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 : π 1 (Σ) − → PSL (2 , C ) is quasi-Fuchsian if Γ ρ = ρ ( π 1 (Σ)) is discrete and its limit set Λ ρ is a Jordan curve. → PSL (2 , C ) qF acts p.d. on H 3 and on Ω ρ ⊂ ∂ H 3 = CP 1 . ρ : π 1 (Σ) − Ω+ Ω+ Ω− Ω− Λ Λ M ρ = H 3 / ∼ quasi-Fuchsian Case Fuchsian Case = Σ × R ; Ω+ Ω ρ / Γ ρ ∼ Γ (X,Y) = Σ ⊔ Σ. 3 H C U Ω+ 3 = Q(X,Y) H Γ (X,Y) Γ (X,Y) Ω− Λ Ω− Γ (X,Y) Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 2 / 19
Convex core of QF Manifolds X Theorem (Bers Simultaneous X’ Uniformization Thm) Y’ QF (Σ) ∼ = T (Σ) × T (Σ) ρ �→ (Ω + ρ / Γ ρ , Ω − ρ / Γ ρ ) = ( X , Y ) Y Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 3 / 19
Convex core of QF Manifolds X Theorem (Bers Simultaneous X’ Uniformization Thm) Y’ QF (Σ) ∼ = T (Σ) × T (Σ) ρ �→ (Ω + ρ / Γ ρ , Ω − ρ / Γ ρ ) = ( X , Y ) Y Every QF 3–manifold M ρ has a convex core C ρ = CH (Λ ρ ) / Γ ρ . The boundary of the convex core ∂ C ρ = ∂ + C ρ ⊔ ∂ − C ρ is a pleated sur- face w/ bending data: ( X ′ , Y ′ ) ∈ T (Σ) × T (Σ) ( λ + , λ − ) ∈ ML (Σ) × ML (Σ) 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) σ H : QF (Σ) − → T (Σ) × T (Σ) ρ �→ ( X ′ , Y ′ ) is a homeo. Conjecture (Thurston) β H : 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) σ H : QF (Σ) − → T (Σ) × T (Σ) ρ �→ ( X ′ , Y ′ ) is a homeo. Theorem (Sullivan) σ H is surjective. Conjecture (Thurston) β H : QF (Σ) \ F (Σ) − → ML (Σ) × ML (Σ) ρ �→ ( λ + , λ − ) is a homeo onto its image. Theorem (Bonahon-Otal) Characterization of Im ( β H ) . Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 4 / 19
Induced Metric and Bending Conjectures in H Conjecture (Thurston) σ H : QF (Σ) − → T (Σ) × T (Σ) ρ �→ ( X ′ , Y ′ ) is a homeo. Theorem (Sullivan) S ⊂ M is a K –surface if it has con- σ H is surjective. stant Gaussian curvature K . Theorem (Labourie) Conjecture (Thurston) ∀ ρ ∈ QF (Σ) , M ρ \ C ρ is foliated by β H : QF (Σ) \ F (Σ) − → K–surfaces with K ∈ ( − 1 , 0) . ML (Σ) × ML (Σ) ρ �→ ( λ + , λ − ) is a homeo onto its image. Theorem (Bonahon-Otal) Characterization of Im ( β H ) . Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 4 / 19
Induced Metric and Bending Conjectures in H Conjecture (Thurston) S ⊂ M is a K –surface if it has con- stant Gaussian curvature K . σ H : QF (Σ) − → T (Σ) × T (Σ) ρ �→ ( X ′ , Y ′ ) is a homeo. Theorem (Labourie) ∀ ρ ∈ QF (Σ) , M ρ \ C ρ is foliated by Theorem (Sullivan) K–surfaces with K ∈ ( − 1 , 0) . σ H is surjective. Theorem (Labourie, Schlenker) Conjecture (Thurston) Let X − , X + ∈ T (Σ) , and let K − , K + ∈ ( − 1 , 0) , ∃ ! ρ ∈ QF (Σ) s.t. β H : QF (Σ) \ F (Σ) − → the induced metric on the ML (Σ) × ML (Σ) ( K ± ) –surface in the lower/upper c. ρ �→ ( λ + , λ − ) is a homeo onto its c. of M ρ \ C ρ is X ± . image. Remark Theorem (Bonahon-Otal) Similar result using III. Characterization of Im ( β H ) . Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 4 / 19
Quasicircles Definition Definition C ⊂ CP 1 is a quasicircle if it is the A quasicircle C ⊂ CP 1 is normalized restriction to RP 1 of a if it contains 0 , 1 and ∞ . quasiconformal homeo of CP 1 . Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 5 / 19
Quasicircles Definition Definition C ⊂ CP 1 is a quasicircle if it is the A quasicircle C ⊂ CP 1 is normalized restriction to RP 1 of a if it contains 0 , 1 and ∞ . quasiconformal homeo of CP 1 . Definition The universal Teichm¨ uller space T is the space of normalized quasisymmetric homeos RP 1 − → RP 1 . Theorem (Bers) QC H ∼ = T . Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 5 / 19
Proof of QC H ∼ = T . Proof. CP 1 \ C = Ω + C ⊔ Ω − C . By Riemann Mapping Theorem Ω ± C are conformally isomorphic to H 2 . By Caratheodory’s theorem any such isom. extend to homeo = ∂ H 2 = RP 1 . C ∼ C ) − 1 ◦ ∂ U + By Ahlfors’ Theorem ϕ C := ( ∂ U − C is quasisymmetric, where C : H 2 ± − → Ω ± are s.t. U ± U ± C ( i ) = i for i = 0 , 1 , ∞ . Theorem (Ahlfors) TFAE: C is quasicircle; U + C extends to quasiconformal map of CP 1 ; ϕ C 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 ) K ( C ) defines Φ C , K : RP 1 − → RP 1 by and S − C , K ) − 1 ◦ ∂ V + Φ C , K = ( ∂ V − C , K where V ± C , K : H 2 ± → S ± K − K ( C ). Note that the case K = − 1 corresponds to ∂ ± CH ( C ). Lemma C , K extend to homeo H 2 ± − 1 V ± → S ± K ( C ) ∪ C. 2 Φ C , 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 ) K ( C ) defines Φ C , K : RP 1 − → RP 1 by and S − C , K ) − 1 ◦ ∂ V + Φ C , K = ( ∂ V − C , K where V ± C , K : H 2 ± → S ± K − K ( C ). Note that the case K = − 1 corresponds to ∂ ± CH ( C ). Lemma C , K extend to homeo H 2 ± − 1 V ± → S ± K ( C ) ∪ C. 2 Φ C , K is quasisymmetric. Conjecture Given K ∈ ( − 1 , 0) , the map Φ · , K : QC H − → T is a bijection. Theorem Surjectivity holds. Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 7 / 19
Surjectivity criterion Proposition Let X = H 3 or X = A d S 3 and let F : QC X − → T s.t. (i) If ( C n ) n ∈ N k–quasicircles converging to k–quasicircle C, then ( F ( C n )) 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 v n . Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 8 / 19
Width of quasicircles Definition C ⊂ ∂ ∞ A d S 3 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 ) < ∞ � CP 1 quasicircle. Proposition 1 C quasicircle ⇒ w ( C ) < ∞ 2 ∃ w 0 s.t. w ( C ) < w 0 ⇒ 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 : H 2 − → H 2 is quasiconformal ⇐ ⇒ the Beltrami coefficient µ f defined by ∂ f /∂ ¯ z = µ f ( z ) ∂ f /∂ z satisfies || µ f || ∞ < 1 ⇒ the complex dilatation K f = sup z ∈ H 2 K f ( z ) = 1+ || µ f || ∞ ⇐ 1 −|| µ f || ∞ < ∞ , where K f ( z ) = 1+ | µ f ( z ) | 1 −| µ f ( z ) | . A or. pr. homeo h : S 1 − → S 1 is quasisymmetric if ∃ M ≥ 1 s. t. � � � � h ( e i ( x + t ) ) − h ( e ix ) 1 � � M ≤ � � � ≤ M , ∀ x ∈ R , ∀ t > 0 . � h ( e ix ) − h ( e i ( x − t ) ) f : RP 1 − Recall: f quasiconformal can be extended to ˆ → RP 1 . 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/ Similarly C AdS QC, K ∈ ( −∞ , − 1), K ( C ) defines Φ C , K : RP 1 − S ± → RP 1 gluing b/ S ± K ( C ) defines C , K ) − 1 ◦ ∂ V + w / Φ C , K = ( ∂ V − Ψ C , K : RP 1 − C , K → RP 1 , V ± C , K : H 2 ± → S ± K − K ( C ). Lemma Lemma 1 V ± C , K extend to homeo 1 V ± C , K extend to homeo H 2 ± − → S ± K ( C ) ∪ C. H 2 ± − → S ± K ( C ) ∪ C. 2 Ψ C , K is quasisymmetric. 2 Φ C , K is quasisymmetric. Sara Maloni (UVa) Convex hulls of quasicircles March 23, 2019 12 / 19
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