Anti-de Sitter 3-manifolds

Chapter IV: Maximal surfaces and AdS 3-manifolds

4.4 Anti-de Sitter 3-manifolds

In this subsection we prove Proposition 4.1.8, which is actually a quick consequence of the proposition immediately below. We work in the PSL(2,R)model throughout.

Proposition 4.4.1. Let𝑉 ⊂ Hbe a domain. The data of a domainΩ ⊂ AdS³and a fibrationΩ→𝑉such that every fiber is a timelike geodesics is equivalent to that of a domain𝑉 ⊂ Hand a locally strictly contracting map𝑔:𝑉 → H.

The proof of the first direction of the equivalence is a straightforward adaptation of the procedure from [GK17, Proposition 7.2]. There, 𝑉 = H, Ω = AdS³, and ℎ is (globally) strictly contracting. We include the proof for the readers convenience.

Proof. The key fact we use is that timelike geodesics𝐿_𝑝

1,𝑞

1 and 𝐿_𝑝

2,𝑞

2 intersect if and only if

𝑑_𝜎(𝑝

1, 𝑝

2) =𝑑_𝜎(𝑞

1, 𝑞

2).

With this in mind, given a locally strictly contracting mapping 𝑔 : 𝑉 → H×H with the properties above, timelike geodesics of the form𝐿_{𝑝,𝑔(𝑝)} and 𝐿_{𝑞,𝑔(𝑞)} never intersect. Thus, the geodesics 𝐿_{𝑝,𝑔(𝑝)} sweep out a connected set Ω ⊂ AdS³ as 𝑝 ranges over𝑉.

We argue thatΩis open. We record that𝑋 ∈ 𝐿_{𝑝,𝑔(𝑝)} if and only if 𝑋⁻¹◦𝑔(𝑝) = 𝑝 .

For small𝜖 > 0, let𝐵_𝜖(𝑝) ⊂ 𝑉 denote the𝜖-ball around 𝑝inH. Let𝐵 ⊂ AdS³be the open ball consisting of isometries𝑌 such that

𝑑_𝜎(𝑝, 𝑌⁻¹𝑔(𝑝)) < (1−Lip(𝑔|𝐵_𝜖(𝑝)))𝜖 . Then for any𝑞 ∈ 𝐵_𝜖(𝑝)and𝑌 ∈𝐵,

𝑑_𝜎(𝑌⁻¹◦𝑔(𝑞), 𝑝) ≤ 𝑑(𝑌⁻¹𝑔(𝑞), 𝑌⁻¹𝑔(𝑝)) +𝑑(𝑌⁻¹𝑔(𝑝), 𝑝) < 𝜖 .

Thus, 𝑌⁻¹𝑔 takes the closure of 𝐵_𝜖(𝑝) to itself, and by the Banach fixed point theorem there is a unique 𝑞 ∈ 𝐵_𝜖(𝑝) such that𝑌 ◦𝑔(𝑞) = 𝑞. So 𝐵 ⊂ Ω. This argument also shows that the fibration fromΩ→𝑉 described by𝐿

𝑓(𝑝), ℎ(𝑝) ↦→ 𝑝is continuous.

For the other direction, any circle fibration Ω → 𝑉 with timelike geodesic fibers determines a map 𝐹 : 𝑉 → H× H by 𝐹(𝑝) = (ℎ(𝑝), 𝑓(𝑝)), where 𝐿_{𝑓(𝑝), ℎ(𝑝)} is the geodesic lying over 𝑝 in Ω. 𝐹 preserves connectedness—using the product structure, [JM18, Theorem 2.2] guarantees it is continuous when 𝑓 is non-constant.

If 𝑓 is a constant 𝑞, then because Ω is open, for any 𝑝 and path from 𝑝 to 𝑞, we can find a continuous path of isometries 𝑟 ↦→ 𝑋_𝑟 such that ℎ(𝑟) = 𝑋⁻¹

𝑟 𝑞. Thus we have continuity here as well. As the timelike geodesics never intersect, 𝑑(𝑓(𝑝), 𝑓(𝑞)) ≠ 𝑑(ℎ(𝑝), ℎ(𝑞)) for 𝑝 ≠ 𝑞. As the diagonal in H× H has codi- mension 2, a connectedness argument shows 𝑑(𝑓(𝑝), 𝑓(𝑞)) < 𝑑(ℎ(𝑝), ℎ(𝑞)) or 𝑑(𝑓(𝑝), 𝑓(𝑞)) > 𝑑(ℎ(𝑝), ℎ(𝑞)) always for 𝑝 ≠ 𝑞. By switching coordinates, we may assume we have the former. This condition ensures that ℎ is injective.

Therefore,𝑔 = 𝑓 ◦ℎ⁻¹is a well-defined locally strictly contracting map. □

Proposition 4.1.8 is just the equivariant version of this: for a pair (𝜌

1, 𝜌

2) with 𝜌

1

acting properly discontinuously on𝑉, we have 𝜌2(𝛾)𝐿_{𝑝,𝑔(𝑝)}𝜌

1(𝛾)⁻¹= 𝐿_𝜌

1(𝛾)𝑝, 𝜌

2(𝛾)𝑔(𝑝), so 𝜌

1× 𝜌

2acts properly discontinuously onΩ and equivariance of the fibration is clear.

It is seen in the proof thatΩ ⊂ AdS³consists of all isometries𝑋 such that 𝑋⁻1◦𝑔 has a fixed point.

Remark 4.4.2. The results here generalize, almost word for word, for quotients of proper domains in the rank 1 Lie groups𝐺 =O(𝑛,1), SO(𝑛,1), SO₀(𝑛,1), and PO(𝑛,1). One can consider the action by left and right multiplication and equivariant 𝐾-fibrations𝐺 ⊃ Ω → 𝑉 ⊂ H^𝑛, 𝑛 ≥ 2, where 𝐾 ⊂ 𝐺 is the maximal compact subgroup. Here the fibers are copies of𝐾, each of the form{𝑋 ∈𝐺 : 𝑋·𝑝 =𝑞}for some 𝑝, 𝑞∈H^𝑛.

Remark 4.4.3. Proposition 4.1.8 applies to non-reductive representations. They have been largely excluded from our discussion because harmonic maps and maximal surfaces do not exist for these representations.

Theorem 4C

Here we give the proof of Theorem 4C. We make use of results from the paper [GK17]. Fix reductive representations𝜌

1, 𝜌

2 :Γ→PSL(2,R) with𝜌

1Fuchsian.

Definition 4.4.4. Let𝑉 ⊂ Hbe a𝜌

1-invariant domain, and 𝑓 :𝑉 → Ha(𝜌

1, 𝜌

2)- equivariant map realizing the minimal Lipschitz constant 𝐿 among equivariant maps. The stretch locus is the set of points 𝑥 ∈ Hsuch that the restriction of 𝑓 to any neighbourhood of𝑥 has Lipschitz constant exactly𝐿and no smaller.

The result below is culled from [GK17, Theorem 1.3 and 5.1]. See the reference for more general statements and details.

Theorem 4.4.5(Guéritaud-Kassel). Assume there exists a(𝜌

1, 𝜌

2)-equivariant map with minimal Lipschitz constant𝐿 =1, and let𝐸be the intersection of all the stretch loci among such maps. Then there exists an “optimal” (𝜌

1, 𝜌

2)-equivariant 1- Lipschitz map whose stretch locus is exactly 𝐸. 𝐸 projects under the action of 𝜌1(Γ)to the convex core for𝜌

1, and is either empty or the union of a lamination and 2-dimensional convex sets with extremal points only in the limit setΛ𝜌

1(Γ) ⊂ 𝜕_∞H.

Proof of Theorem 4C. The equivalence between (1) and (3) is contained in The- orem 4A. Assuming (1) we prove (2). Take any optimal map 𝑔, and the map

˜ 𝐶(H/𝜌

1(Γ)) →H×Hgiven by 𝑝 ↦→ (𝑝, 𝑔(𝑝)). In the case that there exists a pe- ripheral on which𝜌

1is hyperbolic, suppose for the sake of contradiction that there is a choice of𝑔so that the domain extends to give a fibration over a larger subsurface.

From the other direction of Proposition 4.1.8, we obtain a(𝜌

1, 𝜌

2)-equivariant and a locally contracting map defined on the preimage of this subsurface in the univer- sal cover. This implies there is a peripheral 𝛾 with ℓ(𝜌

1(𝛾)) > ℓ(𝜌

2(𝛾)), which contradicts our original Definition 4.1.1.

Now we prove that (2) implies (1). Given such a domain and fibration, from Proposition 4.1.8 we obtain a strictly 1-Lipschitz (𝜌

1, 𝜌

2)-equivariant map defined on ˜𝐶(H/𝜌

1(Γ)). If 𝜌

1has no hyperbolic peripherals, then we get (1) for free and we’re done. So assume there is a peripheral 𝜁 with 𝜌

1(𝜁) hyperbolic. Any 1- Lipschitz map 𝑔 defined inside ˜𝐶(H/𝜌

1(Γ)) extends to a 1-Lipschitz map of the frontier insideH, and hence

ℓ(𝜌

2(𝜁)) ≤ℓ(𝜌

1(𝜁)).

We extend𝑔to all ofHby precomposing with the 1-Lipschitz(𝜌

1, 𝜌

1)-equivariant nearest point projection onto ˜𝐶(H/𝜌

1(Γ)), so we know that the set of globally defined Lipschitz maps is non-empty. From Lemma 4.10 in [GK17] (an application of Arzelà-Ascoli), there exists an optimal(𝜌

1, 𝜌

2)-equivariant Lipschitz map𝑔^′. As for the optimal Lipschitz constant, 𝑔shows 𝐿 ≤ 1, and if 𝐿 < 1 then 𝜌

1× 𝜌

2acts properly discontinuously on the whole AdS³, and hence 𝐿 =1. Applying Theorem 4.4.5, we have a stretch locus𝐸.

𝐸 is contained in the intersection of the stretch loci of 𝑔 and𝑔^′. Since 𝑔 does not maximally stretch in the interior of ˜𝐶(H/𝜌

1(Γ)), 𝐸 is contained in the boundary of

˜ 𝐶(H/𝜌

1(Γ)). If𝐸 is missing the lifts of one boundary component ofC(H/𝜌

1(Γ)), then𝑔^′is strictly contracting inside the half-spaces inHthat project to the infinite funnel bounding this component inH/𝜌

1(Γ). From Proposition 4.1.8, we can thus find a𝜌

1×𝜌

2-equivariant domain that yields a fibration onto the union of the convex core with this funnel, which contradicts our standing assumption. We conclude that the stretch locus is exactly these components, and hence𝑔^′is an almost strictly dominating map.

For the final statement, we use the homeomorphismΨ = Ψ⁰ from Theorem 4B to parametrize the space of representations. We take the domains in AdS³associated

to the spacelike maximal immersions with 0 twist parameter (any one will do). The energy domination implies that they yield proper quotients by Proposition 4.1.8.

Since the harmonic maps for irreducible representations vary continuously with the representation, so do the domains in AdS³. Hence, when we restrict to these classes, Ψparametrizes a deformation space of AdS 3-manifolds. □ To produce more representations that give such incomplete quotients, take an almost strictly dominating pair(𝜌

1, 𝜌

2)(Theorem 4B shows there are many) and an optimal map 𝑔. To relax the condition that all boundary lengths agree, first choose a collection of peripherals, but not all of them. For each of the selected peripherals, there is a geodesic or a horocycle in H/𝜌

1(Γ). We then specify a transversely intersecting geodesic arc that does not intersect any other peripheral geodesic or horocycle, and apply strip deformations toH/𝜌

1(Γ)along these arcs (see [DGK16a], [Sag19, Section 6]). This gives a new hyperbolic surface whose holonomy is a Fuchsian representation 𝑗, and for some 𝜆 < 1, a strictly 𝜆-Lipschitz (𝑗 , 𝜌

1)- equivariant map 𝑔^′. We can extend𝑔 outside of the convex hull of the limit set by using the 1-Lipschitz (𝜌

1, 𝜌

1)-equivariant closest-point projection. We then take the composition𝑔◦𝑔^′and the corresponding circle bundle.

With the main theorems complete, we briefly digress to discuss the topology of the quotients. The quotients naturally acquire an orientation. Since the surface is not compact, the bundle is topologically trivial: BU(1) = CP^∞ and [Σ,CP^∞] = 𝐻2(Σ,Z) =0.

However, the global trivialization is by no means compatible with the AdS structure.

To be precise, the 3-manifold is not “standard” in the sense of [KR85]: its casual double cover does not possess a timelike Killing field. If it did, the holonomy would normalize the isometric flow generated by the Killing field, and it follows from [KR85, pages 237-238] that this is impossible for reductive𝜌

2.