ENERGY OF TWISTED HARMONIC MAPS OF RIEMANN SURFACES

WILLIAM M. GOLDMAN AND RICHARD A. WENTWORTH

Abstract. The energy of harmonic sections of flat bundles of nonpositively curved (NPC) length spaces over Riemann surfaces is a function on Teichmüller space which is a qualitative invariant of the holonomy representation. Adapting ideas of Sacks-Uhlenbeck, Schoen-Yau and Tromba, we show that the energy function is proper for any convex cocompact representation of the fundamental group. As a corollary, the mapping class group acts properly on the subset of convex cocompact representations.

Contents

Introduction 2 Acknowledgements 4 Notation and Terminology 4 1. Flat bundles and harmonic maps 4 2. Bounded geometry 6 3. Existence of harmonic maps 7 4. Properness of the action on T_{S} 8 5. Properness of the energy function 9 6. Action of the mapping class group 12 7. Accidental parabolics 13 8. Speculation 15 References 16

Date: May 5, 2005. 1991 Mathematics Subject Classification. Primary: 57M50; Secondary: 58E20,

53C24.

Key words and phrases. Riemann surface, fundamental group, flat bundle, harmonic map, energy, Teichmüller space, convex cocompact hyperbolic manifold.

Goldman supported in part by NSF grants DMS-0103889 and DMS0405605. Wentworth supported in part by NSF grant DMS-0204496.

Introduction

Let S a closed orientable surface and G a Lie group. This paper discusses an analytic invariant of a representation π_{(}S) ⟶^{ρ} G, and applies to the action of the mapping class group π_{0}Diff^{+}(S) of S on the space of representations Hom(π_{1}(S), G)/G.

We assume G is a reductive real algebraic group with maximal compact subgroup K and symmetric space X = G/K. Suppose that ρ is reductive, that is, a representation whose image is Zariski dense in a reductive subgroup of G. Then according to Corlette [10], for every conformal structure σ on S, there is a ρ-equivariant twisted harmonic map

S̃ ⟶^{f_{ρ,σ}} X,

which is unique up to isometries of X. In particular its energy

E_{ρ}(σ) ∈ ℝ

is well-defined. Letting σ vary over Teichmüller space T_{S} defines a function

T_{S} ⟶^{E_{ρ}} ℝ.

The starting point of our paper is the following result:

Theorem A. Suppose that ρ is convex cocompact. Then E_{ρ} is a proper function on T_{S}.

Recall that a discrete subgroup Γ ⊂ G is convex cocompact if there exists a Γ-invariant closed geodesically convex subset N ⊂ X such that N/Γ is compact. A representation ρ is convex cocompact if ρ is an isomorphism of π_{1}(S) onto a convex cocompact discrete subgroup of G.

From this theorem follows the example which motivated this study. Let ℭ be the subset of Hom(π_{1}(S), G)/G consisting of equivalence classes of convex cocompact representations.

Corollary B. π_{0}Diff^{+}(S) acts properly on ℭ.

When G = PSL(2, ℂ), a convex cocompact representation is quasi-Fuchsian, that is a discrete embedding whose action on S^{2} = ∂H^{3} is topologically conjugate to the action of a discrete subgroup of PSL(2, ℝ). The corollary is just the known fact that π_{0}Diff^{+}(S) acts properly on the space QF_{S} of quasi-Fuchsian embeddings. Bers’s simultaneous uniformization theorem [3] provides a π_{0}Diff^{+}(S)-equivariant homeomorphism

QF_{S} ⟶ T_{S} × T̄_{S}.

Properness of the action of π_{0}Diff^{+}(S) on T_{S} implies properness on on QF_{S}.

However, as first shown by Kleiner and Leeb [25] (see also Quint [37]), convex cocompactness is highly restrictive, only interesting when G has ℝ-rank one. A more general condition guaranteeing properness of the action of π_{0}Diff^{+}(S) is given by the notion of Anosov representations introduced by Labourie [28].

We generalize these results in two directions. First, we extend the isometric actions of surface groups on surface groups to isometric actions on non-positively curved metric spaces as developed by Korevaar-Schoen [26, 27]. Second, following a suggestion of Bruce Kleiner, we consider embeddings of surface groups onto normal subgroups of a convex cocompact group Γ of isometries of an NPC space. The quotient group Q = Γ/ρ(π_{1}(S)) acts on T_{S}. Since E_{ρ} is Q-invariant, it induces a function E′_{ρ} on T_{S}/Q and we show:

Theorem C. The mapping

T_{S}/Q ⟶^{E′_{ρ}} ℝ

is proper.

This generalization was motivated by hyperbolic 3-manifolds fibering over the circle. The hyperbolic 3-manifold determines a representation ρ of the fundamental group of the fiber surface S. Furthermore the monodromy of the fibration determines an automorphism ϕ of π_{1}(S) such that ρ is conjugate to ρ ∘ ϕ.

According to Thurston [44] (see also Otal [35]), Φ is a pseudo-Anosov or hyperbolic mapping class, and generates a proper ℤ-action on T_{S}. In particular every orbit is an infinite discrete subset of T_{S}. Since E_{ρ} is ϕinvariant and constant on each infinite discrete orbit, E_{ρ} is not proper. Kleiner observed that E_{ρ} induces a proper map on the cyclic quotient T_{S}/⟨ϕ⟩.

Properness of the energy function fails for surface group representations containing “accidental parabolics”. Such representations are discrete embeddings mapping some nontrivial simple loop c to a parabolic isometry. One can find a sequence of bounded energy mappings for which the conformal structures σ degenerate as to shorten c, contradicting properness. Using the Marden Tameness Conjecture in Kleinian groups (recently established by Agol [2], Calegari-Gabai [8] and Choi [9]), we obtain a sharp converse to Theorem for discrete embeddings into PSL(2, ℂ):

Theorem D. Let ρ : π_{1}(S) → PSL(2, ℂ) be a discrete embedding. Then E_{ρ} is proper if and only if ρ is convex cocompact (that is, quasi-Fuchsian).

We make no claim to the originality of the results, which can probably be found in the literature, or known to experts. However, we hope that this work will clarify and isolate these ideas and will be useful for researchers in this field.

Acknowledgements

This paper grew out of conversations with many mathematicians over a period of several years. In particular we thank Dick Canary, Joel Hass, Misha Kapovich, Bruce Kleiner, François Labourie, John Loftin, Yair Minsky, Rick Schoen, Bill Thurston, Domingo Toledo, Karen Uhlenbeck, Mike Wolf, and Scott Wolpert for helpful discussions.

Notation and Terminology

If X is a metric space, we denote the distance function by d_{X}. If c is a curve in a length space X, we denote its length by L_{X}(c). We denote by [a] the equivalence class of a, in various contexts. We denote the identity transformation by I. We shall sometimes implicitly assume a fixed basepoint in discussing the fundamental group π_{1}(S) and the corresponding universal covering space S̃ ⟶ S.

1. Flat bundles and harmonic maps

Let S be a closed oriented surface with χ(S) < 0, and let π_{1}(S) be its fundamental group. Let (X, d) be a complete nonpositively curved length space (an NPC space). In this note we discuss an invariant of isometric actions of π_{1}(S) on X.

Choose a universal covering space S̃ ⟶ S with group of deck transformations π_{1}(S). An isometric action of π_{1}(S) on X is a homomorphism π_{1}(S) ⟶^{ρ} G, where G is the isometry group of X. Such a homomorphism defines a flat (G, X)-bundle X_{ρ} over S, whose total space is the quotient S̃ × X by the (diagonal) π-action by deck transformations on S̃ and by ρ on X. Since every flat bundle over a simply connected space is trivial, a section over the universal covering space S̃ is the graph of a mapping S̃ ⟶ X. Sections of the bundle over S correspond to ρ-equivariant mappings

S̃ ⟶^{u} X.

X is contractible (see, for example, Bridson-Haefliger [5]), and therefore sections always exist.

For us, a conformal structure on S will be an almost complex structuresσ on S, that is, an automorphism of the tangent bundle T S satisfying σ^{2} = −I. A Riemannian metric g is in the conformal class of σ if and only if

g(σv_{1}, σv_{2}) = g(v_{1}, v_{2})

for tangent vectors v_{1}, v_{2}.

Choose a conformal structure σ on S. Let S̃ ⟶^{f} X be a continuously differentiable ρ-equivariant mapping. Its differential defines a continuous section df of the vector bundle T^{∗}S̃⊗f^{∗}T X, Choose a Riemannian metric g on S in the conformal class of σ. Denote by g̃ its pullback to S̃ and d̃A the corresponding area form on S̃. Let ∥ · ∥_{g,X} denotes the Hilbert-Schmidt norm with respect to the metric on S̃ induced by g and the metric on X. Define energy density of f with respeect to g on S̃ as

ẽ(f) = ∥df∥_{g,X}^{2} dA,

The energy density ẽ(f) a π_{1}(S)-invariant exterior 2-form on S̃ and hence defines an exterior 2-form, the energy density e(f) on S. Define the energy E_{ρ,g}(f) as its integral over S. Alternatively, E_{ρ,g}(f) is the integral of the energy density on S̃ over a fundamental domain for the π_{1}(S)-action on S̃. Since S is two-dimensional, E_{ρ,g}(f) (if finite) depends only on the conformal structure σ, and we denote it E_{ρ,σ}(f).

When the target X is only a metric space, define the energy density via ball averages of the approximate derivative

\frac{d_{X}(f(x), f(y))}{dS̃(x, y)} .

(See Korevaar-Schoen[26] or Jost [22].)

For finite energy maps the energy density e(f) is a measure which is absolutely continuous with respect to Lebesgue measure. The Radon-Nikodym derivative plays the role of ∥df∥^{2}. For more details, see [26]. Finite energy maps always exist, and [26, Theorem 2.6.4] guarantees an energy minimizing sequence of uniformly Lipschitz equivariant mappings. In addition to providing a definition of energy minimizing maps to metric spaces, this construction also defines a Sobolev completion of the continuously differentiable maps to Riemannian targets that does not appeal to an isometric embedding of X into euclidean space.

Perhaps the most interesting situation occurs when the infimum of the energy is realized. In the context of NPC targets, recall that a map f is called harmonic if it minimizes E_{ρ,σ} among all ρ-equivariant maps of finite energy.

The fundamental existence theorem for harmonic maps to nonpositively curved Riemannian manifolds is due to Eells-Sampson [14]. In the equivariant setting there are various conditions on ρ which guarantee existence. When X is a symmetric space of noncompact type, ρ is said to be reductive if its Zariski closure has trivial unipotent radical. Existence of a harmonic map for reductive ρ was proven by Corlette [10], Donaldson [11], and Jost-Yau [24]. A geometric notion of reductivity involving stabilizers of flat totally geodesic subspaces was used in [28] (see also Jost [22]). Korevaar and Schoen [27] introduced the notion of a proper action (not to be confused with the more standard use of the term proper below) which is the condition that the sublevel sets of the displacement function associated to a generating set of π_{1}(S) are bounded. This condition guarantees the existence of an energy minimizer when X is a general NPC space (see also Jost [23]).

The basic idea goes back to work of Sacks-Uhlenbeck [38] and Schoen-Yau [40]. When ρ is a Fuchsian representation (corresponding to a hyperbolic structure on S), Tromba [45] proved that E_{ρ} is proper and has a unique critical point (necessarily a minimum). When ρ is a quasi-Fuchsian PSL(2, ℂ)-representation, E_{ρ} is proper. Uhlenbeck [46] gave an explicit criterion for when E_{ρ} has a unique minimum. Generally E_{ρ} admits more than one critical point, for quasi-Fuchsian ρ.

2. Bounded geometry

Let γ ∈ G. Its translation length |γ| is defined by:

(2.1)  |γ| := inf_{x∈X} d(x, γx) .

Lemma 2.1. Let Γ ⊂ G be a convex cocompact discrete subgroup. Then ∃ε_{0} > 0 such that |γ| ≥ ε_{0} for all γ ∈ Γ \ {I}.

Proof. Suppose not. Then ∃γ_{i} ∈ Γ such that |γ_{i}| ≠ 0 for all i, and |γ_{i}| → 0. Let N be a closed convex Γ-invariant subset such that N/Γ is compact. Since N is convex and Γ-invariant, ∃x_{i} ∈ N such that

d(x_{i}, γ_{i}x_{i}) ⟶ 0 .

Since N/Γ is compact, ∃λ_{i} ∈ Γ and x ∈ N such that, after passing to a subsequence, λ_{i}x_{i} → x. Set γ̃_{i} = λ_{i}γ_{i}λ_{i}^{−1}. Then

d(λ_{i}x_{i}, γ̃_{i}λx_{i}) ⟶ 0 .

Properness of the action of Γ near x ∈ N implies that for only finitely many i does |γ̃_{i}| = |γ_{i}| This contradicts the assumption that

0 ≠ |γ_{i}| → 0.
□

Lemma 2.2. Let ε_{0} satisfy Lemma 2.1. Let γ_{1}, γ_{2} ∈ Γ and x, y ∈ X. If

• d(x, y) < ε_{0}/2;
• d(γ_{1}x, γ_{2}y) < ε_{0}/2,

then γ_{1} = γ_{2}.

Proof.

|γ_{2}^{−1}γ_{1}| ≤ d(γ_{2}^{−1}γ_{1}x, x)
= d(γ_{1}x, γ_{2}x)
≤ d(γ_{1}x, γ_{2}y) + d(γ_{2}y, γ_{2}x)
= d(γ_{1}x, γ_{2}y) + d(x, y) < ε_{0} .

Now apply Lemma 2.1. □

3. Existence of harmonic maps

Proposition 3.1. Suppose that π_{1}(S) ⟶^{ρ} G is convex cocompact. Then there exists a ρ-equivariant harmonic map S̃ ⟶^{u} X.

We deduce this proposition as an immediate corollary of the following more general proposition, which we state here for later applications.

Proposition 3.2. Suppose that π_{1}(S) ⟶^{ρ} G is an embedding onto a normal subgroup of a convex cocompact subgroup Γ ⊂ Iso(X) such that ρ(π_{1}(S)) has trivial centralizer in Γ. Then there exists a ρ-equivariant harmonic map S̃ ⟶^{u} X.

Proof. For any NPC space X and compact surface S, there exists an energy minimizing sequence u_{i} of uniformly Lipschitz ρ-equivariant mappings S̃ ⟶ X (Korevaar-Schoen [26, Theorem 2.6.4]). Let N ⊂ X be a ρ-invariant convex set such that N/Γ is compact. Projection X ⟶ N decreases distances, and therefore decreases energy. Thus we may assume that the image of u_{i} lies in N.

Fix any point s̃_{0} ∈ S̃ with image s̃ ∈ S. Since N/Γ is compact, after passing to a subsequence, ∃γ_{i} ∈ Γ such that v_{i}(s̃_{0}) converges to a point in N, where

v_{i} := ρ(γ_{i}) ∘ u_{i}.

The v_{i} are uniformly Lipschitz and v_{i}(s̃_{0}) converges. The Arzéla-Ascoli theorem implies that a subsequence of v_{i} converges uniformly on compact subsets of S̃. Choose ε_{0} > 0 satisfying Lemma 2.1. For each compact K ⊂ S̃, there exists I > 0 so that

(3.1)  d(v_{i}(w), v_{j}(w)) < ε_{0}/2

whenever i, j ≥ I and w ∈ K.

Each v_{i} is equivariant with respect to ρ_{i} = ρ ∘ Inn_{γ_{i}}, where Inn_{γ_{i}} denotes the inner automorphism of π_{1}(S) defined by γ_{i}. Fix i, j ≥ I, and set x = v_{i}(s̃_{0}) and y = v_{j}(s̃_{0}).

Choose a finite generating set Π ⊂ π_{1}(S). Applying (3.1) to the finite set K = Πs̃_{0},

d(ρ_{i}(c)x, ρ_{j}(c)y) = d(v_{i}(cs̃_{0}), v_{j}(cs̃_{0})) < ε_{0}/2

whenever c ∈ Π. Since

d(x, y) = d(v_{i}(s̃_{0}), v_{j}(s̃_{0})) < ε_{0}/2 ,

Lemma 2.2 implies ρ_{i}(c) = ρ_{j}(c) for all c ∈ Π. As Π generates π_{1}(S) it follows ρ_{i} = ρ_{j} if i, j ≥ I. Since ρ is injective and the centralizer of π_{1}(S) in Γ is trivial, c_{i} = c_{j} for all i, j ≥ I. Therefore u_{i} itself converges locally uniformly to the desired minimizer. □

4. Properness of the action on T_{S}

Our proof is based on the following classical fact. Using the uniformization theorem, it is a special case of Corollary B.

Lemma 4.1. π_{0}Diff^{+}(S) acts properly on T_{S}.

This theorem is commonly attributed to Fricke. Fricke’s original proof uses the fact that π_{0}Diff^{+}(S) acts isometrically on T_{S} with respect to a metric (say, the Teichmüller metric), and that the marked length spectrum is discrete. Many good expositions of these ideas exist. See Abikoff [1], §2.2, Farb-Margalit [15], Harvey [19], §2.4.1, or Buser [7], §6.5.6 (p.156), Imayoshi-Tanigawa [21],§6.3, Nag [33],§2.7, Theorem II of Bers-Gardiner [4] for this proof.

Another proof follows from the properness of the diffeomorphism group Diff(S) of an arbitrary smooth manifold S on the space Met(S) of Riemannian metrics on S. This properness is due to Ebin [13] Palais (unpublished, and Earle-Eels [12]. (See also Tromba [45].) In dimension two the uniformization theorem identifies T_{S} with the quotient of the Diff(S)-invariant subset Met_{−1}(S) of Met(S) comprising metrics of curvature −1. Since Diff(S) acts properly on Met(S), it acts properly on Met_{−1}(S). Thus

π_{0}Diff^{+}(S) = Diff(S)/Diff^{0}(S)

acts properly on Met_{−1}(S)/Diff^{0}(S), which identifies with T_{S}.

5. Properness of the energy function

We now prove that for ρ convex cocompact, the function E_{ρ} on T_{S} is proper. With little extra effort, we prove a more general theorem (first noticed by Bruce Kleiner), concerning homomorphisms

π_{1}(S) ⟶^{ρ} Γ ⊂ G

where Γ is convex cocompact and ρ(π_{1}(S)) is a normal subgroup Γ_{1}◁Γ. Furthermore we assume that the centralizer of Γ_{1} in Γ is trivial. Let

Γ ⟶^{ψ} Aut(π_{1}(S))

be the homomorphism induced by the inclusion Γ_{1} ↪ Γ and the isomorphism π_{1}(S) ⟶^{ρ} Γ_{1}. By Γ_{1} has trivial centralizer, ψ is injective. Thus ψ induces a monomorphism

Q ↪ Out(π_{1}(S))

where Q := Γ/Γ_{1}. Hence Q acts on T_{S} via (5.5). Furthermore E_{ρ} is

Q-invariant and hence induces a map T_{S}/Q ⟶^{E′_{ρ}} ℝ

Proposition 5.1. The map T_{S}/Q ⟶^{E′_{ρ}} ℝ is proper.

Suppose that [σ_{i}] ∈ T_{S} is a sequence whose image in T_{S}/Q diverges. Suppose further that

E_{ρ}([σ_{i}]) ≤ B

for some constant B > 0, and all i = 1, 2, . . ..

Our assumption that the images of [σ_{i}] diverge in T_{S}/Q means the following. Choose any invariant π_{0}Diff^{+}(S)-invariant metric d_{T} on T_{S}, for example, the Teichmüller metric. (For a survey of invariant metrics on T_{S} see Wolpert’s paper [48] in this volume.) We may assume, for each η ∈ Q, that

(5.1)  d_{T} (ψ(η)[σ_{i}], [σ_{j}]) ≥ 1

for i ≠ j.

By [26] the ρ-equivariant harmonic maps

(S̃, g̃_{i}) ⟶^{u_{i}} X

have a uniform Lipschitz constant K (depending on B), where g̃_{i} denotes the hyperbolic metric on S̃ associated to σ_{i}. In particular, given a closed curve c in S, choose a lift c̃ ⊂ S̃ running from s̃_{0} to [c]s̃_{0}, where [c] ∈ π_{1}(S; s_{0}) denotes the deck transformation corresponding to c. Denote the length of c with respect to the metric g_{i} on S by L_{i}(c). Then

|ρ([c])| ≤ d(u_{i}(s̃_{0}), ρ([c])u_{i}(s̃_{0}))
= d(u_{i}(s̃_{0}), u_{i}(ρ([c])s̃_{0}))
≤ L_{X}(u_{i}(c̃))

≤ KL_{i}(c) . (5.2)

Suppose that c ⊂ Σ is any closed essential curve. Since ρ is injective, the isometry ρ(c) is nontrivial. Let ε_{0} > 0 satisfy Lemma 2.1. Then (5.2) implies

ℓ_{c}(σ_{i}) ≥ ε_{0}/K

where ℓ_{c}(σ) denotes the geodesic length function of c with respect to σ, that is, the length of the unique closed geodesic freely homotopic to c in the hyperbolic metric corresponding to σ.

Mumford’s compactness theorem [32] implies that the conformal structures [σ_{i}] project to a compact subset of the Riemann moduli space T_{S}/π_{0}Diff^{+}(S). Thus [φ_{i}] ∈ π_{0}Diff^{+}(S) and [σ_{∞}] ∈ T_{S} exist such that, after passing to a subsequence,

(5.3)  [φ_{i}][σ_{i}] ⟶ [σ_{∞}].

As Diff(S) acts properly on the set of Riemannian metrics (§4), representatives g_{i} ∈ Met_{−1}(S) and φ_{i} ∈ Diff^{+}(S) exist with φ_{i}(g_{i}) ⟶ g_{∞}, where g_{∞} denotes the hyperbolic metric associated to σ_{∞}. Choose a base point s̃_{0} ∈ S̃ with image s_{0} ∈ S. We may assume that φ_{i}(s_{0}) = s_{0}. Let φ̃_{i} ∈ Diff(S̃) be the unique lift of φ_{i} such that φ̃_{i}(s̃_{0}) = s̃_{0}.

The map

v_{i} := u_{i} ∘ φ̃_{i}^{−1} : S̃ ⟶ X

is harmonic with respect to the metric ϕ_{i}(g_{i}) and equivariant with respect to the homomorphism

ρ ∘ (φ_{i}^{−1}) : π_{1}(S) ⟶ G.

The maps v_{i} are uniformly Lipschitz with respect to the metric g̃_{∞} on S̃ induced from the metric g_{∞} on S. In particular the family {v_{i}} is equicontinuous.

Since N/Γ is compact, ∃γ_{i} ∈ Γ such that all γ_{i}v_{i}(s̃_{0}) lie in a compact subset of X.

By the Arzéla-Ascoli theorem, a subsequence of

w_{i} := γ_{i} ∘ v_{i}

converges uniformly on compact sets. For I sufficiently large,

(5.4)  sup_{z∈S̃} d_{X}(w_{i}(z), w_{j}(z)) < ε_{0}/2

for i, j ≥ I.

Each v_{i} = u_{i} ∘ φ̃_{i}^{−1} is equivariant with respect to ρ ∘ (φ_{i})_{∗}^{−1} and is harmonic with respect to φ_{i}(g_{i}). Thus each w_{i} = γ_{i} ∘ v_{i} is equivariant with respect to

ρ_{i} := ρ ∘ (φ_{i})_{∗}^{−1} ∘ ψ(γ_{i})

and also harmonic with respect to φ_{i}(g_{i}) (since γ_{i} is an isometry).

Fix i, j ≥ I, and let x = w_{i}(s̃_{0}) and y = w_{j}(s̃_{0}). For every c ∈ π_{1}(S), (5.4) implies

d_{X}(ρ_{i}(c)x, ρ_{j}(c)y) = d_{X}(w_{i}(cs̃_{0}), w_{j}(cs̃_{0}))
< ε_{0}/2 .

Since d_{X}(x, y) < ε_{0}/2, Lemma 2.2 implies ρ_{i}(c) = ρ_{j}(c) for all c ∈ π_{1}(S). Since ρ is injective,

ψ(γ_{i}) ∘ (φ_{i})_{∗} = ψ(γ_{j}) ∘ (φ_{j})_{∗}

The Dehn-Nielsen theorem (see Nielsen [34], Stillwell [41]) implies the natural homomorphism

(5.5)  π_{0}Diff^{+}(S) ≅ Out(π_{1}(S)).

is an isomorphism. Its injectivity implies ψ(γ_{i})∘φ_{i} is isotopic to ψ(γ_{j})∘ φ_{j} for all i, j ≥ I. Call this common mapping class [φ]. Thus, for i ≥ I,

(5.6)  ψ(γ)_{i}) ∘ φ_{i})_{∗} = [φ]

Then for i, j ≥ I,

d_{T} (ψ(γ_{i})^{−1}[σ_{i}], ψ(γ_{j})^{−1}[σ_{j}]) = d_{T} ([φ]^{−1}(φ_{i}^{−1})_{∗}[σ_{i}], [φ]^{−1}(φ_{j}^{−1})_{∗}[σ_{j}])
= d_{T} ((φ_{i}^{−1})_{∗}[σ_{i}], (φ_{j}^{−1})_{∗}[σ_{j}])
⟶ 0

by (5.3), contradicting (5.1). Thus E_{ρ} is proper, as claimed.

6. Action of the mapping class group

Corollary B follows from the properness of the action of π_{0}Diff^{+}(S) on T_{S} and a general fact on proper actions on metric spaces. Let X be a metric space and let K(X) denote the space of compact subsets of X, with the Hausdorff metric.

Lemma 6.1. A group Γ of homeomorphisms of X acts properly on X if and only if Γ acts properly on K(X).

Proof. The mapping

X ⟶^{ι} K(X)
x ⟼ {x}

is a proper isometric Γ-equivariant embedding. If Γ acts properly on K(X), then equivariance implies that Γ acts properly on X.

Conversely, suppose that Γ acts properly on X. For any compact subset K ⊂ K(X) of K(X), its union

UK := ⋃_{A∈K} A

is a compact subset of X. For γ ∈ Γ the condition

(6.1)  γ(K) ∩ K ≠ ∅

implies the condition

(6.2)  γ(UK) ∩ UK ≠ ∅.

To show that Γ acts properly on K(X), let K ⊂ K(X) be a compact subset. Since Γ acts properly on X, only finitely many γ ∈ Γ satisfy (6.2), and hence only finitely many γ ∈ Γ satisfy (6.1). Thus Γ acts properly on K(X). □

We now prove Corollary B. Let [ρ] ∈ ℭ. By Theorem A, E_{ρ} is a proper function on T_{S}, and assumes a minimum m_{0}(E_{ρ}) Furthermore

Min(ρ) := {[σ] ∈ T_{S} | E_{ρ}(σ) = m_{0}(E_{ρ})}

is a compact subset of T_{S}, and

ℭ ⟶^{Min} K(T_{S})

is a π_{0}Diff^{+}(S)-equivariant continuous mapping.

Conclusion of Proof of Corollary B . Lemmas 4.1 and 6.1 together imply π_{0}Diff^{+}(S) acts properly on K(T_{S}). By equivariance, π_{0}Diff^{+}(S) acts properly on ℭ. □

7. Accidental parabolics

Now we illustrate with a well-known construction how properness of the energy functional can fail if the action contains non-semisimple isometries. For simplicity, assume in this section that X is a simply connected nonpositively curved complete Riemannian manifold (a Cartan-Hadamard manifold), and let G denote the isometry group of X.

Theorem 7.1. Fix a homomorphism ρ : π_{1}(S) → G. Assume that for some simple closed curve c in S, there is a complete geodesic α in X such that

d_{X}(α(t), ρ[c]α(t)) ≤ Ce^{−δt} ,

for positive constants C, δ and all t ≥ 0. Then the energy functional E_{ρ} is not proper.

Proof. It suffices to construct a family σ_{t}, 0 < t ≤ 1, of conformal structures on S such that the corresponding points in T_{S} diverge as t → 0, and a family u_{t} of ρ-equivariant maps S̃ → X such that E_{ρ,σ_{t}}(u_{t}) is uniformly bounded in t.

Fix an initial conformal structure σ_{1} on S. Let A_{ε} denote a tubular neighborhood of the geodesic representative of c with respect to the hyperbolic metric g_{1} associated to σ_{1}. We denote this geodesic also by c. Let A_{ε}^{±} be the connected components of A_{ε} − c. We furthermore choose A_{ε} such that in the uniformization ℍ of (S, g_{1}), Ã_{ε}^{±} are isometric to the strip

ℝ × [ε, 1/ℓ_{c}(σ_{1})),

where ε_{1} is some positive number, and ℓ_{c}(g_{1}) denotes the length of the geodesic. This realizes the isometry [c] ∈ π_{1}(S) as the isometry

(x, y) ↦ (x + 1, y).

The family σ_{t} is then defined by the “plumbing construction” compare Wolpert [47]).

The key feature is that the conformal structure on S_{ε} = S − A_{ε} is unchanged whereas the conformal structure on A_{ε}^{±} is equivalent to the annulus

A_{t}^{±} := ℝ × [ε, 1/ℓ_{c}(σ_{t}))/(x, y) ∼ (x + 1, y),

where ℓ_{c}(σ_{t}) → 0 as t → 0.

Next, let α_{1} : [0, ∞) → X be a geodesic ray with

d_{X}(α(t), ρ[c]α(t)) ≤ Ce^{−δt}

for all t ≥ 0 and some constants C and δ. Define

α : ℝ × [0, ∞) → X

so that s ↦ α(s, t) is the complete unit speed geodesic satisfying

α(0, t) = α_{1}(t)
α(S_{t}, t) = ρ[c]α_{1}(t)

for some S_{t} ≥ 0. Set

L(t) = (1/δ) log(t/ε),

and notice that S_{L(t)} ≤ Cε/t. Finally, define

[0, 1] × [ε, ∞) ⟶^{β} X
(s, t) ⟼ α(S_{L(t)}s, L(t)) .

Then:

∥∂_{t}β(s, t)∥ ≤ 1/δt

for all 0 ≤ s ≤ 1, by the nonpositive curvature and the fact that

∥∂_{t}β(0, t)∥ = (1/δt)∥α′_{1}(L)∥ = 1/δt.

Also,

∥∂_{s}β(s, t)∥ = S_{L}∥(∂_{s}α)(S_{L}s, L)∥ ≤ Cε/t

We extend β to ℝ × [ε, ∞) equivariantly with respect to the ℤ-action s ↦ s + 1 on the domain and x ↦ ρ(c)x on X. Because of the bounds on the derivatives β has finite energy as an equivariant map.

Choose a finite energy ρ-equivariant map u : (S̃, σ_{1}) → X. Hence, the restriction u_{1}^{c} to S̃_{ε} is also finite energy. By interpolating near the boundary ∂S_{ε} we may assume that u_{1} restricted to the connected components of ∂S̃_{ε} coincides with the geodesic s ↦ α(s, 0). Then for each t, u_{1}^{c} extends to a map u_{t} : S̃ → X by requiring

u_{t}|Ã_{t}^{±} = β|_{ℝ×[ε,1/ℓ_{c}(σ_{t})]} .

Then u_{t} is equivariant and has finite energy with respect to σ_{t}, uniformly in t. This completes the proof. □

For discrete embeddings in G = PSL(2, ℂ), R. Canary and Y. Minsky have explained a partial converse to Theorem A. Namely suppose that ρ is a discrete embedding of a closed surface group π into G. Let M := H^{3}/ρ(π_{1}(S)) be the corresponding hyperbolic 3-manifold.

Then by Marden’s Tameness Conjecture, recently established by Agol [2], Calegari-Gabai [8] and Choi [9], ρ is a strong limit of quasi-Fuchsian representations and either ρ is quasi-Fuchsian (in which case E_{ρ} is proper) or there exist a sequence of pleated surfaces

S ⟶^{f_{n}} M

which exhaust the ends of the hyperbolic 3-manifold M^{3}.

The intrinsic geometry of each f_{n} is that of a totally geodesic surface in H^{3} and therefore its energy (computed with respect to the intrinsic hyperbolic metric) equals

−2πχ(S) = area(S).

Let σ_{n} be the conformal structure underlying this intrinsic metric; then

E_{ρ}(σ_{n}) ≤ −2πχ(S)

is bounded.

However, the corresponding sequence [σ_{n}] ∈ T_{S} tends to ∞. It suffices to show that for some c ∈ π_{1}(S), the geodesic length l_{c}(σ_{n}) in unbounded. Since each pleated surface f_{n} is an isometric map, it suffices to show that for some c, the corresponding closed geodesics c_{n} on f_{n} become arbitrarily long.

Thus E_{ρ} is proper if and only if ρ is quasi-Fuchsian.

8. Speculation

Deformation spaces of flat bundles over a surface S are natural geometric objects upon which the mapping class group of S acts. When G is a compact group, then the action is ergodic (Goldman [17] and Pickrell-Xia [36]. At the other extreme, uniformization identifies the Teichmüller space T_{S} of S with a connected component in the deformation space of flat PSL(2, ℝ)-bundles over S, and π_{0}Diff^{+}(S) acts properly on T_{S}.

In general one expects the dynamics of π_{0}Diff^{+}(S) to intermediate between these two extremes. The purpose of this paper is to survey several conditions which guarantee properness of the π_{0}Diff^{+}(S)-action and discuss several open questions in this subject. In particular we discuss an analytic invariant based on harmonic maps which seems to be closely related to the dynamics of the π_{0}Diff^{+}(S)-action.

As mentioned earlier, convex cocompactness excludes all higher rank examples which do not come from rank one. However it may be possible to replace geodesic convexity of the Riemannian structure by another notion. All that is needed is a compact core N/Γ of the locally symmetric space X/Γ in which all all harmonic mappings S ⟶ X/Γ take values.

For example, when G = SL(3, ℝ), the mapping class group π_{0}Diff^{+}(S) acts properly on the component of Hom(π, G)/G corresponding to convex ℝP^{2}-structures (Goldman [16]). Recently using his notion of Anosov representations, Labourie has proved [28] that for any split real form G, the action of π_{0}Diff^{+}(S) on the Hitchin-Teichmüller component of Hom(π, G)/G (see Hitchin [20]) is proper.

Labourie’s definition is as follows. The unit tangent buncle

US ⟶^{Π} S

induces a central extension of fundamental groups

ℤ ⟶ π_{1}(US) ⟶^{Π_{∗}} π

where the center ℤ of π_{1}(US) corresponds to the fundamental group of the fibers of Π. A representation ρ : π ⟶ G and a linear representation of G on a vector space V defines a flat vector bundle

V_{ρ} ⟶ US

with holonomy representation ρ∘Π_{∗}. Let ξ̃_{t} denote the lift of the vector field on US defining the geodesic flow to the total space 𝕍_{ρ}. Labourie defines an Anosov structure to be a continuous splitting of the vector bundle

V_{ρ} = V_{+} ⊕ V_{0} ⊕ V_{−}

so that vectors in V_{+} (respectively in V_{−}) are exponentially expanded (respectively contracted) under ξ̃_{t}.

Labourie proves [28] that the mapping class group acts properly on all such representations. All known examples of open sets of representations upon which the mapping class group acts properly satisify Labourie’s condition.

In another direction, using ideas generalizing those of Bowditch [6] Tan, Wong and Zhang [43] have shown that the action of π_{0}Diff^{+}(S) on representations satisfying the analogue of Bowditch’s Q-conditions is proper. This also generalizes the properness of the action on the space of quasi-Fuchsian representations.

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Department of Mathematics, University of Maryland, College Park, MD 20742

E-mail address: wmg@math.umd.edu

Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218

E-mail address: wentworth@jhu.edu