An improved bound for Sullivan's convex hull theorem
SSubmitted exclusively to the London Mathematical Society doi:10.1112/0000/000000
An improved bound for Sullivan’s convex hull theorem
M. Bridgeman, R. Canary and A. Yarmola
Abstract
Sullivan showed that there exists K such that if Ω ⊂ ˆ C is a simply connected hyperbolic domain,then there exists a conformally natural K -quasiconformal map from Ω to the boundary Dome(Ω)of the convex hull of its complement which extends to the identity on ∂ Ω. Explicit upper andlower bounds on K were obtained by Epstein, Marden, Markovic and Bishop. We improve onthese bounds, by showing that one may choose K ≤ . Introduction
In this paper we consider the relationship between the Poincar´e metric on a hyperbolic simplyconnected domain Ω in ˆ C = ∂ H and the geometry of the boundary Dome(Ω) of the convex coreof its complement in H . Sullivan [ ] (see also Epstein-Marden [ ]) showed that there exists K > K -quasiconformalmap f : Ω → Dome(Ω) which extends to the identity on ∂ Ω. Epstein, Marden and Markovicprovided upper and lower bounds for the value of K . Theorem 1.1. (Epstein-Marden-Markovic [ , ]) There exists K ≤ . such thatif Ω ⊂ ˆ C is a simply connected hyperbolic domain, then there is a conformally natural K -quasiconformal map f : Ω → Dome(Ω) which extends continuously to the identity on ∂ Ω ⊂ ˆ C . Moreover, one may not choose K ≤ . . We recall that f is said to be conformally natural if whenever A is a conformal automorphismof ˆ C which preserves Ω, then ¯ A ◦ f = f ◦ Ω where ¯ A is the extension of A to an isometry of H . If one does not require that the quasiconformal map f : Ω → Dome(Ω) be conformallynatural, Bishop [ ] obtained a better uniform bound on the quasiconformality constant.Epstein and Markovic [ ] showed that even in this setting one cannot uniformly bound thequasiconformality constant above by 2. Theorem 1.2. (Bishop [ ]) There exists K ≤ . such that if Ω ⊂ ˆ C is a simply connectedhyperbolic domain, then there is a K -quasiconformal map f : Ω → Dome(Ω) which extendscontinuously to the identity on ∂ Ω ⊂ ˆ C . In this paper, we obtain a bound in the conformally natural setting, which improves on bothof these bounds.
Mathematics Subject Classification a r X i v : . [ m a t h . G T ] S e p age 2 of 24 M. BRIDGEMAN, R. CANARY AND A. YARMOLA
Theorem 1.3.
There exists K ≤ . such that if Ω ⊂ ˆ C is a simply connected hyper-bolic domain, then there is a conformally natural K -quasiconformal map f : Ω → Dome(Ω) which extends continuously to the identity on ∂ Ω ⊂ ˆ C .Outline of argument: One may realize Dome(Ω) as the image of a pleated plane P µ : H → H whose bending is encoded by a measured lamination µ . Given L >
0, we define the L -roundness || µ || L of µ to be the least upper bound on the total bending of P µ ( α ) where α is an open geodesicsegment in H of length L . (This generalizes the notion of roundness introduced by Epstein-Marden-Markovic [ ].) Our first bound improves on an earlier bounds of Bridgeman [ , ] onroundness. Theorem 3.1. If L ∈ (0 , − (1)) , µ is a measured lamination on H and P µ is anembedding, then || µ || L ≤ − (cid:18) − sinh (cid:18) L (cid:19)(cid:19) . We then generalize work of Epstein-Marden-Markovic [ , Theorem 4.2, part 2] and anunpublished result of Epstein and Jerrard [ ] which give criteria for P µ to be an embedding. Theorem 4.1.
There exists an increasing function G : (0 , ∞ ) → (0 , π ) with G (1) ≈ . , suchthat if µ is a measured lamination on H such that || µ || L < G ( L ) , then P µ is a bilipschitz embedding which extends continuously to a map ˆ P µ : H ∪ S → H ∪ ˆ C so that ˆ P µ ( S ) is a quasi-circle. With these bounds in place, we may adapt the techniques of Epstein, Marden and Markovic[ , ] to complete the proof of our main result.2. Pleated planes and L -roundness In this section, we recall the definition of the pleated plane associated to a measuredlamination, and introduce the notion of L -roundness.Let G ( H ) be the set of unoriented geodesics on the hyperbolic plane H . One may identify G ( H ) with ( S × S − ∆) / Z . A geodesic lamination on H is a closed subset λ ⊂ G ( H ) whichdoes not contain any intersecting geodesics. A measured lamination µ on H is a non-negativemeasure µ on G ( H ) supported on a geodesic lamination λ = supp( µ ). A geodesic arc α in H is said to be transverse to µ , if it is transverse to every geodesic in the support of µ . If α istransverse to µ , we define i ( µ, α ) = µ (cid:0) { γ ∈ G ( H ) | γ ∩ α (cid:54) = ∅} (cid:1) . If α is not transverse to µ , then it is contained in a geodesic in supp( µ ) and we let i ( µ, α ) = 0.Given a measured lamination µ on H , we may define a pleated plane P µ : H → H , well-defined up to post-composition by an isometry of H . P µ is an isometry on the componentsof H − supp( µ ), which are called flats. If µ is a finite-leaved lamination, then P µ is simplyobtained by bending, consistently rightward, by the angle µ ( l ) along each leaf l of µ . Since anymeasured lamination is a limit of finite-leaved laminations, one may define P µ in general bytaking limits (see [ , Theorem 3.11.9]).If Ω ⊂ ˆ C is a simply connected hyperbolic domain, let Dome(Ω) denote the boundary of theconvex hull of its complement ˆ C − Ω. Thurston [ ] showed that there exists a lamination µ on MPROVED BOUND FOR SULLIVAN’S CONVEX HULL THEOREM
Page 3 of 24 H such that Dome(Ω) = P µ ( H ) and P µ : H → Dome(Ω) is an isometry. (See Epstein-Marden[ , Chapter 1], especially sections 1.11 and 1.12, for a detailed exposition.) Lemma 2.1. If Ω is a hyperbolic domain, there is a lamination µ on H such that P µ is alocally isometric covering map with image Dome(Ω) . For any point p ∈ Dome(Ω), a support plane at p is a totally geodesic plane through p whichis disjoint from the interior of the convex hull of ˆ C − Ω. The exterior angle, denoted ∠ ( P, Q ),between two intersecting support planes P and Q is the angle between their normal vectors ata point of intersection.Let α : [ a, b ] → H be a unit-speed closed geodesic arc. If α ( t ) lies on a leaf l of µ with µ ( l ) > { Q θl } θ ∈ [0 ,µ ( l )] of support planes to Dome(Ω) through P µ ( α ( t )),all of which contain P µ ( l ). In all other cases, Dome(Ω) has a unique support plane at P µ ( α ( t )).One may concatenate all the support planes to points in P µ ( α ([ a, b ])) to obtain a continuousfamily { P t } t ∈ [0 ,k ] of support planes along α , so that P is the leftmost support plane to Dome(Ω)at P µ ( α ( a )) and P k is the rightmost support plane to Dome(Ω) at P µ ( α ( b )). Moreover, thereexists a continuous non-decreasing function q : [0 , k ] → [ a, b ] so that P t is a support plane toDome(Ω) at P µ ( α ( q ( t ))) for all t . If 0 = t < t < · · · < t n = k and P t intersects both P t i − and P t i for all t ∈ [ t i − , t i ], then i ( µ, α ) ≤ n (cid:88) i =1 ∠ ( P t i − , P t i ) . See Section 4 of [ ], especially Lemma 4.1, for a more careful discussion.For a measured lamination µ on H , Epstein, Marden and Markovic [ ] defined the roundness of µ to be || µ || = sup i ( µ, α )where the supremum is taken over all open unit length geodesic arcs in H . The roundnessbounds the total bending of P µ on any segment of length 1 and is closely related to averagebending, which was introduced earlier by the first author in [ ]. In this paper, it will be usefulto consider the L -roundness of a measured lamination for any L > || µ || L = sup i ( α, µ )where now the supremum is taken over all open geodesic arcs of length L in H . We note thatthe supremum over open geodesic arcs of length L , is the same as that over half open geodesicarcs of length L .In [ ], the first author obtained an upper bound on the L -roundness of an embedded pleatedplane. Theorem 2.2. (Bridgeman [ ]) There exists a strictly increasing homeomorphism F : [0 , − (1)] → [ π, π ] such that if µ is a measured lamination on H and P µ is anembedding, then || µ || L ≤ F ( L ) for all L ≤ − (1) . In particular, || µ || ≤ F (1) = 2 π − − (cid:18) (cid:19) ≈ . . age 4 of 24 M. BRIDGEMAN, R. CANARY AND A. YARMOLA
Epstein, Marden and Markovic [ ] provided a criterion guaranteeing that a pleated planeis a bilipschitz embedding. Theorem 2.3. (Epstein-Marden-Markovic [ , Theorem 4.2, part 2]) If µ is a measuredlamination on H such that || µ || ≤ c = 73 , then P µ is a bilipschitz embedding which extendsto an embedding ˆ P µ : H ∪ S → H ∪ ˆ C such that ˆ P µ ( S ) is a quasi-circle. In [ ], Epstein, Marden and Markovic comment “Unpublished work by David Epstein andDick Jerrard should prove that c > . An upper bound on L -roundness for embedded pleated planes In this section, we adapt the techniques of [ ] to obtain an improved bound on the L -roundness of an embedded pleated plane. Theorem 3.1. If L ∈ (0 , − (1)) , µ is a measured lamination on H and P µ is anembedding, then || µ || L ≤ c ( L ) = 2 cos − (cid:18) − sinh (cid:18) L (cid:19)(cid:19) . Proof.
Since F (2 sinh − (1)) = 2 π , Theorem 3.1 follows from Theorem 2.2 when L = 2 sinh − (1) . Therefore, we may assume that
L < − (1).Let α : [0 , L ] → H be a geodesic arc of length L < − (1). Let { P t | t ∈ [0 , k ] } bethe continuous one-parameter family of support planes to α and let q : [0 , k ] → [0 , L ] be thecontinuous non-decreasing map such that P t is a support plane to Dome(Ω) at α ( q ( t )) for all t . We now recall the proof of Lemma 4.3 in [ ]. If P intersects P t for all t ∈ [0 , k ), then i ( α, µ ) ≤ π and we are done. If not, there exists a ∈ (0 , k ) such that P a has an ideal intersectionpoint with P and P t intersects P for all t ∈ (0 , a ). If there exists t ∈ ( a, k ] so that P t isdisjoint from P a , then Lemma 3.2 in [ ] implies that α ([0 , q ( t )]) has length at least 2 sinh − (1),which would be a contradiction. Therefore, if t ∈ ( a, k ], then P a intersects P t . One of thekey arguments in the proof of [ , Lemma 4.3] gives that P must be disjoint from P k (sinceotherwise one could extend α ([0 , P ∪ P k and thenproject onto Dome(Ω) to find a homotopically non-trivial curve on Dome(Ω).)Let φ be the interior angle of intersection between P a and P k . The interior angle ofintersection between P t and P varies continuously from π to 0 as t varies between 0 and a andachieves the value 0 only at a . There exists c ∈ (0 , a ) such that P c has an ideal intersection with P k and P t intersects P k for all t ∈ ( c, a ) (since otherwise we could again argue that i ( µ, α ) ≤ π ).The interior angle of intersection of P t with P k varies from 0 to φ as t varies from c to a . Thus,there exists some b ∈ ( c, a ) such that P b intersects P and P k in the same interior angle θ > , Lemma 4.1], we have i ( µ, α ) ≤ π − θ. Consider the plane R perpendicular to P , P b and P k . Consider the three geodesics g s = P s ∩ R , where s = 0, b or k . Notice that g b intersects both g and g k with interior angle θ .Let ¯ α be the orthogonal projection of α to R . Then ¯ α is a curve in R with ¯ α ( q ( s )) ∈ g s for MPROVED BOUND FOR SULLIVAN’S CONVEX HULL THEOREM
Page 5 of 24 s = 0 , b, k . Let β be the shortest curve joining a point of g to a point on g k which intersects g k . One may easily check that β consists of two geodesic arcs β and β such that β intersects g perpendicularly, β intersects g k perpendicularly and β and β make the same angle with g b at their common point of intersection. Figure 1.
The triangle T and its decomposition Since g and g k do not intersect, β is shortest when the geodesics g and g k have a commonideal point. In this case, the geodesics g , g b and g k form an isosceles triangle T with an idealvertex (see Figure 1). One may apply hyperbolic trigonometry formulae [ , Theorem 7.9.1] and[ , Theorem 7.11.2] to check that in this casecos( θ ) = sinh( (cid:96) ( β ) / . So, in general (cid:96) ( β ) ≥ − (cos( θ )) . Since, by construction, (cid:96) ( β ) ≤ (cid:96) ( α ) = L , we see that L ≥ − (cos( θ ))which implies that θ ≥ cos − (sinh( L/ . age 6 of 24 M. BRIDGEMAN, R. CANARY AND A. YARMOLA
Therefore, i ( µ, α ) ≤ π − − (sinh( L/ − ( − sinh( L/ α of length L . Therefore, the same bound holds for all open geodesicarcs of length L and the result follows.4. A new criterion for embeddednes of pleated planes
In this section, we provide a new criterion which guarantees the embeddedness of a pleatedplane which generalizes earlier work of Epstein-Marden-Markovic [ ] (see Theorem 2.3) andan unpublished result of Epstein-Jerrard [ ] Theorem 4.1.
There exists an increasing function G : (0 , ∞ ) → (0 , π ) , such that if µ is ameasured lamination on H and || µ || L < G ( L ) , then P µ is a bilipschitz embedding which extends continuously to a map ˆ P µ : H ∪ S → H ∪ ˆ C such that ˆ P µ ( S ) is a quasi-circle. Since G (1) ≈ . Corollary 4.2. (Epstein-Jerrard [ ]) If µ is a measured lamination on H such that || µ || < . then P µ is a bilipschitz embedding which extends continuously to a map ˆ P µ : H ∪ S → H ∪ ˆ C such that the image of S is a quasi-circle. We begin by finding an embedding criterion for piecewise geodesics. This portion of theproof follows Epstein and Jerrard’s outline quite closely. Such a criterion is easily translatedinto a criterion for the embeddedness of pleated planes associated to finite-leaved laminations.We then further show that, in the finite-leaved lamination case, the pleated planes are infact quasi-isometric embeddings with uniform bounds on the quasi-isometry constants. Thegeneral case is handled by approximating a general pleated plane by pleated planes associatedto finite-leaved laminations.
Remark 1.
As in [ , Theorem 4.2] we can consider a horocycle C in H and a sequenceof points on C with hyperbolic distance between consecutive points being L . Connectingconsecutive points, one obtains an embedded piecewise geodesic γ in H . Let P µ ( H ) be thepleated plane in H obtained by extending each flat in γ to a flat in H . One may check that || µ || L = 2 sin − (cid:18) tanh (cid:18) L (cid:19)(cid:19) which is the conjectured optimal bound. Since 2 sin − (tanh(1 / ≈ . L = 1. Comparing the bounds for all L ∈ [0 , − (1)], we see they arealso close to optimal (see Figure 2). MPROVED BOUND FOR SULLIVAN’S CONVEX HULL THEOREM
Page 7 of 24
Figure 2. G ( L ) and the conjectured optimal bound − (tanh( L/ on [0 , − (1)]4.1. Piecewise geodesics
Let J be an interval in R containing 0. A continuous map γ : J → H will be called a“piecewise geodesic” if there exists a discrete subset { t i } in J , parameterized by an interval in Z , such that, for all i , t i < t i +1 and γ (( t i , t i +1 )) is a geodesic arc. (If there is a first bendingpoint t r , we let t r − = inf J and if there is a last bending point t s , we define t s +1 = sup J .) Wewill call t i (or γ ( t i )) the bending points of γ . The bending angle φ i at t i is the angle between γ ([ t i − , t i ]) and γ ([ t i , t i +1 )). Let s ( t ) = d H ( γ (0) , γ ( t )) . If L >
0, by analogy with the definition of L -roundness, we may define || γ || L to be thesupremum of the total bending angle in any open subsegment of γ of length L .If t (cid:54) = t i for any i , then let θ ( t ) be the angle between the ray from γ (0) to γ ( t ) and thetangent vector γ (cid:48) ( t ). For i = 1 , . . . , n , we define γ (cid:48) + ( t i ) = lim t → t + i γ (cid:48) ( t ) and γ (cid:48)− ( t i ) = lim t → t − i γ (cid:48) ( t ) . We then choose θ ± ( t i ) to be the angle between the ray from γ (0) to γ ( t ) and the vector γ (cid:48)± ( t ).(Equivalently, we could have defined θ ± ( t i ) to be the angle between the ray from γ (0) to γ ( t )and the geodesic segment γ ([ t i , t i ± )).) Notice that θ ( t ) decreases smoothly on ( t i , t i +1 ) for all i and that | θ + ( t i ) − θ − ( t i ) | ≤ φ i (4.1)for all i .If t (cid:54) = t i for any i , then Lemma 4.4 in Epstein-Marden-Markovic [ ] gives that s (cid:48) ( t ) = cos( θ ( t )) and θ (cid:48) ( t ) = − sin( θ ( t ))tanh( s ( t )) < − sin( θ ( t )) . (4.2)age 8 of 24 M. BRIDGEMAN, R. CANARY AND A. YARMOLA
The hill function of Epstein and Jerrard
A key tool in Epstein and Jerrard’s work is the following hill function h : R → (0 , π ) given by h ( x ) = cos − (tanh( x )) . The defining features of the hill function are that h (cid:48) ( x ) = − sech( x ) = − sin( h ( x )) and h (0) = π . In particular, h is a decreasing homeomorphism.For fixed L >
0, we consider solutions to the equation h (cid:48) ( x ) = h ( x ) − h ( x − L ) L .
Geometrically, we are finding the point on the graph of h such that the tangent line at ( x, h ( x ))intersects the graph at the point ( x − L, h ( x − L )) (see Figure 3). We will show that there isa unique solution x = c ( L ) and that c ( L ) ∈ (0 , L ).Given x ∈ R , the tangent line at ( x, h ( x )) to the graph of h intersects the graph in twopoints ( x, h ( x )) and ( f ( x ) , h ( f ( x )) (except at x = 0 where the points are equal). The function f is continuously differentiable and odd. We define A ( x ) = x − f ( x ), so A is also continuouslydifferentiable and odd. Since A is odd, to show that A is strictly increasing, it suffices toshow that it is strictly increasing on [0 , ∞ ). Suppose that 0 ≤ x < x , and that T and T are the tangent lines to h at x and x . Since h is convex on [0 , ∞ ), T ∩ T = ( x , y ) liesbelow the graph of h and x < x < x . Thus T intersects the graph of h to the left of thepoint of intersection of T with the graph of h . Therefore, f ( x ) < f ( x ) ≤ f (0) = 0 and f isdecreasing. It follows that A ( x ) = x − f ( x ) is increasing and that A ( x ) > x for all x ∈ (0 , ∞ ).The function c is the inverse of A , so c is also continuous differentiable and strictly increasing.Since A ( x ) > x for x > c ( L ) ∈ (0 , L ).Let Θ( L ) = h ( c ( L )) and G ( L ) = h ( c ( L ) − L ) − h ( c ( L )) = − Lh (cid:48) ( c ( L )) . To show G is monotonic, we define B ( x ) = h ( f ( x )) − h ( x ), the difference of the heights ofthe intersection points of the tangent line at ( x, h ( x )) with the graph of h . As h and f areboth strictly decreasing continuous functions, B is strictly increasing and continuous. Since G ( L ) = B ( c ( L )), G is a strictly increasing continuous function.We note that Θ( L ) + G ( L ) = h ( c ( L ) − L ) < π. The following lemma is the key estimate in the proof of Theorem 4.1.
Lemma 4.3. If γ : [0 , ∞ ) → H is piecewise geodesic, L > and || γ || L ≤ G ( L ) , then θ + ( t ) ≤ Θ( L ) + G ( L ) < π for all t > .Proof. We define maps P ± : (0 , ∞ ) → R which are continuous except at the bending points { t i } and whose image lies on the graph of h . Since h is a homeomorphism onto [0 , π ], given t ∈ (0 , ∞ ), we can find a unique g ± ( t ) ∈ R , such that h ( g ± ( t )) = θ ± ( t ) . MPROVED BOUND FOR SULLIVAN’S CONVEX HULL THEOREM
Page 9 of 24We then define P ± ( t ) = ( P ± ( t ) , P ± ( t )) = ( g ± ( t ) , h ( g ± ( t ))) = ( g ± ( t ) , θ ± ( t )) . Note that the functions P + and P − agree except at the bending points. In the intervals, wedenote the common functions by P ( t ), g ( t ), and θ ( t ).Notice that as one moves along the geodesic ray γ , the functions θ ± ( t ) decrease on eachinterval ( t i , t i +1 ) and have vertical jump equal to ψ i = θ + ( t i ) − θ − ( t i ) at each t i . By equation4.1 we have | ψ i | = | θ + ( t i ) − θ − ( t i ) | ≤ φ i . Correspondingly, the point P ± ( t ) move along the graph of h by sliding rightward (anddownward) along ( t i , t i +1 ) and jumping vertically, either upwards or downwards, by ψ i at t i , see Figure 3.We argue by contradiction. Let c = c ( L ), G = G ( L ), and Θ = Θ( L ). Suppose there exists T > θ + ( T ) > Θ + G . Let s = sup { s ∈ (0 , T ] | θ − ( s ) ≤ Θ } . Notice that if s = T , then, since | θ + ( s ) − θ − ( s ) | < G , θ + ( T ) ≤ θ − ( T ) + G ≤ Θ + G which would be a contradiction.Also notice that s = t i for some i , since otherwise θ − is continuous and non-increasing at s , which would contradict the choice of s .If T − s < L , then since θ can only increase at the bending points and the total bending inthe region [ s , T ] is at most G , again θ + ( T ) ≤ θ − ( s ) + G ≤ Θ + G which is a contradiction.So, we may assume that T − s ≥ L . We will use the assumption that θ − ( t ) > Θ on( s , s + L ] to arrive at a contradiction and complete the proof of the lemma.We show that under our hypotheses, P ( T ) cannot lie to the left of ( c ( L ) − L, h ( c ( L ) − L )).The key observation in the proof is that h (cid:48) ( g ( t )) g (cid:48) ( t ) = θ (cid:48) ( t ) < − sin( θ ( t )) = − sin( h ( g ( t ))) = h (cid:48) ( g ( t ))where the middle inequality follows from equation (4.2). Since h (cid:48) ( g ( t )) <
0, we conclude that g (cid:48) ( t ) > t ∈ ( t i , t i +1 ) . Therefore, g ( t i +1 ) − g ( t i ) = g − ( t i +1 ) − g + ( t i ) > t i +1 − t i (4.3)for all i .Let { s = t j , t j +1 , . . . , t j + m } be the bending points in the interval [ s , s + L ). For conve-nience, we redefine t j + m +1 = s + L . Since || γ || L ≤ G , the total vertical jump in the region[ s , s + L ) is at most G , i.e. j + m (cid:88) i = j | θ + ( t i ) − θ − ( t i ) | ≤ G, Since θ + is non-increasing on each interval ( t i , t i +1 ) and θ − ( s ) ≤ Θ, it follows that θ + ( t ) ≤ Θ + G for all t ∈ [ s , s + L ).Let d = min { g + ( t ) | t ∈ [ s , s + L ) } . age 10 of 24 M. BRIDGEMAN, R. CANARY AND A. YARMOLA
Figure 3.
Jumps and slides on the graph of h Notice that as g + is non-decreasing on ( t i , t i +1 ) for all i , there exists a largest k ∈ { j, . . . , j + m } so that g + ( t k ) = d . We further note that d ∈ [ c − L, c ] since θ + ( t ) ∈ [Θ , Θ + G ] for all t ∈ [ s , s + L ). We break the proof into two cases. Case I: d ∈ [ − c, c ] : If d ∈ [ − c, c ] then g + ([ s , s + L ]) ⊆ [ − c, c ]. Since θ − ( t ) ≥ Θ on ( s , s + L ],we have g − (( s , s + L ]) ⊆ [ − c, c ]. Notice that, since h (cid:48) ( x ) = − sin( h ( x )) and h is decreasing,if x ∈ [ − c, c ], then h (cid:48) ( x ) ≤ h (cid:48) ( c ) = − GL .
Therefore, applying (4.3), we see that θ − ( t i +1 ) − θ + ( t i ) ≤ h (cid:48) ( c )( g − ( t i +1 ) − g + ( t i )) = − GL ( g − ( t i +1 ) − g + ( t i )) ≤ − GL ( t i +1 − t i )for all i = j, . . . , j + m . Thus, θ − ( s + L ) − θ − ( s ) = j + m (cid:88) i = j θ + ( t i ) − θ − ( t i ) + j + m (cid:88) i = j θ − ( t i +1 ) − θ + ( t i ) ≤ j + m (cid:88) i = j | θ + ( t i ) − θ − ( t i ) | − (cid:32) j + m (cid:88) i =1 GL ( t i +1 − t i ) (cid:33) ≤ G − GL j + m (cid:88) i =1 ( t i +1 − t i ) = 0This implies that θ − ( s + L ) ≤ Θ, which contradicts the choice of s . Case II: d ∈ [ c − L, , − c ) : If d ∈ [ c − L, − c ), then | h (cid:48) ( g ( t )) | ≥ | h (cid:48) ( d ) | MPROVED BOUND FOR SULLIVAN’S CONVEX HULL THEOREM
Page 11 of 24for all t ∈ [ s , s + L ]. So,( θ + ( t i ) − θ − ( t i +1 )) ≥ | h (cid:48) ( d ) | ( g − ( t i +1 ) − g + ( t i )) ≥ | h (cid:48) ( d ) | ( t i +1 − t i ) (4.4)for all i = j, . . . , j + m. It follows that k − (cid:88) i = i (cid:0) θ + ( t i ) − θ − ( t i +1 ) (cid:1) ≥ | h (cid:48) ( d ) | ( t k − s ) . Thus, since θ + ( t k ) = h ( d ) and θ − ( t j ) ≤ Θ, k (cid:88) i = j (cid:0) θ + ( t i ) − θ − ( t i ) (cid:1) ≥ ( h ( d ) − Θ) + | h (cid:48) ( d ) | ( t k − s )and so, since the total jump on the interval [ s , s + L ) is at most G , j + m (cid:88) i = k +1 θ + ( t i ) − θ − ( t i ) ≤ G − ( h ( d ) − Θ) − | h (cid:48) ( d ) | ( t k − s ) = h ( c − L ) − h ( d ) − | h (cid:48) ( d ) | ( t k − s ) . Since g + ( t k ) = d , g − ( s + L ) = d + (cid:32) j + m (cid:88) i = k g − ( t i +1 ) − g + ( t i ) (cid:33) − (cid:32) j + m (cid:88) i = k +1 g − ( t i ) − g + ( t i ) (cid:33) . Applying inequalities (4.3) and (4.4), we see that g − ( s + L ) > d + (cid:32) j + m (cid:88) i = k t i +1 − t i (cid:33) − | h (cid:48) ( d ) | (cid:32) j + m (cid:88) i = k +1 θ + ( t i ) − θ − ( t i ) (cid:33) > d + ( s + L − t k ) − | h (cid:48) ( d ) | ( h ( c − L ) − h ( d ) − | h (cid:48) ( d ) | ( t k − s ))= d + L − (cid:18) h ( c − L ) − h ( d ) | h (cid:48) ( d ) | (cid:19) . Taking the tangent line at d we note that, since h (cid:48) is negative and decreasing on the interval[ c − L, d ], we have h ( c − L ) ≤ h ( d ) + h (cid:48) ( d )( c − L − d )which implies that 1 h (cid:48) ( d ) ( h ( c − L ) − h ( d )) ≥ c − L − d. Therefore, g − ( s + L ) > d + L + 1 h (cid:48) ( d ) ( h ( c − L ) − h ( d )) ≥ c, so, θ − ( s + L ) ≤ Θ contradicting the definition of s . This final contradiction completes theproof.As a nearly immediate corollary, we obtain an embeddedness criterion for piecewise geodesics. Corollary 4.4. If γ : [0 , ∞ ) → H is a piecewise geodesic, and || γ || L ≤ G ( L ) for some L > , then γ is an embedding. age 12 of 24 M. BRIDGEMAN, R. CANARY AND A. YARMOLA
Proof.
Notice that if the corollary fails, then there exists a piecewise geodesic ray γ : [0 , ∞ ) → H such that || γ || L ≤ G ( L ) and γ (0) = γ ( b ) for some b >
0. (Since if γ ( p ) = γ ( q )for some 0 ≤ p < q , we can instead consider the piecewise geodesic ray γ : [0 , ∞ ) → H where γ ( t ) = γ ( t − p ).) There must exist t i ∈ (0 , b ) so that γ is geodesic on [ t i , b ]. Then, θ + ( t ) = π on ( t i , b ), contradicting Lemma 4.3 above.If µ is a finite-leaved measured lamination on H and α : [0 , ∞ ) is any geodesic ray in H ,then γ = P µ ◦ α is a piecewise geodesic and || γ || L ≤ || µ || L . Since any two points in H can bejoined by a geodesic ray, we immediately obtain an embeddedness criterion for pleated planes. Corollary 4.5. If µ is a finite-leaved measured lamination on H and || µ || L ≤ G ( L ) forsome L > , then P µ : H → H is an embedding. Uniformly bilipschitz embeddings
We next prove that if γ : R → H is a piecewise geodesic and || γ || L < G ( L ), then γ isuniformly bilipschitz. We note that since γ is 1-Lipschitz, we only have to prove a lower bound.This will immediately imply that if µ is a finite-leaved lamination on H and || µ L || < G ( L ),then P µ is a K -bilipschitz embedding. Proposition 4.6. If γ : R → H is a piecewise geodesic such that || γ || L < G ( L ) , then γ is K -bilipschitz where K depends only on L and || γ || L .Proof. We first set our notation. We may assume, without loss of generality, that 0 is nota bending point of γ . Let t = 0 and assume that the bending points in (0 , ∞ ) are indexed byan interval of positive integers beginning with 1 and the ending points in ( −∞ ,
0) are indexedby an interval of negative integers ending with −
1. Let φ i be the bending angle of γ at t i .The following lemma will allow us to reduce to the planar setting. Lemma 4.7.
There exists an embedded piecewise geodesic α : R → H with the samebending points as γ such that (i) if the bending angle of α at a bending point t i is given by φ (cid:48) i , then φ (cid:48) i ≤ φ i , (ii) d ( α (0) , α ( t )) = d ( γ (0) , γ ( t )) for all t , and (iii) there exists a non-decreasing function Ψ : R → ( − π, π ) such that if t > , then Ψ( t ) isthe angle between α ([0 , t ]) and the geodesic joining α (0) to α ( t ) , while if t < , then Ψ( t ) is the angle between α ([ − t , and the geodesic joining α (0) to α ( t ) .Proof. Let f i be the geodesic arc from γ (0) to γ ( t i ) and let T i be the hyperbolic trianglewith vertices γ (0), γ ( t i ), and γ ( t i +1 ) and edges f i , γ ([ t i , t i +1 ]) and f i +1 . We construct α byfirst placing an isometric copy of T in H , so that f is counterclockwise from f . We theniteratively place a copy of T i adjacent to a copy of T i − (so that their interiors are disjoint)along the image of f i for all positive t i . We then place a copy of T − in H so that T − and T intersect along the image of γ (0), so that the images of f and f − lie in a geodesic and theimage of f − is clockwise from f − . We then iteratively place a copy of T − i − next to the copyof T − i for all negative t − i (see Figure 4). MPROVED BOUND FOR SULLIVAN’S CONVEX HULL THEOREM
Page 13 of 24
Figure 4.
The curve α Let α : R → H be the piecewise geodesic traced out by the images of pieces of γ . Then α hasthe same bending points as γ by construction. Moreover, since d ( α (0) , α ( t )) is realized in theisometric copy of T n when t ∈ [ t n , t n +1 ], it is also immediate that d ( α (0) , α ( t )) = d ( γ (0) , γ ( t ))for all t .We next check that the bending angle φ (cid:48) i of α at t i is at most ψ i . We consider the vectors v − n = γ (cid:48)− ( t i ) and v + n = γ (cid:48) + ( t i ) at γ ( t i ). Then the exterior angle φ i is the distance between v − i and v + i in the unit tangent sphere at γ ( t i ). The edge f n defines an axis in the unit sphere. Thepossibilities for gluing T n to T n − are given by the one-parameter family of triangles obtainedby rotating T i about f i . It is then easy to see that the distance is shortest when T i lies in thesame plane as T i − and has disjoint interior Therefore, φ (cid:48) n ≤ φ n . Since || α || L ≤ || γ || L < G ( L ) , Corollary 4.4 implies that α is an embedding.We can now define a continuous non-decreasing function Ψ : R → R so that Ψ(0) = 0 and,if t >
0, then Ψ( t ) is the angle, modulo 2 π , between α ([0 , t ]) and the geodesic joining α (0) to α ( t ), while if t <
0, then Ψ( t ) is the angle between α ([ − t , α (0) to α ( t ).We next show that Ψ( t ) < π for all t >
0. If not, then γ intersects the line g containing α ([0 , t ]). Suppose that α ( b ) ∈ g for some b >
0. Then, consider the piecewise geodesic ˆ α which first traces α ([0 , b ]) backwards and then continues along g forever. Notice that ˆ α is notage 14 of 24 M. BRIDGEMAN, R. CANARY AND A. YARMOLA an embedding. However, || ˆ α || L ≤ || α || L < G ( L ) , so Corollary 4.4 implies that ˆ α is an embedding, which is a contradiction. Similarly, Ψ( t ) > − π for all t <
0. This completes the proof of (3).We notice that it suffices to show that there exists K depending only on L and || γ || L , sothat s ( t ) = d ( γ (0) , γ ( t )) = d ( α (0) , α ( t )) ≥ K | t | for all t ∈ R . Since, if we suppose that γ : R → H is any piecewise geodesic with || γ || L < G ( L )and r < r , then we can consider the new piecewise geodesic γ r : R → H given by γ r ( t ) = γ ( t − r ). Then || γ r || L = || γ || L and s r ( t ) = d ( γ r ( t ) , γ r (0)) ≥ K | t | . It follows that d ( γ ( r ) , γ ( r )) = s r ( r − r ) ≥ K | r − r | Since γ is 1-Lipschitz by definition, it follows immediately that γ is a K -bilipschitz embedding.Since Ψ is monotone and bounded we may defineΨ + ∞ = lim t →∞ Ψ( t ) and Ψ −∞ = lim t →−∞ Ψ( t ) . We now show that α is proper. The basic idea is that, since Ψ is monotonic, then α ([0 , ∞ ))can only accumulate on the geodesic ray (cid:126)r + emanating from α (0) and making angle Ψ + with α ([0 , t ]). If it accumulate at q , then there must be infinitely many segments of α running nearlyparallel to (cid:126)r + and accumulating at some point q on (cid:126)r . However, by Lemma 4.3, no segment of α can be pointing nearly straight back to α (0), so the total length of these segments which are“pointing towards” α (0) is finite. This will allow us to arrive at a contradiction.If α is not proper, then either α | [0 , ∞ ) or α | ( −∞ , is not proper. We may assume ray α | [0 , ∞ ) is not proper. We recall that if t is not a bending point, then θ ( t ) is the angle between α (cid:48) ( t )and the geodesic segment joining α (0) to α ( t ). Lemma 4.3 implies that θ ( t ) ≤ Θ = Θ( L ) + G ( L ) < π for all t . Since α | [0 , ∞ ) is not proper, there is an accumulation point q of α | [0 , ∞ ) on the ray (cid:126)r + emanating from α (0) which makes an angle Φ + ∞ with α ([0 , t ]).We may work in the disk model and assume that α (0) = 0 and α ([0 , t ]) lies in the positivereal axis. If (cid:15) > r, ¯ θ ) by B (cid:15) = [ r ( q ) − (cid:15), r ( q ) + (cid:15) ] × [ θ ( q ) − (cid:15), θ ( q )] ⊂ D . On B (cid:15) we consider the taxicab metric, given by d T (( r , θ ) , ( r , θ )) = | r − r | + | θ − θ | . Wenotice that that d T on B (cid:15) is bilipschitz to the hyperbolic metric. If J = α − ( B (cid:15) ), then J is acountable collection of disjoint arcs. Notice that α ( J ) = α ([0 , ∞ )) ∩ B (cid:15) .Since Ψ is monotonic, the ¯ θ coordinate of α is monotonic, so the total length of α ( J ) in the¯ θ direction is bounded above by (cid:15) . Also the signed length of α ( J ) in the r direction is boundedabove by 2 (cid:15) . Since θ ( t ) ≤ Θ , at all non-bending points, the total length in the negative r -direction is bounded above by (cid:15) tan(Θ ). Therefore, the total length in the positive r -directionis bounded above by (cid:15) + (cid:15) tan(Θ ). It follows that α ( J ) has finite length in the taxicab metricon B (cid:15) . We choose ¯ t ∈ J , so that α ( J ∩ [¯ t, ∞ )) has length, in the taxicab metric, less than (cid:15)/ d B (cid:15) ( α (¯ t ) , q ) < (cid:15)/
4. Therefore, α ( J ∩ [¯ t, ∞ )) ⊂ B (cid:15)/ ( q ) and B (cid:15)/ ( q )) ⊂ B (cid:15) (where B (cid:15)/ ( q ) MPROVED BOUND FOR SULLIVAN’S CONVEX HULL THEOREM
Page 15 of 24is the neighborhood of radius (cid:15)/ q in the taxicab metric on B (cid:15) ). It follows that [¯ t, ∞ ) ⊂ J ,which contradicts the fact that α ([¯ t, ∞ )) has infinite length. Therefore, α must be proper.Since α is proper and Ψ is monotone, α has two unique limit points ξ − and ξ + in S which areendpoints of the geodesic rays from α (0) which make angles Ψ −∞ and Ψ + ∞ with α ([ t − , t ]).Thus, since α is embedded, Ψ + ∞ − Ψ + ∞ ≤ π. Let B = G ( L ) − || µ || L + ∞ − Ψ −∞ ≤ π − B If not, we construct a new piecewise geodesic α : R → H which has a bend of angle G ( L ) −|| µ || L )4 at 0. One then checks that || α || L ≤ || α || L + 3 / G ( L ) − || µ || L ) < G ( L )but α is not an embedding, which would contradict Corollary 4.4.Let g be the geodesic joining ξ − to ξ + . Since Ψ + ∞ − Ψ −∞ ≤ π − B , the visual distancebetween ξ + and ξ − , as viewed from α (0) is at least B . It follows that there exists C , dependingonly on B , so that d ( α (0) , g )) ≤ C . In fact, one may apply Theorem 7.9.1 in Beardon [ ] tocheck that we may choose C = cosh − (cid:18) B/ (cid:19) . Notice that, by considering a reparameterization of α , we can see that the visual distancebetween ξ + and ξ − is at least B as viewed from α ( t ) for any t ∈ R , and thus that α ( t ) lieswithin C of g for any t ∈ R .We next claim there exists K > p : H → g is orthogonal projection, then p ◦ α is a 1-Lipschitz, K -bilipschitz orientation-preserving embedding. The fact that p ◦ α is1-Lipschitz follows immediately from the fact that both p and α are 1-Lipschitz. Let ν be theangle between the orthogonal geodesic h to g through α and the geodesic segment α ([ t − , t ])chosen so that ν > α ( t ) lies on the same side of h as ξ + . Notice that B ≤ ν ≤ π − B + ∞ − Ψ −∞ ≥ π − B . Therefore, the restriction of p ◦ α to [ t − , t ] is anorientation-preserving embedding. We let v be a unit tangent vector at α (0) perpendicular to g . Then || p (cid:48) ( α (0))( v ) || = 1cosh( d ( α (0) , g )) ≥ C ) = sin( B/ α (cid:48) (0) makes an angle at most B/ v || ( p ◦ α ) (cid:48) (0) || ≥ sin( B/ C ) = sin ( B/
2) = 1
K .
Again, by reparameterizing, we may check that if t is a non-bending point, then p ◦ α is anorientation-preserving local homeomorphism at t and that || ( p ◦ α ) (cid:48) ( t ) || ≥ K .
It follows that, for all t , d ( p ( γ (0)) , p ( γ ( t )) ≥ K t. age 16 of 24
M. BRIDGEMAN, R. CANARY AND A. YARMOLA
Therefore, since p is 1-Lipschitz, s ( t ) = d ( α (0) , α ( t )) ≥ d ( p ( γ (0)) , p ( γ ( t ))) ≥ tK We observed earlier that this is enough to guarantee that γ is K -bilipschitz.As an immediate corollary, we obtain a version of Theorem 4.1 for finite-leaved laminations. Corollary 4.8. If µ is a finite-leaved measured lamination on H such that || µ || L < G ( L ) , then P µ is a K -bilipschitz embedding, where K depends only on L and || µ || L . Proof of Theorem 4.1
Suppose that µ is a measured lamination on H with || µ || L < G ( L ). By Lemma 4.6 in Epstein-Marden-Markovic [ ], there exists a sequence { µ n } of finite-leaved measured laminationswhich converges to µ such that || µ n || L = || µ || L for all n . Corollary 4.8 implies that each P µ n isa K -bilipschitz embedding where K depends only on L and || µ || L . The maps { P µ n } convergesuniformly on compact sets to P µ (see [ , Theorem III.3.11.9]), so P µ is also a K -bilipschitzembedding. Therefore, P µ extends continuously to ˆ P µ : H ∪ S ∞ → H ∪ S ∞ and ˆ P µ ( S ) is aquasi-circle. (cid:3) Complex earthquakes
In this section, we use Theorem 4.1 to give improved bounds in results of Epstein-Marden-Markovic which will lead to the improved bound obtained in our main result. We first obtainnew bounds guaranteeing that complex earthquakes extend to homeomorphisms at infinity, seeCorollaries 5.2 and 5.3. Once we have done so, we obtain a generalization of [ , Theorem 4.14]which produces a family of conformally natural quasiconformal maps associated to complexearthquakes with the same support µ which satisfy the bounds obtained in Corollary 5.2 orCorollary 5.3. Finally, we give a version of [ , Theorem 4.3] which gives rise to a family ofquasiregular maps associated to all complex earthquakes with positive bending along µ .If µ is a measured lamination on H , we define E µ : H → H to be the earthquake mapdefined by fixing a component of the complement of µ and left-shearing all other componentsby an amount given by the measure on µ . An earthquake map is continuous except on leavesof µ with discrete measure and extends to a homeomorphism of S .Therefore, any measuredlamination λ on H is mapped to a well-defined measured lamination on H which we denote E µ ( λ ).Given a measured lamination µ on H and z = x + iy ∈ C , we define the complex earthquake C E z = P yE xµ ◦ E xµ : H → H to be the composition of earthquaking along xµ and then bending along the lamination yE xµ ( µ ). The sign of y determines the direction of the bending. By linearity, || yE xµ ( µ ) || L = | y | || E xµ ( µ ) || L . (See Epstein-Marden [ , Chapter 3] or Epstein-Marden-Markovic [ , Section 3] for a detaileddiscussion of complex earthquakes.)The following estimate allows one to bound || E xµ ( µ ) || L . MPROVED BOUND FOR SULLIVAN’S CONVEX HULL THEOREM
Page 17 of 24
Theorem 5.1. (Epstein-Marden-Markovic [ , Theorem 4.12]) Let (cid:96) and (cid:96) be distinctleaves of a measured lamination µ on H . Suppose that α is a closed geodesic segment withendpoints on (cid:96) and (cid:96) and let x = i ( α, µ ) . Let (cid:96) (cid:48) and (cid:96) (cid:48) be the images of (cid:96) and (cid:96) under theearthquake E µ . Then sinh( d ( (cid:96) (cid:48) , (cid:96) (cid:48) )) ≤ e x sinh( d ( (cid:96) , (cid:96) )) and d ( (cid:96) (cid:48) , (cid:96) (cid:48) ) ≤ e x/ d ( (cid:96) , (cid:96) ) . Furthermore, sinh( d ( (cid:96) , (cid:96) )) ≤ e x sinh( d ( (cid:96) (cid:48) , (cid:96) (cid:48) )) and d ( (cid:96) , (cid:96) ) ≤ e x/ d ( (cid:96) (cid:48) , (cid:96) (cid:48) ) . Motivated by this result, Epstein, Marden, and Markovic define the function f ( L, x ) = min (cid:16) Le | x | / , sinh − ( e | x | sinh( L )) (cid:17) . Corollary 4.13 in [ ] generalizes to give: Corollary 5.2. If µ is a measured lamination on H , z = x + iy ∈ C , and L > , then || E xµ ( µ ) || L ≤ (cid:24) f ( L, x ) L (cid:25) || µ || L . Furthermore, if | y | < G ( L ) (cid:108) f ( L,x ) L (cid:109) || µ || L , then C E z extends to an embedding of S into ˆ C . We similarly define g ( L, x ) = max (cid:16) Le −| x | / , sinh − ( e −| x | sinh( L )) (cid:17) . We will show later, see Lemma 7.1, that if 2 tanh( L ) > L then g ( L, x ) = Le −| x | / .Theorem 5.1 and Theorem 4.1 combine to give the following: Corollary 5.3. If µ is a measured lamination on H , z = x + iy ∈ C , and L > , then || E xµ ( µ ) || g ( L,x ) ≤ || µ || L . Furthermore, if | y | < G ( g ( L, x )) || µ || L , then P yE xµ is a bilipschitz embedding and C E z extends to an embedding of S into ˆ C .Proofs: The proofs of Corollaries 5.2 and 5.3 both follow the same outline as the proof of [ ,Corollary 4.13]. Let µ be a measured lamination on H , z = x + iy ∈ C , and L >
A > α is an open geodesic arc in H of length A which is transverseto E xµ ( µ ). Theorem 5.1 guarantees that one can choose an open geodesic arc β in H whichintersects exactly the leaves of µ which correspond to leaves of E xµ which intersect α and hastotal length at most f ( A, x ). Therefore, i ( α, E xµ ( µ )) = i ( β, µ ) ≤ || µ || f ( A,x ) , age 18 of 24 M. BRIDGEMAN, R. CANARY AND A. YARMOLA so || E xµ ( µ ) || A ≤ || µ || f ( A,x ) . (5.1)We begin with the proof of Corollary 5.3. Inequality 5.1 immediately implies that || E xµ ( µ ) || g ( L,x ) ≤ || µ || f ( g ( L,x )) = || µ || L . So, if | y | < G ( g ( L, x )) || µ || L , then || y E xµ || g ( L,x ) < G ( g ( L, x )) . Theorem 4.1 then implies that P yE xµ is a bilipschitz embedding which extends to an embeddingof S into ˆ C . Since E xµ extends to a homeomorphism of S , it follows that C E z extends to anembedding of S into ˆ C . This completes the proof of Corollary 5.3.We now turn to the proof of Corollary 5.2. We can divide a half open geodesic arc in H oflength f ( L, x ) into (cid:100) f ( L, x ) /L (cid:101) half open geodesic arcs of length less than or equal to L , so || E xµ ( µ ) || L ≤ || µ || f ( L,x ) ≤ (cid:24) f ( L, x ) L (cid:25) || µ || L . Therefore, if | y | < G ( L ) (cid:108) f ( L,x ) L (cid:109) || µ || L , then || y E xµ ( µ ) || L < G ( L ) . and we may again use Theorem 4.1 to complete the proof of Corollary 5.2. (cid:3) For all
L >
0, we define Q ( L, x ) = max G ( L ) (cid:108) f ( L,x ) L (cid:109) , G ( g ( L, x )) and T L = int( { x + iy | | y | < Q ( L, x ) } . The following theorem is a direct generalization of Theorem 4.14 in Epstein-Marden-Markovic [ ]. In its proof, we simply replace their use of Corollary 4.13 in [ ] with ourCorollaries 5.2 and 5.3. Theorem 5.4.
Suppose that
L > and µ is a measured lamination on H such that || µ || L =1 . Then, for z ∈ T L , (i) C E z extends to an embedding φ z : S → ˆ C which bounds a region Ω z . (ii) There is a quasiconformal map Φ z : D → Ω z with domain the unit disk and quasicon-formal dilatation K z bounded by K z ≤ | h ( z ) | − | h ( z ) | where h : T → D is a Riemann map taking to .Moreover, Φ z ∪ φ z : D ∪ S → ˆ C is continuous. MPROVED BOUND FOR SULLIVAN’S CONVEX HULL THEOREM
Page 19 of 24(iii) If G is a group of M¨obius transformations preserving µ , then Φ z can be chosen sothat there is a homomorphism ρ z : G → G z where G z is also a group of M¨obiustransformations and Φ z ◦ g = ρ z ( g ) ◦ Φ z for all g ∈ G . Epstein, Marden and Markovic [ ] introduce the theory of complex angle scaling maps anduse them to produce a family of quasiregular mappings indexed by S L = int (cid:26) x + iy ∈ C | y > − . f (1 , x ) (cid:27) so that if | Im( t ) | < . f (1 ,x ) , then Φ t is quasiconformal. (See also the discussion in [ , Section3.4].)We consider the enlarged region T L = int { x + iy ∈ C | y > − Q ( x, L ) } . Given Theorems 4.1 and 5.4, their proof of Theorem 4.3 extends immediately to give:
Theorem 5.5. ([ , Theorem 4.13]) Suppose that
L > , µ is a measured lamination on H with || µ || L = 1 , v > and t = iv ∈ T L . If t ∈ T L , let Ω t be the the image of D underthe map Φ t given by Theorem 5.4. Then there exists a continuous map Ψ : U × Ω t → ˆ C , where U is the upper half-plane, such that (i) Ψ t = id . (ii) For each z ∈ Ω t , Ψ( t, z ) depends holomorphically on t . (iii) For each t ∈ T L , Ψ t can be continuously extended to ∂ Ω t such that Ψ t ◦ Φ t | S = Φ t | S . In particular Ψ : ∂ Ω t → S and Φ t : S → ∂ Ω are inverse homeomorphisms. (iv) If t ∈ T L and Im( t ) > , then Ψ t is injective and Ψ t (Ω ) = Φ t ( D ) = Ω t . (v) If t = u + iv and v > , then Ψ t is locally injective K t -quasiregular mapping where K t − | κ ( t ) | − | κ ( t ) | , | κ ( t ) | = (cid:112) u + ( v − v ) (cid:112) u + ( v + v ) (vi) If G is a group of M¨obius transformations preserving Ω , then there is a homomorphism ρ t : G → G t where G t is also a group of M¨obius transformations, such that Ψ t ◦ g = ρ t ( g ) ◦ Ψ t for all g ∈ G . Quasiconfomal bounds
One can now readily adapt the techniques of proof of Epstein-Marden-Markovic [ , Theorem6.11] to establish: Theorem 6.1. If Ω is a simply connected hyperbolic domain in ˆ C and L > , then there isa conformally natural K -quasiconformal map f : Ω → Dome(Ω) which extends to the identityon ∂ Ω ⊂ ˆ C such that log( K ) ≤ d T L ( ic ( L ) , M. BRIDGEMAN, R. CANARY AND A. YARMOLA where d T L is the Poincar´e metric on the domain T L and c ( L ) = 2 cos − (cid:0) − sinh (cid:0) L (cid:1)(cid:1) . We offer a brief sketch of the proof in order to indicate where our new bounds, as given inTheorems 3.1, 5.4 and 5.5, are used in the argument.We recall that universal Teichm¨uller space U is the space of quasisymmetric homeomorphismsof the unit ciricle S , modulo the action of M¨obius transformations by post-composition (see,for example, Ahlfors [ , Chapter VI]. The Teichm¨uller metric on the space U is defined by d U ( f, g ) = log inf K ( ˆ f − ◦ ˆ g )where the infimum is over all quasiconformal extensions ˆ f and ˆ g of f and g to maps fromthe unit disk to itself and K ( ˆ f − ◦ ˆ g ) is the quasiconformal dilatation of ˆ f − ◦ ˆ g . If Γ isa group of conformal automorphisms of D , we define U (Γ) ⊆ U to be the quasisymmetrichomeomorphisms which conjugate the action of Γ to the action of an isomorphic group ofconformal automorphisms. The Teichm¨uller metric on U (Γ) is defined similarly by consideringextensions which conjugate Γ to a group of conformal automorphisms.Let g : D → ˆ C be a locally injective quasiregular map, i.e. g = h ◦ f where f is a quasicon-formal homeomorphism and h is locally injective and holomorphic on the image of f . We maydefine a complex structure C g on D by pulling back the complex structure on ˆ C via g . Theidentity map defines a quasiconformal homeomorphism ˆ g : D → C g . We then uniformize C g by a conformal map R : C g → D and consider the quasiconformal map R ◦ ˆ g : D → D . Thismap extends to the boundary to give a quasisymmetric map qs ( g ) : S → S .Choose µ so that Dome(Ω) = P cµ ( D ) where || µ || L = 1 and c >
0. We use Theorem 5.4 todefine a map F : T L → U (Γ) , where Γ is the group of conformal automorphisms of H preserving µ . If t ∈ T L , let F ( t ) = qs (Φ t ) . Similarly, we may use Theorem 5.5, with some choice of t = iv ∈ T L , to define a map G : U → U (Γ)by letting G ( t ) = qs (Ψ t ◦ Φ t ) . If t lies in the intersection of the domains of F and G , then even though Φ t and Ψ t ◦ Φ t neednot agree on D , Theorem 5.5 implies that they have the same boundary values and quasi-diskimage Ω t . Therefore F and G agree on the overlap T L ∩ U of their domains. We may combinethe functions to obtain a well-defined function¯ F : T L → U (Γ) . Epstein, Marden, Markovic further show that ¯ F is holomorphic (see [ , Theorem 6.5 andProposition 6.9]).The Kobayashi metric on a complex manifold M is defined to be the largest metric on M with the property that for any holomorphic map f : D → M , f is 1-Lipschitz with respectthe hyperbolic metric on D . Therefore, holomorphic maps between complex manifolds are1-Lipschitz with respect to their Kobayashi metrics. The Teichm¨uller metric agrees with theKobayashi metric on U and U (Γ) (see [ , Chapter 7]). Morever, the Poincar´e metric on anysimply connected domain, in particular T L , agrees with its Kobayashi metric. It follows thenthat for any t ∈ T L , d U (Γ) ( ¯ F ( t ) , ¯ F (0)) ≤ d T L ( t, . MPROVED BOUND FOR SULLIVAN’S CONVEX HULL THEOREM
Page 21 of 24Theorem 3.1 implies that c ≤ c ( L ) = 2 cos − (cid:18) − sinh (cid:18) L (cid:19)(cid:19) so d U (Γ) ( ¯ F ( ic ) , ¯ F (0)) ≤ d T L ( ic, ≤ d T L ( ic ( L ) , . Since C E ic = P cµ and Ω = Ω ic is simply connected, the map g ic = Ψ ic ◦ Φ t is a conformallynatural quasiconformal mapping with image Ω. Moreover, P cµ ◦ g − ic : Ω → Dome(Ω) extendsto the identity on ∂ Ω = ∂ Dome(Ω). (For more details, see the discussion in the proofs of [ ,Theorem 6.11] or [ , Theorem 1.1].)We have that ¯ F ( ic ) = qs ( g ic ) = ( R ◦ g ic ) | S where R : Ω → D is a uniformization map.Therefore, d U (Γ) ( ¯ F ( ic ) , ¯ F (0)) = d U (Γ) ( ¯ F ( ic ) , Id ) = log inf K ( h )where the infimum is taken over all quasiconformal maps from D to D extending ( R ◦ g ic ) | S and conjugating Γ to a group of conformal automorphisms. By basic compactness resultsfor families of quasiconformal maps, this infimal quasiconformal dilatation is achieved by aquasiconformal map h : D → D . If f : Ω → D is given by f = h − ◦ R , then K ( f ) = K ( h ) = d U (Γ) ( ¯ F ( ic ) , ¯ F (0)) ≤ d T L ( ic ( L ) , . Since h and R ◦ g ic are quasiconformal maps with the same extension to ∂ H , they areboundedly homotopic (see, e.g., [ , Lemma 5.10]). So, f is boundedly homotopic to g − ic .Thus, P cµ ◦ f : Ω → Dome(Ω) is boundedly homotopic to P cµ ◦ g − ic . Since P cµ ◦ g − ic extendsto the identity on ∂ Ω, it follows that P cµ ◦ f also extends to the identity on ∂ Ω. Therefore, P cµ ◦ f : Ω → Dome(Ω) is the desired conformally natural K -quasiconformal map whichextends to the identity on ∂ Ω such thatlog( K ) ≤ d T L ( ic ( L ) , . This completes the sketch of the proof of Theorem 6.1.
Remark:
Epstein, Marden and Markovic showed that if Ω is simply connected, then aquasiconformal map between Ω and Dome(Ω) extends to the identity on ∂ Ω if and only if it isboundedly homotopic to the nearest point retraction from Ω to Dome(Ω) (see [ , Theorem5.9]). 7. Derivation of main theorem
In order to complete the proof of our main theorem, Theorem 1.3, it suffices to show thatone can choose
L > d T L ( ic ( L ) , < . . Motivated by computer calculations for various values of L , we choose L = 1 . T L from within, see figure 5. Theapproximation is constructed using MATLAB’s Symbolic Math Toolbox and variable precisionarithmetic. Variable precision arithmetic allows us to compute vertex positions to arbitraryprecision. In particular, we can deduce sign changes to find intervals containing intersectionpoints.We build a step function s ( x ) ≤ Q ( L, x ) as follows; We recall that Q ( L, x ) = max G ( L ) (cid:108) f ( L,x ) L (cid:109) , G ( g ( L, x )) . age 22 of 24 M. BRIDGEMAN, R. CANARY AND A. YARMOLA
Figure 5.
Polygonal approximation of T L We first locate intervals where G ( L ) (cid:100) f ( L,x ) L (cid:101) and G ( g ( L, x )) intersect. For values where G ( L ) (cid:100) f ( L,x ) /L (cid:101) dominates, we bound Q ( L, x ) by truncated decimal expansions (i.e. lower bounds) of values of G ( L ) (cid:100) f ( L,x ) /L (cid:101) , which we compute using variable precision arithmetic.For parts dominated by G ( g ( L, x )), we simplify our computation by using the followinglemma.
Lemma 7.1.
Let L > be the unique positive solution to L ) = L . If L < L ≈ . , then g ( L, x ) = Le −| x | / .Proof. Recall that g ( L, x ) = max (cid:16) Le −| x | / , sinh − ( e −| x | sinh L ) (cid:17) . Let
L < L and consider the function j ( x ) = e x sinh( Le − x/ ). It has a critical point preciselywhen 2 tanh( Le − x/ ) = Le − x/ . Since
L < L , we have Le − x/ < L when x ≥
0, so j has no critical points in the interval[0 , ∞ ). Since j (cid:48) (0) = sinh L − L cosh L > j is increasing on the interval [0 , ∞ ). Therefore, j ( x ) = e x sinh( Le − x/ ) ≥ sinh( L ) = j (0)for all x ≥
0, so Le − x/ ≥ sinh − ( e − x sinh( L ))for all x ≥
0. Thus, g ( L, x ) = Le −| x | / for all x .From our initial analysis of the hill function, we know that G ( t ) is an increasing function on t ∈ [0 , ∞ ). It follows that G ( g ( L, x )) is a decreasing function for x ∈ [0 , ∞ ). Therefore, we canapproximate G ( g ( L, x )) by a step function from below.
MPROVED BOUND FOR SULLIVAN’S CONVEX HULL THEOREM
Page 23 of 24To compute the values of G ( g ( L, x )), recall that G ( t ) = h ( c ( t ) − t ) − h ( c ( t )). The function c ( t ) can be computed to arbitrary precision from the equation t h (cid:48) ( c ( t )) = h ( c ( t )) − h ( c ( t ) − t ) . In particular, variable precision arithmetic can give us truncated decimal expansions of valuesof G ( g ( L, x )). We sample at a collection of points to obtain a step function where G ( g ( L, x ))dominates.We use these computations to build s ( x ) ≤ Q ( L, x ) on some interval [ − a, a ]. Outside of thatinterval, we set s ( x ) = 0. The graph of − s ( x ) gives us the boundary of a polygonal regioncontained in T L .Using the Schwarz-Christoffel mapping toolbox developed by Toby Driscoll [ ], the imagesof the points 0 and 2 cos − (cid:0) − sinh (cid:0) L (cid:1)(cid:1) i are computed under a Riemann mapping of theapproximation of T L to the upper half plane. Computing the hyperbolic distance between theimages provides the result. The Schwarz-Christoffel mapping toolbox provides precision anderror estimates. The error bounds are on the order of 10 − . We found that the optimal bound is given when L is approximately 1 .
48. Using L = 1 . B = c ( L ) i = 2 cos − (cid:18) − sinh (cid:18) L (cid:19)(cid:19) i ≈ . i and e d T L ( ic ( L ) , ≈ . . A truncated version of the output provides the values of G ( L ), HP L (0), and
HP L ( B ), where HP L : T L → H is a Riemann mapping from T L to the upper half-plane. We also have H ( L ) = d H ( HP L (0) , HP L ( B )) and K ( L ) = exp ( d T L ( ic ( L ) , L=1.48G(L) = 1.327185362837166HPL(0) = 0.000007509959438 + 0.009347547230674iHPL(B) = 0.000009420062234 + 0.067016970686742iH(L) = 1.969831901361628K(L) = 7.169471208698489
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