Induced metrics on convex hulls of quasicircles. Sara Maloni - - PowerPoint PPT Presentation

Induced metrics on convex hulls of quasicircles.

Sara Maloni University of Virginia (joint w/ Francesco Bonsante, Jeff Danciger & Jean-Marc Schlenker) March 23, 2019

Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 1 / 19

Quasi-Fuchsian manifolds

ρ: π1(Σ) − → PSL(2, C) is quasi-Fuchsian if Γρ = ρ(π1(Σ)) is discrete and its limit set Λρ is a Jordan curve. ρ: π1(Σ) − → PSL(2, C) qF acts p.d. on H3 and on Ωρ ⊂ ∂H3 = CP1.

Λ Λ Ω+ Ω+ Ω− Ω−

Fuchsian Case quasi-Fuchsian Case

Λ Ω+ Ω−

H

3

Γ(X,Y)

= Q(X,Y)

H C

3

U

Ω− Ω+

Γ(X,Y) Γ(X,Y)

Γ(X,Y)

Mρ = H3/ ∼ = Σ × R; Ωρ/Γρ ∼ = Σ ⊔ Σ.

Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 2 / 19

Convex core of QF Manifolds

Theorem (Bers Simultaneous Uniformization Thm)

QF(Σ) ∼ = T (Σ) × T (Σ) ρ → (Ω+

ρ /Γρ, Ω− ρ /Γρ) = (X, Y )

X Y X’ Y’

Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 3 / 19

Convex core of QF Manifolds

Theorem (Bers Simultaneous Uniformization Thm)

QF(Σ) ∼ = T (Σ) × T (Σ) ρ → (Ω+

ρ /Γρ, Ω− ρ /Γρ) = (X, Y )

Every QF 3–manifold Mρ has a convex core Cρ = CH(Λρ)/Γρ. The boundary of the convex core ∂Cρ = ∂+Cρ ⊔∂−Cρ is a pleated sur- face w/ bending data: (X ′, Y ′) ∈ T (Σ) × T (Σ) (λ+, λ−) ∈ ML(Σ) × ML(Σ)

X Y X’ Y’

Figure: By J. Brock and D. Dumas

Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 3 / 19

Induced Metric and Bending Conjectures in H

Conjecture (Thurston)

σH : QF(Σ) − → T (Σ) × T (Σ) ρ → (X ′, Y ′) is a homeo.

Conjecture (Thurston)

βH : QF(Σ) \ F(Σ) − → ML(Σ) × ML(Σ) ρ → (λ+, λ−) is a homeo onto its image.

Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 4 / 19

Induced Metric and Bending Conjectures in H

Conjecture (Thurston)

σH : QF(Σ) − → T (Σ) × T (Σ) ρ → (X ′, Y ′) is a homeo.

Theorem (Sullivan)

σH is surjective.

Conjecture (Thurston)

βH : QF(Σ) \ F(Σ) − → ML(Σ) × ML(Σ) ρ → (λ+, λ−) is a homeo onto its image.

Theorem (Bonahon-Otal)

Characterization of Im(βH).

Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 4 / 19

Induced Metric and Bending Conjectures in H

Conjecture (Thurston)

σH : QF(Σ) − → T (Σ) × T (Σ) ρ → (X ′, Y ′) is a homeo.

Theorem (Sullivan)

σH is surjective.

Conjecture (Thurston)

βH : QF(Σ) \ F(Σ) − → ML(Σ) × ML(Σ) ρ → (λ+, λ−) is a homeo onto its image.

Theorem (Bonahon-Otal)

Characterization of Im(βH). S ⊂ M is a K–surface if it has con- stant Gaussian curvature K.

Theorem (Labourie)

∀ρ ∈ QF(Σ), Mρ \ Cρ is foliated by K–surfaces with K ∈ (−1, 0).

Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 4 / 19

Induced Metric and Bending Conjectures in H

Conjecture (Thurston)

σH : QF(Σ) − → T (Σ) × T (Σ) ρ → (X ′, Y ′) is a homeo.

Theorem (Sullivan)

σH is surjective.

Conjecture (Thurston)

βH : QF(Σ) \ F(Σ) − → ML(Σ) × ML(Σ) ρ → (λ+, λ−) is a homeo onto its image.

Theorem (Bonahon-Otal)

Characterization of Im(βH). S ⊂ M is a K–surface if it has con- stant Gaussian curvature K.

Theorem (Labourie)

∀ρ ∈ QF(Σ), Mρ \ Cρ is foliated by K–surfaces with K ∈ (−1, 0).

Theorem (Labourie, Schlenker)

Let X−, X+ ∈ T (Σ), and let K−, K+ ∈ (−1, 0), ∃!ρ ∈ QF(Σ) s.t. the induced metric on the (K±)–surface in the lower/upper c.

• c. of Mρ \ Cρ is X±.

Remark

Similar result using III.

Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 4 / 19

Quasicircles

Definition

C ⊂ CP1 is a quasicircle if it is the restriction to RP1 of a quasiconformal homeo of CP1.

Definition

A quasicircle C ⊂ CP1 is normalized if it contains 0, 1 and ∞.

Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 5 / 19

Quasicircles

Definition

C ⊂ CP1 is a quasicircle if it is the restriction to RP1 of a quasiconformal homeo of CP1.

Definition

A quasicircle C ⊂ CP1 is normalized if it contains 0, 1 and ∞.

Definition

The universal Teichm¨ uller space T is the space of normalized quasisymmetric homeos RP1 − → RP1.

Theorem (Bers)

QCH ∼ = T .

Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 5 / 19

Proof of QCH ∼ = T .

Proof.

CP1 \ C = Ω+

C ⊔ Ω− C .

By Riemann Mapping Theorem Ω±

C are conformally isomorphic to H2.

By Caratheodory’s theorem any such isom. extend to homeo C ∼ = ∂H2 = RP1. By Ahlfors’ Theorem ϕC := (∂U−

C )−1 ◦ ∂U+ C is quasisymmetric, where

U±

C : H2± −

→ Ω± are s.t. U±

C (i) = i for i = 0, 1, ∞.

Theorem (Ahlfors)

TFAE: C is quasicircle; U+

C extends to quasiconformal map of CP1;

ϕC is quasisymmetric.

Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 6 / 19

Universal Induced Metric Conjectures

Given C hyp QC and K ∈ [−1, 0), the gluing b/ the K–surfaces S+

K (C)

and S−

K (C) defines ΦC,K : RP1 −

→ RP1 by ΦC,K = (∂V −

C,K)−1 ◦ ∂V + C,K

where V ±

C,K : H2± K −

→ S±

K (C).

Note that the case K = −1 corresponds to ∂±CH(C).

Lemma

1 V ±

C,K extend to homeo H2± −

→ S±

K (C) ∪ C.

2 ΦC,K is quasisymmetric. Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 7 / 19

Universal Induced Metric Conjectures

Given C hyp QC and K ∈ [−1, 0), the gluing b/ the K–surfaces S+

K (C)

and S−

K (C) defines ΦC,K : RP1 −

→ RP1 by ΦC,K = (∂V −

C,K)−1 ◦ ∂V + C,K

where V ±

C,K : H2± K −

→ S±

K (C).

Note that the case K = −1 corresponds to ∂±CH(C).

Lemma

1 V ±

C,K extend to homeo H2± −

→ S±

K (C) ∪ C.

2 ΦC,K is quasisymmetric.

Conjecture

Given K ∈ (−1, 0), the map Φ·,K : QCH − → T is a bijection.

Theorem

Surjectivity holds.

Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 7 / 19

Surjectivity criterion

Proposition

Let X = H3 or X = AdS3 and let F : QCX − → T s.t. (i) If (Cn)n∈N k–quasicircles converging to k–quasicircle C, then (F(Cn))n∈N converges uniformly to F(C). (ii) ∀k, ∃k′ s.t. F(C) k–quasisymm. homeo ⇒ C k′–quasicircle. (iii) Image(F) contains all quasi-Fuchsian elements. Then F is surjective. The proof of this result relies on

Proposition

Any v ∈ T is the limit of uniformly quasi-symm quasi-Fuchsian homeos vn.

Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 8 / 19

Width of quasicircles

Definition

C ⊂ ∂∞AdS3 acausal curve, then its width is w(CH(C)) = sup d(∂−, ∂+).

Proposition (Bonsante-Schlenker)

C quasicircle ⇐ ⇒ w(C) < π

2 .

What about hyperbolic quasicircles?

Proposition

w(C) < ∞ CP1 quasicircle.

Proposition

1 C quasicircle ⇒ w(C) < ∞ 2 ∃w0 s.t. w(C) < w0 ⇒ C quasicircle Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 9 / 19

End

Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 10 / 19

Quasiconformal and Quasisymmetric maps

A or. pr. diffeo f : H2 − → H2 is quasiconformal ⇐ ⇒ the Beltrami coefficient µf defined by ∂f /∂¯ z = µf (z)∂f /∂z satisfies ||µf ||∞ < 1 ⇐ ⇒ the complex dilatation Kf = supz∈H2 Kf (z) = 1+||µf ||∞

1−||µf ||∞ < ∞, where

Kf (z) = 1+|µf (z)|

1−|µf (z)|.

A or. pr. homeo h: S1 − → S1 is quasisymmetric if ∃M ≥ 1 s. t. 1 M ≤

• h(ei(x+t)) − h(eix)

h(eix) − h(ei(x−t))

• ≤ M, ∀x ∈ R, ∀t > 0.

Recall: f quasiconformal can be extended to ˆ f : RP1 − → RP1.

Theorem (Ahlfors–Beurling)

f is quasiconformal if and only if ˆ f is quasisymmetric.

Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 11 / 19

Universal Induced Metric Conjectures for K–surfaces

C hyp QC, K ∈ (−1, 0) gluing b/ S±

K (C) defines ΦC,K : RP1 −

→ RP1 w/ΦC,K = (∂V −

C,K)−1 ◦ ∂V + C,K

V ±

C,K : H2± K −

→ S±

K (C).

Lemma

1 V ±

C,K extend to homeo

H2± − → S±

K (C) ∪ C.

2 ΦC,K is quasisymmetric.

Similarly C AdS QC, K ∈ (−∞, −1), gluing b/ S±

K (C) defines

ΨC,K : RP1 − → RP1,

Lemma

1 V ±

C,K extend to homeo

H2± − → S±

K (C) ∪ C.

2 ΨC,K is quasisymmetric. Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 12 / 19

Universal Induced Metric Conjectures for K–surfaces

C hyp QC, K ∈ (−1, 0) gluing b/ S±

K (C) defines ΦC,K : RP1 −

→ RP1 w/ΦC,K = (∂V −

C,K)−1 ◦ ∂V + C,K

V ±

C,K : H2± K −

→ S±

K (C).

Lemma

1 V ±

C,K extend to homeo

H2± − → S±

K (C) ∪ C.

2 ΦC,K is quasisymmetric.

Conjecture

Given K ∈ (−1, 0), the map Φ·,K : QCH − → T is a bijection.

Theorem

Surjectivity holds. Similarly C AdS QC, K ∈ (−∞, −1), gluing b/ S±

K (C) defines

ΨC,K : RP1 − → RP1,

Lemma

1 V ±

C,K extend to homeo

H2± − → S±

K (C) ∪ C.

2 ΨC,K is quasisymmetric.

Conjecture

Given K ∈ (−∞, −1) the map Ψ·,K : QCAdS − → T is a bijection.

Theorem

Surjectivity holds.

Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 12 / 19

Proof of surjectivity criterion

Proof.

Given v ∈ T , by Proposition 1 ∃vin ∈ T quasi-Fuchsian uniform

• quasisymm. maps s.t. vn −

→ v. By (iii), ∃Cn ∈ QCX s.t. F(Cn) = vn. By (ii), Cn are (k′)–quasicircles for some k′. (vn uniform QS ⇒ Cn uniform QC). Up to passing to a subsequence Cn − → C, C is (k′)–quasicircle and (F(Cn)) − → F(C), so by (i) F(C) = v.

Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 13 / 19

Approximation of quasicircles

Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 14 / 19

Approximation of quasicircles

The proof of this theorem is based on:

Lemma

Given λ ∈ ML(H2) bounded, ∃(λn, Γn) w/ Γn cocompact Fuchsian gp., λn ∈ ML(H2) Γn-inv. s.t. λn − → λ in weak–∗ topology. ∃C > 0 s.t. ∀n λnTh < C.

Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 14 / 19

Anti-de Sitter space AdS3

The anti-de Sitter space AdS3 is a Lorentzian analogue of H3. AdS3 = {x ∈ R4 | x, x(2,2) < 0}/R∗. Its boundary is ∂AdS3 = {x ∈ R4 \ {0} | x, x(2,2) = 0}/R∗. Its isometry group is Isom+(AdS3) = PO0(2, 2). ∃ embeddings H2 ֒ → AdS3.

Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 15 / 19

Anti-de Sitter space AdS3

The anti-de Sitter space AdS3 is a Lorentzian analogue of H3. AdS3 = {x ∈ R4 | x, x(2,2) < 0}/R∗. Its boundary is ∂AdS3 = {x ∈ R4 \ {0} | x, x(2,2) = 0}/R∗. Its isometry group is Isom+(AdS3) = PO0(2, 2). ∃ embeddings H2 ֒ → AdS3. A vector v ∈ TpAdS3 is: space-like if v, v(2,2) > 0. light-like if v, v(2,2) = 0. time-like if v, v(2,2) < 0.

Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 15 / 19

Anti-de Sitter space AdS3

The anti-de Sitter space AdS3 is a Lorentzian analogue of H3. AdS3 = {x ∈ R4 | x, x(2,2) < 0}/R∗ ∼ = PSL(2, R). Its boundary is ∂AdS3 = {x ∈ R4 \ {0} | x, x(2,2) = 0}/R∗ ∼ = RP1 × RP1. Its isometry group is Isom+(AdS3) = PO0(2, 2) ∼ = PSL(2, R) × PSL(2, R). ∃ embeddings H2 ֒ → AdS3. A vector v ∈ TpAdS3 is: space-like if v, v(2,2) > 0. light-like if v, v(2,2) = 0. time-like if v, v(2,2) < 0.

Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 15 / 19

Geometric transitions

H2–structures collapse down to a point. After rescaling, they limit to E2–structures and then transition to S2–structures. (H2, PO(2, 1)) (R2, R2 ⋊ O(2)) (S2, PO(3))

collapse rescale

Jeff Danciger in his thesis studied a similar geometric transition from H3 to AdS3 structure, passing through HP3.

Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 16 / 19

GHMC manifolds

An AdS 3–manifold M is said globally hyperbolic if it contains a Cauchy surface (a space-like surface which intersects every inextendible time-like curve exactly

• nce.)

globally hyperbolic maximal compact (GHMC) if the Cauchy surface is closed and M is maximal (for the inclusion).

Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 17 / 19

GHMC manifolds

An AdS 3–manifold M is said globally hyperbolic if it contains a Cauchy surface (a space-like surface which intersects every inextendible time-like curve exactly

• nce.)

globally hyperbolic maximal compact (GHMC) if the Cauchy surface is closed and M is maximal (for the inclusion). The holonomy of a GHMC AdS 3-manifold is ρ = (ρL, ρR): π1(Σ) − → PSL(2, R) × PSL(2, R). GH(Σ) = {GHMC str. on Σ × R}/isotopy

Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 17 / 19

Convex core of GHMC Manifolds

Theorem (Mess)

GH(Σ) ∼ = T (Σ) × T (Σ) ρ = (ρL, ρR) → (XL, XR)

Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 18 / 19

Convex core of GHMC Manifolds

Theorem (Mess)

GH(Σ) ∼ = T (Σ) × T (Σ) ρ = (ρL, ρR) → (XL, XR) Every GHMC 3–manifold Mρ has a convex core Cρ. The boundary of the convex core ∂Cρ = ∂+Cρ ⊔∂−Cρ is a pleated sur- face w/ bending data: (X +, X −) ∈ T (Σ) × T (Σ) (λ+, λ−) ∈ ML(Σ) × ML(Σ) s.t. X +

E λ+

r

• E λ+

l

• XL

XR X −

E λ−

r

• E λ−

l

• Sara Maloni (UVa)

Convex hulls of quasicircles March 23, 2019 18 / 19

Induced Metric and Bending Conjectures in AdS

Conjecture (Mess)

σAdS : GH(Σ) − → T (Σ) × T (Σ) ρ → (X +, X −) is a homeo.

Conjecture (Mess)

βAdS : GH(Σ) − → ML(Σ) × ML(Σ) ρ → (λ+, λ−) is a homeo onto its image.

Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 19 / 19

Induced Metric and Bending Conjectures in AdS

Conjecture (Mess)

σAdS : GH(Σ) − → T (Σ) × T (Σ) ρ → (X +, X −) is a homeo.

Theorem (Diallo)

σAdS is surjective.

Conjecture (Mess)

βAdS : GH(Σ) − → ML(Σ) × ML(Σ) ρ → (λ+, λ−) is a homeo onto its image.

Theorem (Bonsante-Schlenker)

Characterization of Im(βAdS).

Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 19 / 19

Induced Metric and Bending Conjectures in AdS

Conjecture (Mess)

σAdS : GH(Σ) − → T (Σ) × T (Σ) ρ → (X +, X −) is a homeo.

Theorem (Diallo)

σAdS is surjective.

Conjecture (Mess)

βAdS : GH(Σ) − → ML(Σ) × ML(Σ) ρ → (λ+, λ−) is a homeo onto its image.

Theorem (Bonsante-Schlenker)

Characterization of Im(βAdS). S ⊂ M is a K–surface if it has con- stant Gaussian curvature K.

Theorem (Barbot, B´ eguin, Zeghib)

∀ρ ∈ GH(Σ), Mρ \ Cρ is foliated by K–surfaces with K ∈ (−∞, −1).

Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 19 / 19

Induced Metric and Bending Conjectures in AdS

Conjecture (Mess)

σAdS : GH(Σ) − → T (Σ) × T (Σ) ρ → (X +, X −) is a homeo.

Theorem (Diallo)

σAdS is surjective.

Conjecture (Mess)

βAdS : GH(Σ) − → ML(Σ) × ML(Σ) ρ → (λ+, λ−) is a homeo onto its image.

Theorem (Bonsante-Schlenker)

Characterization of Im(βAdS). S ⊂ M is a K–surface if it has con- stant Gaussian curvature K.

Theorem (Barbot, B´ eguin, Zeghib)

∀ρ ∈ GH(Σ), Mρ \ Cρ is foliated by K–surfaces with K ∈ (−∞, −1).

Conj (Barbot, B´ eguin, Zeghib)

Let X−, X+ ∈ T (Σ), and let K−, K+ ∈ (−∞, −1), ∃!ρ ∈ GH(Σ) s.t. the induced metric on the (K±)–surface in the past/future

• c. c. of Mρ \ Cρ is X±.

Theorem (Tamburelli)

Existence holds.

Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 19 / 19