [12, Conjecture 9.3]) that equivariant harmonic maps associated to Hitchin representations are immersions which would correspond to the set I being empty.

Proof

We equip \(\Xi (S)\) with a choice of real analytic metric. Given a point \(((J,x),\rho ) \in {\widetilde{\Xi }}(S) \times {{\,\mathrm{Hom_{Hit}}\,}}(\Gamma ,G)\) we consider the derivative of \(F'\) in the fibre direction

$$\begin{aligned} dF(J,\cdot ,\rho ) :T_x({\widetilde{\Xi }}(S)_J) \rightarrow \mathrm {im}\,dF(J,\cdot , \rho ). \end{aligned}$$

Because the spaces \(T_x({\widetilde{\Xi }}(S)_J)\) and \(\mathrm {im}\,dF(J,\cdot ,\rho ) \subset T_{F'(J,x,\rho )}X\) are equipped with inner products we can consider the determinant of this map. We let \(d :{\widetilde{\Xi }}(S)\times {{\,\mathrm{Hom_{Hit}}\,}}(\Gamma ,G)\rightarrow \mathbb {R}\) be the map which at a point \(((J,x),\rho )\) is given by the determinant of the above map. Because \(F'\) is real analytic the map d is real analytic as well. We observe that \(I = d^{-1}(0)\) from which the result follows. \(\square \)

In a similar vein we also have the following corollary.

Corollary 2.14

The set

$$\begin{aligned} T&=\{(J,\rho ) \in \mathcal {T}(S)\times {{\,\mathrm{Hom_{Hit}}\,}}(\Gamma ,G) \mid \\&(S,J) \text { can be realised in } X/\rho (\Gamma ) \text { as a branched minimal surface}\} \end{aligned}$$

is a real analytic subvariety of \(\mathcal {T}(S)\times {{\,\mathrm{Hom_{Hit}}\,}}(\Gamma ,G)\).

Proof

For \(J\in \mathcal {T}(S)\) and \(\rho \in {{\,\mathrm{Hom_{Hit}}\,}}(\Gamma ,G)\) we consider the Hopf differential of the harmonic map \(F(J,\rho ) :({\widetilde{S}}, J) \rightarrow X\) which is given by \(\phi _{J,\rho } = (F(J,\rho )^* m)^{2,0}\). Here m is the Riemannian metric of the symmetric space X. The Hopf differential is a holomorphic quadratic differential on \(({\widetilde{S}}, J)\) which vanishes if and only if the harmonic map \(F(J,\rho )\) is a (branched) minimal surface. The Hopf differential \(\phi _{J,\rho }\) is \(\Gamma \)-invariant and descends to S since \(F(J,\rho )\) is \(\rho \)-equivariant. Consider the function \(V :\mathcal {T}(S)\times {{\,\mathrm{Hom_{Hit}}\,}}(\Gamma ,G) \rightarrow \mathbb {R}\) given by the \(L^2\)-norm of \(\phi _{J,\rho }\), namely

$$\begin{aligned} V(J,\rho ) = \int _S \frac{|{\phi _{J,\rho }}|^2}{\sqrt{\det g_J}} |{dz}|^2. \end{aligned}$$

Here \(g_J\) is a metric in the conformal class of J depending real analytically on J. It follows from Proposition 2.12 that this function is real analytic (it is for this reason that we choose the \(L^2\)-norm rather than the \(L^1\)-norm). The statement of the corollary follows from \(T = V^{-1}(0)\). \(\square \)

The space of Fuchsian representations \(\Gamma \rightarrow {{\,\mathrm{SL}\,}}(2,\mathbb {R})\) can be included in \({{\,\mathrm{Hom_{Hit}}\,}}(\Gamma ,G)\) in the following way. There exists an irreducible representation \(\iota :{{\,\mathrm{SL}\,}}(2,\mathbb {R}) \rightarrow G\) that is unique up to conjugation. Then to a Fuchsian representation \(\rho _0 :\Gamma \rightarrow {{\,\mathrm{SL}\,}}(2,\mathbb {R})\) we associate a so-called principal Fuchsian representation \(\rho = \iota \circ \rho _0\) which lies in \({{\,\mathrm{Hom_{Hit}}\,}}(\Gamma ,G)\). This inclusion descends to the natural inclusion of Teichmüller space into the Hitchin component. A principal Fuchsian representation stabilises the totally geodesically embedded copy of \(\mathbb {H}^2\) in G/K given by the inclusion \(\iota ' :{{\,\mathrm{SL}\,}}(2,\mathbb {R})/{{\,\mathrm{SO}\,}}(2) \rightarrow G/K= X\) that is induced by \(\iota \). By uniqueness of equivariant harmonic maps we have that the harmonic map \(({\widetilde{S}},J) \rightarrow X\) equivariant for \(\rho = \iota \circ \rho _0\) is given by the composition \(\iota '\circ f_0\) where \(f_0 :({\widetilde{S}},J) \rightarrow \mathbb {H}^2\) is the unique \(\rho _0\)-equivariant map. It follows from [14, Theorem 11] that \(f_0\) is a diffeomorphism. Hence, equivariant harmonic maps associated to principal Fuchsian representations are immersions. Because being an immersion is an open condition we immediately obtain the following corollary to Proposition 2.12

Corollary 2.15

There exists an open neighbourhood of

$$\begin{aligned} \mathcal {T}(S)\times \{\text {principal Fuchsian representations}\} \subset \mathcal {T}(S)\times {{\,\mathrm{Hom_{Hit}}\,}}(\Gamma ,G) \end{aligned}$$

such that for any pair \((J,\rho )\) in this neighbourhood the \(\rho \)-equivariant harmonic map \(({\widetilde{S}},J) \rightarrow X\) is an immersion.

Proof of the main result

As in the proof of Eells and Lemaire in [4], our main analytical tool will be the implicit function theorem for maps between Banach manifolds. The main difficulty to overcome is that a priori the equivariant harmonic maps are not elements of the same space of mappings. Namely, if \((\rho _t)_t\) is a family of representations, then a \(\rho _t\)-equivariant map is an element of the space \(C^{k,\alpha }(M, X/\rho _t(\Gamma ))\). Since the target manifold is different for each t these spaces are not equal (although they are likely to be diffeomorphic). Our aim is to modify these maps so that they can be seen as elements of a single mapping space. This will be achieved by means of a family of deformation maps which intertwine the representations \(\rho _0\) and \(\rho _t\). By composing with these deformation maps we can view each \(\rho _t\)-equivariant map as element of (a subset of) \(C^{k,\alpha }(M, X/\rho _0(\gamma ))\).

We first fix some notation. We let \((\rho _t)_{t\in D_\epsilon }\) be a real analytic family of representations that is uniformly free and proper. We denote \(X_\epsilon = X\times D_\epsilon \) and by \(\alpha :\Gamma \times X_\epsilon \rightarrow X_\epsilon , \alpha (\gamma )(x,t) = (\rho _t(\gamma )x,t)\) the natural action induced by \((\rho _t)_t\). We fix a base point \(o\in X\) of the symmetric space and denote by \(U_R = \cup _{\gamma \in \Gamma } B(\rho _0(\gamma )o, R)\) the R-neighbourhood of the \(\rho _0(\Gamma )\)-orbit of o. Our deformation maps will be provided by the following proposition.

Proposition 3.1

For every \(R>0\) there exists a \(\delta = \delta (R)>0\) smaller than \(\epsilon \) and family of maps \((\Phi _t :U_R \rightarrow X)_{t\in D_\delta }\) satisfying the following properties:

  1. i

    The induced map \(U_R \times D_\delta \rightarrow X :(x, t) \mapsto \Phi _t(x)\) is real analytic.

  2. ii

    For each \(t\in D_\delta \) the set \(\Phi _t(U_R)\) is open and \(\Phi _t :U_R \rightarrow \Phi _t(U_R)\) is a real analytic diffeomorphism.

  3. iii

    \(\Phi _0 = {{\,\mathrm{id}\,}}:U_R \rightarrow U_R\).

  4. iv

    For each \(t\in D_\delta \) the map \(\Phi _t\) intertwines the actions of \(\rho _0\) and \(\rho _t\), i.e. \(\rho _t(\gamma ) \circ \Phi _t = \Phi _t \circ \rho _0(\gamma )\) for \(\gamma \in \Gamma \).

The content of this proposition is closely related to Ehresmann’s fibration theorem. In fact, when the actions of the representations \(\rho _t\) on X are cocompact Proposition 3.1 can be obtained from it. Consequently the proof of Proposition 3.1 is along the same lines as the proof of the fibration theorem.

We denote by \({{\,\mathrm{pr}\,}}_X :X_\epsilon \rightarrow X\) and \(\pi :X_\epsilon \rightarrow D_\epsilon \) the projections onto the first and second factor of \(X_\epsilon = X \times D_\epsilon \) respectively. By \((t_1,...,t_n)\) we denote the coordinates on \(D_\epsilon \) and also the coordinates on the \(D_\epsilon \) factor in \(X_\epsilon \). So in this notation we have \(d\pi (\frac{\partial }{\partial t_{i}}(x,t)) = \frac{\partial }{\partial t_{i}}(t)\).

We first prove the following lemma.

Lemma 3.2

Let \(R>0\). On an \(\alpha (\Gamma )\)-invariant neighbourhood of \(U_R\times \{0\}\) in \(X_\epsilon \) there exist \(\alpha (\Gamma )\)-invariant real analytic vector fields \(\xi _i\) (for \(i=1,..., n\)) that satisfy \(d\pi (\xi _i(x,t)) = \frac{\partial }{\partial t_{i}}(t)\).

Proof

It is possible to give a more or less explicit construction for such vector fields. However, proving they are indeed real analytic is rather cumbersome. Instead we opt to explicitly construct smooth vector fields which we then approximate by real analytic ones.

We let \(\varphi :[0,\infty ) \rightarrow [0,1]\) be a smooth function satisfying \(\varphi \vert _{[0,R]} \equiv 1\) and \(\varphi \vert _{[R+1, \infty )} \equiv 0\). For \(i=1,...,n\) we define smooth vector fields \(\eta _i\) on \(X_\epsilon \) by

$$\begin{aligned} \eta _i(x,t) = \varphi (d(o,x)) \cdot \frac{\partial }{\partial t_{i}}. \end{aligned}$$

Now let

$$\begin{aligned} \xi _i' = \sum _{\gamma \in \Gamma } (\alpha (\gamma ))_* \eta _i. \end{aligned}$$

The sum on the right hand side is locally finite by the uniform properness assumption on \((\rho _t)_t\). Hence, each \(\xi _i'\) is \(\alpha (\Gamma )\)-invariant smooth vector field on \(X_\epsilon \). We observe that \(d\pi (\eta _i(x,t)) = s(x,t) \frac{\partial }{\partial t_{i}}\) with

$$\begin{aligned} s(x,t) = \sum _{\gamma \in \Gamma } \varphi (d(o, \rho _t(\gamma )^{-1}x)). \end{aligned}$$

On \(B(o,R)\times \{0\}\) we have that \(s(x,t) \ge \varphi (d(o,x)) = 1\) and by \(\alpha (\Gamma )\)-invariance we have that \(s \ge 1\) on \(U_R\times \{0\}\).

We now approximate the smooth vector fields \(\xi _i'\) by real analytic ones. By the uniformly free and proper assumption on \((\rho _t)_{t}\) we have that \(X_\epsilon /\alpha (\Gamma )\) is a real analytic manifold. The vector fields \(\xi _i'\) descend to smooth vector fields. On compact subsets these vector fields can be approximated arbitrarily closely in \(C^0\) norm by real analytic vector fields (see [16] and [13]). The set \(U_R\times \{0\}\) maps to a precompact subset of \(X_\epsilon /\alpha (\Gamma )\). Hence, by pulling back approximating vector fields we see that on a neighbourhood of \(U_R\times \{0\}\) we can approximate \(\xi _i'\) arbitrarily closely by \(\alpha (\Gamma )\)-invariant real analytic vector fields. Let \(\xi _i''\) be such approximating vector fields. For some real analytic functions \(s_i'\) we have \(d\pi (\xi _i''(x,t)) = s_i'(x,t)\frac{\partial }{\partial t_{i}}\). By choosing the approximating vector fields \(\xi _i''\) close enough to \(\xi _i'\) we can arrange that each \(s_i'\) is close to s and hence satisfies \(s_i'>0\) on a neighbourhood of \(U_R\times \{0\}\). For \(i=1,...,n\) we can now define \(\xi _i = \xi _i''/s'\). \(\square \)

Proof of Proposition 3.1

Let \(\xi _i\) for \(i=1,..., n\) be the vector fields constructed in Lemma 3.2. We denote by \(\psi _i^s\) their flows which are defined on a neighbourhood of \(U_R\times \{0\}\). We consider the maximal flow domain for a combination of these flows starting at points in \(X\times \{0\}\), i.e. the set

$$\begin{aligned} \Omega = \{(x,s)\in X\times \mathbb {R}^n \mid \psi _1^{s_1}\circ \cdots \circ \psi _n^{s_n}((x,0)) \text { is defined}\}. \end{aligned}$$

This is an open set containing \(U_R \times \{0\}\). On \(\Omega \) we set

$$\begin{aligned} \Psi (x,(s_1,...,s_n)) = \psi _1^{s_1}\circ \cdots \circ \psi _n^{s_n}((x,0)). \end{aligned}$$

Because \(d\pi (\xi _i) = \frac{\partial }{\partial t_{i}}\) (when defined) we see that

$$\begin{aligned} t \mapsto \pi \circ \Psi (x, (s_1,..., s_{i-1}, t, s_{i+1},..., s_n)) \end{aligned}$$

is an integral line for the vector field \(\frac{\partial }{\partial t_{i}}\). Since these integral lines are unique and \(\pi \circ \Psi (x,0) = 0\) we find \(\pi \circ \Psi (x,s) = s\) when \((x,s)\in \Omega \), i.e. \(\pi \circ \Psi = \pi \). Because the vector fields \(\xi _i\) are \(\alpha (\Gamma )\)-invariant we observe for \(\gamma \in \Gamma \) that

$$\begin{aligned} \Psi (\rho _0(\gamma )x, s)&= \psi _1^{s_1}\circ \cdots \circ \psi _n^{s_n}(\alpha (\gamma )(x,0)) \\ {}&= \alpha (\gamma ) [\psi _1^{s_1}\circ \cdots \circ \psi _n^{s_n}(x,0)] = \alpha (\gamma ) \Psi (x,s) \end{aligned}$$

whenever both sides are defined. We define \(\beta :\Gamma \times \Omega \rightarrow \Omega \) as an action of \(\Gamma \) on \(\Omega \) by \(\beta (\gamma )(x,s) = (\rho _0(\gamma )x, s)\) which is the action of \(\rho _0(\Gamma )\) on X times the trivial action. By the above the we see that the set \(\Omega \) is \(\beta (\Gamma )\)-invariant and \(\Psi \) intertwines the \(\beta \) and \(\alpha \) actions, i.e. \(\alpha (\gamma ) \circ \Psi = \Psi \circ \beta (\gamma )\) for all \(\gamma \in \Gamma \).

On \(X\times \{0\} \cap \Omega \) the map \(\Psi \) is simply the inclusion into \(X_\epsilon \). Combined with the fact that \(\pi \circ \Psi = \pi \) we see for each \((x,0) \in X\times \{0\} \cap \Omega \) the tangent map \(d\Psi \vert _{(x,0)} :T_x X \times T_0\mathbb {R}^n \rightarrow T_x X \times T_0 D_\epsilon \) is the identity map. Hence, we can shrink \(\Omega \) to a smaller open neighbourhood of \(U_R \times \{0\}\) such that \(\Psi \) is a local diffeomorphism on \(\Omega \). By shrinking \(\Omega \) further we can also assume \(\Psi \) to be injective. For if not, then there exist two distinct sequences \((x_n,s_n), (x_n',s_n')\in \Omega \) satisfying \(\Psi (x_n,s_n) = \Psi (x_n', s_n')\) with \(s_n, s_n'\rightarrow 0\) and \(x_n, x_n'\) converging to points x and \(x'\) in \(U_R\). By \(\pi \circ \Psi =\pi \) we see \(s_n=s_n'\). By continuity \(\Psi (x, 0) = \Psi (x',0)\) and because when restricted to \(X\times \{0\}\cap \Omega \) the map \(\Psi \) is an injection we must have \(x=x'\). However, this contradicts the fact that \(\Psi \) is a local diffeomorphism. So we can arrange that \(\Psi \) is a diffeomorphism onto its image. Since \(\Psi \) intertwines \(\beta \) and \(\alpha \) this can be done in such a way that \(\Omega \) is still \(\beta (\Gamma )\)-invariant.

Since \(\Omega \) is a neighbourhood of \(U_R\times \{0\}\) we can, by compactness, find a \(\delta >0\) such that \(\overline{B(o,R)}\times D_\delta \subset \Omega \). By \(\beta (\Gamma )\)-invariance we then have \(U_R \times D_\delta \subset \Omega \). We now define the family of deformation maps \(\Phi _t :U_R \rightarrow X\) as \(\Phi _t(x) = {{\,\mathrm{pr}\,}}_{X}\circ \Psi (x,t)\) for \(t\in D_\delta \). We check that indeed \((\Phi _t)_{t\in D_\delta }\) satisfies Properties (i)-(iv). Property (i) follows since flows of real analytic vector fields are real analytic. Property (ii) follows since \(\Psi :\Omega \rightarrow \Psi (\Omega )\) is a diffeomorphism and satisfies \(\pi \circ \Psi = \pi \) hence induces diffeomorphisms between the fibres \(\pi ^{-1}(t)\cap \Omega \) and \(\pi ^{-1}(t) \cap \Psi (\Omega )\). Property (iii) follows from the fact that \(\Psi \) restricted to \(X\times \{0\}\cap \Omega \) is the inclusion map and Property (iv) follows from the fact that \(\Psi \) intertwines the actions of \(\beta \) and \(\alpha \). \(\square \)

Using the deformation maps the problem of dependence on representations can be reduced to the problem of dependence on metrics on a fixed target manifold. In this case the results of [4] can be used. In their paper Eells and Lemaire only prove smooth dependence so for completeness we prove a version of their result in the real analytic category.

Proposition 3.3

Let MN be real analytic manifolds with M compact. Let \((g_t)_{t\in D_\epsilon }\) and \((h_t)_{t\in D_\epsilon }\) be real analytic families of metrics on M and N, respectively. If \(f_0 :(M,g_0) \rightarrow (N,h_0)\) is a harmonic map such that \(\nabla ^2 E(f_0)\) is non-degenerate, then for every \(k\in \mathbb {N}, 0< \alpha < 1\) there exists a \(\delta >0\) and a unique real analytic map \(F :D_\delta \rightarrow C^{k,\alpha }(M, N)\) such that \(F(0) = f_0\) and each F(t) is a harmonic map \((M,g_t) \rightarrow (N,h_t)\).

Proof

For each \(t\in D_\epsilon \) a \(C^2\) map \(\phi :(M,g_t) \rightarrow (N,h_t)\) is harmonic if and only if \(\tau _t(\phi ) = {{\,\mathrm{tr}\,}}_{g_t} \nabla d\phi =0\) where \(\nabla \) is the connection on \(T^*M \otimes \phi ^* TN\) induced by \(g_t\) and \(h_t\). In local coordinates \((x^i)_i\) on M and \((u^\alpha )_\alpha \) on N, \(\tau _t(\phi )\) is given by

$$\begin{aligned} \tau _t(\phi )^\gamma = (g_t)_{ij} \left\{ \frac{\partial ^2 \phi ^\gamma }{\partial x^i \partial x^j} - (\Gamma _{g_t})_{ij}^k \frac{\partial \phi ^\gamma }{\partial x^k} + (\Gamma _{h_t})_{\alpha \beta }^\gamma (\phi ) \frac{\partial \phi ^\alpha }{\partial x_i}\frac{\partial \phi ^\beta }{\partial x_j} \right\} \end{aligned}$$

here \(\Gamma _{g}\) denote the Christoffel symbols of a metric g. We combine the tension fields for different \(t\in D_\epsilon \) in a map

$$\begin{aligned} \tau :C^{k+2+\alpha }(M,N) \times D_\epsilon \rightarrow TC^{k+\alpha }(M,N). \end{aligned}$$

We claim this map is real analytic. To see this we write \(\tau \) as a composition of two real analytic map. First we consider the second prolongation map

$$\begin{aligned} J :C^{k+2+\alpha }(M,N) \rightarrow C^{k+\alpha }(M, J^2(M,N)) \end{aligned}$$

mapping a map \(\phi :M \rightarrow N\) to its 2-jet \(j^2\phi \). A diffeomorphism between a neighbourhood of the zero section in \(\phi ^*TN\) and a neighbourhood of the image of \(\mathrm {graph}(\phi )\) in \(M\times N\) induces charts of the two mapping spaces modelled on \(\Gamma ^{k+2+\alpha }(\phi ^*TN)\) and \(\Gamma ^{k+\alpha }(J^2(M,\phi ^*TN))\) respectively. In these charts the second prolongation map is a bounded linear map so in particular it is real analytic. Secondly, we consider the map \(T:J^2(M,N) \times D_\epsilon \rightarrow TN\) given in local coordinates (induced by \((x^i)_i\) on M and \((u^\alpha )_\alpha \) on N) by

$$\begin{aligned} (x^i, u^\alpha , u^{\alpha }_{i}, u^{\alpha }_{ij}, t) \mapsto (g_t)_{ij} \left\{ u^\gamma _{ij} - (\Gamma _{g_t})_{ij}^k u^\gamma _{k} + (\Gamma _{h_t})_{\alpha \beta }^\gamma (u) u^\alpha _i u^\beta _j \right\} . \end{aligned}$$

By assumption the coefficients \((g_t)_{ij}, (\Gamma _{g_t})_{ij}^k\) and \((\Gamma _{h_t})_{\alpha \beta }^\gamma \) are real analytic functions so the map T is real analytic. It now follows from the \(\Omega \)-lemma (see Lemma 3.4 below) that T induces a real analytic map

$$\begin{aligned} T_*:&C^{k+\alpha }(M, J^2(M,N))\times D_\epsilon \rightarrow C^{k+\alpha }(M,TN)\\&(\psi ,t) \mapsto T(\psi (\cdot ),t) \end{aligned}$$

The map \(\tau \) is a composition of \(T_*\) and J and is therefore real analytic.

As discussed in [4, p. 35] the partial derivative of \(\tau \) with respect to the first factor \((D_1 \tau )_{(f_0,0)} :T C^{k+2+\alpha }(M,N) \rightarrow T_0 T C^{k+\alpha }(T,N)\) is an isomorphism of Banach spaces precisely when \(\nabla ^2 E(f_0)\) is non-degenerate. Hence, we can apply a real analytic version of the implicit function theorem for Banach spaces (e.g. [17]) to obtain, for \(\delta >0\) small enough, a unique real analytic map \(F :D_\delta \rightarrow C^{k+2+\alpha }(M,N)\) such that \(F(0) = f_0\) and \(\tau _t(F(t)) = 0\) for all \(t\in D_\delta \). \(\square \)

Lemma 3.4

(The \(\Omega \)-Lemma) Let MN and P be real analytic manifolds with M compact. Suppose \(F :N \rightarrow P\) is real analytic. Then F induces a real analytic map

$$\begin{aligned} \Omega _F :C^{k,\alpha }(M,N) \rightarrow C^{k,\alpha }(M, P) :\phi \mapsto F \circ \phi \end{aligned}$$

for all \(k\in \mathbb {N}, 0< \alpha < 1\).

Compare with [1, Theorem 11.3]. Unfortunately, a proof of the real analytic case as stated here does not seem to be available in the literature. We give a sketch of the proof.

Sketch of proof of Lemma 3.4

By following the same steps as in [1] the statement can be reduced to a local version (cf. [1, Theorem 3.7]), i.e. it is enough to prove that if \(K \subset \mathbb {R}^n\) is compact, \(V\subset \mathbb {R}^p\) open and \(F :K \times V \rightarrow \mathbb {R}^q\) real analytic, then \(\Omega _F :C^{k,\alpha }(K, V) \rightarrow C^{k,\alpha }(K, \mathbb {R}^q)\) given by \([\Omega _F(\phi )](x) = F(x, \phi (x))\) is a real analytic map between Banach spaces. To this end we observe that since F is real analytic it can be extended to a complex analytic map \({\widetilde{F}} :U \rightarrow \mathbb {C}^q\) on an open set \(U \subset \mathbb {C}^n \times \mathbb {C}^p\) containing \(K \times V\). Let \({\widetilde{V}} \subset \mathbb {C}^p\) be an open such that \(K \times V \subset K\times {\widetilde{V}} \subset U\). Then \({\widetilde{F}}\) induces a map \(\Omega _{{\widetilde{F}}} :C^{k,\alpha }(K, {\widetilde{V}}) \rightarrow C^{k,\alpha }(K, \mathbb {C}^q)\) between complex Banach spaces which extends \(\Omega _F\). Applying the smooth version of the \(\Omega \)-Lemma yields that \(\Omega _{{\widetilde{F}}}\) is a \(C^1\) map with derivative given by \(D\Omega _{{\widetilde{F}}} = \Omega _{D_2 {\widetilde{F}}}\). Since \({\widetilde{F}}\) is holomorphic we see that this derivative is complex linear. It now follows from [7, Theorem A5.3] that \( \Omega _{{\widetilde{F}}}\) is a complex analytic map. We conclude that \(\Omega _F\), which is a restriction of \(\Omega _{{\widetilde{F}}}\) to \(C^{k,\alpha }(K, V)\), is real analytic.\(\square \)

We can now prove the statement of our main theorem.

Proof of Theorem 2.8

We set \(N = X/\rho _0(\Gamma )\) (recall that by assumption the action of \(\rho _0\) on X is free and proper so N is a manifold). Since \(\rho _0\) is reductive and \(Z_G({{\,\mathrm{im}\,}}\rho _0)\) contains no hyperbolic elements there exists a unique \(\rho _0\)-equivariant harmonic map \(f:{\widetilde{M}} \rightarrow X\). This map descends to a harmonic map \(f :M \rightarrow N\). We denote by \(o'\) the point in N covered by the base point o in X. The set \(U_R\) descends to the set \(V = B(o', R)\) in N. We choose \(R>0\) large enough such that the image of f is contained in \(B(o', R)\).

Let \((\Phi _t)_{t\in D_\delta }\) be the family of deformation maps as in Proposition 3.1. We denote by m the Riemannian metric on the symmetric space X. Define a family of metrics \((h_t)_{t\in D_\delta }\) on \(U_R\) by \(h_t = \Phi _t^* m\). By Property (3.1.i) this is a real analytic family of metrics. We observe for \(\gamma \in \Gamma \) that

$$\begin{aligned} \rho _0(\gamma )^* h_t = \rho _0^*\Phi _t^* m = \Phi _t^* \rho _t(\gamma )^* m = \Phi _t^* m = h_t. \end{aligned}$$

Here we used Property (3.1.iv) and the fact that each \(\rho _t\) acts on X by isometries. We conclude that each \(h_t\) is \(\rho _0(\Gamma )\)-invariant hence the family of metrics descends to a family of metrics, also denoted \((h_t)_{t\in D_\delta }\), on V. By Lemma 2.1 the Hessian \(\nabla ^2E(f)\) is non-degenerate so Proposition 3.3 yields, after shrinking \(\delta \), a unique real analytic map \(G :D_\delta \rightarrow C^{k,\alpha }(M, V)\) such that G(t) is a harmonic map from \((M,g_t)\) to \((V, h_t)\) for each \(t\in D_\delta \). By choosing for each t a \(\rho _0\)-equivariant lift we can view G as a continuous map \(G :D_\delta \rightarrow C^{k,\alpha }({\widetilde{M}}, U_R)\). We define F by composing with the deformation maps, \(F(t) = \Phi _t \circ G(t)\). By Property (3.1.iv) every map F(t) is \(\rho _t\)-equivariant. By construction, each \(\Phi _t\) is an open isometric embedding of \((V, h_t)\) into (Xm) hence each F(t) is also harmonic. Finally, by Property (3.1.i) we see that the map \(F :D_\delta \rightarrow C^{k,\alpha }({\widetilde{M}}, X)\) is continuous and real analytic as a map \(F(\cdot )\vert _\Omega :D_\delta \rightarrow C^{k,\alpha }(\Omega ,X)\). \(\square \)

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Acknowledgements

The author would like to thank Prof. Ursula Hamenstädt for suggesting this problem and her help and encouragement during the project. The author also thanks the anonymous referee for their helpful suggestions and comments. The author is grateful for the financial support received from the Max Planck Institute for Mathematics.

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The author was supported by the IMPRS on Moduli Spaces PhD program of the Max Planck Institute for Mathematics in Bonn.

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Slegers, I. Equivariant harmonic maps depend real analytically on the representation. manuscripta math. (2021). https://doi.org/10.1007/s00229-021-01345-z

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  • DOI: https://doi.org/10.1007/s00229-021-01345-z

Mathematics Subject Classification

  • 58E20