Existence and classification of tame harmonic maps

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

3.5 Existence and classification of tame harmonic maps

as 𝑘 → ∞. However, we can uniformly bound ℓ(ℎ_𝑓

𝑘(𝛿^′)) from above, so this is impossible.

Now, view the punctures on Σ as nodes and double across all punctures that “are opened” to get a noded surfaceΣ^𝑑. Likewise double all surfaces (𝑌 , ℎ_𝑓) ∈ Ψ⁻¹(𝐾) across the boundaries. ℎ_𝑓 extends by reflection and we get a pair (𝑌^𝑑, ℎ^𝑑

𝑓). This provides a map

𝜄:𝑇(Σ, ℓ

1, . . . , ℓ_𝑛) →T₂𝑔+𝑑

1−1,2𝑑

2

that is a diffeomorphism onto its image. By [Ham03, Lemma 3.3] on any𝑆_𝑔,𝑛there is a collection of simple closed curves𝛿

1, . . . , 𝛿

6𝑔−5+2𝑛so that the map L𝑔,𝑛 :T𝑔,𝑛 → R^{6𝑔−5+2𝑛}

given by

[𝑋 , 𝜙] := 𝜒↦→ (ℓ_𝜒(𝛿

1), . . . , ℓ_𝜒(𝛿

6𝑔−5+2𝑛)) is a diffeomorphism onto its image. The compositionL_2𝑔+𝑑

1−1,2𝑑

2◦𝜄takesΨ⁻¹(𝐾) into a compact set, and henceΨ is proper. As discussed above, this completes the

proof. □

3.5 Existence and classification of tame harmonic maps

Proposition 3.5.1. Given the data Σ, 𝑋 , 𝜌 as above, there exists a 𝜌-equivariant harmonic map 𝑓 : ˜Σ → 𝑋.

Proof. Let𝛼: [0, 𝜏] → 𝑋be a constant speed curve with image in the axis of𝜌(𝛾) and so that𝛼(𝜏) = 𝜌(𝛾)𝛼(0). By [Cor92b] there exists a unique harmonic section 𝑠_𝑟 of the pullback bundle𝑖^∗

𝑟𝑋 → Σ𝑟 with boundary values𝛼. Extend 𝑠_𝑟 to Σ via 𝑠_𝑟(𝑥 , 𝑡) = 𝑠_𝑟(𝑥). The𝑠_𝑟 induce equivariant maps 𝑓_𝑟 : ˜Σ → 𝑋, that are harmonic on 𝜋⁻¹(Σ𝑗). We prove the 𝑓_𝑟 converge along a subsequence in the𝐶^∞ topology to an equivariant harmonic map.

Let𝜑be any non-harmonic equivariant map corresponding to a section of𝑖^∗

0𝑋 →Σ^𝑐 with boundary values𝛼. As with 𝑓_𝑟, define 𝜑 on the rest of 𝐷 by 𝜑(𝑥 , 𝑡) = 𝜑(𝑥) and then extend equivariantly to ˜Σ. Let𝛽be the image of𝛼on the geodesic axis of 𝜌(𝛾)and set

𝛽^𝑟

𝑡 := 𝑓_𝑟( [0, 𝜏] × {𝑡}).

Notice that|𝑑 𝜑|𝜎^′ =ℓ(𝛽)/𝜏on𝐶since it has constant speed. For𝑟 > 𝑠, 2𝐸_𝐶

𝑟\𝐶_𝑠(𝜑) =

∫

𝐶𝑟\𝐶𝑠

|𝑑 𝜑|²_𝜎′𝑑 𝑣_𝜎′ =(𝑟−𝑠)ℓ(𝛽)²/𝜏 .

As𝛽is a geodesic arc in a negatively curved space, 𝑠ℓ(𝛽) ≤

∫ ^𝑠

0

ℓ(𝛽^𝑟

𝑡)𝑑 𝑡 , and hence for any𝑟 > 𝑠,

𝐸_𝐶

𝑟\𝐶_𝑠(𝜑) ≤ 1 2

∫ ^𝑟

𝑠

ℓ(𝛽^𝑟

𝑡)²/𝜏 𝑑 𝑡 ≤ 1 2

∫ ^𝑟

𝑠

∫

𝑆1×{𝑡}

|𝑑 𝑓_𝑟|𝑑𝜃 ₂

𝜏⁻¹𝑑 𝑡 ≤ 𝐸_𝐶

𝑟\𝐶_𝑠(𝑓_𝑟). From the non-positive curvature hypothesis 𝑓_𝑟 minimizes energy among maps to 𝑋 with the same equivariant boundary values. In particular,

𝐸_Σ

𝑟(𝑓_𝑟) ≤ 𝐸_Σ

𝑟(𝜑), and moreover

𝐸_Σ

𝑠(𝑓_𝑟) =𝐸_Σ

𝑟(𝑓_𝑟) −𝐸_Σ

𝑟\Σ𝑠(𝑓_𝑟) ≤ 𝐸_Σ

𝑟(𝜑) −𝐸_Σ

𝑟\Σ𝑠(𝜑) =𝐸_Σ

𝑠(𝜑). By a classical PDE estimate (say, from [SY97, page 171]),

sup

𝐷𝑠

𝑒(𝑓_𝑟) =sup

Σ𝑠

𝑒(𝑓_𝑟) ≤ 𝐴_𝑠𝐸_Σ

𝑠+1(𝑓_𝑟) ≤ 𝐴_𝑠𝐸_Σ

𝑠+1(𝜑),

where 𝐴_𝑠 depends on the Ricci curvature ofΣ𝑠+1, the injectivity radius on Σ𝑠, and dist(𝜕Σ𝑠, 𝜕Σ𝑠+1). Since 𝜌 is acting by isometries we get the same bound in all of 𝜋⁻¹(Σ𝑟). Next, we claim there is a compact set𝑂_𝑠 ⊂ 𝑋 such that

𝑓_𝑟(𝐷_𝑠) ⊂𝑂_𝑠

for all𝑟. Appealing to the energy density bound above, it is enough to show that for a fixed point𝑥

0 ∈ 𝐷_𝑠, 𝑓_𝑟(𝑥

0)stays within some compact set as𝑟 → ∞. We find it convenient from here to split cases. Firstly, let us assume that the image of𝜌 does not lie in a parabolic subgroup. Let 𝜉 be a point in the boundary at infinity 𝜕_∞𝑋. There is loop𝛾 : [0, 𝐿] → Σparametrized by arclength such that

𝜌(𝛾) (𝜉) ≠𝜉 .

Chooseℓso that the image of 𝛾 under𝜋lies entirely inΣℓ and let 𝐴_ℓ be a uniform bound on the derivative in𝜋⁻¹(Σℓ). We then have, for𝑟 > ℓ,

𝑑_𝑔(𝜌(𝛾)𝑓_𝑟(𝑥

0), 𝑓_𝑟(𝑥

0)) =𝑑_𝑔(𝑓_𝑟(𝛾(𝑥

0)), 𝑓_𝑟(𝑥

0)) ≤ 𝐴_ℓ𝐿 . This is because lifting 𝛾 to the universal cover gives a path between 𝑥

0and 𝛾 ·𝑥

0

that remains within lifts ofΣℓ. Choose a neighbourhood 𝐵 of𝜉 in 𝑋 ∪𝜕_∞𝑋 such that

𝑑_𝑔(𝐵∩𝑋 , 𝜌(𝛾)𝐵∩𝑋) > 𝐴_ℓ𝐿 . Then 𝑓_𝑟(𝑥

0) cannot enter𝐵, no matter how large𝑟 grows. Via compactness we find a finite number of neighbourhoods(𝐵_𝑘)𝑘 as above that cover the boundary sphere.

Choosing𝑂_𝑠 := 𝑋\(∪𝑘𝑁_𝑘) the claim follows. Notice then that 𝑓_𝑟 takes any lift of Σ𝑠to a compact set:

𝑓_𝑟(𝛾 𝐷_𝑠) ⊂ 𝜌(𝛾)𝑂_𝑠

for all𝑟 , 𝑠. It now follows by a well-known argument, namely an application of the Arzelà-Ascoli theorem and a bootstrap, that a subsequence of the(𝑓_𝑟)𝑟 >0converges uniformly on compact subsets of ˜Σto a harmonic map 𝑓_∞. By equivariance of the

𝑓_𝑟 on𝜋⁻1(Σ𝑟), 𝑓_∞is necessarily equivariant.

We next treat the case where𝜌stabilizes a totally geodesic flat𝐹. 𝐹 is a symmetric space and identifies isometrically as

𝐺/𝐻 := (𝑂(𝑛)⋊ R^𝑛)/𝑂(𝑛). Fix two points 𝑥

0 ∈ 𝐷_𝑠 and 𝑦

0 ∈ 𝐹 and for each 𝑟 choose 𝑔_𝑟 ∈ 𝐺 such that 𝑔_𝑟𝑓_𝑟(𝑥

0) = 𝑦

0. We notice that for any 𝑦 ∈ 𝐹 and 𝛾 ∈ Γ, 𝑑(𝑔_𝑟𝜌(𝛾)𝑔⁻1

𝑟 𝑦, 𝑦) is

uniformly bounded in𝑟. Indeed, 𝑑_𝑔(𝑔_𝑟𝜌(𝛾)𝑔⁻¹

𝑟 𝑦, 𝑦) ≤ 𝑑_𝑔(𝑔_𝑟𝜌(𝛾)𝑔⁻¹

𝑟 𝑦, 𝑔_𝑟𝜌(𝛾)𝑔⁻¹

𝑟 𝑦

0) +𝑑_𝑔(𝑔_𝑟𝜌(𝛾)𝑔⁻¹

𝑟 𝑦

0, 𝑦

0) +𝑑_𝑔(𝑦, 𝑦

0)

=2𝑑_𝑔(𝑦, 𝑦

0) +𝑑_𝑔(𝑔_𝑟𝜌(𝛾)𝑔⁻¹

𝑟 𝑦

0, 𝑦

0)

=2𝑑_𝑔(𝑦, 𝑦

0) +𝑑_𝑔(𝑔_𝑟𝜌(𝛾)𝑓_𝑟(𝑥

0), 𝑔_𝑟𝑓_𝑟(𝑥

0))

=2𝑑_𝑔(𝑦, 𝑦

0) +𝑑_𝑔(𝑔_𝑟𝑓_𝑟(𝛾·𝑥

0), 𝑔_𝑟𝑓_𝑟(𝑥

0))

=2𝑑_𝑔(𝑦, 𝑦

0) +𝑑_𝑔(𝑓_𝑟(𝛾·𝑥

0), 𝑓_𝑟(𝑥

0)),

and we know 𝑓_𝑟has a uniform energy density bound on 𝜌(Γ) ·𝐷_𝑟. By the argument of [JY91, Lemma 2] there is a sequence (𝑟_𝑛)^∞_𝑛=

1 increasing to ∞ and an element 𝑔_∞ ∈𝐺 such that for every𝛾 ∈Γand𝑦 ∈ 𝐹,

𝑛→∞lim 𝑔_𝑟

𝑛𝜌(𝛾)𝑔⁻¹

𝑟_𝑛𝑦 =𝑔_∞𝜌(𝛾)𝑔_∞⁻¹𝑦 . The orbit of the point𝑥

0under the family of maps𝑔_𝑟

𝑛𝑓_𝑟

𝑛 is a singleton, and by our uniform energy bound we see as above that there is a compact set𝑂_𝑠 such that

𝑔_𝑟

𝑛

𝑓_𝑟

𝑛(𝐷_𝑠) ⊂𝑂_𝑠. Arguing as above there is a subsequence along which𝑔_𝑟

𝑛𝑓_𝑟

𝑛converges to a harmonic map 𝑓_∞. Note that 𝑔_𝑟

𝑛𝑓_𝑟

𝑛 is 𝑔_𝑟

𝑛𝜌(Γ)𝑔⁻¹

𝑟𝑛-equivariant, so that 𝑓_∞ is 𝑔_∞𝜌(Γ)𝑔⁻_∞¹- equivariant. Therefore, we may take 𝑓 :=𝑔⁻_∞¹𝑓_∞ as the sought harmonic map. □ We use the ideas above to build a family of harmonic maps, indexed by a real parameter𝜃 ∈ R. We perform afractional Dehn twist on each cylinder𝐶. This is the map given in the cusp coordinates by

𝑥+𝑖 𝑦 ↦→𝑥+𝜃 𝑦+𝑖 𝑦

on𝐶and the identity map on the rest ofΣ. Lift to a map𝑑^𝜃on ˜Σ. The lift commutes with the relevant parabolic isometry. Define 𝑓^𝜃

𝑟 to be the equivariant harmonic map on𝜋⁻1(Σ𝑟) with the same equivariant boundary values as𝜑◦𝑑^𝜃|𝜕 𝐷𝑟. Then extend to agree with𝜑◦𝑑^𝜃on the complement. The derivative matrix of𝑑^𝜃 is

1 𝜃 0 1

! , so that

||𝑑(𝜑◦𝑑^𝜃) || ≤ ||𝑑 𝜑|| (1+𝜃).

Thus onΣ𝑠,

𝐸_Σ

𝑠(𝑓^𝜃

𝑟) ≤ 𝐸_Σ

𝑠(𝑓_𝑟) (1+𝜃)².

By the argument of Proposition 3.5.1 there is a subsequence along which the 𝑓^𝜃

𝑟 ’s converge to a limiting harmonic map 𝑓^𝜃. Of course, 𝑓 = 𝑓0.

We keep the same characters𝛼and𝛽from the proof of the above proposition. Note ℓ(𝛽) =ℓ(𝜌(𝛾)). Define𝜑^𝜃 := 𝜑◦𝑑^𝜃. In local Euclidean coordinates,𝑑^𝜃is harmonic on𝐶. Since∇𝑑 𝜑 =0 on𝐶, the composition is a harmonic map there (see [EL83, Proposition 2.20]).

Lemma 3.5.2. The function𝑧 ↦→𝑑(𝑓^𝜃, 𝜑^𝜃) (𝑧) is uniformly bounded.

Proof. Let𝜓_𝑟 :=𝑑(𝑓^𝜃

𝑟, 𝜑^𝜃). By equivariance, each𝜓_𝑟 descends to a function onΣ. 𝜓_𝑟 =0 onΣ\Σ𝑟, and since𝜓_𝑟 > 0 at some point we know it attains a maximum at a point in the interior ofΣ𝑟. As 𝜓_𝑟 is subharmonic on𝐶_𝑟, sup𝑧∈𝐶𝑟

𝜓_𝑟(𝑧)occurs on

𝜕Σ^𝑐 and moreover𝜓_𝑟 is maximized at a point inΣ^𝑐. Meanwhile, 𝜓_𝑟 → 𝑑²(𝑓^𝜃, 𝜑^𝜃)

uniformly on compacta as𝑟 → ∞. By smoothness,𝜓is uniformly bounded on Σ^𝑐. This implies we have a uniform bound on the𝜓_𝑟’s insideΣ^𝑐 as𝑟 → ∞. Since the relevant maximum is attained insideΣ^𝑐, this bound holds everywhere. □ LetΦ:=Hopf(𝑓^𝜃). The context is clear so we do not include a𝜃in our notation. By equivariance we can viewΦas a holomorphic quadratic differential on any quotient of ˜Σby a subgroup ofΓ.

Lemma 3.5.3. Φhas a pole of order2at the cusp.

Proof. From the infinite energy phenomena,Φeither has a pole of order at least 2 or an essential singularity. The (2,0)component of the pullback metric by 𝜑^𝜃 is a section ofK²that is holomorphic on𝐶. We still denote it by Hopf(𝜑^𝜃).

We compute this differential in𝐶. Choose a local orthonormal basis 𝜕/𝜕 𝑥, 𝜕/𝜕 𝑦 of the relevant tangent spaces so that𝜕 𝜑⁰/𝜕 𝑦 =0 always. Starting with𝜃 =0, we know that in local coordinates

Hopf(𝜑⁰) (𝑧) = 1 4

|𝜕 𝜑⁰/𝜕 𝑥|²− |𝜕 𝜑⁰/𝜕 𝑦|²−2𝑖⟨𝜕 𝜑⁰/𝜕 𝑥 , 𝜕 𝜑⁰/𝜕 𝑦⟩ 𝑑 𝑧².

Since𝜑⁰is constant in the vertical direction Hopf(𝜑⁰) (𝑧) = 1

4

|𝜕 𝜑⁰/𝜕 𝑥|²𝑑 𝑧²=ℓ(𝜌(𝛾))²/4𝜏²𝑑 𝑧². From the chain rule,𝑑 𝜑⁰and𝑑 𝜑^𝜃 admit matrix representations with

𝑑 𝜑⁰=

𝑣 0

, 𝑑 𝜑^𝜃 =

𝑣 𝜃 𝑣

, where𝑣 is a 1×dim𝑋 column vector. Thus,

Hopf(𝜑^𝜃) (𝑧) = 1 4

(|𝜕 𝜑^𝜃/𝜕 𝑥|²− |𝜕 𝜑^𝜃/𝜕 𝑦|²−2𝑖⟨𝜕 𝜑^𝜃/𝜕 𝑥 , 𝜕 𝜑^𝜃/𝜕 𝑦⟩)𝑑 𝑧²

= 1 4

(1−𝜃²−𝑖2𝜃) |𝜕 𝜑⁰/𝜕 𝑥|²𝑑 𝑧². We take the strip conformally to a punctured disk via

𝑧 ↦→ 𝜁(𝑧)=𝑒

2𝜋 𝑖 𝑧 𝜏 ,

taking the point at∞to 0. The transformation law multiplies by−𝜁⁻²𝜏²/4𝜋², and we see that we have a pole of order 2 with residue

−Λ(𝜃)ℓ(𝜌(𝛾))²/16𝜋².

We now compareΦto Hopf(𝜑^𝜃). As𝜑^𝜃 has rank 1, the formula𝐽 = 𝐻−𝐿implies 𝐻(𝜑^𝜃)^1/2= 𝐿(𝜑^𝜃)^1/2= 1

2

𝑒(𝜑^𝜃)^1/2,

so that Hopf(𝜑^𝜃) =𝜎 𝐻(𝜑^𝜃)^1/2𝐿(𝜑^𝜃)^1/2=𝜎 𝑒(𝜑^𝜃)/4. From Young’s inequlaity,

||Φ||=𝜎 𝐻(𝑓)^1/2𝐿(𝑓)^1/2 ≤ 1 2

𝜎 𝑒(𝑓^𝜃),

and hence it is enough to bound𝑒(𝑓^𝜃)by a sublinear function of𝑒(𝜑^𝜃). This is not hard: for any𝑥

0 ∈Σ˜,𝑟

0 > 0, and𝑦 ∈𝐵(𝑥

0, 𝑟

0), 𝑑(𝑓^𝜃(𝑥

0), 𝑓^𝜃(𝑦)) ≤ 𝑑(𝑓^𝜃(𝑥

0), 𝜑^𝜃(𝑥

0)) +𝑑(𝑓^𝜃(𝑦), 𝜑^𝜃(𝑦)) +𝑑(𝜑^𝜃(𝑥

0), 𝜑^𝜃(𝑦))

≤ 𝐴+ sup

𝐵(𝑥

0,𝑟

0)

||𝑑 𝜑^𝜃||𝑑(𝑥

0, 𝑦).

Working in the flat cylinder metric, Cheng’s lemma then gives

||𝑑 𝑓|| (𝑥

0) ≲ 1+𝑟

0

𝑟0

(1+sup||𝑑 𝜑^𝜃||𝑟

0).

In a cusp neighbourhood, the injectivity radius of the flat cylinder metric is uniformly bounded below, and hence we may choose𝑟

0 uniformly bounded below. Squaring

for the energy density gives the desired bound. □

Henceforth, we assume that 𝑋 is CAT(−1). By equivariance, 𝑓^𝜃 and 𝜑^𝜃 induce quotient maps

𝑓_𝛾, 𝜑_𝛾 : ˜Σ/⟨𝛾⟩ → 𝑋/⟨𝜌(𝛾)⟩.

We suppress the𝜃from our notation for convenience. 𝛽projects in the quotient to a core geodesic 𝛽. From the CAT(−1)hypothesis, this is the unique geodesic in the homotopy class. Any𝐷_𝑟/⟨𝛾⟩identifies isometrically with the cylinder

{(𝑥 , 𝑦) =𝑥+𝑖 𝑦 : 0≤ 𝑥 ≤ 𝜏, 𝑎 ≤ 𝑦 ≤𝑟} with the usual identification.

Lemma 3.5.4. There is a translation 𝑅˜ of the geodesic axis of 𝜌(𝛾) such that the mapΣ ∋𝑧↦→ 𝑑(𝑓^𝜃,𝑅˜◦𝜑^𝜃) (𝑧)tends to0as we move into the puncture.

Proof. We defineC_∞to be the infinite cylinder

{(𝑥 , 𝑡) ∈ [0,1] × (−∞,∞) : (0, 𝑡) ∼ (1, 𝑡)}

with the flat metric. Let𝑏_𝑠 : C_∞ → 𝐷/⟨𝛾⟩be the map given by

















(𝑥 , 𝑡) ↦→ (𝑥 , 𝑠) −∞ ≤𝑡 ≤ −𝑠 (𝑥 , 𝑡) ↦→ (𝑥 ,2𝑠+𝑡) −𝑠 ≤ 𝑡 ≤ 𝑠 (𝑥 , 𝑡) ↦→ (𝑥 ,3𝑠) 𝑠 ≤ 𝑡 ≤ ∞.

Then set𝐵_𝑠 := 𝑓_𝛾◦𝑏_𝑠and𝜑_𝑠 :=𝜑_𝛾◦𝑏_𝑠. Both𝐵_𝑠and𝜑_𝑠are harmonic on−𝑠 ≤ 𝑡 ≤ 𝑠 because𝑏_𝑠 is conformal there. From Lemma 3.5.2 the orbit of any point under 𝐵_𝑠 remains in a compact set as𝑠→ ∞. The uniform energy bounds from Lemma 3.5.3 permit us to construct a subsequence along which both 𝐵_𝑠 and 𝜑_𝑠 converge in the 𝐶^∞topology to harmonic maps 𝑓_∞ and𝜑_∞respectively.

Let ℎ denote the harmonic diffeomorphism of the disk whose Hopf differential is Φ. By [Wol91b, Lemma 3.6], the Jacobian 𝐽(ℎ) = 𝐻(ℎ) −𝐿(ℎ) tends to 0 as we approach the puncture. From Proposition 3.3.2, 𝐽(𝑓) → 0 as well. Therefore, 𝐽(𝑓_∞) =0 and necessarily rank𝑑 𝑓_∞ ≤ 1 at each point. By equivariance this is rank 1 in an open set, and by [Sam78, Theorem 3] the image is contained in a geodesic arc. Again by equivariance, the image must then be a closed geodesic arc. There is only one such arc in the quotient, and hence 𝑓_∞maps onto the core geodesic. Lifting 𝑓_∞ and 𝜑_∞ to maps from R² to the axis of 𝜌(𝛾), 𝑓_∞ and 𝜑 differ by a translation along 𝛽. One can justify that last claim by observing that their distance function is

a bounded subharmonic function onR²—hence a constant—and then following the proof of Lemma 3.2.12. Lifting back to ˜Σthis means there is a translation ˜𝑅of the geodesic axis such that for any𝑟 >0,

𝑑(𝑓^𝜃(𝑥 , 𝑠_𝑚 +2𝑡),𝑅˜◦𝜑^𝜃(𝑥 , 𝑠_𝑚 +2𝑡))=𝑑(𝑏_𝑠

𝑚(𝑥 , 𝑡), 𝑅◦𝜑_𝑠

𝑚(𝑥 , 𝑡)) →0 as𝑚 → ∞for−𝑟 ≤ 𝑡 ≤ 𝑟. In particular, the quantities𝑑(𝑓_𝛾(𝑥 , 𝑠_𝑚), 𝑅◦𝜑_𝛾(𝑥 , 𝑠_𝑚)) and𝑑(𝑓_𝛾(𝑥 , 𝑠_𝑚+

1), 𝑅◦𝜑_𝛾(𝑥 , 𝑠_𝑚+

1))are very close to 0. Since the relevant distance function is subharmonic, its maximum on

{(𝑥 , 𝑡) ∈ C_∞ : 𝑠_𝑚 ≤𝑡 ≤ 𝑠_𝑚+

1} is achieved on the boundary. It follows that

𝑑(𝑓_𝛾(𝑥 , 𝑡), 𝑅◦𝜑_𝛾(𝑥 , 𝑡)) →0 as𝑡 → ∞. Returning to the universal cover, we conclude that

𝑑(𝑓^𝜃(𝑧),𝑅˜◦𝜑^𝜃(𝑧)) →0

as we move toward the puncture. □

Proposition 3.5.5. Φhas a pole of order2at the cusp with residue−Λ(𝜃)ℓ(𝜌(𝛾))²/16𝜋². Proof. The lemma above shows

𝑠lim→∞

𝐵_𝑠 = 𝑅◦𝜑_𝛾

in the𝐶⁰topology, and along a subsequence in the𝐶^∞ topology. We prove there is no need to pass to a subsequence. Indeed, if we don’t have 𝐶¹convergence we can pick a subsequence along which our maps are uniformly far from 𝑓_∞in the𝐶¹ norm. One can then use the argument above to pass to a subsequence that converges in the𝐶^∞sense to𝑆◦𝜑_𝛾 for some other rotation𝑆. 𝐶0convergence to𝑅◦𝜑forces 𝑆 = 𝑅, which is a contradiction. Continuing inductively gives𝐶^𝑘 convergence for any 𝑘. The Hopf differential of 𝑓 then converges to Hopf(𝜑^𝜃) as we move into the puncture. The result now follows from the computation in Lemma 3.5.3. □ Uniqueness

Let 𝑓

1 and 𝑓

2 be two harmonic maps whose Hopf differentials have second order poles and such that the residues have the same complex argument𝜈 ∈ (−𝜋, 𝜋).

Lemma 3.5.6. There exists an𝐴_𝑘 > 0such that as𝑦 → ∞, the image of 𝑓_𝑘 remains in an 𝐴_𝑘-neighbourhood of the geodesic axis of 𝜌(𝛾).

Proof. Let𝛽^𝑘

𝑦 be the curve 𝑓_𝑘( [0, 𝜏] × {𝑦}) in the usual coordinates. From Propo- sition 3.3.2 and [Wol91b, page 516], the energy density of 𝑓_𝑘 is uniformly bounded onΣin the flat-cylinder metric. This implies

ℓ(𝛽^𝑘

𝑦) ≤ 𝐴 for all 𝑦 >0. We argue each 𝛽^𝑘

𝑦 becomes trapped close to the geodesic as𝑦 → ∞. If not, there is a subsequence𝑠_𝑗 tending to∞and points 𝑓_𝑘(𝑧_𝑗) ∈ 𝛽^𝑘

𝑠𝑗 such that the closest-point projection onto the geodesic, say 𝑦_𝑗, satisfies

𝑑(𝑓_𝑘(𝑧_𝑗), 𝑦_𝑗) → ∞. Then

ℓ(𝛽^𝑘

𝑠𝑗) ≥ 𝑑(𝑓_𝑘(𝑧_𝑗), 𝑓_𝑘(𝛾·𝑧_𝑗)) =𝑑(𝑓_𝑘(𝑧_𝑗), 𝜌(𝛾)𝑓_𝑘(𝑧_𝑗)).

The right most term blows up as 𝑗 → ∞, and this is a clear contradiction. To verify that last statement, note 𝑓_𝑘(𝑧_𝑗) accumulates along a subsequence to a point 𝜉 ∈ 𝜕_∞𝑋, and since the distance from 𝛽^𝑘

𝑠𝑗 to the geodesic is uniformly bounded below, this is not an endpoint of the geodesic. In particular, the extension of 𝜌(𝛾) to𝜕_∞𝑋 does not fix𝜉, and hence if𝐵^𝑘

𝑠𝑗 is a neighbourhood of𝜉in 𝑋∪𝜕_∞𝑋, 𝑑(𝐵^𝑘

𝑠𝑗 ∩𝑋 , 𝜌(𝛾)𝐵^𝑘

𝑠𝑗 ∩𝑋) → ∞

as 𝑗 → ∞. □

Recall the cylinder C_∞. Let 𝑏^𝑘

𝑠 be the map 𝑏_𝑠 ◦ 𝑓^𝑘

𝛾 : C_∞ → 𝑋/⟨𝜌(𝛾)⟩. Since the energy is controlled and it stays close to the geodesic, 𝑏^𝑘

𝑠 converges along a subsequence to a harmonic map 𝑓_∞^𝑘. By the same argument as in the previous subsection 𝑓_∞^𝑘 has image in a geodesic and from equivariance this must be the core geodesic 𝛽. One can slightly modify an argument as in the previous subsection to check that𝑎^𝑘

𝑠 limits to 𝑓_∞^𝑘 along the whole sequence in the𝐶^∞topology. Moreover 𝑓_𝑘 limits onto the geodesic 𝛽as we go further into the cusp.

Lemma 3.5.7. The residue of 𝑓

1and 𝑓

2is the same.

Proof. LetΦ𝑘 :=Hopf(𝑓_𝑘). In the computations to follow, we use the flat-cylinder metric onΣ. Let 𝛾_𝑦(𝑥) be the curve𝑥 ↦→ 𝑥 +𝑖 𝑦. From the discussion above, the

length of the core geodesic in 𝑋/⟨𝜌(𝛾)⟩ is

𝑦lim→∞

ℓ_𝑔(𝑓_𝑘(𝛾_𝑦)). There are differentialsΦ^′

𝑘 such that

Φ𝑘 =𝑒^{𝑖 𝜈}Φ^′_𝑘.

That is, a differential that differs from Φ𝑘 by a rotation and whose residue at the cusp is real. The pullback metrics can thus be written

𝑓^∗

𝑘𝑔=𝑒(𝑓_𝑘)𝜎^′𝑑 𝑧 𝑑 𝑧+𝑒^{𝑖 𝜈}Φ^′_𝑘+𝑒^{−𝑖 𝜈}Φ^′

𝑘 =𝑒(𝑓_𝑘)𝜎^′𝑑 𝑧 𝑑 𝑧+2ℜ𝑒^{𝑖 𝜈}Φ^′_𝑘. WritingΦ^′

𝑘 =𝜙^′

𝑘(𝑧)𝑑 𝑧2in a local coordinate we know that in the cylinder

|𝜙^′

𝑘|=𝐻(𝑓_𝑘)^1/2𝐿(𝑓_𝑘)^1/2=𝐻(𝑓_𝑘) · 𝐿(𝑓_𝑘)^1/2 𝐻(𝑓_𝑘)^1/2. From [Wol91b, Proposition 3.8], in the strip we can write

Φ𝑘 =

𝑒^{𝑖 𝜈}𝑎^𝑘

−2+𝑒^{𝑖 𝜈}𝑂(𝑒^{−𝐴 𝑦})

𝑑 𝑧², where𝑎^𝑘

−2> 0. From Proposition 3.3.2 and [Wol91b, Lemma 3.6], we also know 𝐿(𝑓_𝑘)

𝐻(𝑓_𝑘) →1

as we move into the puncture. The length of the core geodesic is therefore

𝑦→∞lim

ℓ_𝑔(𝑓_𝑘(𝛾_𝑦)) =_𝑦→∞lim

∫ ^𝜏

0

|| ¤𝛾_𝑦(𝑥) ||𝑓^∗

𝑘𝑔𝑑𝑥

=_𝑦→∞lim

∫ ^𝜏

0

√︁

𝑒(𝑓_𝑘)𝜎^′+2ℜ𝑒^{𝑖 𝜈}𝜙^′𝑑𝑥

=_𝑦→∞lim

∫ ^𝜏

0

√︁

𝐻(𝑓_𝑘) (1+𝐿(𝑓_𝑘)/𝐻(𝑓_𝑘)) +2ℜ𝑒^{𝑖 𝜈}𝜙^′𝑑𝑥

=𝜏

√︃

2|𝑎^𝑘

−2| (1+cos𝜈)

by the dominated convergence theorem. Meanwhile, passing to the quotientH/⟨𝛾⟩ we know the core geodesic has lengthℓ(𝜌(𝛾)). We deduce

ℓ(𝜌(𝛾) =𝜏

√︃

2|𝑎^𝑘

−2| (1+cos𝜈). Since𝜈is fixed, |𝑎^𝑘

−2|does not depend on𝑘. □

Henceforth put𝑎₋

2=𝑎^𝑘

−2(𝑘 =1,2).

Remark 3.5.8. From above we see that the complex argument 𝜈 is related to the twist angle𝜃from the previous subsection by

𝜃= −sin𝑣 1+cos𝑣

. Lemma 3.5.9. The distance function𝑧 ↦→ 𝑑(𝑓

1, 𝑓

2) (𝑧) is bounded.

Proof. It suffices to bound 𝑑(𝑓_∞¹, 𝑓_∞²) as then it is constant and we can lift to the universal cover. By [Wol91b, Proposition 3.8] we can express

Φ𝑘 =

𝑎₋

2𝑒^{𝑖 𝜈}+𝑒^{𝑖 𝜈}𝑂(𝑒^{−𝐴 𝑦}) 𝑑 𝑧² in the cylinder coordinates, where𝑎₋

2is real. Thus, upon taking𝑠→ ∞, the Hopf differential of 𝑓^∞

𝑘 is𝑎₋

2𝑒^{𝑖 𝜈}𝑑 𝑧². That is, the Hopf differentials of 𝑓^∞

1 and 𝑓^∞

2 agree.

We denote this differential byΦ₀, and highlight that theΦ₀-metric is nonsingular.

Set

𝑤0(𝑓_𝑘) = 1 2log𝐻

0(𝑓_𝑘) (𝑧) − 1

2log|Φ₀(𝑧) |. Here 𝐻

0 denotes the holomorphic energy in the Φ₀-metric, and analogously for the other quantities. From above it is clear that 𝐽

0(𝑓_𝑘) = 0 so 𝐻

0(𝑓_𝑘) = 𝐿

0(𝑓_𝑘). From |Φ₀| = 𝐻

0(𝑓_𝑘)^1/2𝐿

0(𝑓_𝑘)^1/2 we see 𝑤

0(𝑓_𝑘) = 0. One can compute 𝑒

0 = 2 cosh 2(𝑤

0(𝑓_𝑘)). In a coordinate𝑧=𝑥+𝑖 𝑦such thatΦ₀ =𝑑 𝑧², 𝑓^∗

𝑘𝑔= (𝑒

0+2)𝑑𝑥²+ (𝑒

0−2)𝑑𝑦²=2𝑑𝑥².

Let 𝛾_ℎ and 𝛾_𝑣 be horizontal and vertical curves for theΦ₀-metric. Explicitly, we mean the tangent vectors for𝛾_ℎ,𝛾_𝑣always evaluate underΦ₀to positive and negative numbers respectively. Then,

ℓ(𝑓_𝑘(𝛾_ℎ))=

∫

𝛾_ℎ

√︁

𝑒0+2𝑑𝑥 , ℓ(𝑓_𝑘(𝛾_𝑣)) =

∫

𝛾_ℎ

√︁

𝑒0−2𝑑𝑦 and we see

ℓ(𝑓_𝑘(𝛾_ℎ)) =2ℓ(𝛾_ℎ), ℓ(𝑓_𝑘(𝛾_𝑣))=0.

Therefore, if𝑣_𝑎 is the tangent vector to the geodesic at a point𝑎 then for all points 𝑧,(𝑑 𝑓_𝑘)𝑧(𝜕_𝑥) =2𝑣_𝑓

𝑘(𝑧) and(𝑑 𝑓_𝑘)𝑧(𝜕_𝑦) =0. In particular, 𝑓_𝑘 is a constant speed map onto the geodesic in the horizontal direction and constant in the vertical direction.

Any two such maps differ by a translation. This establishes the result. □ We apply Lemma 3.2.12 to obtain the uniqueness portion of Theorem 3A. If 𝑓

1 ≠ 𝑓

2, which is only possible if 𝜌 stabilizes a geodesic, then 𝑓

2may be obtained from 𝑓

1

by precomposing with a lift of the translation found in Lemma 3.5.9. The results in this section constitute the proof of Theorem 3A.

3.6 Domination and AdS3-manifolds