Introduction
The Weil–Petersson (WP) metric is a negatively curved Kähler metric on the Teichmüller space Teich g , n \operatorname{Teich}_{g,n} Teich g , n g g g n n n [4 , 25 , 27 ] . It is not complete [26 , 11 ] , and its completion is the augmented Teichmüller space Teich ‾ g , n \overline{\operatorname{Teich}}_{g,n} Teich g , n [7 , 2 Mas76] . The Weil–Petersson distance between two points of Teichmüller space is difficult to compute. Nevertheless, Wolpert [29 , 30 ] gave a simple upper bound for the distance from a point X X X S ( Γ ) S(\Gamma) S ( Γ ) Γ \Gamma Γ
d W P ( X , S ( Γ ) ) ≤ 2 π ∑ γ ∈ Γ ℓ X ( γ ) . d_{\mathrm{WP}}(X,S(\Gamma)) \le\sqrt{2\pi\sum_{\gamma\in\Gamma}\ell_X(\gamma)}. d WP ( X , S ( Γ )) ≤ 2 π γ ∈ Γ ∑ ℓ X ( γ ) . When Γ \Gamma Γ S ( Γ ) S(\Gamma) S ( Γ ) Γ \Gamma Γ [7 , 22 ] , it shows that every point of Teich g , n \operatorname{Teich}_{g,n} Teich g , n g g g n n n [9 , 12 ] . In particular Cavendish and Parlier obtained an upper bound of order g log g \sqrt{g}\log g g log g [12 ] , which was recently shown to be sharp by Wu [34 ] .
Punctured surfaces admit another natural decomposition, into ideal triangles. Whereas a pair of pants is determined up to isometry by its three boundary lengths, all ideal triangles are isometric, so the geometry of the surface is entirely captured by the way the triangles are glued together. Their gluing is determined by the shearing coordinates s e ( X ) s_e(X) s e ( X ) e e e [8 , 14 , 24 ] . Bestvina, Bromberg, Fujiwara and Souto showed that the length of every essential closed curve is a convex function of the shearing coordinates [6 Theorem 1.3] . We want an explicit bound for the WP distance between two surfaces in terms of their shearing coordinates.
We first establish a distance estimate in terms of truncated arc-lengths. After truncating the cusps along horocycles of length 1 / 4 1/4 1/4 e e e T T T X X X δ e ( X ) \delta_e(X) δ e ( X ) § 2.5 . We write
L T ( X ) : = ∑ e ∈ T δ e ( X ) \mathcal{L}_T(X) := \sum_{e\in T} \delta_e(X) L T ( X ) := e ∈ T ∑ δ e ( X ) for the sum of the truncated lengths of the arcs of T T T X X X
Theorem 3.10. Let g , n g,n g , n n ≥ 1 n \ge1 n ≥ 1 2 g − 2 + n > 0 2g-2+n>0 2 g − 2 + n > 0 C < 6 C<6 C < 6 Y , Z ∈ Teich g , n Y,Z\in\operatorname{Teich}_{g,n} Y , Z ∈ Teich g , n T T T
d W P ( Y , Z ) ≤ C n + L T ( Y ) + L T ( Z ) . d_{\mathrm{WP}}(Y,Z) \le C\sqrt{n+\mathcal{L}_T(Y)+\mathcal{L}_T(Z)}. d WP ( Y , Z ) ≤ C n + L T ( Y ) + L T ( Z ) . More precisely, C = 8 + 2 π B < 5.82 C=\sqrt{8+2\pi B}<5.82 C = 8 + 2 π B < 5.82 B = 2 2 + log ( 1 + 2 ) / log 2 B=2\sqrt{2}+\log(1+\sqrt{2})/\log2 B = 2 2 + log ( 1 + 2 ) / log 2 Y Y Y Z Z Z [32 Section 6.1 and Theorem 6.4] .
Shears and truncated lengths are related through Penner’s h h h [23 ] : the shear points of an arc lie on its truncated part, so that ∣ s e ( X ) ∣ ≤ δ e ( X ) − 2 log 4 |s_e(X)|\le\delta_e(X)-2\log4 ∣ s e ( X ) ∣ ≤ δ e ( X ) − 2 log 4 Lemma 2.2 ). Let O T ( X ) \mathcal{O}_T(X) O T ( X ) [16 ] , see § 2.4 . As a consequence of the main theorem we obtain the following upper bound
Corollary 3.11. Let g , n g,n g , n n ≥ 1 n \ge1 n ≥ 1 2 g − 2 + n > 0 2g-2+n>0 2 g − 2 + n > 0 Y , Z ∈ Teich g , n Y,Z\in\operatorname{Teich}_{g,n} Y , Z ∈ Teich g , n T T T
d W P ( Y , Z ) ≤ 21 ( 2 g − 2 + n ) ( log ( 24 ( 2 g − 2 + n ) ) + O T ( Y ) + O T ( Z ) ) . d_{\mathrm{WP}}(Y,Z)\le21\sqrt{(2g-2+n)\left(\log(24(2g-2+n))+\mathcal{O}_T(Y)+\mathcal{O}_T(Z)\right)}. d WP ( Y , Z ) ≤ 21 ( 2 g − 2 + n ) ( log ( 24 ( 2 g − 2 + n )) + O T ( Y ) + O T ( Z ) ) . Kahn–Marković [16 Theorem 2.1] obtained WP distance estimates in terms of oscillation norm and the proportion of edges whose shear exceeds a prescribed threshold, with a constant depending on an upper bound for the individual shears.
Since the sum of all shearing coordinates along arcs ending at the same cusp is zero, we have O T ( X ) ≤ ∑ e ∈ T ∣ s e ( X ) ∣ \mathcal{O}_T(X)\le\sum_{e\in T}|s_e(X)| O T ( X ) ≤ ∑ e ∈ T ∣ s e ( X ) ∣
Corollary 3.12. Let g , n g,n g , n n ≥ 1 n \ge1 n ≥ 1 2 g − 2 + n > 0 2g-2+n>0 2 g − 2 + n > 0 Y , Z ∈ Teich g , n Y,Z\in\operatorname{Teich}_{g,n} Y , Z ∈ Teich g , n T T T
d W P ( Y , Z ) ≤ 21 ( 2 g − 2 + n ) ( log ( 24 ( 2 g − 2 + n ) ) + ∑ e ∈ T ( ∣ s e ( Y ) ∣ + ∣ s e ( Z ) ∣ ) ) . d_{\mathrm{WP}}(Y,Z)\le21\sqrt{(2g-2+n)\left(\log(24(2g-2+n))+\sum_{e\in T}\left(|s_e(Y)|+|s_e(Z)|\right)\right)}. d WP ( Y , Z ) ≤ 21 ( 2 g − 2 + n ) ( log ( 24 ( 2 g − 2 + n )) + e ∈ T ∑ ( ∣ s e ( Y ) ∣ + ∣ s e ( Z ) ∣ ) ) . Outline of the proof
Let Y Y Y Z Z Z T \mathcal{T} T Y Y Y Z Z Z 1 / 4 1/4 1/4 T T T W T ( Y ) W_T(Y) W T ( Y ) W T ( Z ) W_T(Z) W T ( Z ) 2 g + n − 1 2g+n-1 2 g + n − 1 e e e T \mathcal{T} T d ( e ) d(e) d ( e ) d ( e ) d(e) d ( e ) e ∈ T e\in\mathcal{T} e ∈ T P T \mathcal{P}_T P T W T ( Y ) W_T(Y) W T ( Y ) W T ( Z ) W_T(Z) W T ( Z )
As T → ∞ T\to\infty T → ∞ [28 ] , we show that W T ( Y ) W_T(Y) W T ( Y ) W T ( Z ) W_T(Z) W T ( Z ) D ( Y ) D(Y) D ( Y ) D ( Z ) D(Z) D ( Z ) d W P a u x d_{\mathrm{WP}}^{\mathrm{aux}} d WP aux 2 g + n − 1 2g+n-1 2 g + n − 1 D ( Y ) D(Y) D ( Y ) D ( Z ) D(Z) D ( Z )
2 d W P ( Y , Z ) = d W P a u x ( D ( Y ) , D ( Z ) ) ≤ d W P a u x ( D ( Y ) , P T ) + d W P a u x ( P T , D ( Z ) ) . \sqrt{2}\,d_{\mathrm{WP}}(Y,Z)=d_{\mathrm{WP}}^{\mathrm{aux}}(D(Y),D(Z))\leq d_{\mathrm{WP}}^{\mathrm{aux}}(D(Y),\mathcal{P}_T)+d_{\mathrm{WP}}^{\mathrm{aux}}(\mathcal{P}_T,D(Z)). 2 d WP ( Y , Z ) = d WP aux ( D ( Y ) , D ( Z )) ≤ d WP aux ( D ( Y ) , P T ) + d WP aux ( P T , D ( Z )) . We bound each term by passing through the surface W T 0 ( V ) W_{T_0}(V) W T 0 ( V ) T 0 = 2 π T_0=2\pi T 0 = 2 π
2 d W P ( Y , Z ) ≤ d W P a u x ( D ( Y ) , W T 0 ( Y ) ) + d W P a u x ( W T 0 ( Y ) , P T ) + d W P a u x ( P T , W T 0 ( Z ) ) + d W P a u x ( W T 0 ( Z ) , D ( Z ) ) . \begin{aligned}
\sqrt{2}\,d_{\mathrm{WP}}(Y,Z)&\leq d_{\mathrm{WP}}^{\mathrm{aux}}(D(Y),W_{T_0}(Y))+d_{\mathrm{WP}}^{\mathrm{aux}}(W_{T_0}(Y),\mathcal{P}_T)\\
&\quad+d_{\mathrm{WP}}^{\mathrm{aux}}(\mathcal{P}_T,W_{T_0}(Z))+d_{\mathrm{WP}}^{\mathrm{aux}}(W_{T_0}(Z),D(Z)).
\end{aligned} 2 d WP ( Y , Z ) ≤ d WP aux ( D ( Y ) , W T 0 ( Y )) + d WP aux ( W T 0 ( Y ) , P T ) + d WP aux ( P T , W T 0 ( Z )) + d WP aux ( W T 0 ( Z ) , D ( Z )) . The first term is the cost of opening the cylinders along a plumbing deformation (Lemma 3.2) while the second term is bounded by Wolpert’s estimate applied to the pants decomposition P T \mathcal{P}_T P T W T 0 ( Y ) W_{T_0}(Y) W T 0 ( Y ) W T 0 ( Z ) W_{T_0}(Z) W T 0 ( Z ) 2 π ∑ e ∈ T ℓ W T 0 ( V ) ( d ( e ) ) \sqrt{2\pi\sum_{e\in\mathcal{T}}\ell_{W_{T_0}(V)}(d(e))} 2 π ∑ e ∈ T ℓ W T 0 ( V ) ( d ( e )) d ( e ) d(e) d ( e ) e e e ℓ W T 0 ( V ) ( d ( e ) ) ≤ 2 2 δ e ( V ) + 4 log ( 1 + 2 ) \ell_{W_{T_0}(V)}(d(e))\leq2\sqrt{2}\,\delta_e(V)+4\log(1+\sqrt{2}) ℓ W T 0 ( V ) ( d ( e )) ≤ 2 2 δ e ( V ) + 4 log ( 1 + 2 )
The proof combines classical tools, for instance Wolpert used doubles and opened nodes to prove the strict WP convexity of the distance between horocycles of fixed lengths [31 Theorem 6.1] . He also studied shear deformations as limits of opposing twists on opened doubles [33 Sections 4–5] . Mondello gave a formula for the WP Poisson structure in arc-length coordinates on the Teichmüller space of compact hyperbolic surfaces with geodesic boundary by doubling the arcs of a triangulation to obtain a pants decomposition [19 Theorem 4.1] .
Acknowledgments
I am grateful to my supervisor Hugo Parlier for helpful discussions and for his encouragement.
The idea behind this work is to improve the bound on the diameter of the hexagon graph obtained in the first version of [1 ] . My role was that of an archaeologist who knows where to dig: from [1 ] , I identified the precise step needed for this improvement and the part of the literature where it should be found, and I used AI tools as assistants to unearth the argument and to clean and polish it. More precisely, I asked GPT-6-Astra and GPT-6.1-Sol for possible approaches to this step. They suggested a nice strategy, relying on results of Wolpert [28 , 29 ] . I then polished this strategy into the proofs presented here. During this process, I used Claude Opus 5.5 as an assistant to improve the writing, and to help check the clarity and consistency of notation, statements and proofs.
Set up
This section fixes notation and recalls the background material we use. Throughout, Σ \Sigma Σ g g g n n n χ ( Σ ) = 2 − 2 g − n < 0 \chi(\Sigma)=2-2g-n<0 χ ( Σ ) = 2 − 2 g − n < 0
Weil–Petersson distance
The Teichmüller space T e i c h g , n = T e i c h ( Σ ) \mathrm{Teich}_{g,n}=\mathrm{Teich}(\Sigma) Teich g , n = Teich ( Σ ) Σ \Sigma Σ ( X , f ) (X,f) ( X , f ) f : Σ → X f:\Sigma\to X f : Σ → X ( X , f ) ∼ ( Y , g ) (X,f)\sim(Y,g) ( X , f ) ∼ ( Y , g ) g ∘ f − 1 : X → Y g\circ f^{-1}:X\to Y g ∘ f − 1 : X → Y
Let X ∈ T e i c h g , n X\in\mathrm{Teich}_{g,n} X ∈ Teich g , n ρ 2 ∣ d z ∣ 2 \rho^2|dz|^2 ρ 2 ∣ d z ∣ 2 d A X = ρ 2 d x d y dA_X=\rho^2\,dx\,dy d A X = ρ 2 d x d y T e i c h g , n \mathrm{Teich}_{g,n} Teich g , n X X X Q ( X ) Q(X) Q ( X ) X X X B ( X ) B(X) B ( X ) X X X μ ∈ B ( X ) \mu\in B(X) μ ∈ B ( X ) ϕ ∈ Q ( X ) \phi\in Q(X) ϕ ∈ Q ( X ) B ( X ) B(X) B ( X ) Q ( X ) Q(X) Q ( X )
⟨ μ , ϕ ⟩ = ∫ X μ ϕ . \langle\mu,\phi\rangle=\int_X\mu\phi. ⟨ μ , ϕ ⟩ = ∫ X μ ϕ . Denote by N ( X ) ⊂ B ( X ) N(X)\subset B(X) N ( X ) ⊂ B ( X )
N ( X ) = { μ ∈ B ( X ) ∣ ⟨ μ , ϕ ⟩ = 0 for all ϕ ∈ Q ( X ) } . N(X)=\{\mu\in B(X)\mid\langle\mu,\phi\rangle=0\text{ for all }\phi\in Q(X)\}. N ( X ) = { μ ∈ B ( X ) ∣ ⟨ μ , ϕ ⟩ = 0 for all ϕ ∈ Q ( X )} . The tangent space of T e i c h g , n \mathrm{Teich}_{g,n} Teich g , n X X X B ( X ) / N ( X ) B(X)/N(X) B ( X ) / N ( X ) B ( X ) / N ( X ) B(X)/N(X) B ( X ) / N ( X ) μ = ρ − 2 ψ ‾ \mu=\rho^{-2}\overline{\psi} μ = ρ − 2 ψ ψ ∈ Q ( X ) \psi\in Q(X) ψ ∈ Q ( X )
∥ [ μ ] ∥ W P = sup ϕ ∈ Q ( X ) ∖ { 0 } ∣ ⟨ μ , ϕ ⟩ ∣ ∥ ϕ ∥ 2 , ∥ ϕ ∥ 2 2 = ∫ X ∣ ϕ ∣ 2 ρ 2 . \|[\mu]\|_{\mathrm{WP}}=\sup_{\phi\in Q(X)\setminus\{0\}}\frac{|\langle\mu,\phi\rangle|}{\|\phi\|_2},\qquad\|\phi\|_2^2=\int_X\frac{|\phi|^2}{\rho^2}. ∥ [ μ ] ∥ WP = ϕ ∈ Q ( X ) ∖ { 0 } sup ∥ ϕ ∥ 2 ∣ ⟨ μ , ϕ ⟩ ∣ , ∥ ϕ ∥ 2 2 = ∫ X ρ 2 ∣ ϕ ∣ 2 . Writing μ ϕ = ( μ ρ ) ( ϕ / ρ ) \mu\phi=(\mu\rho)(\phi/\rho) μ ϕ = ( μ ρ ) ( ϕ / ρ ) μ ∈ B ( X ) \mu\in B(X) μ ∈ B ( X )
∥ [ μ ] ∥ W P 2 ≤ ∫ X ∣ μ ∣ 2 d A X , (1) \|[\mu]\|_{\mathrm{WP}}^2\leq\int_X|\mu|^2\,dA_X, \tag*{(1)} ∥ [ μ ] ∥ WP 2 ≤ ∫ X ∣ μ ∣ 2 d A X , (1) with equality when μ \mu μ [5 Section 2] .
The augmented Teichmüller space
The Weil–Petersson metric is not complete [26 , 11 ] , its completion, the augmented Teichmüller space T e i c h ‾ g , n \overline{\mathrm{Teich}}_{g,n} Teich g , n Γ \Gamma Γ [7 Mas76, 2 , 29 ] :
T e i c h ‾ g , n = T e i c h g , n ∪ ⋃ Γ ≠ ∅ S ( Γ ) . \overline{\mathrm{Teich}}_{g,n}=\mathrm{Teich}_{g,n}\cup\bigcup_{\Gamma\ne\varnothing}S(\Gamma). Teich g , n = Teich g , n ∪ Γ = ∅ ⋃ S ( Γ ) . The stratum S ( Γ ) S(\Gamma) S ( Γ ) Γ \Gamma Γ Σ ∖ Γ \Sigma\setminus\Gamma Σ ∖ Γ ∏ i T e i c h ( Σ i ) \prod_i\mathrm{Teich}(\Sigma_i) ∏ i Teich ( Σ i ) Σ i \Sigma_i Σ i Σ ∖ Γ \Sigma\setminus\Gamma Σ ∖ Γ Γ = P \Gamma=\mathcal{P} Γ = P S ( P ) S(\mathcal{P}) S ( P ) maximally noded surface . The WP metric extends to T e i c h ‾ g , n \overline{\mathrm{Teich}}_{g,n} Teich g , n S ( Γ ) S(\Gamma) S ( Γ ) Σ ∖ Γ \Sigma\setminus\Gamma Σ ∖ Γ [29 Theorem 2(1), p. 17] . The WP metric near a stratum S ( Γ ) S(\Gamma) S ( Γ )
A point p ∈ T e i c h ‾ ( Σ ) p\in\overline{\mathrm{Teich}}(\Sigma) p ∈ Teich ( Σ ) ( X , f ) (X,f) ( X , f ) f f f α 1 , … , α k \alpha_1,\ldots,\alpha_k α 1 , … , α k X X X
Γ ( p ) = { [ α 1 ] , … , [ α k ] } , \Gamma(p)=\{[\alpha_1],\ldots,[\alpha_k]\}, Γ ( p ) = {[ α 1 ] , … , [ α k ]} , where [ α i ] [\alpha_i] [ α i ] α i \alpha_i α i Σ \Sigma Σ Γ ( p ) \Gamma(p) Γ ( p ) p p p [29 Theorem 13, p. 31] any two points p , q ∈ Teich g , n p,q\in\operatorname{Teich}_{g,n} p , q ∈ Teich g , n S ( Γ ( p ) ∩ Γ ( q ) ) S(\Gamma(p)\cap\Gamma(q)) S ( Γ ( p ) ∩ Γ ( q )) S ( Γ ) S(\Gamma) S ( Γ )
The WP distance to a stratum is controlled by the lengths of the pinched curves [30 ] , see also [29 Corollary 21] . Hence, for every X ∈ Teich g , n X\in\operatorname{Teich}_{g,n} X ∈ Teich g , n Γ \Gamma Γ
d W P ( X , S ( Γ ) ) ≤ 2 π ∑ γ ∈ Γ ℓ X ( γ ) . (2) d_{\mathrm{WP}}(X,S(\Gamma))\leq\sqrt{2\pi\sum_{\gamma\in\Gamma}\ell_X(\gamma)}. \tag*{(2)} d WP ( X , S ( Γ )) ≤ 2 π γ ∈ Γ ∑ ℓ X ( γ ) . (2) When Γ = P \Gamma=\mathcal{P} Γ = P X X X S ( P ) S(\mathcal{P}) S ( P )
The mapping class group Mod ( Σ ) \operatorname{Mod}(\Sigma) Mod ( Σ ) Teich g , n \operatorname{Teich}_{g,n} Teich g , n moduli space M g , n = Teich g , n / Mod ( Σ ) \mathcal{M}_{g,n}=\operatorname{Teich}_{g,n}/\operatorname{Mod}(\Sigma) M g , n = Teich g , n / Mod ( Σ ) Mod ( Σ ) \operatorname{Mod}(\Sigma) Mod ( Σ ) M g , n \mathcal{M}_{g,n} M g , n Teich g , n / Mod ( Σ ) \operatorname{Teich}_{g,n}/\operatorname{Mod}(\Sigma) Teich g , n / Mod ( Σ )
Pants decomposition and Ideal triangulation
A pants decomposition P \mathcal{P} P Σ \Sigma Σ 3 g − 3 + n 3g-3+n 3 g − 3 + n Σ ∖ P \Sigma\setminus\mathcal{P} Σ ∖ P 2 g − 2 + n 2g-2+n 2 g − 2 + n
Assume that Σ \Sigma Σ n ≥ 1 n\geq1 n ≥ 1 ideal triangulation T \mathcal{T} T Σ \Sigma Σ
E : = 6 g − 6 + 3 n E:=6g-6+3n E := 6 g − 6 + 3 n arcs and Σ ∖ T \Sigma\setminus\mathcal{T} Σ ∖ T 4 g − 4 + 2 n 4g-4+2n 4 g − 4 + 2 n c c c v ( c ) v(c) v ( c ) c c c c c c
∑ c v ( c ) = 2 E . \sum_c v(c)=2E. c ∑ v ( c ) = 2 E . Truncating the cusps along disjoint horocycles turns each ideal triangle into a truncated hexagon : a right-angled hexagon whose sides alternate between three geodesic segments, contained in the arcs, and three horocyclic segments, contained in the corners.
Fenchel–Nielsen coordinates, Shearing coordinates
Given a pants decomposition P \mathcal{P} P Fenchel–Nielsen coordinates associate to X ∈ Teich g , n X\in\operatorname{Teich}_{g,n} X ∈ Teich g , n P \mathcal{P} P
F N P : Teich g , n ⟶ ( R > 0 × R ) ∣ P ∣ , X ⟼ ( ℓ X ( γ ) , τ X ( γ ) ) γ ∈ P , \mathrm{FN}_{\mathcal{P}}:\operatorname{Teich}_{g,n}\longrightarrow(\mathbb{R}_{>0}\times\mathbb{R})^{|\mathcal{P}|},\qquad X\longmapsto\bigl(\ell_X(\gamma),\tau_X(\gamma)\bigr)_{\gamma\in\mathcal{P}}, FN P : Teich g , n ⟶ ( R > 0 × R ) ∣ P ∣ , X ⟼ ( ℓ X ( γ ) , τ X ( γ ) ) γ ∈ P , and F N P \mathrm{FN}_{\mathcal{P}} FN P S ( Γ ) S(\Gamma) S ( Γ ) Γ ⊆ P \Gamma\subseteq\mathcal{P} Γ ⊆ P Γ \Gamma Γ [3 ] . When a curve has length zero, there is no twist around it, hence the extended length-twist parameters take value in the space
R ≥ 0 × R { 0 } × R , \frac{\mathbb{R}_{\geq0}\times\mathbb{R}}{\{0\}\times\mathbb{R}}, { 0 } × R R ≥ 0 × R , where the topology around 0 : = [ { 0 } × R ] \mathbf{0} := [\{0\} \times\mathbb{R}] 0 := [{ 0 } × R ] ( ℓ n , τ n ) → 0 (\ell_n,\tau_n) \to\mathbf{0} ( ℓ n , τ n ) → 0 ℓ n → 0 \ell_n \to0 ℓ n → 0 1 S ( Γ ) \mathcal{S}(\Gamma) S ( Γ ) Fenchel–Nielsen coordinates along the curves of P ∖ Γ \mathcal{P} \setminus\Gamma P ∖ Γ P \mathcal{P} P Σ ∖ Γ \Sigma\setminus\Gamma Σ ∖ Γ
Similarly, given an ideal triangulation we can define coordinates on the Teichmüller space. All ideal triangles are isometric to the ideal triangle ( 0 , 1 , ∞ ) (0,1,\infty) ( 0 , 1 , ∞ ) H 2 \mathbb{H}^2 H 2 shear points . Let Δ , Δ ′ \Delta,\Delta' Δ , Δ ′ e e e shear parameter s e ( X ) s_e(X) s e ( X ) e e e Δ \Delta Δ Δ ′ \Delta' Δ ′ e e e
Let T \mathcal{T} T ( g , n ) (g,n) ( g , n ) E = 6 g − 6 + 3 n E=6g-6+3n E = 6 g − 6 + 3 n T \mathcal{T} T c c c c c c c c c shearing coordinates are the shear parameters along all the arcs of T \mathcal{T} T R E \mathbb{R}^{E} R E n n n 6 g − 6 + 2 n 6g-6+2n 6 g − 6 + 2 n [8 , 14 , 24 , 23 ] ,
Teich g , n ⟶ R E , X ⟼ ( s e ( X ) ) e ∈ T . \operatorname{Teich}_{g,n} \longrightarrow\mathbb{R}^{E}, \qquad X \longmapsto(s_e(X))_{e\in\mathcal{T}}. Teich g , n ⟶ R E , X ⟼ ( s e ( X ) ) e ∈ T . Following Kahn and Marković [16 ] , for every cusp c c c c c c e c , 1 , … , e c , v ( c ) e_{c,1},\ldots,e_{c,v(c)} e c , 1 , … , e c , v ( c ) s e c , r ( X ) s_{e_{c,r}}(X) s e c , r ( X ) e c , r e_{c,r} e c , r oscillation norm of X X X T \mathcal{T} T
O T ( X ) : = max c max 1 ≤ j ≤ k ≤ v ( c ) ∣ ∑ r = j k s e c , r ( X ) ∣ . \mathcal{O}_{\mathcal{T}}(X) := \max_{c}\ \max_{1\leq j\leq k\leq v(c)} \left|\sum_{r=j}^{k}s_{e_{c,r}}(X)\right|. O T ( X ) := c max 1 ≤ j ≤ k ≤ v ( c ) max r = j ∑ k s e c , r ( X ) . In particular, the sums of the shears over cyclically consecutive endpoints at c c c O T ( X ) \mathcal{O}_{\mathcal{T}}(X) O T ( X )
Cusp truncations
Each cusp c c c X ∈ Teich g , n X\in\operatorname{Teich}_{g,n} X ∈ Teich g , n
N 2 ( c ) = { z = x + i y ∈ H ∣ y > 1 / 2 } / ( z ↦ z + 1 ) , d s = ∣ d z ∣ y , N_2(c)=\{z=x+iy\in\mathbb{H}\mid y>1/2\}/(z\mapsto z+1), \qquad ds=\frac{|dz|}{y}, N 2 ( c ) = { z = x + i y ∈ H ∣ y > 1/2 } / ( z ↦ z + 1 ) , d s = y ∣ d z ∣ , bounded by a horocycle of length 2 2 2 standard cusp neighborhood . For y 0 > 1 / 2 y_0>1/2 y 0 > 1/2 y 0 y_0 y 0 1 / y 0 1/y_0 1/ y 0
h = 1 4 , a = 1 h = 4 , h=\frac{1}{4}, \qquad a=\frac{1}{h}=4, h = 4 1 , a = h 1 = 4 , and we truncate each cusp along the horocycle { y = a } \{y=a\} { y = a } h h h N h ( c ) = { y > a } N_h(c)=\{y>a\} N h ( c ) = { y > a } [15 , 10 Theorem 4.4.6] . We set
X h = X ∖ ⋃ c N h ( c ) . X^h=X\setminus\bigcup_c N_h(c). X h = X ∖ c ⋃ N h ( c ) . Thus X h X^h X h ∂ X h \partial X^h ∂ X h n n n { y = a } \{y=a\} { y = a } h h h
Let e e e T \mathcal{T} T X X X e e e N h ( c ) N_h(c) N h ( c ) [1 proof of Lemma 8.8] , so e ∩ X h e\cap X^h e ∩ X h ∂ X h \partial X^h ∂ X h truncated length δ e ( X ) \delta_e(X) δ e ( X ) e e e
δ e ( X ) : = ℓ X ( e ∩ X h ) . \delta_e(X):=\ell_X(e\cap X^h). δ e ( X ) := ℓ X ( e ∩ X h ) . This is the truncated length of [1 Definition 2.4] for the empty multicurve and horocycles of length h h h δ e \delta_e δ e Teich g , n \operatorname{Teich}_{g,n} Teich g , n [1 Lemma 3.1] . We set
L T ( X ) : = ∑ e ∈ T δ e ( X ) . \mathcal{L}_{\mathcal{T}}(X) := \sum_{e\in\mathcal{T}} \delta_e(X). L T ( X ) := e ∈ T ∑ δ e ( X ) . Truncated lengths can be expressed in terms of λ \lambda λ h h h [23 Chapter 1, Section 4] . Let h 1 , h 2 h_1,h_2 h 1 , h 2 H \mathbb{H} H e e e δ \delta δ e e e h 1 ∩ e h_1\cap e h 1 ∩ e h 2 ∩ e h_2\cap e h 2 ∩ e h 1 h_1 h 1 h 2 h_2 h 2 λ \lambda λ h 1 , h 2 h_1,h_2 h 1 , h 2
λ ( h 1 , h 2 ) = e δ / 2 . \lambda(h_1,h_2)=e^{\delta/2}. λ ( h 1 , h 2 ) = e δ /2 . A decoration of an ideal triangle Δ ⊂ H \Delta\subset\mathbb{H} Δ ⊂ H v v v Δ \Delta Δ v v v Δ \Delta Δ corner of Δ \Delta Δ v v v h v ( Δ ) h_v(\Delta) h v ( Δ ) h h h h h h λ \lambda λ
Lemma 2.1 ([23 Chapter 1, Lemma 4.9] ). Let Δ \Delta Δ e e e Δ \Delta Δ u u u w w w λ e \lambda_e λ e λ \lambda λ u u u w w w h u ( Δ ) h w ( Δ ) = λ e − 2 h_u(\Delta)h_w(\Delta)=\lambda_e^{-2} h u ( Δ ) h w ( Δ ) = λ e − 2 δ e \delta_e δ e e e e u u u w w w
δ e = log 1 h u ( Δ ) + log 1 h w ( Δ ) . (3) \delta_e=\log\frac{1}{h_u(\Delta)}+\log\frac{1}{h_w(\Delta)}. \tag*{(3)} δ e = log h u ( Δ ) 1 + log h w ( Δ ) 1 . (3) When Δ \Delta Δ T \mathcal{T} T X X X h h h T \mathcal{T} T h h h h h h X ∈ Teich g , n X\in\operatorname{Teich}_{g,n} X ∈ Teich g , n e ∈ T e\in\mathcal{T} e ∈ T
δ e ( X ) ≥ 2 log 1 h = 2 log 4. (4) \delta_e(X)\geq2\log\frac{1}{h}=2\log4. \tag*{(4)} δ e ( X ) ≥ 2 log h 1 = 2 log 4. (4) The following lemma relates shear parameters and truncated lengths.
Lemma 2.2. Let X ∈ Teich g , n X\in\operatorname{Teich}_{g,n} X ∈ Teich g , n c c c v = v ( c ) v=v(c) v = v ( c ) k = 1 , … , v k=1,\ldots,v k = 1 , … , v Δ k \Delta_k Δ k c c c e c , k e_{c,k} e c , k e c , k + 1 e_{c,k+1} e c , k + 1 v v v σ k = s e c , 1 ( X ) + ⋯ + s e c , k ( X ) \sigma_k=s_{e_{c,1}}(X)+\cdots+s_{e_{c,k}}(X) σ k = s e c , 1 ( X ) + ⋯ + s e c , k ( X ) c c c
h c ( Δ k ) = h e σ k ∑ j = 1 v e σ j . (5) h_c(\Delta_k)=h\frac{e^{\sigma_k}}{\sum_{j=1}^{v}e^{\sigma_j}}. \tag*{(5)} h c ( Δ k ) = h ∑ j = 1 v e σ j e σ k . (5) Consequently, every arc e ∈ T e\in\mathcal{T} e ∈ T c c c c ′ c' c ′
∣ s e ( X ) ∣ ≤ δ e ( X ) − 2 log 4 and δ e ( X ) ≤ 2 log 4 + log ( v ( c ) v ( c ′ ) ) + 2 O T ( X ) , |s_e(X)|\leq\delta_e(X)-2\log4\qquad\text{and}\qquad\delta_e(X)\leq2\log4+\log\bigl(v(c)v(c')\bigr)+2\mathcal{O}_{\mathcal{T}}(X), ∣ s e ( X ) ∣ ≤ δ e ( X ) − 2 log 4 and δ e ( X ) ≤ 2 log 4 + log ( v ( c ) v ( c ′ ) ) + 2 O T ( X ) , and in particular L T ( X ) ≤ 2 E ( log ( 8 E ) + O T ( X ) ) \mathcal{L}_{\mathcal{T}}(X)\leq2E\bigl(\log(8E)+\mathcal{O}_{\mathcal{T}}(X)\bigr) L T ( X ) ≤ 2 E ( log ( 8 E ) + O T ( X ) )
Proof. Lift c c c ∞ \infty ∞ H 2 \mathbb{H}^2 H 2 ∞ \infty ∞ z ↦ z + 1 z\mapsto z+1 z ↦ z + 1 c c c { x = x k } \{x=x_k\} { x = x k } x 1 < ⋯ < x v < x 1 + 1 x_1<\cdots<x_v<x_1+1 x 1 < ⋯ < x v < x 1 + 1 e c , k e_{c,k} e c , k { x = x k } \{x=x_k\} { x = x k } { y = a } \{y=a\} { y = a } k = 1 , … , v k=1,\ldots,v k = 1 , … , v Δ k \Delta_k Δ k ( x k , x k + 1 , ∞ ) (x_k,x_{k+1},\infty) ( x k , x k + 1 , ∞ ) x v + 1 = x 1 + 1 x_{v+1}=x_1+1 x v + 1 = x 1 + 1 λ k = x k + 1 − x k \lambda_k=x_{k+1}-x_k λ k = x k + 1 − x k λ 1 + ⋯ + λ v = 1 \lambda_1+\cdots+\lambda_v=1 λ 1 + ⋯ + λ v = 1 λ 0 = λ v \lambda_0=\lambda_v λ 0 = λ v z ↦ λ k 2 z + x k z\mapsto\lambda_k^2z+x_k z ↦ λ k 2 z + x k ( 0 , 1 , ∞ ) (0,1,\infty) ( 0 , 1 , ∞ ) Δ k \Delta_k Δ k Figure 1 . Since the shear points of the former on its vertical sides are i i i 1 + i 1+i 1 + i Δ k \Delta_k Δ k x k + i λ k x_k+i\lambda_k x k + i λ k x k + 1 + i λ k x_{k+1}+i\lambda_k x k + 1 + i λ k { x = x k } \{x=x_k\} { x = x k } λ k − 1 \lambda_{k-1} λ k − 1 λ k \lambda_k λ k
Figure 1. A fan of edges ending in the cusp c c c ∞ \infty ∞ Lemma 2.2 for v ( c ) = 4 v(c)=4 v ( c ) = 4
s e c , k ( X ) = log ( λ k / λ k − 1 ) . s_{e_c,k}(X)=\log(\lambda_k/\lambda_{k-1}). s e c , k ( X ) = log ( λ k / λ k − 1 ) . Hence σ k = s e c , 1 ( X ) + ⋯ + s e c , k ( X ) = log ( λ 1 / λ 0 ) + … log ( λ k / λ k − 1 ) = log ( λ k / λ v ) \sigma_k=s_{e_c,1}(X)+\cdots+s_{e_c,k}(X)=\log(\lambda_1/\lambda_0)+\ldots\log(\lambda_k/\lambda_{k-1})=\log(\lambda_k/\lambda_v) σ k = s e c , 1 ( X ) + ⋯ + s e c , k ( X ) = log ( λ 1 / λ 0 ) + … log ( λ k / λ k − 1 ) = log ( λ k / λ v )
λ k = λ v e σ k . (6) \lambda_k=\lambda_v e^{\sigma_k}. \tag*{(6)} λ k = λ v e σ k . (6) Consequently 1 = ∑ k = 1 v λ k = ∑ k = 1 v λ v e σ k 1=\sum_{k=1}^{v}\lambda_k=\sum_{k=1}^{v}\lambda_v e^{\sigma_k} 1 = ∑ k = 1 v λ k = ∑ k = 1 v λ v e σ k
λ v = 1 ∑ j e σ j . (7) \lambda_v=\frac{1}{\sum_j e^{\sigma_j}}. \tag*{(7)} λ v = ∑ j e σ j 1 . (7) The corner of Δ k \Delta_k Δ k c c c { x + i a : x k ≤ x ≤ x k + 1 } \{x+ia:x_k\leq x\leq x_{k+1}\} { x + ia : x k ≤ x ≤ x k + 1 } λ k \lambda_k λ k a a a
h c ( Δ k ) = ∫ x k x k + 1 d x a = λ k a = h λ k , (8) h_c(\Delta_k)=\int_{x_k}^{x_{k+1}}\frac{dx}{a}=\frac{\lambda_k}{a}=h\lambda_k, \tag*{(8)} h c ( Δ k ) = ∫ x k x k + 1 a d x = a λ k = h λ k , (8) and therefore
h c ( Δ k ) = h λ k = h ⋅ λ v e σ k = h ⋅ e σ k ∑ j = 1 v e σ j . h_c(\Delta_k)=h\lambda_k=h\cdot\lambda_v e^{\sigma_k}=h\cdot\frac{e^{\sigma_k}}{\sum_{j=1}^{v}e^{\sigma_j}}. h c ( Δ k ) = h λ k = h ⋅ λ v e σ k = h ⋅ ∑ j = 1 v e σ j e σ k . Let e ∈ T e\in\mathcal{T} e ∈ T e e e c c c e c , k e_{c,k} e c , k { x = x k } \{x=x_k\} { x = x k } e e e { x = x k } \{x=x_k\} { x = x k } h h h c c c a a a Δ k − 1 \Delta_{k-1} Δ k − 1 Δ k \Delta_k Δ k λ k − 1 , λ k \lambda_{k-1},\lambda_k λ k − 1 , λ k λ k − 1 , λ k ≤ λ 1 + ⋯ + λ v = 1 < a \lambda_{k-1},\lambda_k\leq\lambda_1+\cdots+\lambda_v=1<a λ k − 1 , λ k ≤ λ 1 + ⋯ + λ v = 1 < a N h ( c ) N_h(c) N h ( c )
log a λ k − 1 ≥ log a = log 4 and log a λ k ≥ log 4. \log\frac{a}{\lambda_{k-1}}\geq\log a=\log4\qquad\text{and}\qquad\log\frac{a}{\lambda_k}\geq\log4. log λ k − 1 a ≥ log a = log 4 and log λ k a ≥ log 4. Applying the same argument at the other endpoint of e e e c ′ c' c ′ c ′ = c c'=c c ′ = c N h ( c ′ ) N_h(c') N h ( c ′ ) log 4 \log4 log 4 e ∩ X h e\cap X^h e ∩ X h δ e ( X ) \delta_e(X) δ e ( X ) log 4 \log4 log 4 ∣ s e ( X ) ∣ |s_e(X)| ∣ s e ( X ) ∣ ∣ s e ( X ) ∣ ≤ δ e ( X ) − 2 log 4 |s_e(X)| \leq\delta_e(X) - 2\log4 ∣ s e ( X ) ∣ ≤ δ e ( X ) − 2 log 4
We now prove the upper bound on δ e ( X ) \delta_e(X) δ e ( X ) (5) ,
log 1 h c ( Δ k ) = log 1 h + log ∑ j = 1 v e σ j − σ k \log\frac{1}{h_c(\Delta_k)} = \log\frac{1}{h} + \log\sum_{j=1}^{v} e^{\sigma_j-\sigma_k} log h c ( Δ k ) 1 = log h 1 + log j = 1 ∑ v e σ j − σ k the sum contains the term e σ 0 = 1 e^{\sigma_0}=1 e σ 0 = 1 σ j − σ k \sigma_j-\sigma_k σ j − σ k c c c ∣ σ j − σ k ∣ ≤ O T ( X ) |\sigma_j-\sigma_k| \leq\mathcal{O}_T(X) ∣ σ j − σ k ∣ ≤ O T ( X )
log 1 h ≤ log 1 h c ( Δ k ) ≤ log 1 h + log ( v e O T ( X ) ) ≤ log 1 h + log v ( c ) + O T ( X ) . \log\frac{1}{h} \leq\log\frac{1}{h_c(\Delta_k)} \leq\log\frac{1}{h} + \log\left(v e^{\mathcal{O}_T(X)}\right) \leq\log\frac{1}{h} + \log v(c) + \mathcal{O}_T(X). log h 1 ≤ log h c ( Δ k ) 1 ≤ log h 1 + log ( v e O T ( X ) ) ≤ log h 1 + log v ( c ) + O T ( X ) . Applying this at both endpoints of e e e (3) , yields
δ e ( X ) ≤ 2 log 4 + log ( v ( c ) v ( c ′ ) ) + 2 O T ( X ) . \delta_e(X) \leq2\log4 + \log\bigl(v(c)v(c')\bigr) + 2\mathcal{O}_T(X). δ e ( X ) ≤ 2 log 4 + log ( v ( c ) v ( c ′ ) ) + 2 O T ( X ) . Finally, since ∑ e log ( v ( c ) v ( c ′ ) ) = ∑ c v ( c ) log v ( c ) ≤ 2 E log ( 2 E ) \sum_e \log\bigl(v(c)v(c')\bigr) = \sum_c v(c)\log v(c) \leq2E\log(2E) ∑ e log ( v ( c ) v ( c ′ ) ) = ∑ c v ( c ) log v ( c ) ≤ 2 E log ( 2 E ) T T T
L T ( X ) ≤ 2 E log 4 + 2 E log ( 2 E ) + 2 E O T ( X ) , L_T(X) \leq2E\log4 + 2E\log(2E) + 2E\mathcal{O}_T(X), L T ( X ) ≤ 2 E log 4 + 2 E log ( 2 E ) + 2 E O T ( X ) , where E = 6 g − 6 + 3 n E=6g-6+3n E = 6 g − 6 + 3 n T T T
Plumbing and degenerating families
We recall the plumbing construction of degenerating families of Riemann surfaces from [28 Section 2] , see also [29 Section 3] and [21 Section 2] .
Surfaces with nodes and cusp coordinates
A Riemann surface with nodes R R R C \mathbb{C} C { ( z , w ) ∈ C 2 ∣ z w = 0 , ∣ z ∣ , ∣ w ∣ < 1 } \{(z,w) \in\mathbb{C}^2 \mid zw=0,\ |z|,|w|<1\} {( z , w ) ∈ C 2 ∣ z w = 0 , ∣ z ∣ , ∣ w ∣ < 1 } R R R R ∖ { nodes } R\setminus\{\text{nodes}\} R ∖ { nodes } R R R
Let p p p X X X p p p
N 2 ( p ) = { z = x + i y ∣ x ∈ R / Z , y > 1 / 2 } , ∣ d z ∣ y , N_2(p)=\{z=x+iy\mid x\in\mathbb{R}/\mathbb{Z},\ y>1/2\}, \qquad\frac{|dz|}{y}, N 2 ( p ) = { z = x + i y ∣ x ∈ R / Z , y > 1/2 } , y ∣ d z ∣ , the puncture corresponding to ∞ \infty ∞ z = x + i y z=x+iy z = x + i y p p p p p p
q = e 2 π i z = e 2 π i x e − 2 π y , q=e^{2\pi iz}=e^{2\pi ix}e^{-2\pi y}, q = e 2 π i z = e 2 π i x e − 2 π y , it is a biholomorphism from N 2 ( p ) N_2(p) N 2 ( p ) { q ∈ C ∣ 0 < ∣ q ∣ < e − π } \{q\in\mathbb{C}\mid0<|q|<e^{-\pi}\} { q ∈ C ∣ 0 < ∣ q ∣ < e − π } q = 0 q=0 q = 0 p p p X X X q ( p ) = 0 q(p)=0 q ( p ) = 0 q q q { q ∈ C ∣ ∣ q ∣ < e − π } \{q\in\mathbb{C}\mid|q|<e^{-\pi}\} { q ∈ C ∣ ∣ q ∣ < e − π } ∣ q ∣ = e − 2 π y |q|=e^{-2\pi y} ∣ q ∣ = e − 2 π y { y = y 0 } \{y=y_0\} { y = y 0 } 1 / y 0 1/y_0 1/ y 0 { ∣ q ∣ = e − 2 π y 0 } \{|q|=e^{-2\pi y_0}\} { ∣ q ∣ = e − 2 π y 0 } N h ( p ) = { y > a } N_h(p)=\{y>a\} N h ( p ) = { y > a } X h X^h X h { q ∈ C ∣ 0 < ∣ q ∣ ≤ e − 2 π a } \{q\in\mathbb{C}\mid0<|q|\leq e^{-2\pi a}\} { q ∈ C ∣ 0 < ∣ q ∣ ≤ e − 2 π a } d q = ( 2 π i ) q d z dq=(2\pi i)q\,dz d q = ( 2 π i ) q d z ∣ d q ∣ = 2 π ∣ q ∣ ∣ d z ∣ |dq|=2\pi|q||dz| ∣ d q ∣ = 2 π ∣ q ∣∣ d z ∣ ∣ q ∣ = e − 2 π y |q|=e^{-2\pi y} ∣ q ∣ = e − 2 π y log ∣ q ∣ = − 2 π y < 0 \log|q|=-2\pi y<0 log ∣ q ∣ = − 2 π y < 0 N 2 ( p ) N_2(p) N 2 ( p ) ∣ log ∣ q ∣ ∣ = 2 π y |\log|q||=2\pi y ∣ log ∣ q ∣∣ = 2 π y
∣ d z ∣ y = ∣ d q ∣ 2 π ∣ q ∣ 2 π ∣ log ∣ q ∣ ∣ = ∣ d q ∣ ∣ q ∣ ∣ log ∣ q ∣ ∣ . \frac{|dz|}{y}=\frac{|dq|}{2\pi|q|}\frac{2\pi}{|\log|q||}=\frac{|dq|}{|q|\,|\log|q||}. y ∣ d z ∣ = 2 π ∣ q ∣ ∣ d q ∣ ∣ log ∣ q ∣∣ 2 π = ∣ q ∣ ∣ log ∣ q ∣∣ ∣ d q ∣ . that is, the hyperbolic metric of N 2 ( p ) N_2(p) N 2 ( p ) { q ∈ C ∣ 0 < ∣ q ∣ < 1 } \{q \in\mathbb{C} \mid0 < |q| < 1\} { q ∈ C ∣ 0 < ∣ q ∣ < 1 } { q ∈ C ∣ 0 < ∣ q ∣ < e − π } \{q \in\mathbb{C} \mid0 < |q| < e^{-\pi}\} { q ∈ C ∣ 0 < ∣ q ∣ < e − π } x ↦ x + θ x \mapsto x + \theta x ↦ x + θ θ ∈ R / Z \theta\in\mathbb{R}/\mathbb{Z} θ ∈ R / Z q ↦ e 2 π i θ q q \mapsto e^{2\pi i\theta}q q ↦ e 2 π i θ q [28 Definition 2.1] .
Discs and annuli around paired punctures
Let R R R ν 1 , … , ν k \nu_1,\ldots,\nu_k ν 1 , … , ν k R 0 = R ∖ { ν 1 , … , ν k } R_0=R\setminus\{\nu_1,\ldots,\nu_k\} R 0 = R ∖ { ν 1 , … , ν k } R 0 R_0 R 0 ν j \nu_j ν j ( p j , p j ′ ) (p_j,p'_j) ( p j , p j ′ ) R 0 R_0 R 0 U j ⊂ R 0 U_j\subset R_0 U j ⊂ R 0 U j ′ ⊂ R 0 U'_j\subset R_0 U j ′ ⊂ R 0 p j p_j p j p j ′ p'_j p j ′
q j : U j ⟶ { w ∈ C ∣ 0 < ∣ w ∣ < e − π } , q j ′ : U j ′ ⟶ { w ∈ C ∣ 0 < ∣ w ∣ < e − π } q_j:U_j\longrightarrow\{w\in\mathbb{C}\mid0<|w|<e^{-\pi}\},\qquad q'_j:U'_j\longrightarrow\{w\in\mathbb{C}\mid0<|w|<e^{-\pi}\} q j : U j ⟶ { w ∈ C ∣ 0 < ∣ w ∣ < e − π } , q j ′ : U j ′ ⟶ { w ∈ C ∣ 0 < ∣ w ∣ < e − π } be the cusp coordinates, which are biholomorphisms. For 0 < r 1 < r 2 < e − π 0<r_1<r_2<e^{-\pi} 0 < r 1 < r 2 < e − π
D j ( r 2 ) = { ξ ∈ U j ∣ ∣ q j ( ξ ) ∣ < r 2 } , D j ‾ ( r 1 ) = { ξ ∈ U j ∣ ∣ q j ( ξ ) ∣ ≤ r 1 } , D_j(r_2)=\{\xi\in U_j\mid|q_j(\xi)|<r_2\},\qquad\overline{D_j}(r_1)=\{\xi\in U_j\mid|q_j(\xi)|\le r_1\}, D j ( r 2 ) = { ξ ∈ U j ∣ ∣ q j ( ξ ) ∣ < r 2 } , D j ( r 1 ) = { ξ ∈ U j ∣ ∣ q j ( ξ ) ∣ ≤ r 1 } , A j ( r 1 , r 2 ) = { ξ ∈ U j ∣ r 1 < ∣ q j ( ξ ) ∣ < r 2 } A_j(r_1,r_2)=\{\xi\in U_j\mid r_1<|q_j(\xi)|<r_2\} A j ( r 1 , r 2 ) = { ξ ∈ U j ∣ r 1 < ∣ q j ( ξ ) ∣ < r 2 } the corresponding punctured discs and annulus of R 0 R_0 R 0 p j p_j p j D j ′ ( r 2 ) D'_j(r_2) D j ′ ( r 2 ) D j ′ ‾ ( r 1 ) \overline{D'_j}(r_1) D j ′ ( r 1 ) A j ′ ( r 1 , r 2 ) A'_j(r_1,r_2) A j ′ ( r 1 , r 2 ) U j ′ U'_j U j ′ q j ′ q'_j q j ′ 0 < c < e − π 0<c<e^{-\pi} 0 < c < e − π D j ( c ) D_j(c) D j ( c ) D j ′ ( c ) D'_j(c) D j ′ ( c ) 1 ≤ j ≤ k 1\le j\le k 1 ≤ j ≤ k [10 Theorem 4.4.6] . Their complement
K c = R 0 ∖ ⋃ j = 1 k ( D j ( c ) ∪ D j ′ ( c ) ) K_c=R_0\setminus\bigcup_{j=1}^{k}\left(D_j(c)\cup D'_j(c)\right) K c = R 0 ∖ j = 1 ⋃ k ( D j ( c ) ∪ D j ′ ( c ) ) is a surface with boundary consisting of the 2 k 2k 2 k
{ ξ ∈ U j ∣ ∣ q j ( ξ ) ∣ = c } , { ξ ′ ∈ U j ′ ∣ ∣ q j ′ ( ξ ′ ) ∣ = c } , 1 ≤ j ≤ k . \{\xi\in U_j\mid|q_j(\xi)|=c\},\qquad\{\xi'\in U'_j\mid|q'_j(\xi')|=c\},\qquad1\le j\le k. { ξ ∈ U j ∣ ∣ q j ( ξ ) ∣ = c } , { ξ ′ ∈ U j ′ ∣ ∣ q j ′ ( ξ ′ ) ∣ = c } , 1 ≤ j ≤ k . It is compact when all punctures are paired.
The plumbing construction
Let t = ( t 1 , … , t k ) ∈ C k \mathbf{t}=(t_1,\ldots,t_k)\in\mathbb{C}^k t = ( t 1 , … , t k ) ∈ C k 0 < ∣ t j ∣ < c 2 0<|t_j|<c^2 0 < ∣ t j ∣ < c 2 j j j ∣ t j ∣ / c < c |t_j|/c<c ∣ t j ∣/ c < c j j j
A j = A j ( ∣ t j ∣ / c , c ) , A j ′ = A j ′ ( ∣ t j ∣ / c , c ) , A ˚ j = { w ∈ C ∣ ∣ t j ∣ / c < ∣ w ∣ < c } . A_j=A_j(|t_j|/c,c),\qquad A'_j=A'_j(|t_j|/c,c),\qquad\mathring{A}_j=\{w\in\mathbb{C}\mid|t_j|/c<|w|<c\}. A j = A j ( ∣ t j ∣/ c , c ) , A j ′ = A j ′ ( ∣ t j ∣/ c , c ) , A ˚ j = { w ∈ C ∣ ∣ t j ∣/ c < ∣ w ∣ < c } . Thus A j A_j A j A j ′ A'_j A j ′ p j p_j p j p j ′ p'_j p j ′ A ˚ j \mathring{A}_j A ˚ j q j : A j → A ˚ j q_j:A_j\to\mathring{A}_j q j : A j → A ˚ j q j ′ : A j ′ → A ˚ j q'_j:A'_j\to\mathring{A}_j q j ′ : A j ′ → A ˚ j ι j ( w ) = t j / w \iota_j(w)=t_j/w ι j ( w ) = t j / w C ∖ { 0 } \mathbb{C}\setminus\{0\} C ∖ { 0 } ∣ ι j ( w ) ∣ = ∣ t j ∣ / ∣ w ∣ |\iota_j(w)|=|t_j|/|w| ∣ ι j ( w ) ∣ = ∣ t j ∣/∣ w ∣ A ˚ j \mathring{A}_j A ˚ j ι j ∘ ι j = id \iota_j\circ\iota_j=\operatorname{id} ι j ∘ ι j = id ι j \iota_j ι j A ˚ j \mathring{A}_j A ˚ j ∣ w ∣ = c |w|=c ∣ w ∣ = c ∣ w ∣ = ∣ t j ∣ / c |w|=|t_j|/c ∣ w ∣ = ∣ t j ∣/ c
φ j = ( q j ′ ) − 1 ∘ ι j ∘ q j : A j ⟶ A j ′ \varphi_j=(q'_j)^{-1}\circ\iota_j\circ q_j:A_j\longrightarrow A'_j φ j = ( q j ′ ) − 1 ∘ ι j ∘ q j : A j ⟶ A j ′ is a biholomorphism, and for ξ ∈ A j \xi\in A_j ξ ∈ A j ξ ′ ∈ A j ′ \xi'\in A'_j ξ ′ ∈ A j ′ ξ ′ = φ j ( ξ ) \xi'=\varphi_j(\xi) ξ ′ = φ j ( ξ )
q j ( ξ ) q j ′ ( ξ ′ ) = t j . q_j(\xi)q'_j(\xi')=t_j. q j ( ξ ) q j ′ ( ξ ′ ) = t j . Remove from R 0 R_0 R 0 D j ‾ ( ∣ t j ∣ / c ) \overline{D_j}(|t_j|/c) D j ( ∣ t j ∣/ c ) D j ′ ‾ ( ∣ t j ∣ / c ) \overline{D'_j}(|t_j|/c) D j ′ ( ∣ t j ∣/ c ) 1 ≤ j ≤ k 1\le j\le k 1 ≤ j ≤ k
R t ∗ = R 0 ∖ ⋃ j = 1 k ( D j ‾ ( ∣ t j ∣ / c ) ∪ D j ′ ‾ ( ∣ t j ∣ / c ) ) , R^*_{\mathbf{t}}=R_0\setminus\bigcup_{j=1}^{k}\left(\overline{D_j}(|t_j|/c)\cup\overline{D'_j}(|t_j|/c)\right), R t ∗ = R 0 ∖ j = 1 ⋃ k ( D j ( ∣ t j ∣/ c ) ∪ D j ′ ( ∣ t j ∣/ c ) ) , which contains the pairwise disjoint annuli A j A_j A j A j ′ A'_j A j ′ 1 ≤ j ≤ k 1\le j\le k 1 ≤ j ≤ k j j j ξ ∈ A j \xi\in A_j ξ ∈ A j φ j ( ξ ) ∈ A j ′ \varphi_j(\xi)\in A'_j φ j ( ξ ) ∈ A j ′ ξ ∼ ξ ′ \xi\sim\xi' ξ ∼ ξ ′ q j ( ξ ) q j ′ ( ξ ′ ) = t j q_j(\xi)q'_j(\xi')=t_j q j ( ξ ) q j ′ ( ξ ′ ) = t j R 0 R_0 R 0 t \mathbf{t} t
R t = R t ∗ / ∼ R_t = R_t^*/\sim R t = R t ∗ / ∼ π : R t ∗ → R t \pi: R_t^* \to R_t π : R t ∗ → R t φ j \varphi_j φ j R t ∗ R_t^* R t ∗ R t R_t R t R 0 R_0 R 0
The j j j plumbing collar is C j = π ( A j ) = π ( A j ′ ) C_j = \pi(A_j) = \pi(A'_j) C j = π ( A j ) = π ( A j ′ ) Figure 2 . Every point x ∈ C j x \in C_j x ∈ C j π \pi π ξ ∈ A j \xi\in A_j ξ ∈ A j ξ ′ = φ j ( ξ ) ∈ A j ′ \xi' = \varphi_j(\xi) \in A'_j ξ ′ = φ j ( ξ ) ∈ A j ′ C j C_j C j q j ( x ) : = q j ( ξ ) q_j(x) := q_j(\xi) q j ( x ) := q j ( ξ ) q j ′ ( x ) : = q j ′ ( ξ ′ ) q'_j(x) := q'_j(\xi') q j ′ ( x ) := q j ′ ( ξ ′ ) A j A_j A j q j ( x ) q j ′ ( x ) = t j q_j(x)q'_j(x) = t_j q j ( x ) q j ′ ( x ) = t j R t R_t R t c c c ∣ t j ∣ < c 2 |t_j| < c^2 ∣ t j ∣ < c 2 j j j [28 Remarks in Section 2.4] .
Figure 2. The plumbing collar C j C_j C j
Let β j = { ξ ∈ U j ∣ ∣ q j ( ξ ) ∣ = c } \beta_j = \{\xi\in U_j \mid|q_j(\xi)| = c\} β j = { ξ ∈ U j ∣ ∣ q j ( ξ ) ∣ = c } β j ′ = { ξ ′ ∈ U j ′ ∣ ∣ q j ′ ( ξ ′ ) ∣ = c } \beta'_j = \{\xi' \in U'_j \mid|q'_j(\xi')| = c\} β j ′ = { ξ ′ ∈ U j ′ ∣ ∣ q j ′ ( ξ ′ ) ∣ = c } K c K_c K c A j A_j A j A j ′ A'_j A j ′ K c K_c K c R t R_t R t C j C_j C j ∣ q j ′ ( φ j ( ξ ) ) ∣ = ∣ t j ∣ / ∣ q j ( ξ ) ∣ |q'_j(\varphi_j(\xi))| = |t_j|/|q_j(\xi)| ∣ q j ′ ( φ j ( ξ )) ∣ = ∣ t j ∣/∣ q j ( ξ ) ∣ ( ξ n ) (\xi_n) ( ξ n ) A j A_j A j ∣ q j ( ξ n ) ∣ → c |q_j(\xi_n)| \to c ∣ q j ( ξ n ) ∣ → c q j q_j q j ( ξ n ) (\xi_n) ( ξ n ) β j \beta_j β j ∣ q j ( ξ n ) ∣ → ∣ t j ∣ / c |q_j(\xi_n)| \to|t_j|/c ∣ q j ( ξ n ) ∣ → ∣ t j ∣/ c ( ξ n ) (\xi_n) ( ξ n ) R t ∗ R_t^* R t ∗ A j A_j A j { ∣ q j ∣ = ∣ t j ∣ / c } \{|q_j| = |t_j|/c\} { ∣ q j ∣ = ∣ t j ∣/ c } R t R_t R t ξ n \xi_n ξ n φ j ( ξ n ) ∈ A j ′ \varphi_j(\xi_n) \in A'_j φ j ( ξ n ) ∈ A j ′ ∣ q j ′ ( φ j ( ξ n ) ) ∣ → c |q'_j(\varphi_j(\xi_n))| \to c ∣ q j ′ ( φ j ( ξ n )) ∣ → c β j ′ \beta'_j β j ′ C j ‾ \overline{C_j} C j C j C_j C j R t R_t R t β j ∪ β j ′ \beta_j \cup\beta'_j β j ∪ β j ′
R t = K c ∪ C 1 ‾ ∪ ⋯ ∪ C k ‾ , K c ∩ C j ‾ = β j ∪ β j ′ . R_t = K_c \cup\overline{C_1} \cup\cdots\cup\overline{C_k}, \qquad K_c \cap\overline{C_j} = \beta_j \cup\beta'_j. R t = K c ∪ C 1 ∪ ⋯ ∪ C k , K c ∩ C j = β j ∪ β j ′ . In other words, R t R_t R t K c K_c K c j j j β j \beta_j β j β j ′ \beta'_j β j ′
The local model
The plumbing variety is the complex surface
V = { ( u , v , t ) ∈ C 3 ∣ u v = t , ∣ u ∣ < 1 , ∣ v ∣ < 1 , ∣ t ∣ < 1 } , V = \left\{(u,v,t) \in\mathbb{C}^3 \mid uv=t,\ |u|<1,\ |v|<1,\ |t|<1\right\}, V = { ( u , v , t ) ∈ C 3 ∣ uv = t , ∣ u ∣ < 1 , ∣ v ∣ < 1 , ∣ t ∣ < 1 } , with the holomorphic projection Π : V → { t ∈ C ∣ ∣ t ∣ < 1 } \Pi: V \to\{t \in\mathbb{C} \mid|t|<1\} Π : V → { t ∈ C ∣ ∣ t ∣ < 1 } Π ( u , v , t ) = t \Pi(u,v,t)=t Π ( u , v , t ) = t fiber over t t t
V t = { ( u , v ) ∈ C 2 ∣ u v = t , ∣ u ∣ < 1 , ∣ v ∣ < 1 } . V_t = \left\{(u,v) \in\mathbb{C}^2 \mid uv=t,\ |u|<1,\ |v|<1\right\}. V t = { ( u , v ) ∈ C 2 ∣ uv = t , ∣ u ∣ < 1 , ∣ v ∣ < 1 } . For t ≠ 0 t \ne0 t = 0 ( u , v ) ↦ u (u,v) \mapsto u ( u , v ) ↦ u V t V_t V t { u ∈ C ∣ ∣ t ∣ < ∣ u ∣ < 1 } \{u \in\mathbb{C} \mid|t| < |u| < 1\} { u ∈ C ∣ ∣ t ∣ < ∣ u ∣ < 1 } t = 0 t = 0 t = 0
V 0 = { ( u , 0 ) ∈ C 2 ∣ ∣ u ∣ < 1 } ∪ { ( 0 , v ) ∈ C 2 ∣ ∣ v ∣ < 1 } V_0 = \{(u,0) \in\mathbb{C}^2 \mid|u| < 1\} \cup\{(0,v) \in\mathbb{C}^2 \mid|v| < 1\} V 0 = {( u , 0 ) ∈ C 2 ∣ ∣ u ∣ < 1 } ∪ {( 0 , v ) ∈ C 2 ∣ ∣ v ∣ < 1 } is the union of two discs meeting at the origin: this is the local model of a node. Thus V V V R t R_t R t j j j
C j ⟶ V t j , ξ ⟼ ( q j ( ξ ) , q j ′ ( ξ ) ) , C_j \longrightarrow V_{t_j}, \qquad\xi\longmapsto(q_j(\xi),q'_j(\xi)), C j ⟶ V t j , ξ ⟼ ( q j ( ξ ) , q j ′ ( ξ )) , is a biholomorphism from the collar C j ⊂ R t C_j \subset R_t C j ⊂ R t { ( u , v ) ∈ V t j ∣ ∣ u ∣ < c , ∣ v ∣ < c } \{(u,v) \in V_{t_j} \mid|u| < c,\ |v| < c\} {( u , v ) ∈ V t j ∣ ∣ u ∣ < c , ∣ v ∣ < c } { u ∈ C ∣ ∣ t j ∣ / c < ∣ u ∣ < c } \{u \in\mathbb{C} \mid|t_j|/c < |u| < c\} { u ∈ C ∣ ∣ t j ∣/ c < ∣ u ∣ < c } ( u , v ) ↦ u (u,v) \mapsto u ( u , v ) ↦ u
The model metric of a plumbing collar
By [28 Section 0.1] , for t ≠ 0 t \ne0 t = 0 V t V_t V t { u ∈ C ∣ ∣ t ∣ < ∣ u ∣ < 1 } \{u \in\mathbb{C} \mid|t| < |u| < 1\} { u ∈ C ∣ ∣ t ∣ < ∣ u ∣ < 1 }
d s t = π ∣ log ∣ t ∣ ∣ csc ( π log ∣ u ∣ log ∣ t ∣ ) ∣ d u u ∣ . ds_t = \frac{\pi}{|\log|t||}\csc\left(\frac{\pi\log|u|}{\log|t|}\right)\left|\frac{du}{u}\right|. d s t = ∣ log ∣ t ∣∣ π csc ( log ∣ t ∣ π log ∣ u ∣ ) u d u . and each of the two punctured discs { u ∈ C ∣ 0 < ∣ u ∣ < 1 } \{u \in\mathbb{C} \mid0 < |u| < 1\} { u ∈ C ∣ 0 < ∣ u ∣ < 1 } { v ∈ C ∣ 0 < ∣ v ∣ < 1 } \{v \in\mathbb{C} \mid0 < |v| < 1\} { v ∈ C ∣ 0 < ∣ v ∣ < 1 } V 0 ∖ { ( 0 , 0 ) } V_0 \setminus\{(0,0)\} V 0 ∖ {( 0 , 0 )} d s 0 = ∣ d u ∣ / ( ∣ u ∣ ∣ log ∣ u ∣ ∣ ) ds_0 = |du|/(|u||\log|u||) d s 0 = ∣ d u ∣/ ( ∣ u ∣∣ log ∣ u ∣∣ ) ∣ d v ∣ / ( ∣ v ∣ ∣ log ∣ v ∣ ∣ ) |dv|/(|v||\log|v||) ∣ d v ∣/ ( ∣ v ∣∣ log ∣ v ∣∣ ) u = e 2 π i ( x + i y ) u = e^{2\pi i(x+iy)} u = e 2 π i ( x + i y ) { u ∈ C ∣ ∣ t ∣ < ∣ u ∣ < 1 } \{u \in\mathbb{C} \mid|t| < |u| < 1\} { u ∈ C ∣ ∣ t ∣ < ∣ u ∣ < 1 } { ( x , y ) ∈ ( R / Z ) × R ∣ 0 < y < L } \{(x,y) \in(\mathbb{R}/\mathbb{Z}) \times\mathbb{R} \mid0 < y < L\} {( x , y ) ∈ ( R / Z ) × R ∣ 0 < y < L } L = 1 2 π log 1 ∣ t ∣ L = \frac{1}{2\pi}\log\frac{1}{|t|} L = 2 π 1 log ∣ t ∣ 1 d s t = λ L ( y ) ∣ d z ∣ ds_t = \lambda_L(y)|dz| d s t = λ L ( y ) ∣ d z ∣ L > 0 L > 0 L > 0
λ L ( y ) = π L csc ( π y L ) = 1 y θ sin θ , θ = π y L , 0 < y < L . (9) \lambda_L(y) = \frac{\pi}{L}\csc\left(\frac{\pi y}{L}\right) = \frac{1}{y}\frac{\theta}{\sin\theta}, \qquad\theta= \frac{\pi y}{L}, \qquad0 < y < L. \tag*{(9)} λ L ( y ) = L π csc ( L π y ) = y 1 sin θ θ , θ = L π y , 0 < y < L . (9) This is the complete hyperbolic metric of the flat cylinder { ( x , y ) ∈ ( R / Z ) × R ∣ 0 < y < L } \{(x,y) \in(\mathbb{R}/\mathbb{Z}) \times\mathbb{R} \mid0 < y < L\} {( x , y ) ∈ ( R / Z ) × R ∣ 0 < y < L } y y y ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) d s t ds_t d s t d s 0 = ∣ d z ∣ / y ds_0 = |dz|/y d s 0 = ∣ d z ∣/ y t → 0 t \to0 t → 0 C j C_j C j V t j V_{t_j} V t j d s t j ds_{t_j} d s t j model metric of the j j j
Wolpert approximates the hyperbolic metric of R t R_t R t R 0 R_0 R 0 [28 Sections 3.3–3.4] . The curvature of the resulting grafted metric is − 1 + O ( ∑ j ( log ∣ t j ∣ ) − 2 ) -1 + O\left(\sum_j(\log|t_j|)^{-2}\right) − 1 + O ( ∑ j ( log ∣ t j ∣ ) − 2 ) R t R_t R t [28 Lemma 3.5] . Expansions of the hyperbolic metric of R t R_t R t t j t_j t j [28 Expansion 4.2] and [21 Theorem 4] .
The plumbing family and plumbing coordinates
The surfaces R t R_t R t t t t
Δ c 2 k = { t = ( t 1 , … , t k ) ∈ C k ∣ ∣ t j ∣ < c 2 for 1 ≤ j ≤ k } \Delta_{c^2}^k = \{t = (t_1,\ldots,t_k) \in\mathbb{C}^k \mid|t_j| < c^2 \text{ for } 1 \le j \le k\} Δ c 2 k = { t = ( t 1 , … , t k ) ∈ C k ∣ ∣ t j ∣ < c 2 for 1 ≤ j ≤ k } form a holomorphic family which degenerates to the surface with nodes R R R t = 0 t = 0 t = 0 [28 Sections 0.1 and 2.4] . One may also deform the parts: let μ 1 , … , μ m \mu_1,\ldots,\mu_m μ 1 , … , μ m R 0 R_0 R 0 K c K_c K c R 0 R_0 R 0 m = 3 g − 3 + n − k m = 3g - 3 + n - k m = 3 g − 3 + n − k s ∈ C m s \in\mathbb{C}^m s ∈ C m μ ( s ) = ∑ i = 1 m s i μ i \mu(s) = \sum_{i=1}^m s_i\mu_i μ ( s ) = ∑ i = 1 m s i μ i R 0 μ ( s ) R_0^{\mu(s)} R 0 μ ( s ) t t t R s , t R_{s,t} R s , t ( s , t ) (s,t) ( s , t ) R R R [28 Section 2.4.M] , [29 Section 3] . In this work the parts are fixed, that is s = 0 s = 0 s = 0
Proposition 2.3 ([29 Section 3] ). Let R R R ν 1 , … , ν k \nu_1,\ldots,\nu_k ν 1 , … , ν k R 0 = R ∖ { ν 1 , … , ν k } R_0=R\setminus\{\nu_1,\ldots,\nu_k\} R 0 = R ∖ { ν 1 , … , ν k } 0 < c < e − π 0<c<e^{-\pi} 0 < c < e − π t ∈ ( 0 , c 2 ) k \mathbf{t}\in(0,c^2)^k t ∈ ( 0 , c 2 ) k R t R_{\mathbf{t}} R t R 0 R_0 R 0 t \mathbf{t} t K c K_c K c R R R R t → R R_{\mathbf{t}}\to R R t → R t → 0 \mathbf{t}\to0 t → 0
Proof. By construction, R R R t = 0 \mathbf{t}=0 t = 0 R = R 0 , 0 R=R_{0,0} R = R 0 , 0 ( s , t ) (\mathbf{s},\mathbf{t}) ( s , t )
By [29 Section 3, p. 15] , near R R R ( s , ∣ t ∣ , arg t ) (\mathbf{s},|\mathbf{t}|,\arg\mathbf{t}) ( s , ∣ t ∣ , arg t ) arg t j \arg t_j arg t j R \mathbb{R} R arg t j \arg t_j arg t j 2 π 2\pi 2 π C j C_j C j ∣ t j ∣ → 0 |t_j|\to0 ∣ t j ∣ → 0
Recall from Section 2.4 that, in extended Fenchel–Nielsen coordinates, ( ℓ n , τ n ) → 0 (\ell_n,\tau_n)\to0 ( ℓ n , τ n ) → 0 ℓ n → 0 \ell_n\to0 ℓ n → 0 τ n \tau_n τ n R s , t → R R_{\mathbf{s},\mathbf{t}}\to R R s , t → R s → 0 \mathbf{s}\to0 s → 0 ∣ t ∣ → 0 |\mathbf{t}|\to0 ∣ t ∣ → 0 [29 Section 3, pp. 11–12] .
Now let t ∈ ( 0 , c 2 ) k \mathbf{t}\in(0,c^2)^k t ∈ ( 0 , c 2 ) k R t R_{\mathbf{t}} R t R R R K c K_c K c s = 0 \mathbf{s}=0 s = 0 t → 0 \mathbf{t}\to0 t → 0 ∣ t ∣ → 0 |\mathbf{t}|\to0 ∣ t ∣ → 0 R t → R R_{\mathbf{t}}\to R R t → R
Doubling and the Weil–Petersson distance
Opening the conformal double
Let Y ∈ Teich ( Σ ) Y\in\operatorname{Teich}(\Sigma) Y ∈ Teich ( Σ ) Σ \Sigma Σ ( g , n ) (g,n) ( g , n ) n ≥ 1 n\ge1 n ≥ 1 Y h Y^h Y h h h h z = x + i y z=x+iy z = x + i y c c c Y ‾ \overline{Y} Y Y Y Y c c c z ′ = − z ‾ = − x + i y = x ′ + i y ′ z'=-\overline{z}=-x+iy=x'+iy' z ′ = − z = − x + i y = x ′ + i y ′
Definition of the double
For T > 0 T>0 T > 0
C T = { ( x , s ) ∈ ( R / Z ) × R ∣ 0 ≤ s ≤ T } . C_T=\{(x,s)\in(\mathbb{R}/\mathbb{Z})\times\mathbb{R}\mid0\le s\le T\}. C T = {( x , s ) ∈ ( R / Z ) × R ∣ 0 ≤ s ≤ T } . be the Euclidean cylinder, with the two holomorphic coordinates
w , w ′ : C T ⟶ { ζ ∈ C / Z ∣ a ≤ Im ( ζ ) ≤ a + T } , w,w':C_T\longrightarrow\{\zeta\in\mathbb{C}/\mathbb{Z}\mid a\le\operatorname{Im}(\zeta)\le a+T\}, w , w ′ : C T ⟶ { ζ ∈ C / Z ∣ a ≤ Im ( ζ ) ≤ a + T } , w = x + i ( a + s ) and w ′ = − x + i ( a + T − s ) = − w + i ( 2 a + T ) . w=x+i(a+s)\qquad\text{and}\qquad w'=-x+i(a+T-s)=-w+i(2a+T). w = x + i ( a + s ) and w ′ = − x + i ( a + T − s ) = − w + i ( 2 a + T ) . For each cusp, we glue the circle { s = 0 } \{s=0\} { s = 0 } C T C_T C T { y = a } \{y=a\} { y = a } Y h Y^h Y h ( x , 0 ) ↦ x + i a (x,0)\mapsto x+ia ( x , 0 ) ↦ x + ia z z z Y Y Y Y h Y^h Y h { s = T } \{s=T\} { s = T } { y ′ = a } \{y'=a\} { y ′ = a } Y ‾ h \overline{Y}^h Y h ( x , T ) ↦ − x + i a (x,T)\mapsto-x+ia ( x , T ) ↦ − x + ia z ′ z' z ′ Y ‾ h \overline{Y}^h Y h w w w w ′ w' w ′ z z z Y h Y^h Y h z ′ z' z ′ Y ‾ h \overline{Y}^h Y h { s = 0 } \{s=0\} { s = 0 } { s = T } \{s=T\} { s = T } Figure 3 . The gluings are holomorphic, and we denote by W T ( Y ) W_T(Y) W T ( Y ) ∣ d w ∣ |dw| ∣ d w ∣ { s = T / 2 } \{s=T/2\} { s = T /2 } n n n Y Y Y Y ‾ \overline{Y} Y ( g , n ) (g,n) ( g , n ) W T ( Y ) W_T(Y) W T ( Y ) G = 2 g + ( n − 1 ) ≥ 2 G=2g+(n-1)\ge2 G = 2 g + ( n − 1 ) ≥ 2 2 − 2 G = 2 ( 2 − 2 g − n ) 2-2G=2(2-2g-n) 2 − 2 G = 2 ( 2 − 2 g − n )
Figure 3. Gluing a cylinder C T C_T C T Y h Y^h Y h Y ‾ h \overline{Y}^h Y h
Definition of the marking
Let K K K Σ \Sigma Σ Λ \Lambda Λ K K K K ‾ \overline{K} K K K K Λ \Lambda Λ ( G , 0 ) (G,0) ( G , 0 ) C ⊂ Λ C\subset\Lambda C ⊂ Λ c c c ∂ c K \partial_c K ∂ c K K K K c c c ∂ c K × [ 0 , 1 ] \partial_c K \times[0,1] ∂ c K × [ 0 , 1 ] ∂ c K ⊂ K \partial_c K \subset K ∂ c K ⊂ K ∂ c K × { 0 } \partial_c K \times\{0\} ∂ c K × { 0 } ∂ c K \partial_c K ∂ c K K ‾ \overline{K} K ∂ c K × { 1 } \partial_c K \times\{1\} ∂ c K × { 1 } { v } × [ 0 , 1 ] \{v\} \times[0,1] { v } × [ 0 , 1 ] v ∈ ∂ c K v \in\partial_c K v ∈ ∂ c K ∂ c K × { 1 / 2 } \partial_c K \times\{1/2\} ∂ c K × { 1/2 } d W P d_{\mathrm{WP}} d WP Teich ( Λ ) \operatorname{Teich}(\Lambda) Teich ( Λ ) D ( Y ) ∈ S ( C ) D(Y) \in S(\mathbb{C}) D ( Y ) ∈ S ( C ) Y Y Y Y ‾ \overline{Y} Y Y Y Y Y ‾ \overline{Y} Y Figure 4 .
Figure 4. The surfaces Y Y Y Y ‾ \overline{Y} Y W T ( Y ) W_T(Y) W T ( Y ) D ( Y ) D(Y) D ( Y ) g = 1 g=1 g = 1 n = 2 n=2 n = 2
We now define a marking f Y , T Λ ⟶ W T ( Y ) f_{Y,T\Lambda} \longrightarrow W_T(Y) f Y , T Λ ⟶ W T ( Y ) Y Y Y K K K Y h Y^h Y h f Y : K → Y h f_Y: K \to Y^h f Y : K → Y h c c c f Y f_Y f Y ∂ c K \partial_c K ∂ c K { y = a } \{y=a\} { y = a } β Y , c : ∂ c K → R / Z \beta_{Y,c}: \partial_c K \to\mathbb{R}/\mathbb{Z} β Y , c : ∂ c K → R / Z
f Y ( v ) = β Y , c ( v ) + i a , v ∈ ∂ c K , f_Y(v)=\beta_{Y,c}(v)+ia,\qquad v\in\partial_c K, f Y ( v ) = β Y , c ( v ) + ia , v ∈ ∂ c K , in the normalized coordinate z = x + i y z=x+iy z = x + i y c c c T > 0 T>0 T > 0 f Y , T : Λ → W T ( Y ) f_{Y,T}:\Lambda\to W_T(Y) f Y , T : Λ → W T ( Y ) f Y f_Y f Y K K K f Y f_Y f Y K ‾ \overline{K} K Y ‾ h \overline{Y}^h Y h K ‾ \overline{K} K c c c ( x , s ) (x,s) ( x , s ) C T C_T C T c c c
f Y , T ( v , r ) = ( β Y , c ( v ) , T r ) , ( v , r ) ∈ ∂ c K × [ 0 , 1 ] . f_{Y,T}(v,r)=(\beta_{Y,c}(v),Tr),\qquad(v,r)\in\partial_c K\times[0,1]. f Y , T ( v , r ) = ( β Y , c ( v ) , T r ) , ( v , r ) ∈ ∂ c K × [ 0 , 1 ] . By the gluing rules above, the point ( β Y , c ( v ) , 0 ) (\beta_{Y,c}(v),0) ( β Y , c ( v ) , 0 ) z = β Y , c ( v ) + i a z=\beta_{Y,c}(v)+ia z = β Y , c ( v ) + ia Y h Y^h Y h f Y ( v ) f_Y(v) f Y ( v ) ( β Y , c ( v ) , T ) (\beta_{Y,c}(v),T) ( β Y , c ( v ) , T ) z ′ = − β Y , c ( v ) + i a z'=-\beta_{Y,c}(v)+ia z ′ = − β Y , c ( v ) + ia Y ‾ h \overline{Y}^h Y h f Y ( v ) f_Y(v) f Y ( v ) Y ‾ h \overline{Y}^h Y h z ′ = − z ‾ z'=-\overline{z} z ′ = − z f Y , T f_{Y,T} f Y , T Y Y Y K K K Y ‾ \overline{Y} Y K ‾ \overline{K} K C C C
Lemma 3.1. Let Y ∈ Teich g , n Y\in\operatorname{Teich}_{g,n} Y ∈ Teich g , n n ≥ 1 n\geq1 n ≥ 1 c = e − 2 π a c=e^{-2\pi a} c = e − 2 π a T > 0 T>0 T > 0 W T ( Y ) W_T(Y) W T ( Y ) f Y , T f_{Y,T} f Y , T Y ⊔ Y ‾ Y\sqcup\overline{Y} Y ⊔ Y D ( Y ) D(Y) D ( Y ) n n n
t T = e − 2 π ( 2 a + T ) ∈ ( 0 , c 2 ) t_T=e^{-2\pi(2a+T)}\in(0,c^2) t T = e − 2 π ( 2 a + T ) ∈ ( 0 , c 2 ) at every pair. Consequently W T ( Y ) → D ( Y ) W_T(Y) \to D(Y) W T ( Y ) → D ( Y ) Teich ( Λ ) \operatorname{Teich}(\Lambda) Teich ( Λ ) T → ∞ T \to\infty T → ∞
Proof. Fix a pair of corresponding cusps of Y Y Y Y ‾ \overline{Y} Y q = e 2 π i z q=e^{2\pi iz} q = e 2 π i z q ′ = e 2 π i z ′ q'=e^{2\pi iz'} q ′ = e 2 π i z ′ c = e − 2 π a c=e^{-2\pi a} c = e − 2 π a ∣ q ∣ = e − 2 π y |q|=e^{-2\pi y} ∣ q ∣ = e − 2 π y A = { t T / c < ∣ q ∣ < c } A=\{t_T/c<|q|<c\} A = { t T / c < ∣ q ∣ < c } { a < y < a + T } \{a<y<a+T\} { a < y < a + T } Y Y Y { a < y ′ < a + T } \{a<y'<a+T\} { a < y ′ < a + T } Y ‾ \overline{Y} Y
The coordinates w w w w ′ w' w ′ C T C_T C T z z z z ′ z' z ′ A A A z = w z=w z = w A ′ A' A ′ z ′ = w ′ z'=w' z ′ = w ′ w ′ = − w + i ( 2 a + T ) w'=-w+i(2a+T) w ′ = − w + i ( 2 a + T )
q q ′ = e 2 π i ( w + w ′ ) = e − 2 π ( 2 a + T ) = t T , qq'=e^{2\pi i(w+w')}=e^{-2\pi(2a+T)}=t_T, q q ′ = e 2 π i ( w + w ′ ) = e − 2 π ( 2 a + T ) = t T , which is the same rule as in the plumbing construction. Hence W T ( Y ) = Y h ∪ C T ∪ Y ‾ h W_T(Y)=Y^h\cup C_T\cup\overline{Y}^h W T ( Y ) = Y h ∪ C T ∪ Y h Y ⊔ Y ‾ Y\sqcup\overline{Y} Y ⊔ Y { ∣ q ∣ ≤ t T / c } \{|q|\le t_T/c\} { ∣ q ∣ ≤ t T / c } { ∣ q ′ ∣ ≤ t T / c } \{|q'|\le t_T/c\} { ∣ q ′ ∣ ≤ t T / c } A A A A ′ A' A ′ q q ′ = t T qq'=t_T q q ′ = t T t T t_T t T f Y , T f_{Y,T} f Y , T Y Y Y Y ‾ \overline{Y} Y K c = Y h ⊔ Y ‾ h K_c=Y^h\sqcup\overline{Y}^h K c = Y h ⊔ Y h
Since a = 4 a=4 a = 4 c = e − 8 π ≤ e − π c=e^{-8\pi}\le e^{-\pi} c = e − 8 π ≤ e − π t T = c 2 e − 2 π T ∈ ( 0 , c 2 ) t_T=c^2e^{-2\pi T}\in(0,c^2) t T = c 2 e − 2 π T ∈ ( 0 , c 2 ) T → ∞ T\to\infty T → ∞ t T → 0 t_T\to0 t T → 0 Proposition 2.3 gives W T ( Y ) → D ( Y ) W_T(Y)\to D(Y) W T ( Y ) → D ( Y ) Teich ( Λ ) \operatorname{Teich}(\Lambda) Teich ( Λ )
The next lemma bounds the WP length along the path T ↦ W T ( Y ) T\mapsto W_T(Y) T ↦ W T ( Y )
Lemma 3.2. For every T > 0 T>0 T > 0
∥ d d T W T ( Y ) ∥ W P 2 ≤ 2 π n T 3 and d W P a u x ( W T ( Y ) , D ( Y ) ) ≤ 2 2 π n T . \left\|\frac{d}{dT}W_T(Y)\right\|_{\mathrm{WP}}^2\le\frac{2\pi n}{T^3}
\qquad\text{and}\qquad
d_{\mathrm{WP}}^{\mathrm{aux}}(W_T(Y),D(Y))\le2\sqrt{\frac{2\pi n}{T}}. d T d W T ( Y ) WP 2 ≤ T 3 2 π n and d WP aux ( W T ( Y ) , D ( Y )) ≤ 2 T 2 π n . Proof. Fix T > 0 T>0 T > 0 C T , 1 , … , C T , n C_{T,1},\ldots,C_{T,n} C T , 1 , … , C T , n W T ( Y ) W_T(Y) W T ( Y ) Y Y Y ζ T , k = x + i s \zeta_{T,k}=x+is ζ T , k = x + i s x ∈ R / Z x\in\mathbb{R}/\mathbb{Z} x ∈ R / Z 0 ≤ s ≤ T 0\le s\le T 0 ≤ s ≤ T C T , k C_{T,k} C T , k C T , k C_{T,k} C T , k w = ζ T , k + i a w=\zeta_{T,k}+ia w = ζ T , k + ia w ′ = − ζ T , k + i ( a + T ) w'=-\zeta_{T,k}+i(a+T) w ′ = − ζ T , k + i ( a + T ) z z z Y Y Y Y h Y^h Y h z ′ z' z ′ Y ‾ h \overline{Y}^h Y h { s = 0 } \{s = 0\} { s = 0 } { s = T } \{s = T\} { s = T } v T : [ 0 , T ] → [ 0 , 1 ] v_T\colon[0,T]\to[0,1] v T : [ 0 , T ] → [ 0 , 1 ]
v T ( s ) = { 0 , 0 ≤ s ≤ T / 4 , 2 s / T − 1 / 2 , T / 4 ≤ s ≤ 3 T / 4 , 1 , 3 T / 4 ≤ s ≤ T . v_T(s)=
\begin{cases}
0, & 0\le s\le T/4,\\
2s/T-1/2, & T/4\le s\le3T/4,\\
1, & 3T/4\le s\le T.
\end{cases} v T ( s ) = ⎩ ⎨ ⎧ 0 , 2 s / T − 1/2 , 1 , 0 ≤ s ≤ T /4 , T /4 ≤ s ≤ 3 T /4 , 3 T /4 ≤ s ≤ T . and for ∣ ϵ ∣ < T / 2 |\epsilon|<T/2 ∣ ϵ ∣ < T /2 ϕ ϵ ( s ) = s + ϵ v T ( s ) \phi_\epsilon(s)=s+\epsilon v_T(s) ϕ ϵ ( s ) = s + ϵ v T ( s ) F ϵ : W T ( Y ) → W T + ϵ ( Y ) F_\epsilon\colon W_T(Y)\to W_{T+\epsilon}(Y) F ϵ : W T ( Y ) → W T + ϵ ( Y )
ζ T + ϵ , k ( F ϵ ( x + i s ) ) = x + i ϕ ϵ ( s ) on C T , k , k = 1 , … , n , \zeta_{T+\epsilon,k}(F_\epsilon(x+is))=x+i\phi_\epsilon(s)\quad\text{on }C_{T,k},\ k=1,\ldots,n, ζ T + ϵ , k ( F ϵ ( x + i s )) = x + i ϕ ϵ ( s ) on C T , k , k = 1 , … , n , and by the identity on Y h Y^h Y h Y h ‾ \overline{Y^h} Y h Figure 5 . On { 0 ≤ s ≤ T / 4 } \{0\le s\le T/4\} { 0 ≤ s ≤ T /4 } ϕ ϵ ( s ) = s \phi_\epsilon(s)=s ϕ ϵ ( s ) = s w ∘ F ϵ = w w\circ F_\epsilon=w w ∘ F ϵ = w { 3 T / 4 ≤ s ≤ T } \{3T/4\le s\le T\} { 3 T /4 ≤ s ≤ T } ϕ ϵ ( s ) = s + ϵ \phi_\epsilon(s)=s+\epsilon ϕ ϵ ( s ) = s + ϵ w ′ = − ζ T + ϵ , k + i ( a + T + ϵ ) w'=-\zeta_{T+\epsilon,k}+i(a+T+\epsilon) w ′ = − ζ T + ϵ , k + i ( a + T + ϵ ) C T + ϵ , k C_{T+\epsilon,k} C T + ϵ , k
Figure 5. The map F ϵ : W T → W T + ϵ F_\epsilon\colon W_T\to W_{T+\epsilon} F ϵ : W T → W T + ϵ
w ′ ( F ϵ ( x + i s ) ) = − x + i ( a + T + ϵ − ( s + ϵ ) ) = − x + i ( a + T − s ) = w ′ ( x + i s ) . w'(F_\epsilon(x+is))=-x+i\bigl(a+T+\epsilon-(s+\epsilon)\bigr)=-x+i(a+T-s)=w'(x+is). w ′ ( F ϵ ( x + i s )) = − x + i ( a + T + ϵ − ( s + ϵ ) ) = − x + i ( a + T − s ) = w ′ ( x + i s ) . Thus, near each gluing circle, F ϵ F_\epsilon F ϵ Y h Y^h Y h Y h ‾ \overline{Y^h} Y h F ϵ F_\epsilon F ϵ F ϵ F_\epsilon F ϵ F ϵ ∘ f Y , T F_\epsilon\circ f_{Y,T} F ϵ ∘ f Y , T f Y , T + ϵ f_{Y,T+\epsilon} f Y , T + ϵ f Y f_Y f Y K K K K ‾ \overline{K} K F ϵ F_\epsilon F ϵ Y h Y^h Y h Y h ‾ \overline{Y^h} Y h c c c ( v , r ) (v,r) ( v , r ) { x = β Y , c ( v ) } \{x=\beta_{Y,c}(v)\} { x = β Y , c ( v )} C T + ϵ C_{T+\epsilon} C T + ϵ s = ϕ ϵ ( T r ) s=\phi_\epsilon(Tr) s = ϕ ϵ ( T r ) s = ( T + ϵ ) r s=(T+\epsilon)r s = ( T + ϵ ) r 0 0 0 T + ϵ T+\epsilon T + ϵ r r r 0 0 0 1 1 1 F ϵ F_\epsilon F ϵ W T ( Y ) W_T(Y) W T ( Y ) W T + ϵ ( Y ) W_{T+\epsilon}(Y) W T + ϵ ( Y ) W T ( Y ) W_T(Y) W T ( Y ) τ ↦ W τ ( Y ) \tau\mapsto W_\tau(Y) τ ↦ W τ ( Y ) Teich ( Λ ) \operatorname{Teich}(\Lambda) Teich ( Λ ) τ > 0 \tau>0 τ > 0
We compute the Beltrami coefficient of F ϵ F_\epsilon F ϵ C T , k C_{T,k} C T , k ζ = ζ T , k = x + i s \zeta=\zeta_{T,k}=x+is ζ = ζ T , k = x + i s F ϵ ( ζ ) = x + i ϕ ϵ ( s ) F_\epsilon(\zeta)=x+i\phi_\epsilon(s) F ϵ ( ζ ) = x + i ϕ ϵ ( s ) ∂ ζ = 1 2 ( ∂ x − i ∂ s ) \partial_\zeta=\frac{1}{2}(\partial_x-i\partial_s) ∂ ζ = 2 1 ( ∂ x − i ∂ s ) ∂ ζ ˉ = 1 2 ( ∂ x + i ∂ s ) \partial_{\bar\zeta}=\frac{1}{2}(\partial_x+i\partial_s) ∂ ζ ˉ = 2 1 ( ∂ x + i ∂ s ) ∂ ζ F ϵ = 1 2 ( 1 + ϕ ϵ ′ ( s ) ) \partial_\zeta F_\epsilon=\frac{1}{2}(1+\phi_\epsilon'(s)) ∂ ζ F ϵ = 2 1 ( 1 + ϕ ϵ ′ ( s )) ∂ ζ ˉ F ϵ = 1 2 ( 1 − ϕ ϵ ′ ( s ) ) \partial_{\bar\zeta}F_\epsilon=\frac{1}{2}(1-\phi_\epsilon'(s)) ∂ ζ ˉ F ϵ = 2 1 ( 1 − ϕ ϵ ′ ( s ))
∂ ζ ˉ F ϵ ∂ ζ F ϵ = 1 − ϕ ϵ ′ ( s ) 1 + ϕ ϵ ′ ( s ) . \frac{\partial_{\bar\zeta}F_\epsilon}{\partial_\zeta F_\epsilon}
=
\frac{1-\phi_\epsilon'(s)}{1+\phi_\epsilon'(s)}. ∂ ζ F ϵ ∂ ζ ˉ F ϵ = 1 + ϕ ϵ ′ ( s ) 1 − ϕ ϵ ′ ( s ) . Let M T , k = { T / 4 ≤ s ≤ 3 T / 4 } ⊂ C T , k M_{T,k}=\{T/4\le s\le3T/4\}\subset C_{T,k} M T , k = { T /4 ≤ s ≤ 3 T /4 } ⊂ C T , k C T , k C_{T,k} C T , k M T = M T , 1 ∪ ⋯ ∪ M T , n M_T=M_{T,1}\cup\cdots\cup M_{T,n} M T = M T , 1 ∪ ⋯ ∪ M T , n ϕ ε = 1 + 2 ε / T \phi_\varepsilon=1+2\varepsilon/T ϕ ε = 1 + 2 ε / T M T M_T M T ϕ ε ′ = 1 \phi'_\varepsilon=1 ϕ ε ′ = 1 F ε F_\varepsilon F ε
μ ε = − ε T + ε d ζ ˉ T , k d ζ T , k on M T , k , for k = 1 , … , n , (10) \mu_\varepsilon=-\frac{\varepsilon}{T+\varepsilon}\frac{d\bar{\zeta}_{T,k}}{d\zeta_{T,k}}\quad\text{on }M_{T,k},\text{ for }k=1,\ldots,n, \tag*{(10)} μ ε = − T + ε ε d ζ T , k d ζ ˉ T , k on M T , k , for k = 1 , … , n , (10) and μ ε = 0 \mu_\varepsilon=0 μ ε = 0 M T M_T M T F ε F_\varepsilon F ε ∣ μ ε ∣ < 1 |\mu_\varepsilon|<1 ∣ μ ε ∣ < 1 ε = 0 \varepsilon=0 ε = 0 W T ( Y ) W_T(Y) W T ( Y ) F 0 F_0 F 0 − ε / ( T + ε ) -\varepsilon/(T+\varepsilon) − ε / ( T + ε ) ε \varepsilon ε ε = 0 \varepsilon=0 ε = 0 − 1 / T -1/T − 1/ T W T ( Y ) W_T(Y) W T ( Y ) τ ↦ W τ ( Y ) \tau\mapsto W_\tau(Y) τ ↦ W τ ( Y )
μ F ( T ) = − 1 T d ζ ˉ T d ζ T , \mu_F(T)=-\frac{1}{T}\frac{d\bar{\zeta}_T}{d\zeta_T}, μ F ( T ) = − T 1 d ζ T d ζ ˉ T , where d ζ ˉ T d ζ T = d ζ ˉ T , k d ζ T , k \frac{d\bar{\zeta}_T}{d\zeta_T}=\frac{d\bar{\zeta}_{T,k}}{d\zeta_{T,k}} d ζ T d ζ ˉ T = d ζ T , k d ζ ˉ T , k M T , k M_{T,k} M T , k k = 1 , … , n k=1,\ldots,n k = 1 , … , n M T M_T M T (1) ,
∥ d d T W T ( Y ) ∥ W P 2 ≤ ∫ W T ( Y ) ∣ μ F ( T ) ∣ 2 d A = 1 T 2 ∑ k = 1 n area ( M T , k ) , (11) \left\|\frac{d}{dT}W_T(Y)\right\|_{\mathrm{WP}}^2\le\int_{W_T(Y)}|\mu_F(T)|^2\,dA=\frac{1}{T^2}\sum_{k=1}^{n}\operatorname{area}(M_{T,k}), \tag*{(11)} d T d W T ( Y ) WP 2 ≤ ∫ W T ( Y ) ∣ μ F ( T ) ∣ 2 d A = T 2 1 k = 1 ∑ n area ( M T , k ) , (11) where d A dA d A W T ( Y ) W_T(Y) W T ( Y )
To bound area ( M T , k ) \operatorname{area}(M_{T,k}) area ( M T , k ) W T ( Y ) W_T(Y) W T ( Y ) C T , k ∘ = { 0 < s < T } ⊂ C T , k C^\circ_{T,k}=\{0<s<T\}\subset C_{T,k} C T , k ∘ = { 0 < s < T } ⊂ C T , k [26 Section 2] . The open cylinder C T , k ∘ C^\circ_{T,k} C T , k ∘ { ζ ∈ C ∣ 0 < Im ( ζ ) < T } \{\zeta\in\mathbb{C}\mid0<\operatorname{Im}(\zeta)<T\} { ζ ∈ C ∣ 0 < Im ( ζ ) < T } ζ ↦ ζ + 1 \zeta\mapsto\zeta+1 ζ ↦ ζ + 1 C T , k ∘ C^\circ_{T,k} C T , k ∘ ζ ↦ ω = e π ζ / T = e π x / T e i π s / T \zeta\mapsto\omega=e^{\pi\zeta/T}=e^{\pi x/T}e^{i\pi s/T} ζ ↦ ω = e π ζ / T = e π x / T e iπ s / T H \mathbb{H} H arg ω = π s / T ∈ ( 0 , π ) \arg\omega=\pi s/T\in(0,\pi) arg ω = π s / T ∈ ( 0 , π ) d ω = π T ω d ζ d\omega=\frac{\pi}{T}\omega\,d\zeta d ω = T π ω d ζ Im ( ω ) = e π x / T sin ( π s / T ) = ∣ ω ∣ sin ( π s / T ) \operatorname{Im}(\omega)=e^{\pi x/T}\sin(\pi s/T)=|\omega|\sin(\pi s/T) Im ( ω ) = e π x / T sin ( π s / T ) = ∣ ω ∣ sin ( π s / T ) ∣ d ω ∣ / Im ( ω ) |d\omega|/\operatorname{Im}(\omega) ∣ d ω ∣/ Im ( ω ) H \mathbb{H} H
∣ d ω ∣ Im ( ω ) = π T ∣ ω ∣ ∣ d ζ ∣ ∣ ω ∣ sin ( π s / T ) = λ T ( s ) ∣ d ζ ∣ , where λ T ( s ) = π T csc ( π s T ) , 0 < s < T . \frac{|d\omega|}{\operatorname{Im}(\omega)}
=\frac{\frac{\pi}{T}|\omega|\,|d\zeta|}{|\omega|\sin(\pi s/T)}
=\lambda_T(s)|d\zeta|,\qquad\text{where }\lambda_T(s)=\frac{\pi}{T}\csc\left(\frac{\pi s}{T}\right),\quad0<s<T. Im ( ω ) ∣ d ω ∣ = ∣ ω ∣ sin ( π s / T ) T π ∣ ω ∣ ∣ d ζ ∣ = λ T ( s ) ∣ d ζ ∣ , where λ T ( s ) = T π csc ( T π s ) , 0 < s < T . Since λ T \lambda_T λ T s s s T T T ζ ↦ ζ + 1 \zeta\mapsto\zeta+1 ζ ↦ ζ + 1 C T , k ∘ C^\circ_{T,k} C T , k ∘ ζ = x + i s \zeta=x+is ζ = x + i s W T ( Y ) W_T(Y) W T ( Y ) ρ ( ζ ) ∣ d ζ ∣ \rho(\zeta)|d\zeta| ρ ( ζ ) ∣ d ζ ∣ d A = ρ ( ζ ) 2 d x d s dA=\rho(\zeta)^2\,dx\,ds d A = ρ ( ζ ) 2 d x d s C T , k ∘ ↪ W T ( Y ) C^\circ_{T,k}\hookrightarrow W_T(Y) C T , k ∘ ↪ W T ( Y ) H \mathbb{H} H W T ( Y ) W_T(Y) W T ( Y )
ρ ( ζ ) ≤ λ T ( s ) on C T , k ∘ . \rho(\zeta)\le\lambda_T(s)\quad\text{on }C^\circ_{T,k}. ρ ( ζ ) ≤ λ T ( s ) on C T , k ∘ . Hence
area ( M T , k ) ≤ ∫ 0 1 ∫ T / 4 3 T / 4 π 2 T 2 csc 2 ( π s T ) d s d x = π T [ − cot ( π s T ) ] T / 4 3 T / 4 = 2 π T , (12) \operatorname{area}(M_{T,k})\le\int_0^1\int_{T/4}^{3T/4}\frac{\pi^2}{T^2}\csc^2\left(\frac{\pi s}{T}\right)\,ds\,dx=\frac{\pi}{T}\left[-\cot\left(\frac{\pi s}{T}\right)\right]_{T/4}^{3T/4}=\frac{2\pi}{T}, \tag*{(12)} area ( M T , k ) ≤ ∫ 0 1 ∫ T /4 3 T /4 T 2 π 2 csc 2 ( T π s ) d s d x = T π [ − cot ( T π s ) ] T /4 3 T /4 = T 2 π , (12) and by combining (11) and (12) we obtain:
∥ d d T W T ( Y ) ∥ W P ≤ ( 1 T 2 ∑ k = 1 n area ( M T , k ) ) 1 / 2 ≤ 2 π n T 3 . \left\|\frac{d}{dT}W_T(Y)\right\|_{\mathrm{WP}}\le\left(\frac{1}{T^2}\sum_{k=1}^{n}\operatorname{area}(M_{T,k})\right)^{1/2}\le\sqrt{\frac{2\pi n}{T^3}}. d T d W T ( Y ) WP ≤ ( T 2 1 k = 1 ∑ n area ( M T , k ) ) 1/2 ≤ T 3 2 π n . Consequently, for T ≤ T ′ T\le T' T ≤ T ′
d W P ( W T ( Y ) , W T ′ ( Y ) ) ≤ ∫ T T ′ ∥ d d t W t ( Y ) ∥ W P d t ≤ ∫ T T ′ 2 π n t − 3 / 2 d t = 2 2 π n T − 2 2 π n T ′ ≤ 2 2 π n T . d_{\mathrm{WP}}(W_T(Y),W_{T'}(Y))\le\int_T^{T'}\left\|\frac{d}{dt}W_t(Y)\right\|_{\mathrm{WP}}\,dt\le\int_T^{T'}\sqrt{2\pi n}\,t^{-3/2}\,dt
=2\sqrt{\frac{2\pi n}{T}}-2\sqrt{\frac{2\pi n}{T'}}\le2\sqrt{\frac{2\pi n}{T}}. d WP ( W T ( Y ) , W T ′ ( Y )) ≤ ∫ T T ′ d t d W t ( Y ) WP d t ≤ ∫ T T ′ 2 π n t − 3/2 d t = 2 T 2 π n − 2 T ′ 2 π n ≤ 2 T 2 π n . Hence ( W t ( Y ) ) t ≥ T (W_t(Y))_{t\geq T} ( W t ( Y ) ) t ≥ T D ( Y ) D(Y) D ( Y ) T ′ → ∞ T'\to\infty T ′ → ∞
d W P a u x ( W T ( Y ) , D ( Y ) ) ≤ 2 2 π n T . d_{\mathrm{WP}}^{\mathrm{aux}}(W_T(Y),D(Y))\leq2\sqrt{\frac{2\pi n}{T}}. d WP aux ( W T ( Y ) , D ( Y )) ≤ 2 T 2 π n . Lengths of the doubled arcs
Set
T 0 = π a 2 = 2 π , b = a + T 0 2 = a + π a 4 = 4 + π , T_0=\frac{\pi a}{2}=2\pi,\qquad b=a+\frac{T_0}{2}=a+\frac{\pi a}{4}=4+\pi, T 0 = 2 π a = 2 π , b = a + 2 T 0 = a + 4 π a = 4 + π , and write W T 0 ( Y ) W_{T_0}(Y) W T 0 ( Y ) C T 0 , k C_{T_0,k} C T 0 , k W T 0 ( Y ) W_{T_0}(Y) W T 0 ( Y ) w = x + i y w=x+iy w = x + i y y = a + s ∈ [ a , a + T 0 ] y=a+s\in[a,a+T_0] y = a + s ∈ [ a , a + T 0 ] { y = a } \{y=a\} { y = a } z = x + i y z=x+iy z = x + i y Y Y Y { 1 / 2 < y ≤ a } ⊂ Y h \{1/2<y\leq a\}\subset Y^h { 1/2 < y ≤ a } ⊂ Y h { s = T 0 / 2 } \{s=T_0/2\} { s = T 0 /2 } C T 0 , k C_{T_0,k} C T 0 , k { y = b } \{y=b\} { y = b } w ′ = − x + i y ′ w'=-x+iy' w ′ = − x + i y ′ y ′ = a + T 0 − s = 2 b − y y'=a+T_0-s=2b-y y ′ = a + T 0 − s = 2 b − y
Let e e e T \mathcal{T} T Y Y Y c 1 c_1 c 1 c 2 c_2 c 2 e e e c 1 = c 2 c_1=c_2 c 1 = c 2 j = 1 , 2 j=1,2 j = 1 , 2 z j = x + i y z_j=x+iy z j = x + i y Y Y Y c j c_j c j z ˉ j = − z j ‾ \bar z_j=-\overline{z_j} z ˉ j = − z j Y ‾ \overline{Y} Y c j c_j c j w j = x + i y w_j=x+iy w j = x + i y w ˉ j = − x + i y ′ \bar w_j=-x+iy' w ˉ j = − x + i y ′ W T 0 ( Y ) W_{T_0}(Y) W T 0 ( Y ) c j c_j c j e ∩ Y h e\cap Y^h e ∩ Y h δ e ( Y ) \delta_e(Y) δ e ( Y ) ∂ Y h \partial Y^h ∂ Y h z 1 = x 1 + i a z_1=x_1+ia z 1 = x 1 + ia { y = a } \{y=a\} { y = a } c 1 c_1 c 1 z 2 = x 2 + i a z_2=x_2+ia z 2 = x 2 + ia { y = a } \{y=a\} { y = a } c 2 c_2 c 2 e e e [1 proof of Lemma 8.8] , near its endpoint at c j c_j c j e ∩ Y h e\cap Y^h e ∩ Y h { x = x j , 1 / 2 < y ≤ a } \{x=x_j,\ 1/2<y\leq a\} { x = x j , 1/2 < y ≤ a } z j z_j z j
We denote by d ( e ) d(e) d ( e ) W T 0 ( Y ) W_{T_0}(Y) W T 0 ( Y )
the segment e ∩ Y h e\cap Y^h e ∩ Y h z 1 = x 1 + i a z_1=x_1+ia z 1 = x 1 + ia z 2 = x 2 + i a z_2=x_2+ia z 2 = x 2 + ia
the vertical segment { x = x 2 , a ≤ y ≤ a + T 0 } \{x=x_2,\ a\leq y\leq a+T_0\} { x = x 2 , a ≤ y ≤ a + T 0 } c 2 c_2 c 2 w 2 w_2 w 2 w 2 = x 2 + i a w_2=x_2+ia w 2 = x 2 + ia w 2 ′ = − x 2 + i a w'_2=-x_2+ia w 2 ′ = − x 2 + ia
the mirror image of e ∩ Y h e\cap Y^h e ∩ Y h Y ‾ h \overline{Y}^h Y h z ˉ 2 = − x 2 + i a \bar z_2=-x_2+ia z ˉ 2 = − x 2 + ia z ˉ 1 = − x 1 + i a \bar z_1=-x_1+ia z ˉ 1 = − x 1 + ia
the vertical segment { x = x 1 , a ≤ y ≤ a + T 0 } \{x=x_1,\ a\leq y\leq a+T_0\} { x = x 1 , a ≤ y ≤ a + T 0 } c 1 c_1 c 1 w 1 w_1 w 1 w ˉ 1 = − x 1 + i a \bar w_1=-x_1+ia w ˉ 1 = − x 1 + ia w 1 = x 1 + i a w_1=x_1+ia w 1 = x 1 + ia
See Figure 6 . If c 1 = c 2 c_1=c_2 c 1 = c 2 x 1 ≠ x 2 x_1\neq x_2 x 1 = x 2
Figure 6. The doubled arc d ( e ) d(e) d ( e ) W T 0 ( Y ) W_{T_0}(Y) W T 0 ( Y ) e e e T \mathcal{T} T c 1 c_1 c 1 c 2 c_2 c 2
Lemma 3.3 (Doubled arc estimate). Let Y ∈ T e i c h g , n Y\in\mathrm{Teich}_{g,n} Y ∈ Teich g , n n ≥ 1 n\geq1 n ≥ 1 T \mathcal{T} T Y Y Y T 0 = 2 π T_0=2\pi T 0 = 2 π e e e T \mathcal{T} T W T 0 ( Y ) W_{T_0}(Y) W T 0 ( Y ) d ( e ) d(e) d ( e )
ℓ W T 0 ( Y ) ( d ( e ) ) ≤ 2 2 δ e ( Y ) + 4 log ( 1 + 2 ) . \ell_{W_{T_0}(Y)}(d(e))\leq2\sqrt{2}\,\delta_e(Y)+4\log(1+\sqrt{2}). ℓ W T 0 ( Y ) ( d ( e )) ≤ 2 2 δ e ( Y ) + 4 log ( 1 + 2 ) . Proof. We construct a conformal metric g ∗ g_* g ∗ W T 0 ( Y ) W_{T_0}(Y) W T 0 ( Y ) g W g_W g W W T 0 ( Y ) W_{T_0}(Y) W T 0 ( Y ) g ∗ g_* g ∗ d ( e ) d(e) d ( e ) ζ \zeta ζ g W = ρ ( ζ ) ∣ d ζ ∣ 2 g_W=\rho(\zeta)|d\zeta|^2 g W = ρ ( ζ ) ∣ d ζ ∣ 2 g ∗ = ρ ∗ ( ζ ) 2 ∣ d ζ ∣ 2 g_*=\rho_*(\zeta)^2|d\zeta|^2 g ∗ = ρ ∗ ( ζ ) 2 ∣ d ζ ∣ 2 J ⊂ W T 0 ( Y ) J\subset W_{T_0}(Y) J ⊂ W T 0 ( Y ) 2 n 2n 2 n { y = a } \{y=a\} { y = a } { y ′ = a } \{y'=a\} { y ′ = a } J = ∂ Y h ∪ ∂ Y ‾ h J=\partial Y^h\cup\partial\overline{Y}^h J = ∂ Y h ∪ ∂ Y h
On Y h Y^h Y h Y ‾ h \overline{Y}^h Y h g ∗ = 2 g Y g_*=2g_Y g ∗ = 2 g Y g ∗ = 2 g Y ‾ g_*=2g_{\overline{Y}} g ∗ = 2 g Y g Y g_Y g Y g Y ‾ g_{\overline{Y}} g Y Y Y Y Y ‾ \overline{Y} Y g ∗ = ρ ∗ ( y ) 2 ∣ d w ∣ 2 g_*=\rho_*(y)^2|dw|^2 g ∗ = ρ ∗ ( y ) 2 ∣ d w ∣ 2
ρ ∗ ( y ) = 1 a sec ( b − y a ) , a ≤ y ≤ a + T 0 . \rho_*(y)=\frac{1}{a}\sec\left(\frac{b-y}{a}\right),\qquad a\leq y\leq a+T_0. ρ ∗ ( y ) = a 1 sec ( a b − y ) , a ≤ y ≤ a + T 0 . Note that for a ≤ y ≤ a + T 0 a\leq y\leq a+T_0 a ≤ y ≤ a + T 0 ∣ b − y ∣ / a ≤ T 0 / ( 2 a ) = π / 4 < π / 2 |b-y|/a\leq T_0/(2a)=\pi/4<\pi/2 ∣ b − y ∣/ a ≤ T 0 / ( 2 a ) = π /4 < π /2 cos ( ( b − y ) / a ) ≥ cos ( π / 4 ) > 0 \cos((b-y)/a)\geq\cos(\pi/4)>0 cos (( b − y ) / a ) ≥ cos ( π /4 ) > 0 ρ ∗ \rho_* ρ ∗ 2 b = 2 a + T 0 2b=2a+T_0 2 b = 2 a + T 0 y ∈ [ a , a + T 0 ] y\in[a,a+T_0] y ∈ [ a , a + T 0 ] 2 b − y ∈ [ a , a + T 0 ] 2b-y\in[a,a+T_0] 2 b − y ∈ [ a , a + T 0 ] y ↦ 2 b − y y\mapsto2b-y y ↦ 2 b − y { y = b } \{y=b\} { y = b } { y = a } \{y=a\} { y = a } { y = a + T 0 } = { y ′ = a } \{y=a+T_0\}=\{y'=a\} { y = a + T 0 } = { y ′ = a } sec \sec sec
ρ ∗ ( 2 b − y ) = 1 a sec ( b − ( 2 b − y ) a ) = 1 a sec ( y − b a ) = 1 a sec ( b − y a ) = ρ ∗ ( y ) . \rho_*(2b-y)=\frac{1}{a}\sec\left(\frac{b-(2b-y)}{a}\right)=\frac{1}{a}\sec\left(\frac{y-b}{a}\right)=\frac{1}{a}\sec\left(\frac{b-y}{a}\right)=\rho_*(y). ρ ∗ ( 2 b − y ) = a 1 sec ( a b − ( 2 b − y ) ) = a 1 sec ( a y − b ) = a 1 sec ( a b − y ) = ρ ∗ ( y ) . Moreover the mirror coordinate is w ′ = − x + i y ′ w'=-x+iy' w ′ = − x + i y ′ y ′ = 2 b − y y'=2b-y y ′ = 2 b − y d w ′ = − d w dw'=-dw d w ′ = − d w ∣ d w ′ ∣ = ∣ d w ∣ |dw'|=|dw| ∣ d w ′ ∣ = ∣ d w ∣
g ∗ = ρ ∗ ( y ) 2 ∣ d w ∣ 2 = ρ ∗ ( 2 b − y ′ ) 2 ∣ d w ′ ∣ 2 = ρ ∗ ( y ′ ) 2 ∣ d w ′ ∣ 2 . g_*=\rho_*(y)^2|dw|^2=\rho_*(2b-y')^2|dw'|^2=\rho_*(y')^2|dw'|^2. g ∗ = ρ ∗ ( y ) 2 ∣ d w ∣ 2 = ρ ∗ ( 2 b − y ′ ) 2 ∣ d w ′ ∣ 2 = ρ ∗ ( y ′ ) 2 ∣ d w ′ ∣ 2 . Thus, by construction, g ∗ g_* g ∗ { 1 / 2 < y ≤ a } ⊂ Y h \{1/2<y\leq a\}\subset Y^h { 1/2 < y ≤ a } ⊂ Y h Y Y Y g Y = ∣ d z ∣ 2 / y 2 g_Y=|dz|^2/y^2 g Y = ∣ d z ∣ 2 / y 2
g ∗ = 2 g Y = 2 ∣ d z ∣ 2 y 2 = ( 2 y ) 2 ∣ d z ∣ 2 . g_*=2g_Y=\frac{2|dz|^2}{y^2}=\left(\frac{\sqrt{2}}{y}\right)^2|dz|^2. g ∗ = 2 g Y = y 2 2∣ d z ∣ 2 = ( y 2 ) 2 ∣ d z ∣ 2 . Since w = z w=z w = z w w w z z z g ∗ g_* g ∗ ρ ∗ ( y ) 2 ∣ d w ∣ 2 \rho_*(y)^2|dw|^2 ρ ∗ ( y ) 2 ∣ d w ∣ 2 { 1 / 2 < y ≤ a + T 0 } \{1/2<y\leq a+T_0\} { 1/2 < y ≤ a + T 0 }
ρ ∗ ( y ) = { 2 / y , 1 / 2 < y ≤ a , 1 a sec ( b − y a ) , a ≤ y ≤ a + T 0 , \rho_*(y)=
\begin{cases}
\sqrt{2}/y, & 1/2<y\leq a,\\
\frac{1}{a}\sec\left(\frac{b-y}{a}\right), & a\leq y\leq a+T_0,
\end{cases} ρ ∗ ( y ) = { 2 / y , a 1 sec ( a b − y ) , 1/2 < y ≤ a , a ≤ y ≤ a + T 0 , and, by symmetry, g ∗ = ρ ∗ ( y ′ ) 2 ∣ d w ′ ∣ 2 g_*=\rho_*(y')^2|dw'|^2 g ∗ = ρ ∗ ( y ′ ) 2 ∣ d w ′ ∣ 2 { y ′ = a } \{y'=a\} { y ′ = a } ρ ∗ ( y ′ ) = 2 / y ′ \rho_*(y')=\sqrt{2}/y' ρ ∗ ( y ′ ) = 2 / y ′ { 1 / 2 < y ′ ≤ a } ⊂ Y ‾ h \{1/2<y'\leq a\}\subset\overline{Y}^h { 1/2 < y ′ ≤ a } ⊂ Y h y = a y=a y = a 2 / a \sqrt{2}/a 2 / a ρ ∗ \rho_* ρ ∗ { y = a } \{y=a\} { y = a } { y ′ = a } \{y'=a\} { y ′ = a } g ∗ g_* g ∗ W T 0 ( Y ) W_{T_0}(Y) W T 0 ( Y ) W T 0 ( Y ) W_{T_0}(Y) W T 0 ( Y ) Y h Y^h Y h Y ‾ h \overline{Y}^h Y h C T 0 , 1 , … , C T 0 , n C_{T_0,1},\ldots,C_{T_0,n} C T 0 , 1 , … , C T 0 , n J J J ρ ∗ \rho_* ρ ∗ J J J
Claim 3.4. On each of the surfaces Y h Y^h Y h Y ‾ h \overline{Y}^h Y h C T 0 , k C_{T_0,k} C T 0 , k log ρ ∗ \log\rho_* log ρ ∗ W T 0 ( Y ) W_{T_0}(Y) W T 0 ( Y ) log ρ ∗ \log\rho_* log ρ ∗ C 1 C^1 C 1 W T 0 ( Y ) W_{T_0}(Y) W T 0 ( Y ) J J J log ρ ∗ \log\rho_* log ρ ∗
Proof of Claim 3.4. By symmetry it suffices to consider Y h Y^h Y h { y = a } \{y=a\} { y = a } Y h Y^h Y h ρ ∗ \rho_* ρ ∗ 2 g Y 2g_Y 2 g Y ∂ Y h \partial Y^h ∂ Y h 2 / y \sqrt{2}/y 2 / y y > 0 y>0 y > 0 C T 0 , k C_{T_0,k} C T 0 , k 1 a sec ( b − y a ) \frac{1}{a}\sec\left(\frac{b-y}{a}\right) a 1 sec ( a b − y ) ∣ b − y ∣ < π a / 2 |b-y|<\pi a/2 ∣ b − y ∣ < π a /2 [ a , a + T 0 ] [a,a+T_0] [ a , a + T 0 ] b − a = π a / 4 b-a=\pi a/4 b − a = π a /4 y = a y=a y = a ρ ∗ \rho_* ρ ∗ 2 / a \sqrt{2}/a 2 / a y y y ∂ x log ρ ∗ = 0 \partial_x\log\rho_*=0 ∂ x log ρ ∗ = 0 y y y y = a y=a y = a Y h Y^h Y h ( log ρ ∗ ) ′ ( y ) = − 1 / y (\log\rho_*)'(y)=-1/y ( log ρ ∗ ) ′ ( y ) = − 1/ y − 1 / a -1/a − 1/ a ( log ρ ∗ ) ′ ( y ) = − 1 a tan ( b − y a ) (\log\rho_*)'(y)=-\frac{1}{a}\tan\left(\frac{b-y}{a}\right) ( log ρ ∗ ) ′ ( y ) = − a 1 tan ( a b − y ) − 1 a tan ( π / 4 ) = − 1 / a -\frac{1}{a}\tan(\pi/4)=-1/a − a 1 tan ( π /4 ) = − 1/ a
The curvature of g ∗ = ρ ∗ 2 ∣ d ζ ∣ 2 g_*=\rho_*^2|d\zeta|^2 g ∗ = ρ ∗ 2 ∣ d ζ ∣ 2 K ∗ = − ρ ∗ − 2 Δ log ρ ∗ K_*=-\rho_*^{-2}\Delta\log\rho_* K ∗ = − ρ ∗ − 2 Δ log ρ ∗ Δ = ∂ x 2 + ∂ y 2 \Delta=\partial_x^2+\partial_y^2 Δ = ∂ x 2 + ∂ y 2 ζ = x + i y \zeta=x+iy ζ = x + i y Y h Y^h Y h Y ‾ h \overline{Y}^h Y h K ∗ = − 1 / 2 K_*=-1/2 K ∗ = − 1/2 g ∗ g_* g ∗ ρ ∗ \rho_* ρ ∗ y y y
Δ log ρ ∗ = ( log ρ ∗ ) ′ ′ ( y ) = 1 a 2 sec 2 ( b − y a ) = ρ ∗ ( y ) 2 , \Delta\log\rho_*=(\log\rho_*)''(y)=\frac{1}{a^2}\sec^2\left(\frac{b-y}{a}\right)=\rho_*(y)^2, Δ log ρ ∗ = ( log ρ ∗ ) ′′ ( y ) = a 2 1 sec 2 ( a b − y ) = ρ ∗ ( y ) 2 , so K ∗ = − 1 K_*=-1 K ∗ = − 1 K ∗ ≥ − 1 K_*\geq-1 K ∗ ≥ − 1 W T 0 ( Y ) ∖ J W_{T_0}(Y)\setminus J W T 0 ( Y ) ∖ J
Δ log ρ ∗ ≤ ρ ∗ 2 on W T 0 ( Y ) ∖ J . (13) \Delta\log\rho_*\leq\rho_*^2 \quad\text{on } W_{T_0}(Y)\setminus J. \tag*{(13)} Δ log ρ ∗ ≤ ρ ∗ 2 on W T 0 ( Y ) ∖ J . (13) Note that on J J J log ρ ∗ \log\rho_* log ρ ∗ C 1 C^1 C 1 y = a y=a y = a Y h Y^h Y h 1 a 2 sec 2 ( π / 4 ) = 2 / a 2 \frac{1}{a^2}\sec^2(\pi/4)=2/a^2 a 2 1 sec 2 ( π /4 ) = 2/ a 2 Δ log ρ ∗ \Delta\log\rho_* Δ log ρ ∗ J J J (13) holds on each component of $W_{T_0}(Y)\setminus J.
The hyperbolic metric of W T 0 ( Y ) W_{T_0}(Y) W T 0 ( Y ) − 1 -1 − 1 Δ log ρ = ρ 2 \Delta\log\rho=\rho^2 Δ log ρ = ρ 2 g W g_W g W g ∗ g_* g ∗ W T 0 ( Y ) W_{T_0}(Y) W T 0 ( Y )
g W = e 2 u g ∗ , g_W=e^{2u}g_*, g W = e 2 u g ∗ , that is ρ 2 = ρ ∗ 2 e 2 u \rho^2=\rho_*^2e^{2u} ρ 2 = ρ ∗ 2 e 2 u ζ \zeta ζ g W = ρ ( ζ ) 2 ∣ d ζ ∣ 2 g_W=\rho(\zeta)^2|d\zeta|^2 g W = ρ ( ζ ) 2 ∣ d ζ ∣ 2 g ∗ = ρ ∗ ( ζ ) 2 ∣ d ζ ∣ 2 g_*=\rho_*(\zeta)^2|d\zeta|^2 g ∗ = ρ ∗ ( ζ ) 2 ∣ d ζ ∣ 2 ξ \xi ξ ∣ d ζ / d ξ ∣ |d\zeta/d\xi| ∣ d ζ / d ξ ∣ ρ / ρ ∗ \rho/\rho_* ρ / ρ ∗ u = log ( ρ / ρ ∗ ) u=\log(\rho/\rho_*) u = log ( ρ / ρ ∗ ) W T 0 ( Y ) W_{T_0}(Y) W T 0 ( Y ) log ρ \log\rho log ρ W T 0 W_{T_0} W T 0 u = log ρ − log ρ ∗ u=\log\rho-\log\rho_* u = log ρ − log ρ ∗ C 1 C^1 C 1 W T 0 W_{T_0} W T 0 Y h Y^h Y h Y ‾ h \overline{Y}^h Y h C T 0 , 1 , … , C T 0 , n C_{T_0,1},\ldots,C_{T_0,n} C T 0 , 1 , … , C T 0 , n J J J u u u C 1 C^1 C 1
Claim 3.5. We have u ≤ 0 u\leq0 u ≤ 0 W W W g W ≤ g ∗ g_W\leq g_* g W ≤ g ∗
Proof of Claim 3.5. By (13) , on W T 0 ( Y ) ∖ J W_{T_0}(Y)\setminus J W T 0 ( Y ) ∖ J
Δ u = ρ 2 − Δ log ρ ∗ ≥ ρ 2 − ρ ∗ 2 = ρ ∗ 2 ( e 2 u − 1 ) . (14) \Delta u=\rho^2-\Delta\log\rho_*\geq\rho^2-\rho_*^2=\rho_*^2(e^{2u}-1). \tag*{(14)} Δ u = ρ 2 − Δ log ρ ∗ ≥ ρ 2 − ρ ∗ 2 = ρ ∗ 2 ( e 2 u − 1 ) . (14) Since W T 0 ( Y ) W_{T_0}(Y) W T 0 ( Y ) u u u M M M p ∈ W T 0 ( Y ) p\in W_{T_0}(Y) p ∈ W T 0 ( Y ) M > 0 M>0 M > 0
If p ∉ J p\notin J p ∈ / J u u u p p p Δ u ( p ) ≤ 0 \Delta u(p)\leq0 Δ u ( p ) ≤ 0 p p p (14) gives Δ u ( p ) ≥ ρ ∗ ( p ) 2 ( e 2 M − 1 ) > 0 \Delta u(p)\geq\rho_*(p)^2(e^{2M}-1)>0 Δ u ( p ) ≥ ρ ∗ ( p ) 2 ( e 2 M − 1 ) > 0
If p ∈ J p \in J p ∈ J p p p { y = a } \{y=a\} { y = a } p = x 0 + i a p=x_0+ia p = x 0 + ia w w w z z z u + u_+ u + u u u C T 0 , k C_{T_0,k} C T 0 , k p p p C T 0 , k C_{T_0,k} C T 0 , k { y = a } \{y=a\} { y = a } u u u C 1 C^1 C 1 p p p ∂ x u ( p ) = ∂ y u ( p ) = 0 \partial_xu(p)=\partial_yu(p)=0 ∂ x u ( p ) = ∂ y u ( p ) = 0 x ↦ u + ( x , a ) x\mapsto u_+(x,a) x ↦ u + ( x , a ) x 0 x_0 x 0 ∂ x 2 u + ( p ) ≤ 0 \partial_x^2u_+(p)\leq0 ∂ x 2 u + ( p ) ≤ 0 y ↦ u + ( x 0 , y ) y\mapsto u_+(x_0,y) y ↦ u + ( x 0 , y ) y ≥ a y\geq a y ≥ a y = a y=a y = a ∂ y 2 u + ( p ) ≤ 0 \partial_y^2u_+(p)\leq0 ∂ y 2 u + ( p ) ≤ 0 u + ( x 0 , y ) > M u_+(x_0,y)>M u + ( x 0 , y ) > M y > a y>a y > a a a a Δ u + ( p ) ≤ 0 \Delta u_+(p)\leq0 Δ u + ( p ) ≤ 0 y → a + y\to a^+ y → a + (14) gives Δ u + ( p ) ≥ ρ ∗ ( p ) 2 ( e 2 M − 1 ) > 0 \Delta u_+(p)\geq\rho_*(p)^2(e^{2M}-1)>0 Δ u + ( p ) ≥ ρ ∗ ( p ) 2 ( e 2 M − 1 ) > 0
Therefore u ≤ 0 u\leq0 u ≤ 0 g W ≤ g ∗ g_W\leq g_* g W ≤ g ∗
The closed geodesic freely homotopic to d ( e ) d(e) d ( e ) g W g_W g W
ℓ W T 0 ( Y ) ( d ( e ) ) ≤ ℓ g W ( d ( e ) ) ≤ ℓ g ∗ ( d ( e ) ) . \ell_{W_{T_0}(Y)}(d(e))\leq\ell_{g_W}(d(e))\leq\ell_{g_*}(d(e)). ℓ W T 0 ( Y ) ( d ( e )) ≤ ℓ g W ( d ( e )) ≤ ℓ g ∗ ( d ( e )) . Since g ∗ = 2 g Y g_*=2g_Y g ∗ = 2 g Y Y h Y^h Y h g ∗ = 2 g Y ‾ g_*=2g_{\overline{Y}} g ∗ = 2 g Y Y ‾ h \overline{Y}^h Y h e ∩ Y h e\cap Y^h e ∩ Y h g ∗ g_* g ∗ 2 δ e ( Y ) \sqrt{2}\,\delta_e(Y) 2 δ e ( Y ) j = 1 , 2 j=1,2 j = 1 , 2 { x = x j , a ≤ y ≤ a + T 0 } \{x=x_j,\ a\leq y\leq a+T_0\} { x = x j , a ≤ y ≤ a + T 0 } c j c_j c j y = Im ( w j ) y=\operatorname{Im}(w_j) y = Im ( w j ) g ∗ g_* g ∗
∫ a a + T 0 1 a sec ( b − y a ) d y = ∫ − π / 4 π / 4 sec t d t = 2 log ( 1 + 2 ) , \int_a^{a+T_0}\frac{1}{a}\sec\left(\frac{b-y}{a}\right)\,dy=\int_{-\pi/4}^{\pi/4}\sec t\,dt=2\log(1+\sqrt{2}), ∫ a a + T 0 a 1 sec ( a b − y ) d y = ∫ − π /4 π /4 sec t d t = 2 log ( 1 + 2 ) , Summing these four contributions gives ℓ g ∗ ( d ( e ) ) = 2 2 δ e ( Y ) + 4 log ( 1 + 2 ) \ell_{g_*}(d(e))=2\sqrt{2}\,\delta_e(Y)+4\log(1+\sqrt{2}) ℓ g ∗ ( d ( e )) = 2 2 δ e ( Y ) + 4 log ( 1 + 2 )
Compatible markings and a common maximally noded surface
In this section we construct a pants decomposition P T P_{\mathcal T} P T Λ \Lambda Λ f Y , T 0 f_{Y,T_0} f Y , T 0 d ( e ) d(e) d ( e ) W T 0 W_{T_0} W T 0 e ∈ T e\in\mathcal T e ∈ T P T P_{\mathcal T} P T P T P_{\mathcal T} P T Teich ( Λ ) \operatorname{Teich}(\Lambda) Teich ( Λ ) § 3.1 that Λ \Lambda Λ K K K K ‾ \overline{K} K ∂ K × [ 0 , 1 ] \partial K\times[0,1] ∂ K × [ 0 , 1 ] W T ( Y ) W_T(Y) W T ( Y ) f Y , T : Λ → W T ( Y ) f_{Y,T}\colon\Lambda\to W_T(Y) f Y , T : Λ → W T ( Y ) K K K Y h Y^h Y h K ‾ \overline{K} K Y ‾ h \overline{Y}^h Y h Λ \Lambda Λ
Fix pairwise disjoint simple arcs a e ⊂ K a_e\subset K a e ⊂ K e ∈ T e\in\mathcal T e ∈ T ∂ K \partial K ∂ K a e a_e a e e e e a e a_e a e Σ \Sigma Σ e e e e ∈ T e\in\mathcal T e ∈ T d Λ ( e ) ⊂ Λ d_\Lambda(e)\subset\Lambda d Λ ( e ) ⊂ Λ a e a_e a e a ‾ e ⊂ K ‾ \overline{a}_e\subset\overline{K} a e ⊂ K a e a_e a e d Λ ( e ) d_\Lambda(e) d Λ ( e )
Lemma 3.6. Let T \mathcal T T Σ \Sigma Σ d Λ ( e ) ⊂ Λ d_\Lambda(e)\subset\Lambda d Λ ( e ) ⊂ Λ e ∈ T e\in\mathcal T e ∈ T
For every Y ∈ Teich ( Σ ) Y\in\operatorname{Teich}(\Sigma) Y ∈ Teich ( Σ ) e ∈ T e\in\mathcal T e ∈ T f Y , T 0 ( d Λ ( e ) ) f_{Y,T_0}(d_\Lambda(e)) f Y , T 0 ( d Λ ( e )) W T 0 ( Y ) W_{T_0}(Y) W T 0 ( Y ) d ( e ) d(e) d ( e ) § 3.2 .
The curves d Λ ( e ) d_\Lambda(e) d Λ ( e ) e ∈ T e\in\mathcal T e ∈ T P T P_{\mathcal T} P T Λ \Lambda Λ S ( P T ) ⊂ Teich ( Λ ) S(P_{\mathcal T})\subset\operatorname{Teich}(\Lambda) S ( P T ) ⊂ Teich ( Λ ) P T P_{\mathcal T} P T
Proof. (1) Let c 1 c_1 c 1 c 2 c_2 c 2 e e e d ( e ) d(e) d ( e ) § 3.2 ). We orient e e e c 1 c_1 c 1 c 2 c_2 c 2 a e a_e a e v 1 ∈ ∂ 1 K v_1\in\partial_1K v 1 ∈ ∂ 1 K v 2 ∈ ∂ 2 K v_2\in\partial_2K v 2 ∈ ∂ 2 K f Y , T 0 ( d Λ ( e ) ) f_{Y,T_0}(d_\Lambda(e)) f Y , T 0 ( d Λ ( e )) f Y ( a e ) ⊂ Y h f_Y(a_e)\subset Y^h f Y ( a e ) ⊂ Y h f Y ( v 2 ) f_Y(v_2) f Y ( v 2 ) f Y ( a e ) f_Y(a_e) f Y ( a e ) Y ‾ h \overline{Y}^h Y h
::: the vertical segment of the cylinder at f Y ( v 1 ) f_Y(v_1) f Y ( v 1 ) d ( e ) d(e) d ( e ) e ∩ Y h e \cap Y^h e ∩ Y h a e a_e a e e e e ( γ t ) t ∈ [ 0 , 1 ] (\gamma_t)_{t\in[0,1]} ( γ t ) t ∈ [ 0 , 1 ] γ 0 = f Y ( a e ) \gamma_0=f_Y(a_e) γ 0 = f Y ( a e ) γ 1 = e ∩ Y h \gamma_1=e\cap Y^h γ 1 = e ∩ Y h Y h Y^h Y h { y = a } \{y=a\} { y = a } c 1 c_1 c 1 { y = a } \{y=a\} { y = a } c 2 c_2 c 2 z j z_j z j z j ′ z'_j z j ′ w j w_j w j w j ′ w'_j w j ′ j = 1 , 2 j=1,2 j = 1 , 2 d ( e ) d(e) d ( e ) γ t \gamma_t γ t
z 1 = x 1 ( t ) + i a and z 2 = x 2 ( t ) + i a , z_1=x_1(t)+ia \qquad\text{and}\qquad z_2=x_2(t)+ia, z 1 = x 1 ( t ) + ia and z 2 = x 2 ( t ) + ia , where x 1 , x 2 : [ 0 , 1 ] → R / Z x_1,x_2:[0,1]\to\mathbb{R}/\mathbb{Z} x 1 , x 2 : [ 0 , 1 ] → R / Z t ∈ [ 0 , 1 ] t\in[0,1] t ∈ [ 0 , 1 ] δ t \delta_t δ t W T 0 ( Y ) W_{T_0}(Y) W T 0 ( Y )
(a) the path γ t \gamma_t γ t Y h Y^h Y h z 1 = x 1 ( t ) + i a z_1=x_1(t)+ia z 1 = x 1 ( t ) + ia z 2 = x 2 ( t ) + i a z_2=x_2(t)+ia z 2 = x 2 ( t ) + ia
(b) the vertical segment { x = x 2 ( t ) , a ≤ y ≤ a + T 0 } \{x=x_2(t),\ a\le y\le a+T_0\} { x = x 2 ( t ) , a ≤ y ≤ a + T 0 } c 2 c_2 c 2 w 2 = x 2 ( t ) + i a w_2=x_2(t)+ia w 2 = x 2 ( t ) + ia w 2 ′ = − x 2 ( t ) + i a w'_2=-x_2(t)+ia w 2 ′ = − x 2 ( t ) + ia
(c) the mirror image of γ t \gamma_t γ t Y h ‾ \overline{Y^h} Y h z 2 ′ = − x 2 ( t ) + i a z'_2=-x_2(t)+ia z 2 ′ = − x 2 ( t ) + ia z 1 ′ = − x 1 ( t ) + i a z'_1=-x_1(t)+ia z 1 ′ = − x 1 ( t ) + ia
(d) the vertical segment { x = x 1 ( t ) , a ≤ y ≤ a + T 0 } \{x=x_1(t),\ a\le y\le a+T_0\} { x = x 1 ( t ) , a ≤ y ≤ a + T 0 } c 1 c_1 c 1 w 1 ′ = − x 1 ( t ) + i a w'_1=-x_1(t)+ia w 1 ′ = − x 1 ( t ) + ia w 1 = x 1 ( t ) + i a w_1=x_1(t)+ia w 1 = x 1 ( t ) + ia
Each piece ends where the next one starts, since w j w_j w j w j ′ w'_j w j ′ z j z_j z j z j ′ z'_j z j ′ z j = x j ( t ) + i a z_j=x_j(t)+ia z j = x j ( t ) + ia z j ′ = − x j ( t ) + i a z'_j=-x_j(t)+ia z j ′ = − x j ( t ) + ia δ t \delta_t δ t δ t \delta_t δ t t t t t = 0 t=0 t = 0 γ 0 = f Y ( a e ) \gamma_0=f_Y(a_e) γ 0 = f Y ( a e ) x j ( 0 ) = β Y , c j ( v j ) x_j(0)=\beta_{Y,c_j}(v_j) x j ( 0 ) = β Y , c j ( v j ) δ 0 = f Y , T 0 ( d Λ ( e ) ) \delta_0=f_{Y,T_0}(d_\Lambda(e)) δ 0 = f Y , T 0 ( d Λ ( e )) t = 1 t=1 t = 1 γ 1 = e ∩ Y h \gamma_1=e\cap Y^h γ 1 = e ∩ Y h x j ( 1 ) = x j x_j(1)=x_j x j ( 1 ) = x j δ 1 = d ( e ) \delta_1=d(e) δ 1 = d ( e ) f Y , T 0 ( d Λ ( e ) ) f_{Y,T_0}(d_\Lambda(e)) f Y , T 0 ( d Λ ( e )) d ( e ) d(e) d ( e )
(2) The curves d Λ ( e ) d_\Lambda(e) d Λ ( e ) e ∈ T e\in\mathcal{T} e ∈ T a e a_e a e Λ \Lambda Λ K K K a e a_e a e H τ H_\tau H τ τ \tau τ T \mathcal{T} T ∂ K \partial K ∂ K τ \tau τ K ‾ \overline{K} K H τ ‾ \overline{H_\tau} H τ I × [ 0 , 1 ] I\times[0,1] I × [ 0 , 1 ] I I I I ⊂ ∂ H τ I\subset\partial H_\tau I ⊂ ∂ H τ I ‾ ⊂ ∂ H τ ‾ \overline{I}\subset\partial\overline{H_\tau} I ⊂ ∂ H τ
P τ = H τ ∪ H τ ‾ ∪ (three rectangles) , P_\tau=H_\tau\cup\overline{H_\tau}\cup\text{(three rectangles)}, P τ = H τ ∪ H τ ∪ (three rectangles) , one for each triangle τ \tau τ T \mathcal{T} T P τ = A ∪ B P_\tau=A\cup B P τ = A ∪ B A = H τ ⊔ H τ ‾ A=H_\tau\sqcup\overline{H_\tau} A = H τ ⊔ H τ B B B A ∩ B A\cap B A ∩ B 1 1 1
χ ( P τ ) = χ ( A ) + χ ( B ) − χ ( A ∩ B ) = 2 + 3 − 6 = − 1. \chi(P_\tau)=\chi(A)+\chi(B)-\chi(A\cap B)=2+3-6=-1. χ ( P τ ) = χ ( A ) + χ ( B ) − χ ( A ∩ B ) = 2 + 3 − 6 = − 1. Write e 1 , e 2 , e 3 e_1,e_2,e_3 e 1 , e 2 , e 3 τ \tau τ I k I_k I k H τ H_\tau H τ a e k a_{e_k} a e k a e k + 1 a_{e_{k+1}} a e k + 1 H τ H_\tau H τ a e 1 , I 1 , a e 2 , I 2 , a e 3 , I 3 a_{e_1},I_1,a_{e_2},I_2,a_{e_3},I_3 a e 1 , I 1 , a e 2 , I 2 , a e 3 , I 3 Figure 7 . Note that two sides of τ \tau τ P τ P_\tau P τ I k I_k I k H τ H_\tau H τ H τ ‾ \overline{H_\tau} H τ I k × [ 0 , 1 ] I_k\times[0,1] I k × [ 0 , 1 ] P τ P_\tau P τ H τ H_\tau H τ H τ ‾ \overline{H_\tau} H τ a e k a_{e_k} a e k I k − 1 I_{k-1} I k − 1 I k I_k I k P τ P_\tau P τ a e k a_{e_k} a e k a e k a_{e_k} a e k I k × [ 0 , 1 ] I_k\times[0,1] I k × [ 0 , 1 ] a e k a_{e_k} a e k I k I_k I k a ‾ e k \overline{a}_{e_k} a e k H τ ‾ \overline{H_\tau} H τ I k − 1 × [ 0 , 1 ] I_{k-1}\times[0,1] I k − 1 × [ 0 , 1 ] I k − 1 I_{k-1} I k − 1 a e k a_{e_k} a e k d Λ ( e k ) d_\Lambda(e_k) d Λ ( e k ) P τ P_\tau P τ χ ( P τ ) = − 1 \chi(P_\tau)=-1 χ ( P τ ) = − 1 P τ P_\tau P τ
Figure 7. The pair of pants P τ P_{\tau} P τ τ \tau τ e 1 , e 2 , e 3 e_1,e_2,e_3 e 1 , e 2 , e 3
Since every component of Λ \Lambda Λ d Λ ( e ) d_\Lambda(e) d Λ ( e ) P τ P_\tau P τ d Λ ( e ) d_\Lambda(e) d Λ ( e )
Figure 8. A pants decomposition P T P_{\mathcal{T}} P T Y Y Y T \mathcal{T} T
Hence the curves d Λ ( e ) d_{\Lambda}(e) d Λ ( e ) P T P_{\mathcal{T}} P T Λ \Lambda Λ
Remark 3.7. The curves d Λ ( e ) d_{\Lambda}(e) d Λ ( e ) K K K a e a_e a e Y ∈ Teich ( Σ ) Y \in\operatorname{Teich}(\Sigma) Y ∈ Teich ( Σ ) P T P_{\mathcal{T}} P T Λ \Lambda Λ P T P_{\mathcal{T}} P T Y Y Y
Corollary 3.8. For every Y ∈ Teich ( Σ ) Y \in\operatorname{Teich}(\Sigma) Y ∈ Teich ( Σ )
d W P a u x ( D ( Y ) , P T ) ≤ 2 n + 2 π ( 2 2 L T ( Y ) + 4 E log ( 1 + 2 ) ) . d_{\mathrm{WP}}^{\mathrm{aux}}(D(Y),P_{\mathcal{T}}) \le2\sqrt{n}+\sqrt{2\pi\left(2\sqrt{2}\mathcal{L}_{\mathcal{T}}(Y)+4E\log(1+\sqrt{2})\right)}. d WP aux ( D ( Y ) , P T ) ≤ 2 n + 2 π ( 2 2 L T ( Y ) + 4 E log ( 1 + 2 ) ) . Proof. By Lemma 3.6, the curve f Y , T 0 ( d Λ ( e ) ) f_{Y,\mathcal{T}_0}(d_{\Lambda}(e)) f Y , T 0 ( d Λ ( e )) d ( e ) d(e) d ( e ) W T 0 ( Y ) W_{\mathcal{T}_0}(Y) W T 0 ( Y ) Teich ( Λ ) \operatorname{Teich}(\Lambda) Teich ( Λ ) W T 0 ( Y ) W_{\mathcal{T}_0}(Y) W T 0 ( Y ) P T P_{\mathcal{T}} P T
d W P a u x ( W T 0 ( Y ) , P T ) ≤ 2 π ∑ e ∈ T ℓ W T 0 ( Y ) ( d ( e ) ) , d_{\mathrm{WP}}^{\mathrm{aux}}(W_{\mathcal{T}_0}(Y),P_{\mathcal{T}}) \le\sqrt{2\pi\sum_{e\in\mathcal{T}}\ell_{W_{\mathcal{T}_0}(Y)}(d(e))}, d WP aux ( W T 0 ( Y ) , P T ) ≤ 2 π e ∈ T ∑ ℓ W T 0 ( Y ) ( d ( e )) , and Lemma 3.3 gives
d W P a u x ( W T 0 ( Y ) , P T ) ≤ 2 π ( 2 2 L T ( Y ) + 4 E log ( 1 + 2 ) ) . (15) d_{\mathrm{WP}}^{\mathrm{aux}}(W_{\mathcal{T}_0}(Y),P_{\mathcal{T}}) \le\sqrt{2\pi\left(2\sqrt{2}\mathcal{L}_{\mathcal{T}}(Y)+4E\log(1+\sqrt{2})\right)}. \tag*{(15)} d WP aux ( W T 0 ( Y ) , P T ) ≤ 2 π ( 2 2 L T ( Y ) + 4 E log ( 1 + 2 ) ) . (15) By Lemma 3.2, for T = T 0 = 2 π T=T_0=2\pi T = T 0 = 2 π
d W P a u x ( W T 0 ( Y ) , D ( Y ) ) ≤ 2 2 π n / T 0 = 2 n . (16) d_{\mathrm{WP}}^{\mathrm{aux}}(W_{T_0}(Y),D(Y))\leq2\sqrt{2\pi n/T_0}=2\sqrt{n}. \tag*{(16)} d WP aux ( W T 0 ( Y ) , D ( Y )) ≤ 2 2 π n / T 0 = 2 n . (16) Thus, by the triangle inequality, from (15) and (16) we obtain
d W P a u x ( D ( Y ) , P T ) ≤ d W P a u x ( D ( Y ) , W T 0 ( Y ) ) + d W P a u x ( W T 0 ( Y ) , P T ) ≤ 2 n + 2 π ( 2 2 L T ( Y ) + 4 E log ( 1 + 2 ) ) . \begin{aligned}
d_{\mathrm{WP}}^{\mathrm{aux}}(D(Y),P_T)&\leq d_{\mathrm{WP}}^{\mathrm{aux}}(D(Y),W_{T_0}(Y))+d_{\mathrm{WP}}^{\mathrm{aux}}(W_{T_0}(Y),P_T)\\
&\leq2\sqrt{n}+\sqrt{2\pi\left(2\sqrt{2}\mathcal{L}_T(Y)+4E\log(1+\sqrt{2})\right)}.
\end{aligned} d WP aux ( D ( Y ) , P T ) ≤ d WP aux ( D ( Y ) , W T 0 ( Y )) + d WP aux ( W T 0 ( Y ) , P T ) ≤ 2 n + 2 π ( 2 2 L T ( Y ) + 4 E log ( 1 + 2 ) ) . The distance in the auxiliary stratum
Let Y , Z ∈ Teich ( Σ ) Y,Z\in\operatorname{Teich}(\Sigma) Y , Z ∈ Teich ( Σ ) D ( Y ) D(Y) D ( Y ) D ( Z ) D(Z) D ( Z ) S ( C ) ⊂ Teich ( Λ ) S(C)\subset\operatorname{Teich}(\Lambda) S ( C ) ⊂ Teich ( Λ ) C ⊂ Λ C\subset\Lambda C ⊂ Λ D ( Y ) D(Y) D ( Y ) D ( Z ) D(Z) D ( Z ) [20 Section 2.4] .
Lemma 3.9. For every Y , Z ∈ Teich ( Σ ) Y,Z\in\operatorname{Teich}(\Sigma) Y , Z ∈ Teich ( Σ )
d W P a u x ( D ( Y ) , D ( Z ) ) = 2 d W P ( Y , Z ) . d_{\mathrm{WP}}^{\mathrm{aux}}(D(Y),D(Z))=\sqrt{2}\,d_{\mathrm{WP}}(Y,Z). d WP aux ( D ( Y ) , D ( Z )) = 2 d WP ( Y , Z ) . Proof. The surface Λ ∖ C \Lambda\setminus C Λ ∖ C K K K Σ \Sigma Σ K ‾ \overline{K} K Σ ‾ \overline{\Sigma} Σ § 2.2 , the stratum S ( C ) S(C) S ( C ) Teich ( Σ ) × Teich ( Σ ‾ ) \operatorname{Teich}(\Sigma)\times\operatorname{Teich}(\overline{\Sigma}) Teich ( Σ ) × Teich ( Σ ) f Y , T f_{Y,T} f Y , T Y Y Y Y ‾ \overline{Y} Y K K K K ‾ \overline{K} K D ( Y ) D(Y) D ( Y ) ( Y , Y ‾ ) (Y,\overline{Y}) ( Y , Y ) D ( Z ) D(Z) D ( Z ) ( Z , Z ‾ ) (Z,\overline{Z}) ( Z , Z ) [29 Theorem 2(1), p. 17] , the restriction of the WP metric of Teich ( Λ ) \operatorname{Teich}(\Lambda) Teich ( Λ ) S ( C ) S(C) S ( C ) Teich ( Σ ) \operatorname{Teich}(\Sigma) Teich ( Σ ) Teich ( Σ ‾ ) \operatorname{Teich}(\overline{\Sigma}) Teich ( Σ ) d C d_C d C S ( C ) S(C) S ( C ) D ( Y ) D(Y) D ( Y ) D ( Z ) D(Z) D ( Z ) S ( C ) S(C) S ( C ) § 2.2 ), that is
d W P a u x ( D ( Y ) , D ( Z ) ) = d C ( ( Y , Y ‾ ) , ( Z , Z ‾ ) ) . (17) d_{\mathrm{WP}}^{\mathrm{aux}}(D(Y),D(Z))=d_C((Y,\overline{Y}),(Z,\overline{Z})). \tag*{(17)} d WP aux ( D ( Y ) , D ( Z )) = d C (( Y , Y ) , ( Z , Z )) . (17) Let σ : Teich ( Σ ) → Teich ( Σ ‾ ) \sigma:\operatorname{Teich}(\Sigma)\to\operatorname{Teich}(\overline{\Sigma}) σ : Teich ( Σ ) → Teich ( Σ ) X ↦ X ‾ X\mapsto\overline{X} X ↦ X X ‾ \overline{X} X X X X ζ = x + i y \zeta=x+iy ζ = x + i y X X X ξ = − ζ ‾ = − x + i y = u + i v \xi=-\overline{\zeta}=-x+iy=u+iv ξ = − ζ = − x + i y = u + i v X ‾ \overline{X} X ∣ d ξ ∣ 2 = ∣ d ζ ∣ 2 |d\xi|^2=|d\zeta|^2 ∣ d ξ ∣ 2 = ∣ d ζ ∣ 2 X ‾ \overline{X} X X X X ρ ‾ 2 ∣ d ξ ∣ 2 = ρ 2 ∣ d ζ ∣ 2 \overline{\rho}^{\,2}|d\xi|^2=\rho^2|d\zeta|^2 ρ 2 ∣ d ξ ∣ 2 = ρ 2 ∣ d ζ ∣ 2
Q ( X ) ⟶ Q ( X ‾ ) , φ = φ ( ζ ) d ζ 2 ⟼ φ ‾ = φ ( ζ ) ‾ d ζ ‾ 2 = φ ( − ξ ‾ ) d ξ 2 , B ( X ) ⟶ B ( X ‾ ) , μ = μ ( ζ ) d ζ ‾ d ζ ⟼ μ ‾ = μ ( ζ ) ‾ d ζ ‾ d ζ = μ ( − ξ ‾ ) d ξ d ξ ‾ . (18) \begin{aligned}
Q(X)&\longrightarrow Q(\overline{X}),&\quad\varphi=\varphi(\zeta)\,d\zeta^2&\longmapsto\overline{\varphi}=\overline{\varphi(\zeta)}\,d\overline{\zeta}^{\,2}=\varphi(-\overline{\xi})\,d\xi^2,\\
B(X)&\longrightarrow B(\overline{X}),&\quad\mu=\mu(\zeta)\frac{d\overline{\zeta}}{d\zeta}&\longmapsto\overline{\mu}=\overline{\mu(\zeta)}\frac{d\overline{\zeta}}{d\zeta}=\mu(-\overline{\xi})\frac{d\xi}{d\overline{\xi}}. \tag*{(18)}
\end{aligned} Q ( X ) B ( X ) ⟶ Q ( X ) , ⟶ B ( X ) , φ = φ ( ζ ) d ζ 2 μ = μ ( ζ ) d ζ d ζ ⟼ φ = φ ( ζ ) d ζ 2 = φ ( − ξ ) d ξ 2 , ⟼ μ = μ ( ζ ) d ζ d ζ = μ ( − ξ ) d ξ d ξ . (18) The first map is well-defined since ξ ↦ φ ( − ξ ‾ ) \xi\mapsto\varphi(-\overline{\xi}) ξ ↦ φ ( − ξ ) X ‾ \overline{X} X X ‾ ‾ = X \overline{\overline{X}}=X X = X d ξ = − d ζ ‾ d\xi=-d\overline{\zeta} d ξ = − d ζ ξ \xi ξ X ‾ \overline{X} X φ ‾ = φ ( ζ ) d ξ 2 \overline{\varphi}=\varphi(\zeta)\,d\xi^2 φ = φ ( ζ ) d ξ 2 μ ‾ = μ ( ζ ) d ξ ‾ d ξ \overline{\mu}=\mu(\zeta)\frac{d\overline{\xi}}{d\xi} μ = μ ( ζ ) d ξ d ξ
∫ X ‾ μ ‾ φ ‾ = ∫ X ‾ μ ( − ξ ‾ ) ‾ φ ( − ξ ‾ ) ‾ d u d v = ∫ X μ ( ζ ) ‾ φ ( ζ ) ‾ d x d y = ∫ X μ φ ‾ . (19) \int_{\overline{X}}\overline{\mu}\,\overline{\varphi}=\int_{\overline{X}}\overline{\mu(-\overline{\xi})}\,\overline{\varphi(-\overline{\xi})}\,du\,dv=\int_X\overline{\mu(\zeta)}\,\overline{\varphi(\zeta)}\,dx\,dy=\int_X\overline{\mu\varphi}. \tag*{(19)} ∫ X μ φ = ∫ X μ ( − ξ ) φ ( − ξ ) d u d v = ∫ X μ ( ζ ) φ ( ζ ) d x d y = ∫ X μ φ . (19) and
∥ φ ‾ ∥ 2 2 = ∫ X ‾ ∣ φ ( − ξ ‾ ) ∣ 2 ρ ( − ξ ‾ ) ‾ − 2 d u d v = ∫ X ‾ ∣ φ ( − ξ ‾ ) ∣ 2 ρ ( − ξ ‾ ) − 2 d u d v = ∫ X ∣ φ ( ζ ) ∣ 2 ρ ( ζ ) − 2 d x d y = ∥ φ ∥ 2 2 . (20) \begin{aligned}
\|\overline{\varphi}\|_2^2&=\int_{\overline{X}}|\varphi(-\overline{\xi})|^2\overline{\rho(-\overline{\xi})}^{-2}\,du\,dv=\int_{\overline{X}}|\varphi(-\overline{\xi})|^2\rho(-\overline{\xi})^{-2}\,du\,dv\\
&=\int_X|\varphi(\zeta)|^2\rho(\zeta)^{-2}\,dx\,dy=\|\varphi\|_2^2. \tag*{(20)}
\end{aligned} ∥ φ ∥ 2 2 = ∫ X ∣ φ ( − ξ ) ∣ 2 ρ ( − ξ ) − 2 d u d v = ∫ X ∣ φ ( − ξ ) ∣ 2 ρ ( − ξ ) − 2 d u d v = ∫ X ∣ φ ( ζ ) ∣ 2 ρ ( ζ ) − 2 d x d y = ∥ φ ∥ 2 2 . (20) The differential of σ \sigma σ [ μ ] [\mu] [ μ ] [ μ ‾ ] [\overline{\mu}] [ μ ] μ ∈ B ( X ) \mu\in B(X) μ ∈ B ( X ) ( X t ) t (X_t)_t ( X t ) t Teich ( Σ ) \operatorname{Teich}(\Sigma) Teich ( Σ ) X 0 = X X_0=X X 0 = X μ ( t ) \mu(t) μ ( t ) X X X μ ( 0 ) = 0 \mu(0)=0 μ ( 0 ) = 0 μ ˙ ( 0 ) = μ \dot{\mu}(0)=\mu μ ˙ ( 0 ) = μ t = 0 t=0 t = 0 [ μ ] [\mu] [ μ ] w w w X t X_t X t ζ \zeta ζ ∂ ζ ‾ w = μ ( t ) ∂ ζ w \partial_{\overline{\zeta}}w=\mu(t)\partial_\zeta w ∂ ζ w = μ ( t ) ∂ ζ w w ^ = − w ‾ \widehat{w}=-\overline{w} w = − w X t ‾ \overline{X_t} X t ζ = − ξ ‾ \zeta=-\overline{\xi} ζ = − ξ ∂ ξ ‾ = − ∂ ζ \partial_{\overline{\xi}}=-\partial_\zeta ∂ ξ = − ∂ ζ ∂ ξ = − ∂ ζ ‾ \partial_\xi=-\partial_{\overline{\zeta}} ∂ ξ = − ∂ ζ f f f ∂ ζ ‾ f ‾ = ∂ ζ f ‾ \partial_{\overline{\zeta}}\overline{f}=\overline{\partial_\zeta f} ∂ ζ f = ∂ ζ f ∂ ζ ‾ f ‾ ‾ = ∂ ζ f \partial_{\overline{\zeta}}\overline{\overline{f}}=\partial_\zeta f ∂ ζ f = ∂ ζ f
∂ ξ ‾ w ^ = − ∂ ζ ( − w ‾ ) = ∂ ζ w ‾ = μ ( t ) ‾ ∂ ζ w ‾ = μ ( t ) ‾ ∂ ζ ‾ w ‾ = μ ( t ) ‾ ∂ ξ w ^ . \partial_{\overline{\xi}}\widehat{w}=-\partial_\zeta(-\overline{w})=\partial_\zeta\overline{w}=\overline{\mu(t)}\,\partial_\zeta\overline{w}=\overline{\mu(t)}\,\partial_{\overline{\zeta}}\overline{w}=\overline{\mu(t)}\,\partial_\xi\widehat{w}. ∂ ξ w = − ∂ ζ ( − w ) = ∂ ζ w = μ ( t ) ∂ ζ w = μ ( t ) ∂ ζ w = μ ( t ) ∂ ξ w . Thus σ ( X t ) = X t ‾ \sigma(X_t)=\overline{X_t} σ ( X t ) = X t μ ( t ) ‾ \overline{\mu(t)} μ ( t ) X ‾ \overline{X} X t = 0 t=0 t = 0 μ ‾ \overline{\mu} μ ( σ ( X t ) ) t (\sigma(X_t))_t ( σ ( X t ) ) t t = 0 t=0 t = 0 [ μ ‾ ] [\overline{\mu}] [ μ ] d σ ( [ μ ] ) = [ μ ‾ ] d\sigma([\mu])=[\overline{\mu}] d σ ([ μ ]) = [ μ ] μ ∈ N ( X ) \mu\in N(X) μ ∈ N ( X ) μ ‾ ∈ N ( X ‾ ) \overline{\mu}\in N(\overline{X}) μ ∈ N ( X )
Consequently the map σ \sigma σ μ ∈ B ( X ) \mu\in B(X) μ ∈ B ( X )
∥ d σ ( [ μ ] ) ∥ W P = ∥ [ μ ‾ ] ∥ W P = sup ψ ∈ Q ( X ‾ ) ∖ { 0 } ∣ ∫ X ‾ μ ‾ ψ ∣ ∥ ψ ∥ 2 = sup φ ∈ Q ( X ) ∖ { 0 } ∣ ∫ X μ ‾ φ ‾ ∣ ∥ φ ‾ ∥ 2 (21) \left\|d\sigma([\mu])\right\|_{\mathrm{WP}}=\left\|[\overline{\mu}]\right\|_{\mathrm{WP}}=\sup_{\psi\in Q(\overline{X})\setminus\{0\}}\frac{\left|\int_{\overline{X}}\overline{\mu}\,\psi\right|}{\|\psi\|_2}=\sup_{\varphi\in Q(X)\setminus\{0\}}\frac{\left|\int_X\overline{\mu}\,\overline{\varphi}\right|}{\|\overline{\varphi}\|_2} \tag*{(21)} ∥ d σ ([ μ ]) ∥ WP = ∥ [ μ ] ∥ WP = ψ ∈ Q ( X ) ∖ { 0 } sup ∥ ψ ∥ 2 ∫ X μ ψ = φ ∈ Q ( X ) ∖ { 0 } sup ∥ φ ∥ 2 ∫ X μ φ (21) = sup φ ∈ Q ( X ) ∖ { 0 } ∣ ∫ X μ φ ∣ ∥ φ ∥ 2 = ∥ [ μ ] ∥ W P . (22) =\sup_{\varphi\in Q(X)\setminus\{0\}}\frac{\left|\int_X\mu\varphi\right|}{\|\varphi\|_2}=\|[\mu]\|_{\mathrm{WP}}. \tag*{(22)} = φ ∈ Q ( X ) ∖ { 0 } sup ∥ φ ∥ 2 ∫ X μ φ = ∥ [ μ ] ∥ WP . (22) Hence, since σ \sigma σ X ‾ ↦ X \overline{X}\mapsto X X ↦ X d W P ( Y , Z ‾ ) = d W P ( Y , Z ) d_{\mathrm{WP}}(Y,\overline{Z})=d_{\mathrm{WP}}(Y,Z) d WP ( Y , Z ) = d WP ( Y , Z ) d = d W P ( Y , Z ) d=d_{\mathrm{WP}}(Y,Z) d = d WP ( Y , Z )
Now we compute d C ( ( Y , Y ‾ ) , ( Z , Z ‾ ) ) d_C((Y,\overline{Y}),(Z,\overline{Z})) d C (( Y , Y ) , ( Z , Z )) γ ( t ) = ( α ( t ) , β ( t ) ) \gamma(t)=(\alpha(t),\beta(t)) γ ( t ) = ( α ( t ) , β ( t )) t ∈ [ 0 , 1 ] t\in[0,1] t ∈ [ 0 , 1 ] S ( C ) S(\mathbb{C}) S ( C ) ( Y , Y ‾ ) (Y,\overline{Y}) ( Y , Y ) ( Z , Z ‾ ) (Z,\overline{Z}) ( Z , Z ) S ( C ) = Teich ( Σ ) × Teich ( Σ ‾ ) S(\mathbb{C})=\operatorname{Teich}(\Sigma)\times\operatorname{Teich}(\overline{\Sigma}) S ( C ) = Teich ( Σ ) × Teich ( Σ ) ( α ˙ , β ˙ ) (\dot{\alpha},\dot{\beta}) ( α ˙ , β ˙ ) ( ∥ α ˙ ∥ W P 2 + ∥ β ˙ ∥ W P 2 ) 1 / 2 (\|\dot{\alpha}\|_{\mathrm{WP}}^2+\|\dot{\beta}\|_{\mathrm{WP}}^2)^{1/2} ( ∥ α ˙ ∥ WP 2 + ∥ β ˙ ∥ WP 2 ) 1/2 α \alpha α Y Y Y Z Z Z Teich ( Σ ) \operatorname{Teich}(\Sigma) Teich ( Σ ) ℓ ( α ) ≥ d \ell(\alpha)\ge d ℓ ( α ) ≥ d β \beta β Y ‾ \overline{Y} Y Z ‾ \overline{Z} Z Teich ( Σ ‾ ) \operatorname{Teich}(\overline{\Sigma}) Teich ( Σ ) σ − 1 ∘ β \sigma^{-1}\circ\beta σ − 1 ∘ β Y Y Y Z Z Z σ \sigma σ ℓ ( β ) = ℓ ( σ − 1 ∘ β ) ≥ d \ell(\beta)=\ell(\sigma^{-1}\circ\beta)\ge d ℓ ( β ) = ℓ ( σ − 1 ∘ β ) ≥ d A = ∥ α ˙ ∥ W P A=\|\dot{\alpha}\|_{\mathrm{WP}} A = ∥ α ˙ ∥ WP B = ∥ β ˙ ∥ W P B=\|\dot{\beta}\|_{\mathrm{WP}} B = ∥ β ˙ ∥ WP 2 ( A 2 + B 2 ) − ( A + B ) 2 = ( A − B ) 2 ≥ 0 2(A^2+B^2)-(A+B)^2=(A-B)^2\ge0 2 ( A 2 + B 2 ) − ( A + B ) 2 = ( A − B ) 2 ≥ 0 A 2 + B 2 ≥ ( A + B ) / 2 \sqrt{A^2+B^2}\ge(A+B)/\sqrt{2} A 2 + B 2 ≥ ( A + B ) / 2
ℓ ( γ ) = ∫ 0 1 A 2 + B 2 d t ≥ 1 2 ( ∫ 0 1 A d t + ∫ 0 1 B d t ) = ℓ ( α ) + ℓ ( β ) 2 ≥ 2 d 2 = 2 d . \ell(\gamma)=\int_0^1\sqrt{A^2+B^2}\,dt\ge\frac{1}{\sqrt{2}}\left(\int_0^1A\,dt+\int_0^1B\,dt\right)=\frac{\ell(\alpha)+\ell(\beta)}{\sqrt{2}}\ge\frac{2d}{\sqrt{2}}=\sqrt{2}d. ℓ ( γ ) = ∫ 0 1 A 2 + B 2 d t ≥ 2 1 ( ∫ 0 1 A d t + ∫ 0 1 B d t ) = 2 ℓ ( α ) + ℓ ( β ) ≥ 2 2 d = 2 d . Taking the infimum over γ \gamma γ
d C ( ( Y , Y ‾ ) , ( Z , Z ‾ ) ) ≥ 2 d . (23) d_C((Y,\overline{Y}),(Z,\overline{Z}))\ge\sqrt{2}d. \tag*{(23)} d C (( Y , Y ) , ( Z , Z )) ≥ 2 d . (23) Let η > 0 \eta>0 η > 0 α \alpha α Y Y Y Z Z Z Teich ( Σ ) \operatorname{Teich}(\Sigma) Teich ( Σ ) ℓ ( α ) < d + η \ell(\alpha)<d+\eta ℓ ( α ) < d + η d d d γ ( t ) = ( α ( t ) , σ ( α ( t ) ) ) \gamma(t)=(\alpha(t),\sigma(\alpha(t))) γ ( t ) = ( α ( t ) , σ ( α ( t ))) ( Y , Y ‾ ) (Y,\overline{Y}) ( Y , Y ) ( Z , Z ‾ ) (Z,\overline{Z}) ( Z , Z ) S ( C ) S(\mathbb{C}) S ( C ) ( α ˙ , d σ ( α ˙ ) ) (\dot{\alpha},d\sigma(\dot{\alpha})) ( α ˙ , d σ ( α ˙ )) ∥ d σ ( α ˙ ) ∥ W P = ∥ α ˙ ∥ W P \|d\sigma(\dot{\alpha})\|_{\mathrm{WP}}=\|\dot{\alpha}\|_{\mathrm{WP}} ∥ d σ ( α ˙ ) ∥ WP = ∥ α ˙ ∥ WP
∥ γ ˙ ∥ = ∥ α ˙ ∥ W P 2 + ∥ d σ ( α ˙ ) ∥ W P 2 = 2 ∥ α ˙ ∥ W P \|\dot{\gamma}\|=\sqrt{\|\dot{\alpha}\|_{\mathrm{WP}}^2+\|d\sigma(\dot{\alpha})\|_{\mathrm{WP}}^2}=\sqrt{2}\|\dot{\alpha}\|_{\mathrm{WP}} ∥ γ ˙ ∥ = ∥ α ˙ ∥ WP 2 + ∥ d σ ( α ˙ ) ∥ WP 2 = 2 ∥ α ˙ ∥ WP and
ℓ ( γ ) = 2 ℓ ( α ) < 2 ( d + η ) . \ell(\gamma)=\sqrt{2}\ell(\alpha)<\sqrt{2}(d+\eta). ℓ ( γ ) = 2 ℓ ( α ) < 2 ( d + η ) . Hence d C ( ( Y , Y ‾ ) , ( Z , Z ‾ ) ) < 2 ( d + η ) d_C((Y,\overline{Y}),(Z,\overline{Z}))<\sqrt{2}(d+\eta) d C (( Y , Y ) , ( Z , Z )) < 2 ( d + η ) η > 0 \eta>0 η > 0
d C ( ( Y , Y ‾ ) , ( Z , Z ‾ ) ) ≤ 2 d . (24) d_C((Y,\overline{Y}),(Z,\overline{Z}))\le\sqrt{2}d. \tag*{(24)} d C (( Y , Y ) , ( Z , Z )) ≤ 2 d . (24) By (23) and (24) , we obtain d C ( ( Y , Y ‾ ) , ( Z , Z ‾ ) ) = 2 d d_C((Y,\overline{Y}),(Z,\overline{Z}))=\sqrt{2}d d C (( Y , Y ) , ( Z , Z )) = 2 d (17) this gives d W P a u x ( D ( Y ) , D ( Z ) ) = 2 d W P ( Y , Z ) d_{\mathrm{WP}}^{\mathrm{aux}}(D(Y),D(Z))=\sqrt{2}\,d_{\mathrm{WP}}(Y,Z) d WP aux ( D ( Y ) , D ( Z )) = 2 d WP ( Y , Z )
Proof of the main theorem
Theorem 3.10. Let g , n g,n g , n n ≥ 1 n \ge1 n ≥ 1 2 g − 2 + n > 0 2g-2+n>0 2 g − 2 + n > 0 C < 6 C<6 C < 6 Y , Z ∈ Teich g , n Y,Z\in\operatorname{Teich}_{g,n} Y , Z ∈ Teich g , n T T T
d W P ( Y , Z ) ≤ C n + L T ( Y ) + L T ( Z ) . d_{\mathrm{WP}}(Y,Z)\le C\sqrt{n+\mathcal{L}_{T}(Y)+\mathcal{L}_{T}(Z)}. d WP ( Y , Z ) ≤ C n + L T ( Y ) + L T ( Z ) . Proof. Let Y , Z ∈ Teich g , n Y,Z\in\operatorname{Teich}_{g,n} Y , Z ∈ Teich g , n T T T W T 0 ( Y ) W_{T_{0}}(Y) W T 0 ( Y ) W T 0 ( Z ) W_{T_{0}}(Z) W T 0 ( Z ) f Y , T 0 f_{Y,T_{0}} f Y , T 0 f Z , T 0 f_{Z,T_{0}} f Z , T 0 § 3.1 . By Lemma 3.9,
2 d W P ( Y , Z ) = d W P a u x ( D ( Y ) , D ( Z ) ) . (25) \sqrt{2}\,d_{\mathrm{WP}}(Y,Z)=d_{\mathrm{WP}}^{\mathrm{aux}}(D(Y),D(Z)). \tag*{(25)} 2 d WP ( Y , Z ) = d WP aux ( D ( Y ) , D ( Z )) . (25) By Lemma 3.6 and Remark 3.7, the maximally noded surface P T ∈ Teich ( Λ ) P_{T}\in\operatorname{Teich}(\Lambda) P T ∈ Teich ( Λ ) Y Y Y Z Z Z
d W P a u x ( D ( Y ) , D ( Z ) ) ≤ d W P a u x ( D ( Y ) , P T ) + d W P a u x ( P T , D ( Z ) ) . (26) d_{\mathrm{WP}}^{\mathrm{aux}}(D(Y),D(Z))\le d_{\mathrm{WP}}^{\mathrm{aux}}(D(Y),P_{T})+d_{\mathrm{WP}}^{\mathrm{aux}}(P_{T},D(Z)). \tag*{(26)} d WP aux ( D ( Y ) , D ( Z )) ≤ d WP aux ( D ( Y ) , P T ) + d WP aux ( P T , D ( Z )) . (26) Let V ∈ { Y , Z } V\in\{Y,Z\} V ∈ { Y , Z } (4) , every arc e e e T T T δ e ( V ) ≥ 2 log 4 \delta_{e}(V)\ge2\log4 δ e ( V ) ≥ 2 log 4 L T ( V ) ≥ 2 E log 4 \mathcal{L}_{T}(V)\ge2E\log4 L T ( V ) ≥ 2 E log 4 E ≤ L T ( V ) / ( 2 log 4 ) E\le\mathcal{L}_{T}(V)/(2\log4) E ≤ L T ( V ) / ( 2 log 4 )
2 2 L T ( V ) + 4 E log ( 1 + 2 ) ≤ ( 2 2 + 2 log ( 1 + 2 ) log 4 ) L T ( V ) = B L T ( V ) , 2\sqrt{2}\mathcal{L}_{T}(V)+4E\log(1+\sqrt{2})\le\left(2\sqrt{2}+\frac{2\log(1+\sqrt{2})}{\log4}\right)\mathcal{L}_{T}(V)=B\mathcal{L}_{T}(V), 2 2 L T ( V ) + 4 E log ( 1 + 2 ) ≤ ( 2 2 + log 4 2 log ( 1 + 2 ) ) L T ( V ) = B L T ( V ) , where B = 2 2 + log ( 1 + 2 ) / log 2 B=2\sqrt{2}+\log(1+\sqrt{2})/\log2 B = 2 2 + log ( 1 + 2 ) / log 2
d W P a u x ( D ( V ) , P T ) ≤ 2 n + 2 π B L T ( V ) . (27) d_{\mathrm{WP}}^{\mathrm{aux}}(D(V),P_{T})\le2\sqrt{n}+\sqrt{2\pi B\mathcal{L}_{T}(V)}. \tag*{(27)} d WP aux ( D ( V ) , P T ) ≤ 2 n + 2 π B L T ( V ) . (27) Combining (25) , (26) and (27) for V = Y V=Y V = Y V = Z V=Z V = Z 2 \sqrt{2} 2
d W P ( Y , Z ) ≤ 1 2 ( 4 n + 2 π B ( L T ( Y ) + L T ( Z ) ) ) = 2 2 n + π B ( L T ( Y ) + L T ( Z ) ) . \begin{aligned}
d_{\mathrm{WP}}(Y,Z)\le\frac{1}{\sqrt{2}}\left(4\sqrt{n}+\sqrt{2\pi B}\left(\sqrt{\mathcal{L}_{T}(Y)}+\sqrt{\mathcal{L}_{T}(Z)}\right)\right) \\
&=2\sqrt{2}\sqrt{n}+\sqrt{\pi B}\left(\sqrt{\mathcal{L}_{T}(Y)}+\sqrt{\mathcal{L}_{T}(Z)}\right).
\end{aligned} d WP ( Y , Z ) ≤ 2 1 ( 4 n + 2 π B ( L T ( Y ) + L T ( Z ) ) ) = 2 2 n + π B ( L T ( Y ) + L T ( Z ) ) . By the Cauchy–Schwarz inequality,
d W P ( Y , Z ) ≤ 8 + 2 π B n + L T ( Y ) + L T ( Z ) . d_{\mathrm{WP}}(Y,Z)\le\sqrt{8+2\pi B}\sqrt{n+\mathcal{L}_{T}(Y)+\mathcal{L}_{T}(Z)}. d WP ( Y , Z ) ≤ 8 + 2 π B n + L T ( Y ) + L T ( Z ) . Finally B < 4.1 B<4.1 B < 4.1 C = 8 + 2 π B < 5.811 < 6 C=\sqrt{8+2\pi B}<5.811<6 C = 8 + 2 π B < 5.811 < 6 g g g n n n Y Y Y Z Z Z
We now deduce Corollary 3.11 from Theorem 3.10 and Lemma 2.2.
Corollary 3.11. Let g , n g,n g , n n ≥ 1 n\ge1 n ≥ 1 2 g − 2 + n > 0 2g-2+n>0 2 g − 2 + n > 0 Y , Z ∈ Teich g , n Y,Z\in\operatorname{Teich}_{g,n} Y , Z ∈ Teich g , n T T T
d W P ( Y , Z ) ≤ 21 ( 2 g − 2 + n ) ( log ( 24 ( 2 g − 2 + n ) ) + O T ( Y ) + O T ( Z ) ) , d_{\mathrm{WP}}(Y,Z)\le21\sqrt{(2g-2+n)\left(\log(24(2g-2+n))+\mathcal{O}_{T}(Y)+\mathcal{O}_{T}(Z)\right)}, d WP ( Y , Z ) ≤ 21 ( 2 g − 2 + n ) ( log ( 24 ( 2 g − 2 + n )) + O T ( Y ) + O T ( Z ) ) , where O T \mathcal{O}_{T} O T [16 ] , defined in § 2.4 .
Proof of Corollary 3.11. Set m = 2 g − 2 + n m=2g-2+n m = 2 g − 2 + n m ≥ 1 m\ge1 m ≥ 1 E = 6 g − 6 + 3 n = 3 m E=6g-6+3n=3m E = 6 g − 6 + 3 n = 3 m V ∈ { Y , Z } V\in\{Y,Z\} V ∈ { Y , Z }
L T ( V ) ≤ 2 E ( log ( 8 E ) + O T ( V ) ) = 6 m ( log ( 24 m ) + O T ( V ) ) . \mathcal{L}_{T}(V)\le2E\left(\log(8E)+\mathcal{O}_{T}(V)\right)=6m\left(\log(24m)+\mathcal{O}_{T}(V)\right). L T ( V ) ≤ 2 E ( log ( 8 E ) + O T ( V ) ) = 6 m ( log ( 24 m ) + O T ( V ) ) . Moreover n = m + 2 − 2 g ≤ m + 2 ≤ 3 m n=m+2-2g\le m+2\le3m n = m + 2 − 2 g ≤ m + 2 ≤ 3 m m ≥ 1 m\ge1 m ≥ 1 3 < log 24 ≤ log ( 24 m ) 3<\log24\le\log(24m) 3 < log 24 ≤ log ( 24 m ) n ≤ m log ( 24 m ) n\le m\log(24m) n ≤ m log ( 24 m )
n + L T ( Y ) + L T ( Z ) ≤ 13 m log ( 24 m ) + 6 m ( O T ( Y ) + O T ( Z ) ) n+\mathcal{L}_{T}(Y)+\mathcal{L}_{T}(Z)\le13m\log(24m)+6m\left(\mathcal{O}_{T}(Y)+\mathcal{O}_{T}(Z)\right) n + L T ( Y ) + L T ( Z ) ≤ 13 m log ( 24 m ) + 6 m ( O T ( Y ) + O T ( Z ) ) ≤ 13 m ( log ( 24 m ) + O T ( Y ) + O T ( Z ) ) . \le13m\left(\log(24m)+\mathcal{O}_{T}(Y)+\mathcal{O}_{T}(Z)\right). ≤ 13 m ( log ( 24 m ) + O T ( Y ) + O T ( Z ) ) . By Theorem 3.10, with C = 8 + 2 π B < 5.811 C = \sqrt{8 + 2\pi B} < 5.811 C = 8 + 2 π B < 5.811
d W P ( Y , Z ) ≤ C 13 m ( log ( 24 m ) + O T ( Y ) + O T ( Z ) ) , d_{\mathrm{WP}}(Y,Z) \le C\sqrt{13}\sqrt{m\left(\log(24m)+\mathcal{O}_{\mathcal{T}}(Y)+\mathcal{O}_{\mathcal{T}}(Z)\right)}, d WP ( Y , Z ) ≤ C 13 m ( log ( 24 m ) + O T ( Y ) + O T ( Z ) ) , and C 13 < 5.811 ⋅ 3.606 < 21 C\sqrt{13} < 5.811 \cdot3.606 < 21 C 13 < 5.811 ⋅ 3.606 < 21
Corollary 3.12. Let g , n g,n g , n n ≥ 1 n \ge1 n ≥ 1 2 g − 2 + n > 0 2g-2+n>0 2 g − 2 + n > 0 Y , Z ∈ Teich g , n Y,Z \in\operatorname{Teich}_{g,n} Y , Z ∈ Teich g , n T \mathcal{T} T
d W P ( Y , Z ) ≤ 21 ( 2 g − 2 + n ) ( log ( 24 ( 2 g − 2 + n ) ) + ∑ e ∈ T ( ∣ s e ( Y ) ∣ + ∣ s e ( Z ) ∣ ) ) . d_{\mathrm{WP}}(Y,Z) \le21\sqrt{(2g-2+n)\left(\log(24(2g-2+n))+\sum_{e\in\mathcal{T}}\left(\lvert s_e(Y)\rvert+\lvert s_e(Z)\rvert\right)\right)}. d WP ( Y , Z ) ≤ 21 ( 2 g − 2 + n ) ( log ( 24 ( 2 g − 2 + n )) + e ∈ T ∑ ( ∣ s e ( Y )∣ + ∣ s e ( Z )∣ ) ) . Proof. Fix X ∈ Teich g , n X \in\operatorname{Teich}_{g,n} X ∈ Teich g , n c c c a r = s e c , r ( X ) a_r=s_{e_{c,r}}(X) a r = s e c , r ( X ) 1 ≤ r ≤ v ( c ) 1\le r\le v(c) 1 ≤ r ≤ v ( c ) § 2.4 . Since the arcs are ideal one has ∑ r = 1 v ( c ) a r = 0 \sum_{r=1}^{v(c)}a_r=0 ∑ r = 1 v ( c ) a r = 0 I I I c c c
∣ ∑ r ∈ I a r ∣ = 1 2 ∣ ∑ r ∈ I a r − ∑ r ∉ I a r ∣ ≤ 1 2 ∑ r = 1 v ( c ) ∣ a r ∣ ≤ ∑ e ∈ T ∣ s e ( X ) ∣ . \left|\sum_{r\in I}a_r\right|=\frac{1}{2}\left|\sum_{r\in I}a_r-\sum_{r\notin I}a_r\right|\le\frac{1}{2}\sum_{r=1}^{v(c)}|a_r|\le\sum_{e\in\mathcal{T}}|s_e(X)|. r ∈ I ∑ a r = 2 1 r ∈ I ∑ a r − r ∈ / I ∑ a r ≤ 2 1 r = 1 ∑ v ( c ) ∣ a r ∣ ≤ e ∈ T ∑ ∣ s e ( X ) ∣. The last inequality holds because each arc has at most two endpoints at c c c c c c I I I O T ( X ) ≤ ∑ e ∈ T ∣ s e ( X ) ∣ \mathcal{O}_{\mathcal{T}}(X)\le\sum_{e\in\mathcal{T}}|s_e(X)| O T ( X ) ≤ ∑ e ∈ T ∣ s e ( X ) ∣ Y Y Y Z Z Z Email address: marie.abadie@uni.lu
