Uniformizing surfaces via discrete harmonic maps

We show that for any closed surface of genus greater than one and for any finite weighted graph filling the surface, there exists a hyperbolic metric which realizes the least Dirichlet energy harmonic embedding of the graph among a fixed homotopy class and all hyperbolic metrics on the surface. We give explicit examples of such hyperbolic surfaces as a refinement of the Nielsen realization problem for the mapping class groups.


Introduction
The Koebe-Andreev-Thurston theorem states that any skeleton of triangulation for an oriented compact surface arises as the contact graph of a circle packing on a geometrized surface ([Thu78, Section 13.7]; see also [CdV91b]). In the case of a closed Riemann surface of genus greater than one, one obtains a hyperbolic metric which realizes given combinatorial structure by a circle pattern on the surface. The hyperbolic metric obtained is unique up to isometries, and this theorem is also called discrete uniformization theorem [Ste05,Chapter 4]. In this article, we offer yet another way of endowing a hyperbolic metric on a surface in a manner that given weighted graph is realized as the image of a harmonic map with the least energy among a fixed homotopy class as well as all the hyperbolic metrics.
Let X = (V, E, m E ) be a finite weighted graph with m E : E → (0, ∞). We identify each edge e with the unit interval [0, 1], and denote by e its reversed edge. We understand E the set of all oriented edges e, e ∈ E, and the weight function m E is symmetric, i.e., m E (e) = m E (e).
For any closed Riemann surface (S, G) equipped with a hyperbolic metric G, let f : X → (S, G) be a piecewise smooth map, i.e., the restriction f e : [0, 1] → (S, G) is piecewise smooth for each edge e. We define the energy of f by for e ∈ E and t ∈ [0, 1]. It is known that the energy E G (f ) attains the minimum by a harmonic map h G among the set of all piecewise smooth maps homotopic to f (Section 2.1). A harmonic map h from X into a hyperbolic surface (S, G) is a map such that h e is a (constant speed) geodesic for each e ∈ E, and satisfies the balanced condition e∈Ex m E (e) dh e dt (0) = 0, for each x ∈ V , where E x = {e ∈ E : e 0 = x}, the set of edges emanating from x. Moreover, if a harmonic map h G is not homotopic to a point nor to a closed curve in (S, G), then h G is unique as a map due to the negative curvature of (S, G). Then, regarding the energy E G (h G ) of the harmonic map h G as a function of hyperbolic metrics G on S, we may further minimize E G (h G ) given a fixed homotopy class C of f . A central question we consider is whether the energy E G (f ) attains the minimum for a pair (h 0 , G 0 ) of a map h 0 : X → S in C and a hyperbolic metric G 0 , or not. We show that under some necessary topological condition on f : X → S, the joint minimum of E G (f ) does exist.
Furthermore, we identify the hyperbolic metric G when the underlying weighted graph X admits an automorphism group which is compatible with the mapping class group of S.
Note that for i = 1, 2, any two pairs (h i , G i ) of a hyperbolic metric G i on S and a harmonic map h i : X → (S, G i ) in C have the same energy E G1 (h 1 ) = E G2 (h 2 ) if there exists an isometry ϕ : (S, G 1 ) → (S, G 2 ) homotopic to the identity map on S and h 2 = ϕ • h 1 . It holds that if a pair (h, G) attains the minimum of the energy E G (h), then the hyperbolic surface (S, G) is unique up to isometries homotopic to the identity map on S under some condition on the fixed homotopy class C as we see below.
We say that a continuous map f : X → S fills S if it is injective and the complement of the image f is a disjoint union of (topological) disks. For example, any skeleton of triangulation of S gives rise to such X and f . Let us denote the set of all piecewise smooth maps homotopic to f by C = [f ], and the space of hyperbolic metrics on S by M −1 (S). We define Note that for any piecewise smooth map f : X → (S, G), we have E G (f ) < ∞, and any continuous map f is homotopic to a piecewise smooth map.
Theorem 1.1. Let X = (V, E, m E ) be a connected finite weighted graph, and S be a closed Riemann surface of genus greater than one. If f : X → S is a continuous map which fills S, then there exists a pair (h 0 , G 0 ) which attains the minimum of E on C × M −1 (S), where C is the homotopy class of f . Moreover, the hyperbolic surface (S, G 0 ) is unique up to isometries homotopic to the identity map on S.
In fact, we prove a more general theorem when f induces a surjective homomorphism f * : π 1 (X, x 0 ) → π 1 (S, f (x 0 )) in Theorem 2.6. The unique hyperbolic metric G 0 in Theorem 1.1 realizes the least energy harmonic embedding of X into the hyperbolic surface (S, G 0 ) determined by the weighted graph X and the fixed homotopy class of maps f : X → S.
The problem of minimizing energy (1) was studied by Colin de Verdière [CdV91a] for embedding of finite weighted graphs into surfaces. The problem of minimizing energy also in hyperbolic metrics was suggested by Kotani and Sunada [KS01, p.7, Section 2] and has been reiterated by Sunada [Sun13,p.124, Section 7.7]. They showed the corresponding result in the case of flat tori (of dimension at least 2); for a connected finite weighted graph X, and for an n-dimensional torus T n , if f : X → T n induces a surjective homomorphism from π 1 (X) to π 1 (T n ), then there exists a pair of flat metric G 0 on T n and a harmonic map h 0 : X → (T n , G 0 ) homotopic to f such that the corresponding energy functional E attains the minimum on C × M 0 (T n ), where C = [f ] and M 0 (T n ) is the space of flat metrics on T n whose volume is normalized to 1 (and such a pair is unique up to translations on T n ). In fact, they proved the result by constructing a flat metric explicitly by using the 1-homology group of graph X, and called the resulting harmonic map h 0 into (T n , G 0 ) (or, its equivariant lift on an abelian covering of X into R n ) the standard realization of X. Then, they proposed the problem in the case of closed hyperbolic surfaces whether such an energy functional has a minimum as a non-Euclidean analogue to their result in dimension 2. It seems that a direct construction of such a pair of metric and harmonic map into closed hyperbolic surfaces would not work by a non-linear nature of target spaces.
Our approach to show the existence of a minimum pair of energy functional E for surfaces is that we define a function E C associated to E on the Teichmüller space T (S). Namely, since the energy E G (f ) attains the minimum at a harmonic map h G in the homotopy class C of f , the function G → E G (h G ) is defined on the space of hyperbolic metrics M −1 (S). This function naturally yields a function on the Teichmüller space T (S) of S; it is not the the moduli space of S to work on because we evaluate the energy by harmonic maps in the fixed homotopy class. Recalling that the Teichmüller space T (S) consists of isotopy classes of pairs ((Σ, G Σ ), ϕ) of a hyperbolic surface (Σ, G Σ ) and a diffeomorphism (a marking) ϕ : S → Σ, we write G = ϕ * G Σ the pull-back metric of G Σ on S. Then, the function is well-defined on T (S) (Section 2.2). We show that the function E C on T (S) is strictly convex with respect to the Weil-Petersson metric (Theorem 2.3) and proper if f induces a surjective homomorphism from π 1 (X) to π 1 (S) (Theorem 2.5). This shows that a minimum of the function E C on T (S) exists uniquely and we show that it also yields a minimum pair of the original energy functional E.
Actually, the result we show is a discrete analogue of a result by Yamada; he proved that if the domain is a general closed Riemannian manifold M and f : M → (S, G) is a continuous map to a closed hyperbolic surface inducing a surjective homomorphism between fundamental groups, then the Dirichlet energy functional evaluated at a (unique) harmonic map homotopic to f is proper and strictly convex on the Teichmüller space with respect to the Weil-Petersson metric [Yam99, Theorem 3.2.1] (see also expositions [Yam14] and [Yam17], and the recent paper [KWZ18]). We mostly follow the strategy by Yamada to show the convexity and the properness of energy functional E C ; but we give an independent proof for the convexity that enables us to establish an explicit Hessian formula of E C (Theorem 3.9). It is also known that such a functional on the Teichmüller space T (S) has various applications to, for example, the Nielsen realization problem for the mapping class groups and the Steinness of T (S) ( [Ker83], [Tro92] and [Wol87]). Our construction provides a family of proper convex energy functionals on T (S) from finite weighted graphs; this also solves the Nielsen realization problem (Remark 5.2). Moreover, since the functional is defined in terms of a finite graph, in many cases, we are able to find explicit hyperbolic surfaces as minima of the energy functionals E C as fixed points of finite subgroups in the mapping class groups.
More precisely, let Aut(X) be the group of automorphisms σ of a finite weighted graph X = (V, E, m E ), where σ is a bijection to V to itself, preserves edges with σe = σe and m E (σe) = m E (e) for e ∈ E. Recall that the mapping class group Mod(S) is the group of isotopy classes of orientationpreserving homeomorphisms. For any continuous map f : X → S, we define a subgroup of Mod(S) by where f 1 ≃ f 2 means that f 1 and f 2 are homotopic.
, gives an isomorphism of groups. Moreover, if h 0 : X → (S, ϕ * 0 G 0 ) is the unique harmonic map in the homotopy class C = [f ], then for any ϕ in G 0 , there exists σ [ ϕ] in Aut(X) such that ϕ The proof is given in Section 5. Theorem 1.2 shows that several classical examples of hyperbolic surfaces arise as minima of some energy functionals E C defined for some finite weighted graphs X. We discuss examples associated to pairs of pants decompositions and triangle tessellations in Section 6. Let us give one simple example: for any integer g ≥ 2, consider the regular 4g-gon F with each inner angle π/(2g). Identifying opposite pairs of sides in the (induced) orientation-reversing way, we obtain a closed surface of genus g endowed with a hyperbolic metric, and denote it by (S, G reg ). We consider the bouquet graph X = (V, E, m E ) with equal positive weight m E ≡ m, where the underlying graph consists of a single vertex and 2g self-loops. If we have the regular 4g-gon F and take the center of F as a vertex and 2g (constant speed) geodesic segments passing through the center such that each line has two extremes lying on the midpoints in each pair of opposite sides, then we obtain an embedding map f : X → (S, G reg ) (Figure 1). Note that f fills the surface S since the complement of the image f is an (open) 4g-gon. In this case, f : X → (S, G reg ) is a harmonic map; the image is a union of 2g closed geodesics at a point, and thus the balanced condition is automatically satisfied. We claim that the hyperbolic surface (S, G reg ) realizes the unique minimizer of E C for C = [f ] with the harmonic map f : X → (S, G reg ). Indeed, for the regular 4g-gon F in the hyperbolic plane H 2 , the counter-clock wise rotation about the center of F induces an element ϕ of the mapping class group of the surface S of order 4g. Theorem 1.2 implies that if a surface with a harmonic map h 0 homotopic to f realizes the minimizer of E C for C = [f ], then ϕ is realized as an isometry and preserves the image of h 0 . Then, the hyperbolic surface has a fundamental domain which is a regular 4g-gon in H 2 and thus it is (S, G reg ). Moreover, since the harmonic map h 0 is homotopic to f , the uniqueness of harmonic map implies that h 0 = f . Therefore (S, G reg ) gives the unique minimizer of E C for C = [f ] with the harmonic map f as claimed.
Let us point out that we use the uniqueness of harmonic map in a fixed homotopy class in order to determine the minimizing hyperbolic surface of energy functional; this is an advantage of a harmonic map and its associated energy functional over a sum of length functionals for a filling family of closed curves studied in [Wol87].
The organization of this paper is the following: in Section 2, we introduce and study the main energy functional E C on the Teichmüller space; first we review results that we use on harmonic maps from finite weighted graphs into closed surfaces, following Kotani and Sunada [KS01], and then state results on the convexity (Theorem 2.3) and the properness (Theorem 2.5) of E C , and show the main result (a generalized form Theorem 2.6 and Theorem 1.1) by combining these two theorems. In Section 3, we give the proof of Theorem 2.3 and also give a formula for the Hessian of E C (Theorem 3.9), and in Section 4, we prove Theorem 2.5. In Section 5, we discuss the action of a finite subgroup of the mapping class group on the Teichmüller space and prove Theorem 1.2. In Section 6, we provide examples which are minima of some energy functionals E C by using Theorem 1.2. In Appendix, we prove technical results: the first and second variational formulas for finite weighted graphs and the smooth dependency of harmonic maps along a smooth one parameter family of metrics in a general closed Riemannian manifold target.
2. The energy functional on the Teichmüller space 2.1. Discrete harmonic maps. Let S be a closed Riemann surface of genus greater than one endowed with a hyperbolic metric G, and X = (V, E, m E ) be a finite weighted graph. Theorems by Colin de Verdière [CdV91a] and by Kotani and Sunada in [KS01] imply that for any continuous map f : X → S, the minimum of energy E G is achieved by a harmonic map h among all piecewise smooth maps C homotopic to f . Moreover, h is a harmonic map in C if and only if h attains the minimum of the energy E G in C. We give a proof adapted to our setting for the sake of convenience.
Theorem 2.1 (Théorème 1 in [CdV91a] and Theorem 2.2, 2.3 and 2.5 in [KS01]). Fix a hyperbolic metric G in S and a (not necessarily connected) finite weighted graph X = (V, E, m E ). For any continuous map f : X → (S, G), let C = [f ] be the set of all piecewise smooth maps homotopic to f .
(i) There exists a map h such that the energy E G is the minimum in C. Moreover, if a map h attains the minimum of E G in C, then h is a harmonic map. (ii) For any harmonic maps h 0 and h 1 in C, If the image of f on each connected component of X is not homotopic to a point nor a closed circle, then there exists a unique harmonic map h in C (as a map). Moreover, the Hessian Hess EG of the energy functional E G at h is non-degenerate.
Proof. Let C geo be the set of piecewise geodesic maps in C, i.e., maps restricted to each edge e is a geodesic. Note that C geo is non-empty because we find a map f geo ∈ C geo for any piecewise smooth map f ∈ C so that f (x) = f geo (x) for any x ∈ V , and furthermore E G (f geo ) ≤ E G (f ). If we endow the C 1 -topology on C geo , then the energy functional E G restricted to C geo is continuous and proper (i.e., for each R ≥ 0, the set {f ∈ C geo : E G (f ) ≤ R} is compact) on C geo . Thus, there exists a piecewise geodesic map h which attains the minimum of E G in C.
Note that a map h : X → (S, G) is a harmonic map if and only if for any smooth variations f s for s ∈ (−ε, ε), ε > 0 and f 0 = h, we have dE G (f s )/ds| s=0 = 0 by the first variation formula (33) in Lemma A.1 since h satisfies that where we write ∂ t h e (t) := (dh e /dt)(t) for e ∈ E and t ∈ [0, 1]. Hence if h attains the minimum of E G in C, then such h is always a harmonic map. This proves (i).
For any two harmonic maps h 0 and h 1 in C, there exists a homotopy f s : X → S so that f s (x) for s ∈ [0, 1] is a (unique) geodesic from h 0 (x) to h 1 (x) in the hyperbolic surface (S, G) for any x ∈ X by perturbing the given homotopy if necessary. We denote the variational vector field of f s by V s e (t) := d ds f s,e (t) and T s e := ∂ t f s,e (t). Then, ∇ V s e V s e = 0 for the Levi-Civita connection ∇, and hence, the second variational formula (34) in Lemma A.1 shows that for any s ∈ [0, 1] since (S, G) has constant sectional curvature −1, where (V s e ) ⊥ means the orthogonal projection of V s e normal to T s e if T s s = 0, and 0 otherwise. Since h 0 and h 1 are harmonic maps, we have dE Furthermore, in fact, we have d 2 E G (f s )/ds 2 ≡ 0 on [0, 1], and by (2), ∇ T s e V s e = 0 and since (S, G) has negative sectional curvature, V s e and T s e are proportional for any s ∈ [0, 1]. If the image of f on each connected component X 0 of X is not homotopic to a point nor a closed circle, then for any s ∈ [0, 1] and in each connected component X 0 = (V 0 , E 0 ), there exists a vertex x ∈ V 0 for which {T s e } e∈E 0 x contains at least two linearly independent vectors in T fs(x) S since f s is homotopic to f ; and thus V s e ≡ 0 for all e ∈ E 0 x . In fact, V s e ≡ 0 for all e ∈ E by parallel transports along the image of edges since ∇ T s e V s e = 0 and each X 0 is connected, and this holds for any s ∈ [0, 1]. Therefore h 0 = h 1 . Moreover, Hess EG (V, V ) = (d 2 /ds 2 )| s=0 E G (f s ) = 0 implies V e ≡ 0 for all e ∈ E. We conclude the proof of (iii).
The following proposition guarantees that for a given closed hyperbolic surface (S, G) and a harmonic map h : X → (S, G), harmonic maps h s depends smoothly on a smooth change of hyperbolic metrics G s . We give the proof in a more general setting in Proposition B.1 in Appendix.
Proposition 2.2. Let X be a connected finite weighted graph, (S, G) be a closed hyperbolic surface and h : X → (S, G) be a harmonic map. Suppose that the image of h is not a point nor a closed geodesic. Then, for any ε > 0 and for any smooth family of hyperbolic metrics {G s } s∈(−ε,ε) with G 0 = G in M −1 (S), there exists a unique family of maps h s : X → S for s ∈ (−ε, ε) with h 0 = h such that h s : X → (S, G s ) is a harmonic map for every s ∈ (−ε, ε) and h s is smooth with respect to the variable s.
Proof. If the image of h is not homotopic to a point nor a closed geodesic, then Theorem 2.1 (iii) implies that for each s ∈ (−ε, ε) there exists a unique harmonic map h s : X → (S, G s ) homotopic to h and the Hessian of E G at h s is non-degenerate. Hence by Proposition B.1, since G s ∈ M −1 (S) for all s ∈ (−ε, ε), for each G s and for the harmonic map h s : X → (S, G s ), there exists an open set U around G s in M −1 (G) and a smooth map h : U → C ∞ (X, S) (the space of piecewise smooth map from X into S) such that h(G ′ ) : X → (S, G ′ ) is a harmonic map for all G ′ ∈ U and h(G s ) = h s . Since for each s ∈ (−ε, ε), the harmonic map h s : X → (S, G s ) is unique in the homotopy class, covering the curve {G s } s∈(−ε,ε) by such open sets, we obtain the claim.
2.2. The energy functional on the Teichmüller space. Recall that the Teichmüller space of S consists of equivalence classes of pairs ((Σ, G Σ ), ϕ) of a hyperbolic surface (Σ, G Σ ) and a diffeomorphism (a marking) ϕ : S → Σ, where ((Σ 1 , G 1 ), ϕ 1 ) ∼ ((Σ 2 , G 2 ), ϕ 2 ) if and only if ϕ 2 • ϕ −1 1 is homotopic to an isometry from (Σ 1 , G 1 ) to (Σ 2 , G 2 ). We discuss the standard topology in T (S) (see e.g., [FM12, Section 10.3, p.269]). The mapping class group Mod(S) is the group of isotopy classes of orientation-preserving diffeomorphisms on S. The group Mod(S) acts on the Teichmüller space by the change of markings The energy E G (f ) attains the minimum at a harmonic map h G in the homotopy class C of f . The function G → E G (h G ) on the space of hyperbolic metrics M −1 (S) naturally defines a function on the Teichmüller space T (S) of S. For a point [(Σ, G Σ ), ϕ] of the Teichmüller space T (S), we write ϕ * G Σ the pull-back metric of G Σ on S by ϕ : S → (Σ, G Σ ). If h 1 : X → (S, ϕ * 1 G 1 ) is harmonic and ((Σ 1 , G 1 ), ϕ 1 ) ∼ ((Σ 2 , G 2 ), ϕ 2 ), then the identity map on S is homotopic to an isometry ι : (S, ϕ * 1 G 1 ) → (S, ϕ * 2 G 2 ), and ι • h 1 : X → (S, ϕ * 2 G 2 ) is also a harmonic map homotopic to h 1 ; hence for any harmonic map h 2 : X → (S, ϕ * 2 G 2 ) homotopic to h 1 , one has by Theorem 2.1 (i) and (ii). We define Note that E C is a function on the Teichmüller space We show that the function E C on T (S) is strictly convex with respect to the Weil-Petersson metric; we will give the proof in Section 3. Recall that a continuous map f : X → S fills S if there exists an injective map f 1 : X → S homotopic to f such that the complement of the image f 1 is a disjoint union of disks.
Proof. There exists an injective map f 1 homotopic to f such that the complement of the image f 1 is a disjoint union of disks. Then, for each disk, choosing a point from the interior, we take a homotopy on the disk such that it pushes outside of a small neighborhood of the point to the boundary and remains identity on the boundary. Patching these homotopies together, we obtain a homotopy on the surface.
For any point x 0 in X, and any loop γ based at f 1 (x 0 ) in S, up to a small perturbation of γ if necessary, composing the homotopy constructed, we obtain a loop confined in the boundaries of disks. Since f 1 is injective, there is a loop c in the graph X based at x 0 , whose image by f 1 is homotopic to the original loop γ relative to f 1 (x 0 ) on the surface. This shows that the loop c . Thus, f 1 * : π 1 (X, x 0 ) → π 1 (S, f 1 (x 0 )) is surjective, and since f 1 is homotopic to f , we conclude the claim.
If we fix the homotopy class C of f such that f induces a surjective homomorphism from π 1 (X) to π 1 (S), then the energy functional E C is proper; we shall give the proof in Section 4.
Theorem 2.5. Let X = (V, E, m E ) be a finite weighted graph, and S be a closed Riemann surface of genus greater than one. Let C be the homotopy class of a continuous map f : In general, we are not able to remove the condition that f induces a surjective homomorphism from π 1 (X) to π 1 (S) in Theorem 2.5. For example, if we have a simple closed curve in S as an image of f , then one is able to make the length of closed geodesic homotopic to f arbitrary small by changing hyperbolic metrics in S.
Theorem 2.6. Let X = (V, E, m E ) be a finite weighted graph, and S be a closed Riemann surface of genus greater than one. If C = [f ] is the homotopy class of f which induces a surjective homomorphism from π 1 (X) to π 1 (S), then the energy functional E C has a unique minimum point in the Teichmüller space T (S).
Proof. Theorem 2.5 implies that there exists a minimum of E C on T (S) if f induces a surjective homomorphism from π 1 (X) to π 1 (S). Since any two points in T (S) can be connected by a Weil-Peterson geodesic by a theorem by Wolpert [Wol87, Corollary 5.6], Theorem 2.3 shows that a minimum is unique.
Proof of Theorem 1.1. Lemma 2.4 and Theorem 2.6 implies that there exists a unique point in the homotopy class C and G = ϕ * G Σ , the pair (h G , G) of a harmonic map h G in C and the hyperbolic metric G on S attains the minimum E G (h G ). Note that the hyperbolic metric G ∈ M −1 (S) is unique up to isometries homotopic to the identity on S, we conclude the claim.

Convexity of the energy functional
In this section, we compute the second derivative of the energy functional E C for C = [f ]. The argument is inspired by Wolf, who showed the convexity of the length functional of a closed curve along the Weil-Petersson geodesic [Wol12]. We discuss the energy of curves and extend it to any finite weighted graphs. First, we recall some differential geometric aspects and facts that we use on the Teichmüller space, following [Yam99], [Yam14] and [Wol12].
3.1. Hessian formula of the energy functional. Let Met(S) be the set of all smooth metrics on S. Then, M −1 (S) is the subset of Met(S) consisting of metrics of constant sectional curvature K = −1. The tangent spaces of Met(S) is regarded as where Diff 0 (S) is the identity component of the group of diffeomorphisms on S.
The deformations arising from Diff 0 (S) are given by the set of Lie derivatives L X G of G for smooth vector fields X on S. The tangent space to T (S) at [G] is identified with where for a local coordinate (x 1 , x 2 ) on S, we abbreviate ∂/∂x 1 and ∂/∂x 2 by ∂ 1 and ∂ 2 , respectively. We denote the matrix representation of any symmetric (0, 2)-tensor Q by Lemma 3.1. Let (S, G) be a hyperbolic surface of constant sectional curvature K = −1 and Q be a symmetric (0, 2)-tensor on S such that tr G Q = 0 and δ G Q = 0. Then, we have the following: Proof. Take an isothermal coordinate (U, z = x 1 + √ −1x 2 ) so that the metric G is expressed by G(z) = ρ(z)|dz| 2 = ρ(x 1 , x 2 )((dx 1 ) 2 + (dx 2 ) 2 ) for a positive real function ρ on U . Note that the Christoffel symbol Γ k ij for the Levi-Civita connection on any isothermal coordinate satisfies Using δ G Q = 0, Q 22 = −Q 11 and Q 12 = Q 21 , we obtain Q 11,1 = −Q 12,2 and Q 11,2 = Q 12,1 , hence, Q 11 − √ −1Q 12 is a holomorphic function on U since it satisfies the Cauchy-Riemann equation. This shows (i).
Let us show (ii). If Q 2 G ≡ 0 or equivalently Q ≡ 0, then the statement is obvious. We may assume the zeros of the function Q 2 G is isolated since Q 11 − √ −1Q 12 is holomorphic by (i). Take any p ∈ S with Q 2 G (p) = 0 and an isothermal coordinate (U, since Q is symmetric and trace free tensor. Taking the logarithm of this equation, we have log Q 2 G = log |Q| 2 − 2 log ρ + log 2, where we set |Q| 2 := |Q 11 − √ −1Q 12 | 2 = Q 2 11 + Q 2 12 , and hence, we obtain On the other hand, in the isothermal coordinate, the sectional curvature K is given by By using the formula as required, and conclude the proof of (ii).
It turns out that Φ defines a global section of (T 1,0 S) * ⊗ (T 1,0 S) * and the section is holomorphic by Lemma 3.1 (1). The section Φ is a holomorphic quadratic differential on the Riemann surface S. One verifies that there exists a one-to-one correspondence between Q and Φ.
We define an L 2 -pairing on T G Met(S) by Recall that the energy functional E C for [G s ] is given by where and henceforth we use the short hand notation ∂ t = d/dt on [0, 1] for the sake of simplicity.
We shall consider the first and second derivatives of E C ([G s ]). First, for any fixed s ∈ (−ε, ε), we have since h s : X → (S, G s ) is harmonic for any s ∈ (−ε, ε) and the energy E Gs (h u ) is minimum at u = s by Theorem 2.1 (i) and (ii). Therefore we have that where we have used (4) in the last equality. The above computation yields a characterization of critical points for E C . We prove the following proposition regarding the first variation of E C although we will not make use of it in this paper.
for any holomorphic quadratic differentials Φ on (S, G).

Proof. It follows that [G] ∈ T (S) is a critical point of E C if and only if for any harmonic map
for any symmetric (0, 2)-tensors Q such that tr G0 Q = 0 and δ G0 Q = 0 from (4) and (5). In any isothermal coordinate (U, where we regard h 0 (t) as a complex valued function in the right hand side. Since there is a one-to-one correspondence between Q ∈ T [G] T (S) and Φ holomorphic quadratic differentials on (S, G), where Φ(z) = ϕ(z)dz 2 on U for some holomorphic function ϕ (Remark 3.2), and if Φ is a holomorphic quadratic differential, then √ −1Φ is also so; thus we obtain the claim.
Differentiating (5) in s once more, we obtain Here, the second term of the right hand side of (6) is arranged as follows due to the harmonicity: Proof. Differentiating (4) with respect to the variable s at s = 0, we have where we used the following identity in the second equality: On the other hand, a similar computation shows that (9), we obtain (7). Substituting (7) to (6), we have that The first term in the right hand side of (10) can be computed by using the differential geometry of Teichmüller space (without any assumption for h s ).
for some (auxiliary) vector filed Z on S.
Thus, the rest is to compute the second derivative of E G0 (h s ). For each edge e ∈ E, let T s e (t) := ∂ t h s,e (t), and T e (t) = T 0 e (t) when s = 0 for simplicity.
Definition 3.6. For any velocity vector Q := (∂ s G s )| s=0 , we define a 1-form on S along h 0,e (t) by Q e := Q(T e , · ) for any e ∈ E, and we denote its G 0 -metric dual by Q ♯ e , that is, Q ♯ e satisfies that G 0 (Q ♯ e , ·) = Q e . Furthermore, we define a vector field along h 0,e (t) by and V e ≡ 0 if T e G0 = 0 for any e ∈ E.
The following proposition holds for any horizontal lift of a smooth curve in T (S) (not necessarily a Weil-Petersson geodesic).
Proposition 3.7. Let {[G s ]} s∈(−ε,ε) for ε > 0 be any smooth curve in T (S). For any horizontal lift {G s } s∈(−ε,ε) of {[G s ]} s∈(−ε,ε) with velocity vector Q := (∂/∂s)G s | s=0 , and for any smooth family of harmonic maps h s : X → (S, G s ) for s ∈ (−ε, ε) with the variational vector field V , we have where V ⊥ e denotes the G 0 -normal component of V e with respect to T e if T e = 0 and 0 otherwise, and V e is the vector field on (S, G 0 ) defined in Definition 3.6.
Remark 3.8. By the formula given in Lemma 3.12 in the next subsection, the variational vector field V e (t) along h 0,e (t) is determined by the velocity vector Q = (∂ s G s )| s=0 and the second order ordinary differential equation with initial vector V e (x) and ∇ Te V e (x) for x = h 0,e (0), where R is the curvature tensor of (S, G 0 ). Note that the variational vector filed V is determined uniquely if we take a Weil-Petersson geodesic {[G s ]} s∈(−ε,ε) and an initial harmonic map h 0 : X → S whose image is not a point nor closed circle by the uniqueness of the harmonic map. Moreover, the vector V e (t) defined by (11) is a solution of the ordinary differential equation with initial vector V e (h 0,e (0)) if T e G0 = 0.
We postpone the proof of Proposition 3.7 for the moment. Then, for any Weil-Petersson geodesic {[G s ]} s∈(−ε,ε) in T (S), the second variational formula (10) becomes by Propositions 3.5 and 3.7, and using the fact due to the harmonicity of h 0 (see (4) for a similar identity). More generally, we obtain the following Hessian formula: where h 0 : X → (S, G 0 ) is any smooth harmonic map, T e (t) := ∂ t h 0,e (t), V e (t) := ∂ r h r,e (t) for the family of harmonic maps h r : X → (S, G r,0 ), V e (t) are defined by (11) relative to the variable r, and W e (t), W e (t) are defined analogously relative to the variable s. In the integral, P, Q G0 stands for G 0 (P, Q).
One can check this formula by using (14), and thus, we omit the proof. Now, we give a proof of strictly convexity of the energy functional E C .
Combining (15) with (14), we have  for each edge e ∈ E. Moreover, we simply write T e (t) := T 0 e (t) at s = 0. Since h s : X → (S, G s ) is a family of harmonic maps, we may assume the following: • h s,e (t) is a constant speed geodesic in (S, G s ) for each e ∈ E, namely, we have along h s,e (t) for any s ∈ (−ε, ε), where ∇ s is the Levi-Civita connection of (S, G s ). We simply write ∇ = ∇ 0 . The parameter t is proportional to the arc-length parameter of h s,e (t), i.e., the norm T s e (t) Gs is constant along h s,e (t) in t ∈ [0, 1] for each e ∈ E and each s ∈ (−ε, ε). for any x ∈ V and any s ∈ (−ε, ε).
The second variational formula (34) in Appendix implies that for the harmonic map h 0 : X → (S, G 0 ), where R is the curvature tensor of (S, G 0 ). Then, the right hand side becomes .
The balanced condition (19) implies the following: Lemma 3.11. It holds that Proof. Denote the origin and the terminal of edge e ∈ E by o(e) and t(e), respectively. Since for T e = −T e , and m E (e) = m E (e), we have that On the other hand, the balanced condition (19) implies that for any s since V s e (h s (x)) is independent of the choice of e ∈ E x . Differentiating (23) with respect to s at s = 0, we see where the second equality is due to the balanced condition of h 0 , the independence of ∇ Ve V s e | s=0 (x) of e ∈ E x and [V s e , T s e ] = 0. Combining this with (22), we obtain the lemma.
Recall that, for the velocity vector Q := (∂ s G s )| s=0 , we have defined the vector field Q ♯ e along h 0,e (t) such that Q(T e , ·) = G 0 (Q ♯ e , ·) (Definition 3.6). The following formula uses the property (18).
The second term is 0 since the balanced condition (19) implies that The integrand of the first term for each e ∈ E is Letting J 0 be the complex structure on (S, G 0 ), we have for e ∈ E with T e = 0, 2 , and note that by Definition 3.6 for e ∈ E with T e = 0, and by (21), Substituting these to (27), we obtain Since (S, G 0 ) is a surface of constant sectional curvature −1, we have where V ⊥ e is the orthonormal projection with respect to T e for T e = 0 and 0 otherwise, and we complete the proof of Proposition 3.7.

Properness of the energy functional
We will establish the properness of the energy functional E C for C = [f ] if f : X → S induces a surjective homomorphism between the fundamental groups. First we show the following lemma (which will also be used in the next Section 5).
Lemma 4.1. Assume that f : X → S is a continuous map which induces a surjective homomorphism from π 1 (X) to π 1 (S). If ϕ 1 , ϕ 2 are orientation-preserving homeomorphisms on S and ϕ 1 • f and ϕ 2 • f are homotopic, then ϕ 1 and ϕ 2 are isotopic on S.
Proof. Let ϕ := ϕ −1 2 • ϕ 1 , then the maps ϕ • f and f are homotopic. For any x 0 ∈ X, let γ be the path from f (x 0 ) to ϕ • f (x 0 ) given by the homotopy between ϕ • f and f . Then, there exists a homeomorphism β γ homotopic to the identity map id on S such that for [α] ∈ π 1 (S, ϕ • f (x 0 )), where γ denotes the reversed path of γ. (Such an β γ is obtained first when γ is a short enough path on S, and in general divide γ into finitely many short enough paths and composite homeomorphisms homotopic to the identity.) Since f induces a surjective homomorphism f * : π 1 (X, and since ϕ • f and f are homotopic, β γ * • ϕ * coincides with id * given by id on S. Since S is a K(π 1 (S), 1)-space, β γ • ϕ and id are homotopic fixing f (x 0 ) [Hat02, Proposition 1B.9], and since β γ is homotopic to id, ϕ is homotopic to id; hence ϕ 1 and ϕ 2 are homotopic. It is known that two orientation-preserving homeomorphisms which are homotopic on a closed Riemann surface S are isotopic [Eps66, Theorem 6.3], therefore ϕ 1 and ϕ 2 are in fact isotopic on S.
Proof of Theorem 2.5. For any R ≥ 0, let us denote the sublevel set by Assume that Sub(R) = ∅. Taking a harmonic map h G : X → (S, G) in the homotopy class C = [f ], we write Let l(h G,e ) be the length of h G,e : [0, 1] → (S, G) for each e ∈ E. Since h G is harmonic and piecewise geodesic, we have for each e ∈ E, The Cauchy-Schwarz inequality gives For given closed Riemann surface S, we take a finite collection of simple closed curves {γ 1 , . . . , γ N } on S such that the union γ 1 ∪ · · · ∪ γ N fills S, i.e., the complement of this union is a disjoint union of disks. (For example, one may take simple closed curves corresponding to the free homotopy classes of the standard set of generators of π 1 (S), where N is twice the genus of S.) Then, if γ is any simple closed curve which is not homotopic to a point, then there exists a γ i such that the geometric intersection number between free homotopy classes of γ and γ i is not zero.
Since f * : π 1 (X, x 0 ) → π 1 (S, f (x 0 )) is surjective, we choose a collection of closed paths {c 1 , . . . , c N } in X such that f (c i ) is freely homotopic to γ i for each i = 1, . . . , N . Let C max := max i=1,...,N |c i |, where |c i | is the number of edges in c i . (One may just take C max = 1 if f fills the surface in the following inequality.) Then, for any c i , we have for any [G] ∈ Sub(R). For any closed hyperbolic surface (S, G), let inj((S, G)) be the injectivity radius. Taking a closed geodesic γ inj in (S, G) such that the length realizes twice the injectivity radius l(γ inj ) = 2 inj((S, G)), we have a γ i such that the geometric intersection number is non zero in their free homotopy classes. Let γ i,G be a (unique) closed geodesic freely homotopic to γ i , then γ i,G has the shortest length among the free homotopy class of γ i since (S, G) is a hyperbolic surface, and we have Then, the Collar Lemma (e.g., [FM12,Lemma 13.6]) implies that is an embedded annulus, and thus Therefore, letting , and thus up to passing to a subsequence there exists a hyperbolic surface [(S, G ⋆ )] in M ε(R) (S) such that [(S, ϕ * i G i )] converges to [(S, G ⋆ )] in M(S). Moreover, there exists a sequence of (orientation-preserving) diffeomorphisms ϕ i on S to itself and ϕ ⋆ := id : i ] converges to [(S, G ⋆ ), ϕ ⋆ ] in the Teichmüller space T (S), and (ϕ i • ϕ −1 i ) * G i converges to G ⋆ in the C ∞ -topology: for any ε > 0, for all large enough i, we have Let h i : X → (S, ϕ * i G i ) be a harmonic map in the homotopy class C. The maps may not be homotopic to h i ). We note that ϕ i • h i : X → (S, G ⋆ ) are equicontinuous; this follows from (28) and (29). Thus by the Ascoli-Arzelà theorem after passing to a subsequence if necessary, ϕ i • h i converges to a continuous map h ⋆ : X → (S, G ⋆ ) uniformly. Therefore for all large enough i, j, the maps ϕ i • h i and ϕ j • h j are homotopic.
Then, the condition that ϕ i • h i and ϕ j • h j are in the same homotopy class inducing surjective homomorphisms π 1 (X) → π 1 (S) implies that ϕ i and ϕ j are isotopic on S. Indeed, since both h i and h j are homotopic to f , ϕ i • f and ϕ j • f are homotopic for all large enough i, j, and Lemma 4.1 implies that ϕ i and ϕ j are isotopic for all large enough i, j.
For any ε > 0, and for all large enough i, we have (29) and

Actions by finite subgroups of the mapping class groups
Recall that Aut(X) is the group of automorphisms σ of a finite weighted graph X = (V, E, m E ), where σ is a bijection from V to itself, preserves edges with σe = σe and m E (σe) = m E (e) for e ∈ E, and also recall that the mapping class group Mod(S) is the group of isotopy classes of orientation-preserving homeomorphisms Homeo + (S). We define a subgroup of Mod(S) for any continuous map f : X → S by Proof. Let us define a subgroup of Aut(X) by This map π is well-defined; indeed, for σ ∈ Aut(X), if there are ϕ 1 , ϕ 2 ∈ Homeo + (S) such that ϕ 1 • f ≃ f • σ and ϕ 2 • f ≃ f • σ, then ϕ 1 • f ≃ ϕ 2 • f . Since f : X → S induces a surjective homomorphism from π 1 (X) to π 1 (S), Lemma 4.1 implies that ϕ 1 and ϕ 2 are isotopic on S, i.e., Proof of Theorem 1.2. By Lemma 2.4, we assume that f induces a surjective homomorphism from π 1 (X) to π 1 (S). Then Lemma 5.1 implies that G [f ] (S) is a finite group. Let us simply write G = G [f ] (S), and define Since E C is strictly convex with respect to the Weil-Petersson metric as well as proper by Theorem 2.3 and 2.5, and Mod(S) acts on T (S) as isometries relative to the Weil-Petersson metric, each function E C • [ϕ] for [ϕ] ∈ G, and its convex combination E C are also so. Hence E C has a unique minimizer [( Σ 0 , G 0 ), ϕ 0 ] in T (S). Note that E C is invariant under the action of G, and thus the minimizer is fixed by G. This implies that G acts on (S, ϕ * 0 G 0 ) as isometries. Each [ϕ] ∈ G has a unique representative as an isometry of (S, ϕ * 0 G 0 ) since two isometries which are isotopy (or, homotopy) must coincide. Let us define G 0 as the group generated by these isometries.
So far, we have obtained a unique minimizer [( Σ 0 , G 0 ), ϕ 0 ] for E C and a group G 0 of isometries on (S, induces an isomorphism of groups. Then, we shall show that a point in T (S) is the unique minimizer of E C if and only if it is the unique minimizer of E C .
Suppose that [(Σ 0 , G 0 ), ϕ 0 ] is the unique minimizer of E C . Let h 0 : X → (S, ϕ * 0 G 0 ) be the unique harmonic map in the homotopy class C = [f ]. Then it is fixed by G [f ] (S): indeed, for any Take a diffeomorphism ϕ as a representative of [ϕ], and a unique harmonic map , and the uniqueness of minimizer for As a consequence, we have On the other hand, if [( Σ 0 , G 0 ), ϕ 0 ] is the unique minimizer of E C , then it is fixed by Therefore Recall that the group G 0 acts on (S, ϕ * 0 G 0 ) as isometries. Since for each ϕ by the uniqueness of harmonic map in the same homotopy class by Theorem 2.1 (iii). We complete the proof.
Remark 5.2. Note that for any finite subgroup G of Mod(S), there exists a hyperbolic surface (S, G) and a group G of isometries of (S, G) such that the natural map G → G, ϕ → [ϕ] gives an isomorphism of groups. This is known as the Nielsen realization problem (a theorem by Kerckhoff [Ker83]; for the history and a proof, we refer to [Tro92, Section 6.4], which is regarded as a continuous counterpart to our setting). The proof which we have given above also provides another proof of this problem by using the functional E C ; note that the first paragraph of the proof of Theorem 1.2 works for any finite subgroup G of Mod(S).

Examples
We give some examples where we are able to find the unique minimizer of energy functional E C in a fairly explicit way. The first class of examples we discuss is a hyperbolic surface constructed by gluing right-angled hexagons in the hyperbolic plane H 2 . Given a specific map f : X → S, we determine each hexagon and how they are combined to form the hyperbolic surface (S, G) where the harmonic map into (S, G) homotopic to f realizes the minimum of E C with C = [f ]. We discuss the case of closed surface S 2 of genus 2 with a map f : X → S 2 in Section 6.1, and generalize that construction to closed surface of higher genus in Section 6.2. Another class of examples is provided by classical triangle tessellations on the hyperbolic plane H 2 . We discuss general triangle tessellations in Section 6.3, and in particular show that Klein's surface of genus 3 arises as the minimizer for some explicit energy functional. In all cases, we apply Theorem 1.2; we make use of the group G [f ] (S) which we have introduced in order to reduce the dimension of parameter space (the Teichmüller space). 6.1. A simple example for a closed surface of genus 2. A closed Riemann surface of genus two S 2 is obtained by two pairs of pants P and P ′ . A pair of pants is topologically a compact connected surface of genus zero with three boundary components which are circles. Each pair of pants P (resp. P ′ ) is obtained by two hexagons H and H (resp. H ′ and H ′ ), where a set of every other sides and a set of corresponding sides are identified. We glue two pairs of pants together by identifying three pairs of boundary circles, and obtain the surface S 2 . When we endow S 2 with a hyperbolic metric G, we realize each hexagon as an isometric copy of a right-angled hexagon whose sides are geodesics in the hyperbolic plane H 2 . Note that by the Gauss-Bonnet theorem, any right-angled hexagon in H 2 has the area π = 3.1415 · · · .
We define a finite graph whose edges are sides of hexagons in the surface S 2 , and vertices are the points at intersections of sides. Let X = (V, E, m E ) be the corresponding weighted graph with unit weight on edges m E ≡ 1. The map f we consider is the natural embedding of X into S 2 . Then, the map f fills the surface S 2 . Therefore there exists a unique minimizer (S 2 , G 0 ) for the energy functional E C with C = [f ] by Theorem 1.1.
The group G [f ] (S 2 ) has three elements ϕ rot , ϕ inv and ϕ ex , where ϕ rot is given by the rotation of order 3 corresponding to a cyclic permutation of glued three boundary components of two pairs of pants, ϕ inv is given by an involution of order 2 corresponding to exchanging two pairs of pants, and ϕ ex is given by an involution of order 2 exchanging hexagons H and H, as well as H ′ and H ′ simultaneously (Figure 2). If a hyperbolic surface (S 2 , G 0 ) realizes the minimum of E C with a harmonic map h 0 in the homotopy class C = [f ], then Theorem 1.2 implies that (S 2 , G 0 ) admits isometries corresponding to ϕ rot , ϕ inv and ϕ ex , and the image of h 0 which consists of geodesic segments has to be invariant under their actions (as a set). This requires that four hexagons which are complements of the image of h 0 are isometric copies of a single right-angled hexagon H 0 whose every other sides are geodesic segments of the same length. Indeed, first each pair of pants P (resp. P ′ ) has three boundary components which are all (isometric) closed geodesics since h 0 is harmonic. Second, every other sides in each hexagon have the same length by the action of ϕ rot , two pairs of pants P and P ′ are isometric by the action of ϕ inv , and the hexagons H and H (resp. H ′ and H ′ ) are isometric by the action of ϕ ex . Then, the sides of these hexagons in the surface realizes the image of the harmonic map h 0 .
This process reduces the space of parameters to be determined; originally it has the (real) dimension 6g − 6 = 6 (the dimension of Teichmüller space of S 2 where g = 2), but now it is as large as the space of parameters to determine H 0 -it has the dimension 1.
Let H * be a right-angled hexagon whose sides are geodesic segments in H 2 with one distinguished vertex (a mark). We equip H * with the induced orientation from H 2 . From the mark, we denote every other sides in the counter-clockwise by c 1 , c 2 and c 3 , and remaining sides by d 1 , d 2 and d 3 (Figure 3). Let H * be the hexagon given by a reflection of H * . We glue H * and H * along d 1 , d 2 Figure 3. A marked right-angled hexagon in the hyperbolic plane. and d 3 with their corresponding copies, and obtain a hyperbolic pair of pants P 0 . Let c 1 , c 2 and c 3 be the boundary circles corresponding to c 1 , c 2 and c 3 . Let P ′ 0 be a reflected isometric copy of P 0 , and we glue P 0 and P ′ 0 by paring c i and its corresponding copy for i = 1, 2, 3. The hyperbolic metric on the resulting surface is determined by the hexagon H * whose sides are c i and d i for Then we shall find the corresponding unique minimizer for the weights m c and m d .
Since s > 0 and t > 0 we may assume ∂ s F = 0 and ∂ t F = 0 in the following discussion. By using the method of Lagrange multiplier, we deduce that if (s, t) is the minimizer of the above function, then One can check that the function H(s, t(s)) is strictly increasing for s ∈ (0, ∞) and the range coincides with (0, ∞). Hence there exists a unique solution (s, t(s)) of (32) for any M ∈ (0, ∞). This gives the unique minimizer of the energy functional E C for the weight (m d , m c ). If M = 1, then s = t = log(2 + √ 3), namely, the corresponding hexagon is the regular hexagon. If M → 0 or M → ∞, then (s, t) → (0, ∞) or (s, t) → (∞, 0), respectively, i.e., if we change the ratio M continuously from 1 to 0 or from 1 to ∞, then the corresponding hexagon tends to an ideal triangle.
6.2. Closed surfaces of genus greater than one. Let S g be a closed oriented surface of genus g ≥ 2. We consider a graph X and a continuous map f : X → S g such that the complement of the image of f gives a decomposition into 2(g − 1)-pairs of pants. More precisely, for each i = 1, . . . , g − 1, let H i be an oriented hexagon in the plane, H i be a reflected copy of H i , and P i = H i ∪ H i be a pair of pants where we identify a set of every other sides of H i with the corresponding set of sides of H i . Let P ′ i be a homeomorphic copy of P i . For 1 ≤ i ≤ g − 1, we glue P i and P ′ i by identifying two pairs of boundary components so that P i ∪ P ′ i is a compact connected oriented surface of genus 1 with two boundary components, which we denote by δ i , δ i . Then, we glue all pairs of pants by identifying δ i and δ i+1 in the orientation-reversing way, where the indices i are modulo g − 1, and obtain a closed oriented surface S g of genus g. Let X be a finite graph whose edges are the (identified) sides of hexagons, and vertices are the (identified) corners of hexagons in S g . We denote by X = (V, E, m E ) the corresponding weighted graph with unit weights m E ≡ 1, and by f g : X → S g the natural embedding. Note that f g fills S g for every g ≥ 2. We shall find the unique minimizer of the energy functional E C(g) for C(g) = [f g ].
Actually, for any g ≥ 2, we show that the unique minimizer is obtained by gluing 4(g − 1) isometric copies of regular right-angled hexagons. Consider an elements ϕ ex of order 2 and an element ϕ rot of order g − 1 in Mod(S g ) such that ϕ ex exchanges two hexagons H i and H i (resp. H ′ i and H ′ i ) in each P i (resp. P ′ i ) simultaneously in the orientation-preserving way, and ϕ rot sends P i ∪ P ′ i and P i+1 ∪ P ′ i+1 for i mod g − 1. Note that ϕ ex , ϕ rot ∈ G [fg ] (S g ). If we have a hyperbolic surface (S g , G g,0 ) realizing the unique minimizer of E C(g) , then the image of the unique harmonic map in the class C(g) is invariant under the action of isometric group realizing G [fg ] (S g ) by Theorem 1.2. This implies that P i ∪ P ′ i are isometric for all i = 1, . . . , g − 1, and two boundary components of P i ∪ P ′ i are closed geodesics which are isometric for each i = 1, . . . , g − 1. Moreover, two hexagons H i and H i (resp. H ′ i and H ′ i ) in P i (resp. P ′ i ) are isometric for each i = 1, . . . , g − 1. Since sides of hexagons are in the image of the harmonic map, the sides are geodesic arcs, and such arcs are contained in closed geodesics; since there are exactly four hexagons at each corner in the surface, each hexagon is right-angled. Hence we may cut out P 1 ∪ P ′ 1 and glue two boundary components of P 1 ∪ P ′ 1 by orientation-reversing way. The resulting surface (S 2 , G * ) is a hyperbolic surface of genus 2 endowed with the image of f 2 . Note that the total energy E C(g) is the sum of squared lengths of sides of hexagons and all P i ∪ P ′ i are isometric. We observe that if (S g , G g,0 ) is the unique minimizer of E C(g) if and only if (S 2 , G * ) is the unique minimizer of E C(2) . Since the unique minimizer of E C(2) is given by gluing four isometric copies of regular right-angled hexagons in Section 6.1, the hyperbolic surface (S g , G g,0 ) is obtained by gluing 4(g − 1) isometric copies of regular right-angled hexagons. 6.3. Closed surfaces associated with triangle tessellations. Let ∆ be a geodesic triangle in H 2 with inner angles π/p, π/q and π/r for positive integers p, q and r with p −1 +q −1 +r −1 < 1. We consider the associated triangle group T (p, q, r), and a subgroup Γ of index 2 with a fundamental domain ∆ ∪ ∆ (where ∆ is a reflected copy of ∆). There exists a finite index, torsion-free normal subgroup Γ 0 of Γ ([Sti92, Theorem in Section 8.6]; in general, this follows from Selberg's lemma). The quotient space (S, G) = Γ\H 2 is a closed hyperbolic surface. The group Γ/Γ 0 acts on (S, G) by isometry with a fundamental domain which is an isometric copy of ∆ ∪ ∆; this gives a triangle tessellation of (S, G). We consider the finite graph X = (V, E, m E ) where edges arise as sides of triangles in (S, G) and vertices are points of intersections of sides with unit weight m E ≡ 1, and the natural embedding map f : X → S. Note that f fills the surface S. We claim that (S, G) attains the minimum of E C with C = [f ].
If (S, G 0 ) attains the minimum of E C , then the group G [f ] (S) is realized as an isometry group G 0 of (S, G 0 ) by Theorem 1.2, and G 0 has to contain G 0 which is an isomorphic copy of Γ/Γ 0 . The quotient space (S, G 0 )/G 0 is a hyperbolic orbifold, which is obtained by gluing the sides of two copies of a triangle ∆ ′ (Figure 5). Since G 0 is isomorphic to Γ/Γ 0 and is realized in G 0 via G [f ] (S), the triangle ∆ ′ is isometric to ∆. Moreover, since (S, G)/(Γ/Γ 0 ) is isometric to this orbifold and Figure 5. A hyperbolic orbifold corresponding to the triangle of inner angles π p . π q , π r .
G 0 is isomorphic to Γ/Γ 0 , the surface (S, G 0 ) is isometric to (S, G). Therefore (S, G) realizes the minimum of As a special case, let us consider the triangle group T (2, 3, 7). For a subgroup Γ of T (2, 3, 7) with index 2, there is a regular 14-gon with all inner angle 2π/7 in the plane H 2 as a fundamental domain of Γ. The quotient by this subgroup Γ is a hyperbolic surface known as the Klein quartic surface (e.g., [Sti92, Example 2 in Section 7.3, p.176]). We denote the surface by (S 3 , G Klein ). This is a surface of genus 3 with the isometry group G 0 of order 168, which is the largest possible among hyperbolic surfaces of genus 3, and such a surface is unique up to isometry. (Recall that the order of isometry group for a closed hyperbolic surface of genus g ≥ 2 is at most 84(g − 1).) Note that an isometric copy of a pair of triangles ∆ and ∆ is a fundamental domain of this isometry group G 0 in (S 3 , G Klein ). If we consider the graph X and the map f : X → S 3 given by sides of triangles in (S 3 , G Klein ) as above, the surface (S 3 , G Klein ) realizes the unique minimizer of the energy functional E C with C = [f ].
In this case, the group G [f ] (S 3 ) is isomorphic to G 0 . Indeed, first the group G [f ] (S 3 ) has to contain an isomorphic copy of G 0 . If G [f ] (S 3 ) is realized as an isometry group of some hyperbolic surface by Theorem 1.2, then that surface has to be isometric to (S 3 , G Klein ) since G 0 and thus G [f ] (S 3 ) have the largest order of isometry group among hyperbolic surfaces of genus 3, and this shows that |G [f ] (S 3 )| = |G 0 |; hence G [f ] (S 3 ) and G 0 are isomorphic. In particular, (S 3 , G Klein ) is the unique minimizer of E C with C = [f ].
Next, we have that d 2 ds 2 E G (f s ) = not found the argument adapted to our setting, we describe the setup in a self-contained manner in the following.
For any piecewise C k -map f : X → M , let us denote the vector space consisting of C k -vector fields along f by C k (X, f −1 T M ). We understand that every v in C k (X, f −1 T M ) is in C k on each interior (0, 1) of edges and continuous on X. We define a norm on C k (X, f −1 T M ) by where E 0 is the set of unoriented edges, for each edge e ∈ E 0 , v e is the restriction of v and ∇ i ∂tfe denotes the i-th covariant derivative along f e relative to the metric G. Then, (C k (X, f −1 T M ), · k ) becomes a Banach space.
Let where ∂ t f e = (d/dt)f e . Note that the set C k bal (X, M ) is not empty since there always exists a harmonic map, which satisfies the balanced condition. For any f ∈ C k bal (X, M ), let us define the Banach space as a closed subspace of C k (X, f −1 T M ).
Lemma B.2. For any integer k ≥ 1, C k bal (X, M ) is a submanifold of C k (X, M ) and for any C kmetric G on M and any f ∈ C k bal (X, M ), the tangent space of C k bal (X, M ) at f is isomorphic to the Banach space V k f .
Proof. Fix arbitrary f ∈ C k bal (X, M ). Since V k f is a closed subspace of C k (X, f −1 T M ), we can induce the norm · k to it so that V k f becomes a Banach space. For any v ∈ V k f , we setf := exp f v ∈ C k (X, M ). We take a geodesic normal coordinate (U, for each e ∈ E x , and hence, we obtain since f ∈ C k bal (X, M ). By (41), if v ∈ V k f , thenf = exp f v satisfies the balanced condition. Thus, we obtain a map exp f : V k f ∩ U → exp f (U) ∩ C k bal (X, M ) for an open neighborhood U of f in C k (X, M ), and this map is injective if U is small enough since it is a restriction of the exponential map exp f to V k f . We shall show the map is surjective, i.e., for anyf ∈ exp f (U) ∩ C k bal (X, M ), the corresponding vector field v := exp −1 f (f ) ∈ C k (X, f −1 T M ) is actually contained in V k f . By (41), it is sufficient to show that (dexp) (x,ve(0)) (0, w) = 0 implies w = 0. Indeed, we have (dexp) (f (x),ve(0)) (0, w) = d(exp f (x) ) ve(0) (w), and hence, if U is small enough so that for any v ∈ U satisfies that v k < ε for sufficiently small ε > 0 (for instance, we may take ε as inj(M ) > 0), then d(exp x ) ve(0) is injective, and w = 0; as required. This implies the lemma.
We define the map We take any smooth curve s → (f s , G s ) in U bal × Met k+2 (M ) through (h, G) at s = 0 for s ∈ (−ε, ε) and for ε > 0. Let us write τ (f s , G s ) = (τ s,e ) e∈E0 for s ∈ (−ε, ε). Then, by the equations (24) and (25) given in the proof of Lemma 3.12 (Note that the equations (24) and (25) hold for any Riemannian manifold as a target manifold), we see on each e ∈ E 0 , dτ s,e ds s=0 = ∇ Te ∇ Te V e + R(V e , T e )T e + d ds s=0 ∇ Gs Te T e , where T e := ∂ t h e and V e := (∂ s f s,e )| s=0 . Note that the final term in the right hand side depends only on G s and the initial condition h. where we used Lemma 3.11 in the last equality which follows from the assumption f s ∈ C k+2 bal (X, M ). The assumption in the statement implies that Hess EG is non-degenerate at h, we obtain V = 0, and thus dτ | U bal ,o is injective.
Next we show that dτ | U bal ,o is surjective. For any given W = (W e ) e∈E0 where W e ∈ C k ([0, 1], h −1 e T M ) for each e ∈ E 0 , we consider the second order ordinary differential equations ∇ Te ∇ Te V e + R(V e , T e )T e = W e (43) on each e ∈ E 0 with the condition that V is in V k+2 h , i.e., V solves (43) on each interior (0, 1) of edges e, and is continuous on X satisfying the balanced condition. Let us consider the Hilbert space H 1,2 h as the completion of V k+2 h endowed with the inner product h . This implies that V solves (43) on each interior (0, 1) of edges e and V e is in C k+2 on (0, 1) by taking smooth functions ϕ e whose supports are included in (0, 1) for each e ∈ E 0 . Then, integration by parts gives