Introduction - Infinite energy harmonic maps and AdS 3-manifolds

Chapter III: Infinite energy harmonic maps and AdS 3-manifolds

3.1 Introduction

Harmonic maps play a special role in the theory of geometric structures on manifolds.

The existence results of Donaldson, Corlette, and Labourie link the purely algebraic data of a matrix representation of a discrete group to a geometric object—an equivariant harmonic map between manifolds—realising the prescribed transformations.

In this chapter we generalize their work to a non-compact setting and apply it to the study of domination between representations.

Let Γ be a discrete group and for 𝑘 = 1,2 let 𝜌_𝑘 : Γ → Isom(𝑋_𝑘, 𝑔_𝑘) be representations into the isometry groups of Riemannian manifolds (𝑋_𝑘, 𝑔_𝑘). A function

𝑓 : 𝑋

1→ 𝑋

2is(𝜌

1, 𝜌

2)-equivariant if for all𝛾 ∈Γand𝑥 ∈ 𝑋

1, 𝑓(𝜌

1(𝛾) ·𝑥) = 𝜌

2(𝛾) · 𝑓(𝑥). 𝜌1 dominates 𝜌

2 if there exists a 1-Lipschitz (𝜌

1, 𝜌

2)-equivariant map. The domination is strictif the Lipschitz constant can be made strictly smaller than 1. The translation length of an isometry𝛾 of a metric space (𝑋 , 𝑑)is

ℓ(𝛾)= inf

𝑥∈𝑋

𝑑(𝑥 , 𝛾·𝑥). 𝜌1dominates𝜌

2in length spectrumif there is a𝜆∈ [0,1] such that ℓ(𝜌

2(𝛾)) ≤𝜆ℓ(𝜌

1(𝛾))

for all 𝛾 ∈ Γ. This domination is strict if 𝜆 < 1. From the definitions, (strict) domination implies (strict) domination in length spectrum.

Domination is essential to understanding complete manifolds locally modeled on 𝐺 =PO(𝑛,1)₀: a geometrically finite representation𝜌

1: Γ→ 𝐺strictly dominates 𝜌2 : Γ→ 𝐺 if and only if the (𝜌

1, 𝜌

2)-action on𝐺 by left and right multiplication is properly discontinous [GK17]. For 𝑛 = 2 these are the anti-de Sitter (AdS) 3-manifolds, and for 𝑛 = 3 we have the 3-dimensional complex holomorphic- Riemannian3-manifolds of constant non-zero curvature (see [DZ09] for details).

Anti-de Sitter space

The exposition here is minimal, and for more information, we suggest the recent survey [BS20]. Denote by R^𝑛,² the real vector spaceR^𝑛+² equipped with the non- degenerate bilinear form

𝑞_𝑛,

2(𝑥) =

𝑛

∑︁

𝑖=1

𝑥_𝑖𝑦_𝑖−𝑥_𝑛+

1𝑦_𝑛+

1−𝑥_𝑛+

2𝑦_𝑛+

1. We define

H^𝑛,¹= {𝑥 ∈R^𝑛,²:𝑞_𝑛,

2(𝑥) =−1}.

The quadric H^𝑛,¹ ⊂ R^𝑛,² is a smooth connected submanifold of dimension 𝑛+1, and each tangent space 𝑇_𝑥H^𝑛,¹ identifies with the 𝑞_𝑛,

2-orthogonal complement of the linear span of𝑥inR^𝑛,². The restriction of𝑞_𝑛,

2to such a tangent space is a non- degenerate bilinear form of signature(𝑛,1), and this induces a Lorentzian metric on H^𝑛,¹of constant curvature−1 on non-degenerate 2-planes. H^𝑛,¹identifies with the Lorentzian symmetric space𝑂(𝑛,2)/𝑂(𝑛,1), where𝑂(𝑛,1) embeds into 𝑂(2,2) as the stabilizer of the standard basis vector𝑒_𝑛.

The center of𝑂(𝑛,2) is {±𝐼}, where𝐼 is the identity matrix. The Klein model of AdS^𝑛+¹is the quotient

AdS^𝑛⁺¹=H^𝑛,¹/{±𝐼},

with the Lorentzian metric induced from H^𝑛,¹. This also identifies as the space of negative (timelike) directions inR^𝑛,²,

AdS^𝑛+¹={[𝑥] ∈RP^𝑛+¹: 𝑞_𝑛,

2(𝑥) <0}.

A tangent vector 𝑣 ∈ 𝑇_𝑥H²^,¹ is timelike, lightlike, and spacelike if 𝑞_𝑛,

2(𝑣 , 𝑣) < 0, 𝑞_𝑛,

2(𝑣 , 𝑣) = 0, and𝑞_𝑛,

2(𝑣 , 𝑣) > 0 respectively, and likewise for AdS³. The causal character of a geodesic curve is constant, and correspondingly we call geodesics timelike, lightlike, or spacelike if every tangent vector is timelike, lightlike, or spacelike.

AdS^𝑛⁺¹space arose in physics: anti-de Sitter metrics are exact solutions of the Ein- stein field equations (in which the only term in the stress-energy tensor is a negative cosmological constant). Now there are more modern applications in physics. In this part of the thesis we study 3-dimensional anti-de Sitter space. The low dimensional AdS³appears in physics—for instance, see the work of Witten [Wit89], [Wit07].

On AdS3-manifolds

In dimension 3, there is another model of AdS³: the Lie group PSL(2,R). The determinant form 𝑞 = (−det) defines a signature (2,1) bilinear form on the lie algebra𝔰𝔩(2,R) =𝑇_[_𝐼_]PSL(2,R) (it is a multiple of the Killing form). Translating to each tangent space via the group multiplication, we obtain a Lorentzian metric that is isometric to AdS³. The space and time-orientation preserving component of the isometry group is PSL(2,R) ×PSL(2,R), acting via the left and right multiplication:

(𝑔, ℎ) ·𝑥 =𝑔𝑥 ℎ⁻¹.

An AdS 3-manifold is a Lorentzian 3-manifold of constant curvature−1. Equiva- lently, such a manifold is locally isometrically modelled on PSL(2,R).

There are two main lines of research in AdS³:

1. globally hyperbolic maximally compact AdS 3-manifolds, as studied by Mess [Mes07] and developed by many others [KS07] (see [BS20] and the references therein).

2. and the study of properly discontinuous group actions on AdS³. Some of the main works are [KR85], [Kli96], [Sal00], [Kas10], [GKW15], [DGK16a], [DGK16b], [DT16], [Tho17], and [Tho18].

Our interest here is in the latter. More generally there is an interest in properly discontinuous group actions on Clifford-Klein forms, and study that has its roots in some famous conjectures about affine geometry.

We give a brief overview of some aspects. If an AdS 3-manifold is geodesically complete, meaning geodesics run for all time, then it comes from a proper quotient of PSLg(2,R)with respect to the lift of the action above. Goldman showed that the space of closed AdS 3-manifolds is larger than originally expected [Gol85], and Kulkarni and Raymond took up the problem of understanding all geodesically complete AdS 3-manifolds [KR85]. Among other things, they proved [KR85, Theorem 5.2] that any torsion-free discrete group acting properly discontinuously on AdS³ is of the form

Γ𝜌

1, 𝜌

2 ={(𝜌

1(𝛾), 𝜌

2(𝛾)) :𝛾 ∈ Γ}, whereΓ is a the fundamental group of a surface, and 𝜌

1, 𝜌

2 : Γ → PSL(2,R) are representations, with at least one of them Fuchsian. This is generalized for actions on rank 1 Lie groups in [Kas08].

Remark 3.1.1. Shortly after, Klingler [Kli96] proved that closed Lorentzian manifolds of constant curvature are geodesically complete. Thus, the completeness assumption can be dropped in the work of Kulkarni and Raymond on closed AdS 3-manifolds.

The natural next step is to understand whichΓ𝜌

1, 𝜌

2 act properly discontinuously. In the cocompact case, Salein observed it is sufficient [Sal00], and Kassel proved it is necessary [Kas10] that 𝜌

1 strictly dominates 𝜌

2(defined below). Guéritaud and Kassel extended these results to surfaces with punctures and higher dimensional hyperbolic spaces [GK17].

When a group Γ acts on a manifold with no reference to a representation 𝜌

1, we may just write 𝜌

2-equivariant. By the Selberg lemma, we only need to consider torsion-free groups.

Theorem 3.1.2 (Guéritaud-Kassel, Theorem 1.8 in [GK17]). A finitely generated discrete groupΓ𝜌

1, 𝜌

2 acts properly discontinuously and without torsion if and only if𝜌

1is Fuchsian and strictly dominates𝜌

2, up to interchanging𝜌

1and𝜌

2. The quotient is a Seifert-fibered AdS3-manifold over the hyperbolic surfaceH/𝜌

1(Γ) such that the circle fibers are timelike geodesics.

In the PSL(2,R)model, timelike geodesics are all of the form 𝐿_𝑝,𝑞 ={𝑋 ∈PSL(2,R) : 𝑋· 𝑝 =𝑞},

where (𝑝, 𝑞) range overH×H. These are topological circles and have Lorentzian length𝜋.

There was an open question: is every non-Fuchsian representation 𝜋

1(𝑆_𝑔) → PSL(2,R) strictly dominated by a Fuchsian one? This question was answered by Deroin-Tholozan in [DT16] and Guéritaud-Kassel-Wolf in [GKW15], using different methods. Deroin-Tholozan actually proved a more general result.

Theorem 3.1.3(Deroin-Tholozan, Theorem A in [DT16]). Let(𝑋 , 𝜈)be a CAT(−1) Hadamard manifold with isometry group𝐺and 𝜌 : 𝜋

1(𝑆_𝑔) → 𝐺a representation, 𝑔 ≥ 2. Then𝜌is strictly dominated by a Fuchsian representation, unless it stabilizes a totally geodesic copy ofHon which the action is Fuchsian.

Marché and Wolff use the domination result to answer a question of Bowditch and resolve the Goldman conjecture in genus 2 [MW16].

Tholozan completed the story for closed 3-manifolds in [Tho17].

Theorem 3.1.4 (Tholozan, Theorem 1 in [Tho17]). Fix 𝑔 ≥ 2 and a CAT(−1) Hadamard manifold (𝑋 , 𝜈) with isometry group𝐺. The space of dominating pairs withinT𝑔×Rep^{𝑛 𝑓}(𝜋

1(𝑆_𝑔), 𝐺)is homeomorphic to T𝑔×Rep^{𝑛 𝑓}(𝜋

1(𝑆_𝑔), 𝐺), where Rep^{𝑛 𝑓}(𝜋

1(𝑆_𝑔), 𝐺)is the space of representations that do not stabilize a totally geodesic copy ofHon which the action is Fuchsian.

The homeomorphism is fiberwise in the sense that for each𝜌 ∈ Rep^{𝑛 𝑓}(𝜋

1(𝑆_𝑔), 𝐺), it restricts to a homeomorphism fromT𝑔× {𝜌} → 𝑈 × {𝜌}, where𝑈 ⊂ T𝑔 is an open subset. The key point is that when (𝑋 , 𝜈) = (H, 𝜎), this is the deformation space of closed AdS quotients of PSL(2,R). The components of the deformation space are thus organized according to Euler numbers.

Results

Henceforth a manifold that is “complete, finite volume” is implicitly understood to be non-compact. A Hadamard manifold (𝑋 , 𝑔)is CAT(−𝜅), 𝜅 ≥ 0, if all sectional curvatures are≤ −𝜅. See [BH99] for information on CAT(−𝜅)metric spaces. When describing a fundamental group we suppress dependence on a basepoint. We often identify the fundamental group with the group of deck transformations without a change in notation. If Γ acts isometrically on a Riemannian manifold and 𝜌 is a representation, an (id, 𝜌)-equivariant map is simply called 𝜌-equivariant. The functionΛ:R→Cgiven by

Λ(𝜃) =(1−𝜃²) −𝑖2𝜃

will frequently appear in the chapter. We record here that as 𝜃increases from−∞

to∞, the complex argument ofΛ(𝜃) decreases from𝜋to−𝜋.

All of the other relevant definitions and ambiguities will be discussed in later sections. Our first result generalizes the work of Donaldson [Don87], Corlette [Cor88], and Labourie [Lab91] and may also be regarded as an equivariant extension of [Wol91b, Theorem 3.11]. Naturally, a portion of our analysis resembles that of Wolf.

Theorem 3A. Let Σ = Σ/Γ˜ be a complete finite volume hyperbolic surface and (𝑋 , 𝑔)a Hadamard manifold. Let𝜌 :Γ→ Isom(𝑋 , 𝑔)be a reductive representation.

There exists a𝜌-equivariant harmonic map 𝑓 : ˜Σ → 𝑋.

If we assume𝑋is CAT(−1), we may construct 𝑓 so that if𝛾is a peripheral isometry and𝜃 ∈R, the Hopf differentialΦhas the following behaviour at the corresponding cusp:

• if𝜌(𝛾)is parabolic or elliptic,Φhas a pole of order at most 1 and

• if𝜌(𝛾)is hyperbolic,Φhas a pole of order 2 with residue

−Λ(𝜃)ℓ(𝜌(𝛾))²/16𝜋².

Suppose that𝜌 does not fix a point on𝜕_∞𝑋. Then all harmonic maps whose Hopf differentials have poles of order at most 2 at the cusps are of this form. If𝜌stabilizes a geodesic, then any other harmonic map with the same asymptotic behaviour differs by a translation along that geodesic.

Regarding domination, the next theorem is the main result of this chapter.

Theorem 3B. Let Σ = Σ/Γ˜ be a complete finite volume hyperbolic orbifold and (𝑋 , 𝑔) a CAT(−1) Hadamard manifold. Let 𝜌 : Γ→ Isom(𝑋 , 𝑔) be any representation. There exists a geometrically finite representation 𝑗_Σ dominating𝜌 in length spectrum. If𝜌 is reductive, then 𝑗_Σ dominates𝜌 in the traditional sense. There is a family of convex cocompact Fuchsian representations strictly dominating 𝑗_Σ. Given a peripheral isometry𝛾,

• if𝜌(𝛾)is not hyperbolic, then 𝑗_Σ(𝛾)is parabolic and

• if𝜌(𝛾)is hyperbolic, 𝑗_Σ(𝛾)is hyperbolic with the same translation length.

In general 𝑗_Σwill not strictly dominate𝜌. This will be discussed in detail in Section 3.6. If 𝑋 = H and 𝜌 is Fuchsian with no elliptic monodromy it will follow from the proof that 𝑗_Σ = 𝜌. For holonomy representations of closed surfaces, Thurston observed in [Thu98, Proposition 2.1] that strict domination contradicts the Gauss- Bonnet theorem and is therefore impossible.

Most of the proof of Theorem 3B is devoted to constructing 𝑗_Σ. To upgrade to a strictly dominating representation we perform a strip deformation, a procedure introduced by Thurston [Thu98] and further developed in [DGK16a].

Setting𝑋 =Hin Theorem 3B, from [GK17, Theorem 1.8] we obtain:

Theorem 3C. Let Σ = Σ/Γ˜ be a complete finite volume hyperbolic orbifold and 𝜌 : Γ → PSL₂(R) any representation. Then there is a Fuchsian representation 𝑗_Σ dominating 𝜌 and a family of convex cocompact representations (𝑗^𝛼

Σ) strictly dominating 𝑗_Σ such that

(𝜌× 𝑗^𝛼

Σ) (Γ) ⊂PSL₂(R) ×PSL₂(R)

admits a properly discontinuous action on PSL₂(R) preserving the Lorentz metric of constant curvature−1. If𝛾 ∈Γis elliptic and𝜌(𝛾)has smaller order than𝛾, then the action is torsion free as well. Consequently there exists a geometrically finite AdS 3-manifold Seifert-fibered overH/𝑗^𝛼

Σ(Γ).

Note that ifΣis a manifold, the torsion condition always holds.

As an intermediate step in the proof of Theorem 3B, we obtain a result of independent interest. LetΣbe a complete finite volume hyperbolic surface with𝑛punctures and let 𝑇(Σ, 𝑝

1, . . . , 𝑝_𝑑

, ℓ_𝑑

1+1, . . . , ℓ_𝑛) denote the subspace of the Fricke-Teichmüller space ofΣ consisting of holonomies of hyperbolic surfaces with𝑑

1ordered punctures and 𝑑

2 ordered geodesic boundary components of lengthℓ_𝑑

1+1, . . . , ℓ_𝑑

2 > 0.

Let (𝜃_𝑘)^𝑛

𝑘=𝑑

1+1 ⊂ Rand𝑃 := (ℓ_𝑘, 𝜃_𝑘)^𝑛

𝑘=𝑑

1+1. Denote by𝑄(Σ, 𝑃) the space of holomorphic quadratic differentials onΣwith poles of order at most one at the punctures corresponding to cusps and poles of order 2 with residue

−Λ(𝜃_𝑘)ℓ²

𝑘/16𝜋²

for each puncture labelled by ℓ_𝑘. From the results in [Wol91b], for each point in𝑇(Σ, 𝑝

1, . . . , 𝑝_𝑑

1, ℓ

1, . . . , ℓ_𝑑

2), there is a unique homotopic harmonic diffeomor- phismℎ_𝑓 :Σ → 𝑆whose Hopf differential lives in𝑄(Σ, 𝑃).

Theorem 3D. LetΣ be a finite volume hyperbolic surface. The map Ψ :𝑇(Σ, 𝑝

1, . . . , 𝑝_𝑑

, ℓ1, . . . , ℓ_𝑑

2) →𝑄(Σ, 𝑃) given by [𝑆, 𝑓] ↦→Hopf(ℎ_𝑓)is a homeomorphism.

We expect the above result is known to experts, but could not find a proof in the literature. Hence we supply our own. The parametrization of the Teichmüller space of a closed surface by holomorphic quadratic differentials goes back to Sampson, Schoen-Yau, and Wolf (see [Wol89] for the full result). The case of Teichmüller spaces of punctured surfaces, corresponding to differentials with a pole of order at

most 1, was completed by Lohkamp [Loh91]. In [Gup17], Gupta parametrizedwild Teichmüller spacesby certain equivalence classes of holomorphic differentials with poles of order at least 3. Theorem 3D thus completes a description of the space of meromorphic quadratic differentials over a Riemann surface in terms of harmonic diffeomorphisms.

We end this subsection by presenting quick corollaries of Theorem 3B, unrelated to the rest of the chapter. When 𝑋 is a CAT(−1) Hadamard manifold and 𝜌 : Γ → Isom(𝑋 , 𝑔) is geometrically finite, thelimit set of 𝜌(Γ) is the set of limit points of Γ· 𝑧in 𝜕_∞𝑋 for a fixed point 𝑧 in 𝑋. It is a standard exercise to confirm that this does not depend on the point 𝑧. When 𝑋 =PSL₂(R) and 𝜌 is Fuchsian, the limit set is either the full circle 𝜕_∞H or a Cantor set. Thecritical exponent 𝛿(𝜌) is the smallest constant𝑠such that the Poincaré series

∑︁

𝛾∈Γ

𝑒^{−𝑠 𝑑(}^{𝑧, 𝜌(𝛾)·𝑧)}

converges, and it coincides with the Hausdorff dimension of the limit set (see [Coo93] for a proof). The analogue of the following result is known for closed surfaces and is observed in [DT16], but to the author’s knowledge it is new in our context.

Corollary 3E. Let Σ = Σ/Γ˜ be a complete finite volume hyperbolic orbifold and (𝑋 , 𝑔)a CAT(−1)Hadamard manifold. Let𝜌 :Γ→Isom(𝑋 , 𝑔)be a geometrically finite representation. There is a Fuchsian representation 𝑗_Σ such that the Hausdorff dimension of the limit set of 𝜌(Γ) is bounded below by that of 𝑗_Σ. 𝑗_Σ has the following property around a peripheral𝛾:

• if𝜌(𝛾)is not hyperbolic, then 𝑗_Σ(𝛾)is parabolic and

• if𝜌(𝛾)is hyperbolic, then 𝑗_Σ(𝛾) is hyperbolic withℓ(𝑗_Σ(𝛾)) =ℓ(𝜌(𝛾)). The Hausdorff dimension can be estimated and sometimes fully understood from the monodromy around the punctures. For instance, if Σ is a pair of pants and 𝜌 takes the cuffs to isometries with lengths𝑎, 𝑏, 𝑐 >0, then the Hausdorff dimension of the limit set of 𝑗_Σ occurs as a zero of a certainSelberg zeta function

𝑍_{𝑎,𝑏,𝑐}(𝑠)=Ö

𝛾∈Γ

∞

𝑚=0

1−𝑒⁻⁽^𝑠⁺^𝑚⁾^ℓ⁽^𝛾⁾

The map𝛾 ↦→ℓ(𝛾)is determined entirely by𝑎, 𝑏, 𝑐. These zeroes can be computed efficiently (see [PV17] for details).

Outline and strategy of proof

In the next section we introduce the relevant definitions and notations in the representation theory of discrete groups, and we prove some preparatory results about equivariant harmonic maps for surfaces with punctures. In Section 3.3 we prove the energy domination lemma, which says that the energy of an equivariant harmonic map is bounded above by that of a special harmonic diffeomorphism of the disk with the same Hopf differential. As is standard in this field, we argue via an analysis of the Bochner formula. This estimate is a central technical results of this paper, and is instrumental in proving Theorem 3B.

In Section 4.4 we prove Theorem 3D using classical techniques from the theory of harmonic maps. Section 3.5 is devoted to the proof of Theorem 3A. Infinite energy harmonic maps were constructed for some special cases in [Wol91b], [Sim90], [JZ97], and [KM08]. Consequently, there is nothing truly novel in the proof of the general existence result—it is an amalgamation of known ideas. The real work is done in studying the behaviour and uniqueness of the harmonic maps. We combine the energy estimate from Section 3/3 with Theorem 3D to control the energy locally, as well as a distance comparison to a special non-harmonic map to understand the directions in which our map should expand and contract.

In Section 3.6, we attempt to follow the approach of [DT16] to prove domination in the compact case. We take an equivariant harmonic map 𝑓 from Theorem 3A and choose a harmonic diffeomorphism ℎ from Σ to the convex core of some geometrically finite hyperbolic surface 𝑁 that has the same Hopf differential as 𝑓. From our energy estimates, 𝑓 ◦ℎ⁻¹ is 1-Lipschitz and intertwines 𝜌 with the holonomy of𝑁, but it is not strictly 1-Lipschitz (this issue does not occur in [DT16]).

We introduce strip deformations to strictly dominate the holonomy of𝑁, completing the proof of Theorem 3B.

Other recent work

Shortly after a preprint of the paper was posted to the arXiv, Gupta-Su proved the same domination result for representations to PSL₂(C) [GS20]. Their proof is different: they straighten the pleated plane determined by the Fock-Goncharov coordinates associated to a framed representation, and then use strip deformations.

Acknowledgments

It is a pleasure to thank my advisor, Professor Vladimir Marković, for his support, insight, patience, and guidance. I would also like to thank Qiongling Li and Peter

Smillie for their interest and helpful conversations, as well as François Guéritaud for graciously answering some questions over email. Finally, I would like to thank my friend Arian Jadbabaie for helping create the figure on Inkscape.

3.2 Representations of discrete groups

Dalam dokumen Harmonic maps of Riemann surfaces and applications in geometry (Halaman 37-46)