Quasiconformal Extensions to Space of Weierstrass-Enneper Lifts
QQUASICONFORMAL EXTENSIONS TO SPACE OFWEIERSTRASS-ENNEPER LIFTS
M. CHUAQUI, P. DUREN, AND B. OSGOOD
To the memory of Professor F.W. Gehring
Abstract.
The Ahlfors-Weill extension of a conformal mapping of the disk is generalizedto the Weierstrass-Enneper lift of a harmonic mapping of the disk to a minimal surface,producing homeomorphic and quasiconformal extensions to space. The extension is definedthrough the family of best M¨obius approximations to the lift applied to a bundle of Euclideancircles orthogonal to the disk. Extension of the planar harmonic map is also obtained subjectto additional assumptions on the dilatation. The hypotheses involve bounds on a generalizedSchwarzian derivative for harmonic mappings in terms of the hyperbolic metric of the diskand the Gaussian curvature of the minimal surface. Hyperbolic convexity plays a crucialrole. Introduction If f is an analytic, locally injective function its Schwarzian derivative is Sf = (cid:18) f (cid:48)(cid:48) f (cid:48) (cid:19) (cid:48) − (cid:18) f (cid:48)(cid:48) f (cid:48) (cid:19) . We owe to Nehari [16] the discovery that the size of the Schwarzian derivative of f is relatedto its injectivity, and to Ahlfors and Weill [3] the discovery of an allied, stronger phenomenonof quasiconformal extension. We state the combined results as follows: Theorem 1 (Nehari, Ahlfors-Weill) . Let f be analytic and locally injective in the unit disk, D .(a) If (1) | Sf ( z ) | ≤ − | z | ) , z ∈ D , then f is injective in D .(b) If for some ρ < , (2) | Sf ( z ) | ≤ ρ (1 − | z | ) , z ∈ D , then f has a ρ − ρ -quasiconformal extension to C . Mathematics Subject Classification.
Primary 30C99; Secondary 31A05, 53A10.
Key words and phrases.
Harmonic mapping, Schwarzian derivative, curvature, minimal surface.The authors were supported in part by FONDECYT Grant a r X i v : . [ m a t h . C V ] A p r M. CHUAQUI, P. DUREN, AND B. OSGOOD
A remarkable aspect of Ahlfors and Weill’s theorem is the explicit formula they give forthe extension. They need the stronger inequality (2) to show, first of all, that the extendedmapping has a positive Jacobian and is hence a local homeomorphism. Global injectivitythen follows from the monodromy theorem and quasiconformality from a calculation of thedilatation. The topological argument cannot get started without (2), but a different approachin [8] shows that the same Ahlfors-Weill formula still provides a homeomorphic extensioneven when f satisfies the weaker inequality (1) and f ( D ) is a Jordan domain. As to thelatter requirement, if f satisfies (1) then f ( D ) fails to be a Jordan domain only when f ( D )is a parallel strip or the image of a parallel strip under a M¨obius transformation, as shownby Gehring and Pommerenke [14].In earlier work, [4], [6], we introduced a Schwarzian derivative for harmonic mappings inthe plane and we established an injectivity criterion analogous to (1) for the Weierstrass-Enneper lift of a harmonic mapping of D to a minimal surface. For a very interestinggeneralization we also call attention to the important paper of D. Stowe, [19]. In this paperwe show that homeomorphic and quasiconformal extensions also obtain in this more generalsetting under a hypothesis analogous to (2). The construction is a geometric generalizationof the Ahlfors-Weill formula and extends the lift not just to the plane but to all of space.To state our results we need some terminology and notation for harmonic mappings; werefer to [11] for more details. Let D denote the unit disk in the complex plane and let f : D → C be harmonic. As is customary, we write f = h + ¯ g where g and h are analytic.If | h (cid:48) | + | g (cid:48) | (cid:54) = 0 and the dilatation ω = g (cid:48) /h (cid:48) is the square of a meromorphic function, thenthere is a lift ˜ w = ˜ f ( z ) of f mapping D onto a minimal surface Σ in R . The function ˜ f iscalled the Weierstrass-Enneper parametrization of Σ. Its three components are themselvesharmonic functions and ˜ f is a conformal mapping of D onto Σ with conformal metric˜ f ∗ ( | d ˜ w | ) = e σ ( z ) | dz | , e σ = | h (cid:48) | + | g (cid:48) | , on D . Then (cid:104) ∂ x ˜ f , ∂ x ˜ f (cid:105) = (cid:104) ∂ y ˜ f , ∂ y ˜ f (cid:105) = e σ , (cid:104) ∂ x ˜ f , ∂ y ˜ f (cid:105) = 0 , z = x + iy, where (cid:104)· , ·(cid:105) denotes the Euclidean inner product. The Gaussian curvature of Σ at a point˜ f ( z ) is K ( ˜ f ( z )) = − e − σ ( z ) ∆ σ ( z ) . As introduced in [4], the Schwarzian derivative of ˜ f is(3) S ˜ f = 2( ∂ zz σ − ( ∂ z σ ) ) . This becomes the familiar Schwarzian when ˜ f is analytic and Σ ⊂ C , where then σ = log | ˜ f (cid:48) | .The principal result of this paper is the following generalization of the Ahlfors-Weill the-orem. Theorem 2.
Let ≤ ρ ≤ . Suppose ˜ f satisfies (4) |S ˜ f ( z ) | + e σ ( z ) | K ( ˜ f ( z )) | ≤ ρ (1 − | z | ) , z ∈ D . Then ˜ f is injective. If ρ < then ˜ f has a k ( ρ ) -quasiconformal extension E ˜ f to R . If ρ = 1 and ∂ Σ is a Jordan curve then E ˜ f is a homeomorphism. UASICONFORMAL EXTENSIONS TO SPACE OF WEIERSTRASS-ENNEPER LIFTS 3
That ˜ f is injective in D was proved in [6] in even greater generality, so the point here isthe extension. It was also proved in [6] that if ˜ f satisfies (4) with ρ = 1 then f and ˜ f havespherically continuous extensions to ∂ D . Furthermore, we know exactly when ∂ Σ fails to bea Jordan curve in R , namely when either ˜ f is analytic and ˜ f ( D ) is the M¨obius image of aparallel strip, or when ˜ f maps D into a catenoid and ∂ Σ is pinched by a Euclidean circleon the surface. In either case, there is a Euclidean circle C on Σ and a point p ∈ C with˜ f ( ζ ) = p = ˜ f ( ζ ) for a pair of points ζ , ζ ∈ ∂ D . Furthermore, equality holds in (4) with ρ = 1 along ˜ f − ( C \ { p } ), and because of this a function satisfying the stronger inequalitywith ρ < ρ = 1 is always injective on ∂ D .It follows from properties of the Schwarzian and from Schwarz’s lemma that if ˜ f satisfies(4) then so does ˜ f ◦ M for any M¨obius transformation M of D onto itself. Note also thatthe condition trivially entails a bound on the curvature,(5) e σ ( z ) | K ( ˜ f ( z )) | ≤ ρ (1 − | z | ) . The extension E ˜ f is defined in equation (9) in Section 2. It is constructed by setting upa correspondence between two fibrations of space by Euclidean circles, one based on D andthe other on Σ. Fundamental properties of these fibrations rely on the convexity relativeto the hyperbolic metric of a real-valued function, denoted U ˜ f and defined in Section 2.1,naturally associated with conformal mappings of D into R ; it is the pullback under ˜ f of thesquare root of what can naturally be regarded as the Poincar´e metric of Σ. The argumentsrely on comparison theorems for differential equations. Of particular interest is the use of aSchwarzian derivative for curves introduced by Ahlfors, [2].The correspondence between the two fibrations that defines E ˜ f is via p (cid:55)→ M ˜ f ( p, ζ ), p ∈ R , using the family M ˜ f ( p, ζ ) of best M¨obius approximations to ˜ f parametrized by ζ ∈ D . Sections 3 and 4 study best M¨obius approximations in some detail and provideformulas and properties that underly the proof of quasiconformality of E ˜ f in Section 5.Moreover, in Section 3 we show that restricting the extension E ˜ f to C yields a reflection R across ∂ Σ with a formula quite like the Ahlfors-Weill reflection. In particular, R sewsthe reflected surface, Σ ∗ = R (Σ), onto Σ along the boundary. Then the topological sphereΣ ∪ Σ ∗ is a quasisphere, being the image of C under the quasiconformal mapping E ˜ f of R ,and ∂ Σ is a spatial quasicircle, being the image of ∂ D . When f is analytic all aspects ofthe construction and the theorem reduce to the classical results, including the bound for thequasiconformality of E ˜ f which becomes k ( ρ ) = (1 + ρ ) / (1 − ρ ).Our analysis of the quasiconformality of E ˜ f is very much influenced by C. Epstein’s insight-ful treatment of the classical theorems in [12], which relies on aspects of hyperbolic geometryof the upper half-space and parallel flow. However, as will be apparent, the nonzero curvatureof Σ is a considerable complication and a new approach is necessary.As a corollary of this work, in Section 6 we will derive a sufficient condition for quasi-conformal extension of planar harmonic mappings f = h + ¯ g . This is perhaps closer tothe original Ahlfors-Weill result in that we obtain simultaneously an injectivity criterion forharmonic mappings together with a quasiconformal extension. M. CHUAQUI, P. DUREN, AND B. OSGOOD
Theorem 3.
Suppose f = h + ¯ g is a locally injective harmonic mapping of D whose lift ˜ f satisfies (4) for a ρ < and whose dilatation ω satisfies sup ζ ∈ D (cid:112) | ω ( ζ ) | < − √ ρ √ ρ , ζ ∈ D . Then f is injective and has a quasiconformal extension to C given by F ( ζ ) = f ( ζ ) , ζ ∈ D f ( ζ ∗ ) + (1 − | ζ ∗ | ) h (cid:48) ( ζ ∗ )¯ ζ ∗ − (1 − | ζ ∗ | ) ∂ z σ ( ζ ∗ ) + (1 − | ζ ∗ | ) g (cid:48) ( ζ ∗ ) ζ ∗ − (1 − | ζ ∗ | ) ∂ ¯ z σ ( ζ ∗ ) , ζ ∗ = 1¯ ζ , , ζ / ∈ D . This is essentially the Ahlfors-Weill formula applied to h and ¯ g separately, and becomesthe classical formula exactly when f is analytic. The condition on ω makes certain that thereflected surface Σ ∗ is locally a graph. This is explained in Section 6.We are grateful to many people for their comments. Especially, an earlier version ofthis paper concentrated only on the reflection R and a two-dimensional extension, and wewere encouraged to develop the techniques presented here that give the extension to space.Finally, we were colleagues and friends of Fred Gehring, and we respectfully dedicate thispaper to his memory.2. Circle Bundles, Convexity, and Critical Points
This section introduces the central notions through which the extension E ˜ f of ˜ f to spaceis defined: bundles of Euclidean circles orthogonal to D and to Σ = ˜ f ( D ), respectively, andtheir correspondence via the family of best M¨obius approximations to ˜ f .As a general configuration, let B be a smooth, open surface in R , and consider a family C ( B ) of Euclidean circles C p , at most one of which is a Euclidean line, indexed by p ∈ B ,having the geometric properties:( i ) C p is orthogonal to B at p and C p ∩ B = { p } ;( ii ) if p (cid:54) = p then C p ∩ C p = ∅ ;( iii ) (cid:83) p ∈ B C p = R \ ∂B .We refer to p ∈ C p as the base point. If B is unbounded then there is no line in C ( B ), for aline would meet B at its base point and at the point at infinity contrary to ( i ).If such a configuration is possible for a given B , it is proper to refer to C ( B ) as a circlebundle with base space B . The model example is B = D with C ζ , ζ ∈ D , the Euclideancircle in R that is orthogonal to D and that passes through ζ and ζ ∗ = 1 / ¯ ζ . When ζ = 0the circle is a line. Next, if T is a M¨obius transformation of R then T ( D ) supports such acircle bundle, and simply(6) C ( T ( D )) = T ( C ( D )) . Under the assumptions of Theorem 2 one can push forward C ( D ) to get a circle bundle ofthe same type on Σ by means of a family of M¨obius transformations p (cid:55)→ M ˜ f ( p, ζ ) of p ∈ R ,approximating ˜ f at each ζ ∈ D . They are defined as follows. Let ˜ w = ˜ f ( ζ ). We require firstthat M ˜ f ( · , ζ ) maps C to the tangent plane T ˜ w (Σ) to Σ at ˜ w with M ˜ f ( ζ, ζ ) = ˜ w . Next, forany smooth curve ψ ( t ) in D with ψ (0) = ζ we further require that the orthogonal projectionof the curve ˜ f ( ψ ( t )) to T ˜ w (Σ) has second order contact at ˜ w with the curve M ˜ f ( ψ ( t ) , ζ ).This is possible for harmonic mappings and by a parameter count M ˜ f ( p, ζ ) is uniquely UASICONFORMAL EXTENSIONS TO SPACE OF WEIERSTRASS-ENNEPER LIFTS 5 determined; see Section 3, where we will also provide a formula for M ˜ f ( z, ζ ), for z ∈ C ,that is amenable to our calculations.Now let C (Σ) be the family of circles(7) C ˜ w = M ˜ f ( C ζ , ζ ) , ζ ∈ D , ˜ w = ˜ f ( ζ ) . The main work of this section is then to prove:
Theorem 4. If ˜ f satisfies (4) and is injective on ∂ D then C (Σ) satisfies properties ( i ) , ( ii ) and ( iii ) above. Clearly if T is a M¨obius transformation of R then T (Σ) also supports such a circle bundleand as in (6)(8) C ( T (Σ)) = T ( C (Σ)) . We now define the extension E ˜ f : R → R of ˜ f by(9) E ˜ f ( p ) = (cid:40) M ˜ f ( p, ζ ) , p ∈ C ζ , ˜ f ( p ) , p ∈ ∂ D , deferring discussion of the correspondence of ∂ D and ∂ Σ to the next section. Note that if p ∈ D then E ˜ f ( p ) = ˜ f ( p ). The disjointness of the circles C ˜ w guarantees that E ˜ f is injective(when ∂ Σ is a Jordan curve) and the fact that R = (cid:83) ˜ w ∈ Σ C ˜ w ∪ ∂ Σ guarantees that it issurjective. It is obviously a homeomorphism, even real analytic, off ∂ D .2.1. Hyperbolic Convexity, the Auxiliary Function U ˜ f , and Ahlfors’ Schwarzian. The proof of Theorem 4 is analytic and relies on the hyperbolic convexity and a study ofthe critical points of the following function. For a conformal mapping Φ : D → R withconformal metric e τ ( z ) | dz | on D define(10) U Φ( z ) = 1 (cid:112) (1 − | z | ) e τ ( z ) , z ∈ D . From Schwarz’s lemma, if M is any M¨obius transformation of D onto itself then(11) U (Φ ◦ M ) = ( U Φ) ◦ M. We will be considering the critical points of U Φ, for various Φ, and the importance of (11) isthat we can shift a critical point to be located at the origin, and not introduce any additionalcritical points.
Lemma 1. If ˜ f satisfies (4) and T is any M¨obius transformation of R then U ( T ◦ ˜ f ) ishyperbolically convex. If ˜ f is injective on ∂ D then U ( T ◦ ˜ f ) has at most one critical point in D . To explain the terms, a real-valued function u on D is hyperbolically convex if(12) ( u ◦ γ ) (cid:48)(cid:48) ( s ) ≥ γ ( s ) in D , where s is the hyperbolic arclength parameter. Aspecial case of Theorem 4 in [6] tells us that when ˜ f satisfies (4) the function U ˜ f is hyper-bolically convex. The principle is that an upper bound for the Schwarzian leads to a lowerbound for the Hessian of U ˜ f , and this leads to (12) when U ˜ f is restricted to a geodesic. It is M. CHUAQUI, P. DUREN, AND B. OSGOOD important more generally, as in the present lemma, that convexity holds for U ( T ◦ ˜ f ) where T is a M¨obius transformation of R . The mapping T ◦ ˜ f is typically not harmonic, but it isstill conformal and in (10) the function e τ for U ( T ◦ ˜ f ) is(13) e τ = ( | T (cid:48) | ◦ ˜ f ) e σ . To motivate the functions U ˜ f , and U ( T ◦ ˜ f ), let λ D ( z ) | dz | = 1(1 − | z | ) | dz | be the Poincar´e metric for D and, supposing that ˜ f is injective, let λ | d ˜ w | be the conformalmetric on Σ with ˜ f ∗ ( λ | d ˜ w | ) = λ D | dz | , so that ˜ f is an isometry. Since ˜ f ∗ ( | d ˜ w | ) = e σ | dz | we have(14) ( λ Σ ◦ ˜ f )( z ) = 1(1 − | z | ) e σ ( z ) or λ Σ ◦ ˜ f = e − σ λ D , and(15) U ˜ f = ( λ Σ ◦ ˜ f ) / . If f is analytic and injective in D and the plane domain Ω = f ( D ) replaces the minimalsurface Σ, then λ Σ = λ Ω is the Poincar´e metric for Ω. It is reasonable to consider λ Σ as thePoincar´e metric of Σ in the case of minimal surfaces. In [9] it was shown that the hyperbolicconvexity of λ / T (Ω) for any M¨obius transformation T is a necessary and sufficient conditionfor a function to satisfy the Nehari injectivity condition (1). The first part of Lemma 1 isthe analogous result for harmonic maps of the sufficient condition.The proof of Lemma 1 employs a version of the Schwarzian for curves introduced byAhlfors in [2]. Let ϕ : ( a, b ) → R be of class C with ϕ (cid:48) ( x ) (cid:54) = 0. As a generalization of thereal part of the analytic Schwarzian, Ahlfors defined(16) S ϕ = (cid:104) ϕ (cid:48)(cid:48)(cid:48) , ϕ (cid:48) (cid:105)(cid:107) ϕ (cid:48) (cid:107) − (cid:104) ϕ (cid:48)(cid:48) , ϕ (cid:48) (cid:105) (cid:107) ϕ (cid:48) (cid:107) + 32 (cid:107) ϕ (cid:48)(cid:48) (cid:107) (cid:107) ϕ (cid:48) (cid:107) . If T is a M¨obius transformation of R then(17) S ( T ◦ ϕ ) = S ϕ, a crucial invariance property.Ahlfors’ interest was in the relation of S ϕ to the change in cross ratio under ϕ , whileanother geometric property of S ϕ was discovered by Chuaqui and Gevirtz in [7]. Namely, if v = (cid:107) ϕ (cid:48) (cid:107) then(18) S ϕ = (cid:18) v (cid:48) v (cid:19) (cid:48) − (cid:18) v (cid:48) v (cid:19) + 12 v κ , where κ is the curvature of the curve x (cid:55)→ ϕ ( x ). We will need the connection between S and the Schwarzian for harmonic maps, namely S ˜ f ( x ) ≤ Re {S f ( x ) } + e σ ( x ) | K ( ˜ f ( x )) | , − < x < UASICONFORMAL EXTENSIONS TO SPACE OF WEIERSTRASS-ENNEPER LIFTS 7 see Lemma 1 in [6]. Thus if f satisfies (4) then(19) S ˜ f ( x ) ≤ ρ (1 − x ) , − < x < . We proceed with:
Proof of Lemma 1.
To show U ( T ◦ ˜ f ) is hyperbolically convex it suffices to show that U ( T ◦ ˜ f ) (cid:48)(cid:48) ( s ) ≥ − , x ∈ ( − ,
1) let ϕ ( x ) = ( T ◦ ˜ f )( x ). From (17) and(19), S ϕ ( x ) = S ˜ f ( x ) ≤ ρ (1 − x ) . Next let(20) v ( x ) = | ϕ (cid:48) ( x ) | = e τ ( x ) . From (18),(21) (cid:18) v (cid:48) ( x ) v ( x ) (cid:19) (cid:48) − (cid:18) v (cid:48) ( x ) v ( x ) (cid:19) ≤ S ϕ ( x ) ≤ ρ (1 − x ) . Let 2 P denote the left-hand side, so that(22) 2 P ( x ) ≤ ρ (1 − x ) , − < x < . The function V = v − / satisfies the differential equation(23) V (cid:48)(cid:48) + P V = 0and the function(24) W ( x ) = V ( x ) √ − x is precisely U ( T ◦ ˜ f ) restricted to − < x <
1. If we give − < x < s = 12 log 1 + x − x , x ( s ) = e s − e s + 1 , x (cid:48) ( s ) = 1 − x ( s ) , a calculation produces d ds W = (cid:18) − x ) − P ( x ) (cid:19) (1 − x ) W ( x ) , x = x ( s ) , and this is nonnegative by (22).For the second part of the lemma, suppose that U ( T ◦ ˜ f ) has two critical points. Composing˜ f with a M¨obius transformation of D onto itself we may locate these at 0 and a , 0 < a < U ( T ◦ ˜ f ) in D , and the same must be trueof U ( T ◦ ˜ f )( x ) for 0 ≤ x ≤ a . Hence U ( T ◦ ˜ f ) is constant on [0 , a ] and thus constant on( − ,
1) because it is real analytic there.It follows that the function v ( x ) = e τ ( x ) is a constant multiple of 1 / (1 − x ) . But then V ( x ) = v ( x ) − / is constant multiple of √ − x , and from the differential equation (23) we M. CHUAQUI, P. DUREN, AND B. OSGOOD conclude that p ( x ) = 1 / (1 − x ) . In turn, from (18) and (21) this forces the curvature κ ofthe curve x (cid:55)→ ( T ◦ ˜ f )( x ) to vanish identically. Thus T ◦ ˜ f maps the interval ( − ,
1) onto aline with speed | ϕ (cid:48) ( x ) | = v ( x ) = 1 / (1 − x ), and so ϕ (1) = ϕ ( −
1) = ∞ . This violates theassumption that ˜ f , hence T ◦ ˜ f , is injective on ∂ D . (cid:3) Critical Points of U ˜ f . The crucial connection between critical points of U ( T ◦ ˜ f ) andthe circles in C (Σ) is that inversion can be used to produce a critical point exactly when thecenter of inversion is on the circle. In fact, this is an analytical characterization of the circlesin C ( D ) and in C (Σ).We denote M¨obius inversion by(25) I q ( p ) = p − q (cid:107) p − q (cid:107) , p ∈ R , with (cid:107) DI q ( p ) (cid:107) = 1 (cid:107) p − q (cid:107) . Lemma 2.
A point q ∈ R lies on the circle C ˜ w ∈ C (Σ) , ˜ w = ˜ f ( ζ ) , if and only if U ( I q ◦ ˜ f ) has a critical point at ζ .Proof. The statement also applies to the bundle C ( D ) by taking ˜ f to be the identity. Considerthis case first, and assume further that ζ = 0. The lemma then says that q ∈ C , the verticalline in R through the origin, if and only if U I q ( z ) = (cid:107) z − q (cid:107) (cid:112) − | z | has a critical point at 0. This is easy to check by direct calculation. The result for ζ ∈ D follows from this and from (11) letting M be a M¨obius transformation of the disk mapping0 to ζ , since M (cid:48) does not vanish in D and the extension of M to space maps C to C ζ .Now take a general ˜ f , fix ζ ∈ D , and let ˜ w = ˜ f ( ζ ). Observe that U ( I ˜ q ◦ ˜ f ) has a criticalpoint at ζ if and only if U ( I ˜ q ◦ M ˜ f ( · , ζ )) has as well, because M ˜ f ( · , ζ ) and ˜ f agree at ζ tofirst order. Suppose ˜ q ∈ ˜ C ˜ w with ˜ q = M ˜ f ( q, ζ ), q ∈ C ζ . As a M¨obius transformation of R the mapping I ˜ q ◦ M ˜ f ( · , ζ ) sends q to ∞ , and so up to an affine transformation it is I q . Butfrom the first part of the lemma U I q has a critical point at ζ . This proves necessity, and theargument can be reversed to prove sufficiency. (cid:3) Knowing how to produce a critical point, we now show what happens when there is one(and only one). Lemma 3.
Let ˜ f satisfy (4) and be injective on ∂ D . Let T be a M¨obius transformation of R . The following are equivalent: ( i ) U ( T ◦ ˜ f ) has a critical point. ( ii ) ( T ◦ ˜ f )( D ) is bounded. ( iii ) U ( T ◦ ˜ f )( re iθ ) is eventually increasing along each radius [0 , e iθ ) . ( iv ) U ( T ◦ ˜ f )( z ) → ∞ as | z | → .Proof. If ( iv ) holds there is an interior minimum so ( iv ) = ⇒ ( i ) is immediate.Suppose ( i ) holds. We follow the notation in Lemma 1. We may assume the critical pointis at the origin. The value U ( T ◦ ˜ f )(0) is the absolute minimum for U ( T ◦ ˜ f ) in D and so e τ ( z ) ≤ e τ (0) − | z | , z ∈ D . UASICONFORMAL EXTENSIONS TO SPACE OF WEIERSTRASS-ENNEPER LIFTS 9
Thus τ remains finite in D and ∞ cannot be a point on ( T ◦ ˜ f )( D ).To show that ( T ◦ ˜ f )( D ) is bounded we first work along [0 , W ( x ) = U ( T ◦ ˜ f )( x ) in (24) cannot be constant because 0 is the unique criticalpoint. Hence if x ( s ) is the hyperbolic arclength parametrization of [0 ,
1) with x (0) = 0 dds W ( x ( s )) ≥ a, W ( x ( s )) ≥ as + b, for some a, b > s ≥ s >
0. From this v ( x ) = 1 V ( x ) ≤ − x ) (cid:0) a log x − x + b (cid:1) = − a ddx (cid:32) a log x − x + b (cid:33) . Therefore (cid:90) e τ ( x ) dx = (cid:90) v ( x ) dx < ∞ , with a bound depending only on a , b , and s , so ( T ◦ ˜ f )(1) is finite.This argument can be applied on every radius [0 , e iθ ), and by compactness the corre-sponding numbers a θ , b θ , s θ can be chosen positive independent of θ . This proves that T ◦ ˜ f is bounded, and hence that ( i ) = ⇒ ( ii ).For ( ii ) = ⇒ ( iii ) we can first rotate and assume e iθ = 1. In the notation above, we needto show for some x > W ( x ) is increasing for x ≤ x < T by an inversion, so to simplify the notation let ˜ f = T ◦ ˜ f and U ˜ f ( z ) = ((1 − | z | ) e τ ( z ) ) − / . For a q ∈ R to be determined let˜ f = I q ◦ ˜ f , U ˜ f ( z ) = 1 (cid:112) (1 − | z | ) e ν ( z ) , ν ( z ) = τ ( z ) − log | ˜ f ( z ) − q | . Let W ( x ) = U ˜ f ( x ), x ∈ ( − , W ( x ( s )) is convex, where s is thehyperbolic arclength parameter. Now(26) ∇ ν (0) = ∇ τ (0) + 2 | q | ( (cid:104) ∂ x ˜ f (0) , q (cid:105) , (cid:104) ∂ y ˜ f (0) , q (cid:105) ) . But also ∇U ˜ f (0) = − e − ν (0) / ∇ ν (0) , and from this equation and (26) it is clear we can choose q to make W (cid:48) (0) = a > . Convexity then ensures W ( x ( s )) ≥ as .To work back to W , write(27) ˜ f = ˜ f | ˜ f | + q, whence (cid:107) D ˜ f (cid:107) = (cid:107) D ˜ f (cid:107)(cid:107) ˜ f (cid:107) , and W = W (cid:107) ˜ f (cid:107) . The assumption we make in ( ii ) is that ˜ f ( D ) = ( T ◦ ˜ f )( D ) is bounded, and (27) thus impliesthat (cid:107) ˜ f (cid:107) ≥ δ >
0. Therefore W ( x ( s )) ≥ aδs . By convexity, there is an x > W ( x )is increasing for x ≤ x <
1. This completes the proof that ( ii ) = ⇒ ( iii ).Finally, if ( iii ) holds then for each θ there exists 0 < r θ < ∂∂r U ( T ◦ ˜ f )( r θ e iθ ) ≥ a θ > . By compactness the r θ can be chosen bounded away from 1 and the a θ bounded away from 0.By hyperbolic convexity, along the tail of each radius U ( T ◦ ˜ f )( r ( s ) e iθ ) is uniformly boundedbelow by a linear function of the hyperbolic arclength parameter s , which tends to ∞ as r = r ( s ) → (cid:3) We now have:
Proof of Theorem 4.
For ( i ), orthogonality is obvious, and suppose C ˜ w meets Σ at a secondpoint ˜ w (cid:48) . Then the inversion I ˜ w (cid:48) , which takes ˜ w (cid:48) to ∞ , produces a critical point for U ( I ˜ w (cid:48) ◦ ˜ f )at ζ = ˜ f − ( ˜ w ) by Lemma 2. But by Lemma 3, I ˜ w (cid:48) (Σ) is bounded.For ( ii ), if there is a point ˜ w ∈ C ˜ w ∩ ˜ C ˜ w then the inversion I ˜ w produces critical points for U ( I ˜ w ◦ ˜ f ) at the two distinct points ζ = ˜ f − ( ˜ w ) and ζ = ˜ f − ( ˜ w ), contradicting Lemma1. Finally for ( iii ), by definition the base points ˜ w ∈ Σ for the circles C ˜ w cover Σ. Suppose q (cid:54)∈ Σ. Then under inversion I q (Σ) is bounded. Therefore by Lemma 3, I q ◦ ˜ f has a criticalpoint, and by Lemma 2 the point q is on some circle C ˜ w , ˜ w ∈ Σ. (cid:3) Remark.
The differential equations argument in Lemma 1 is a version of what we have called‘relative convexity’ in other work, [5], [6]. See also the paper of Aharonov and Elias [1]. Therelation between critical points of the Poincar´e metric and the Ahlfors-Weill extension wasthe subject of [8].3.
Best M¨obius Approximations, I: Reflection across ∂ Σ and theAhlfors-Weill Extension To study the extension E ˜ f we need an expression for the best M¨obius approximations.The first condition on M ˜ f is that p (cid:55)→ M ˜ f ( p, ζ ) maps C to the tangent plane T ˜ w (Σ), where˜ w = ˜ f ( ζ ) = M ˜ f ( ζ, ζ ). Let N be a unit normal vector field along Σ. At each point ˜ w ∈ Σwe write H w (Σ) for the hyperbolic (upper) half-space over T ˜ w (Σ) determined by N ˜ w . Then p (cid:55)→ M ˜ f ( p, ζ ) is an isometry of H with H w (Σ), but the fact that these half-spaces varyalong Σ, unlike when Σ is planar, is at the root of the complications in our analysis.In appropriate coordinates on the range we can take T ˜ w (Σ) = C and regard M ˜ f ( z, ζ ), z ∈ C , as an ordinary complex M¨obius transformation of C . With this convention, z (cid:55)→M ˜ f ( z, ζ ), z = x + iy , depends on six real parameters, each depending on ζ , and oncethese are determined so is M ˜ f ( p, ζ ) for p ∈ R . Specifying M ˜ f ( ζ, ζ ) = ˜ w fixes two of theparameters. Next, let ψ ( t ) be a smooth curve in D with ψ (0) = ζ . To match ˜ f and M ˜ f along ψ to first order at ˜ w it suffices to have ∂ x ˜ f ( ζ ) = ∂ x M ˜ f ( ζ, ζ ) because, using that˜ f and M ˜ f are conformal, the same will then be true of the y -derivatives. It takes two UASICONFORMAL EXTENSIONS TO SPACE OF WEIERSTRASS-ENNEPER LIFTS 11 more real parameters in M ˜ f to ensure this. We can use the final two parameters to makethe orthogonal projection of ∂ xx ˜ f ( ζ ) onto T ˜ w (Σ) equal to ∂ xx M ˜ f ( ζ, ζ ), and because ˜ f and M ˜ f are harmonic (as functions of x and y ) the second y -derivatives also agree. Finally,a calculation using again the conformality of ˜ f and of M ˜ f shows that ∂ xy M ˜ f ( ζ, ζ ) agreeswith the tangential component of ∂ xy ˜ f ( ζ ).Requiring equality of the various derivatives of ˜ f and M ˜ f is an alternate way of defining M ˜ f and can be put to use to develop a formula for M ˜ f ( z, ζ ) for z = x + iy ∈ C . We havefound that the most convenient expression is(28) M ˜ f ( z, ζ ) = ˜ f ( ζ ) + Re { m ( z, ζ ) } ∂ ξ ˜ f ( ζ ) + Im { m ( z, ζ ) } ∂ η ˜ f ( ζ ) , ζ = ξ + iη, where(29) m ( z, ζ ) = z − ζ − ∂ ζ σ ( ζ )( z − ζ ) . Note the two special values(30) m ( ζ, ζ ) = 0 , m ( ζ ∗ , ζ ) = 1 − | ζ | ¯ ζ − ∂ ζ σ ( ζ )(1 − | ζ | ) . We verify that (28) meets the requirements in the preceding paragraph.Immediately M ˜ f ( ζ, ζ ) = ˜ f ( ζ ), from m ( ζ, ζ ) = 0. Next, with ζ fixed and z = x + iy varying, differentiate the right-hand side of (28) with respect to x and set z = ζ . As(31) ∂ x m ( z, ζ ) = 1(1 − ∂ ζ σ ( ζ )( z − ζ )) , we have simply ∂ x m ( z, ζ ) | z = ζ = 1, thus ∂ x M ˜ f ( z, ζ ) | z = ζ = 1 · ∂ ξ ˜ f ( ζ ) . Taking the second x -derivative we first have, ∂ xx m ( z, ζ ) = 2 ∂ ζ σ ( ζ )(1 − ∂ ζ σ ( ζ )( z − ζ )) , whence ∂ xx m ( z, ζ ) | z = ζ = 2 ∂ ζ σ ( ζ ) , and ∂ xx M ˜ f ( z, ζ ) | z = ζ = 2 Re { ∂ ζ σ ( ζ ) } ∂ ξ ˜ f ( ζ ) + 2 Im { ∂ ζ σ ( ζ ) } ∂ η ˜ f ( ζ )= ∂ ξ σ ( ζ ) ∂ ξ ˜ f ( ζ ) − ∂ η σ ( ζ ) ∂ η ˜ f ( ζ ) . Next, projecting onto ∂ ξ ˜ f ( ζ ) gives (cid:104) ∂ xx M ˜ f ( z, ζ ) | z = ζ , ∂ ξ ˜ f ( ζ ) (cid:105) = ∂ ξ σ ( ζ ) (cid:104) ∂ ξ ˜ f ( ζ ) , ∂ ξ ˜ f ( ζ ) (cid:105) = e σ ( ζ ) ∂ ξ σ ( ζ ) . On the other hand, projecting ∂ ξξ ˜ f ( ζ ) onto ∂ ξ ˜ f ( ζ ) we get (cid:104) ∂ ξξ ˜ f ( ζ ) , ∂ ξ ˜ f ( ζ ) (cid:105) = 12 ∂ ξ (cid:104) ∂ ξ ˜ f , ∂ x ˜ f ) (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) ζ = 12 ∂ ξ ( e σ )( ζ ) = e σ ( ζ ) ∂ ξ σ ( ζ ) , as we should. Similarly, using (cid:104) ∂ ξξ ˜ f ( ζ ) , ∂ η ˜ f ( ζ ) (cid:105) = −(cid:104) ∂ ηη ˜ f ( ζ ) , ∂ η ˜ f ( ζ ) (cid:105) = − ∂ η (cid:104) ∂ η ˜ f , ∂ η ˜ f (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) ζ = − e σ ( ζ ) ∂ η σ ( ζ ) we again get the correct equality. This completes the verification of (28).As a point of reference we want to see the form that (28) takes when ˜ f is analytic. In thatcase σ ( ζ ) = log | ˜ f (cid:48) ( ζ ) | , and ∂ ζ σ ( ζ ) = 12 ˜ f (cid:48)(cid:48) ( ζ )˜ f (cid:48) ( ζ ) , and(32) m ( z, ζ ) = z − ζ − ( z − ζ ) ˜ f (cid:48)(cid:48) ( ζ )˜ f (cid:48) ( ζ ) . Then using the Cauchy-Riemann equations,(33) M ˜ f ( z, ζ ) = ˜ f ( ζ ) + m ( z, ζ ) ˜ f (cid:48) ( ζ ) . For derivatives of M ˜ f ( z, ζ ) with respect to z we obtain: ∂ z M ˜ f ( z, ζ ) = ˜ f (cid:48) ( ζ ) (cid:32) − ( z − ζ ) ˜ f (cid:48)(cid:48) ( ζ )˜ f (cid:48) ( ζ ) (cid:33) ,∂ zz M ˜ f ( z, ζ ) = ˜ f (cid:48)(cid:48) ( ζ ) (cid:32) − ( z − ζ ) ˜ f (cid:48)(cid:48) ( ζ )˜ f (cid:48) ( ζ ) (cid:33) , showing second order contact between ˜ f and M ˜ f when z = ζ .3.1. Reflection Across ∂ Σ . The existence of the bundle C (Σ) allows us to define a reflec-tion of Σ across its boundary. If ˜ w ∈ Σ the circle C ˜ w intersects the tangent plane T ˜ w (Σ)orthogonally at a diametrically opposite point ˜ w ∗ outside Σ, and we write(34) ˜ w ∗ = R ( ˜ w ) , for this correspondence. Equivalently, if ˜ w = ˜ f ( ζ ) = M ˜ f ( ζ, ζ ) then(35) ˜ w ∗ = M ˜ f ( ζ ∗ , ζ ) = E ˜ f ( ζ ∗ ) , ζ ∗ = 1 / ¯ ζ. In this section we will show that R fixes ∂ Σ pointwise.From ˜ w ∗ = M ˜ f ( ζ ∗ , ζ ) and m ( ζ, ζ ) = 0 we obtain(36) ˜ w ∗ − ˜ w = Re { m ( ζ ∗ , ζ ) } ∂ x ˜ f ( ζ ) + Im { m ( ζ ∗ , ζ ) } ∂ y ˜ f ( ζ ) . Moreover, from (10) we have(37) ∂ z log U ˜ f ( ζ ) = ¯ ζ − ∂ z σ ( ζ )(1 − | ζ | )2(1 − | ζ | ) = 12 m ( ζ ∗ , ζ ) , and so we also obtain(38) (cid:107) ˜ w ∗ − ˜ w (cid:107) = e σ ( ζ ) (cid:107)∇ log U ˜ f ( ζ ) (cid:107) . This is the length of the diameter of C ˜ w and we want to see that it tends to 0 as ˜ w approaches ∂ Σ. We formulate the result as:
UASICONFORMAL EXTENSIONS TO SPACE OF WEIERSTRASS-ENNEPER LIFTS 13
Theorem 5.
Let d denote the spherical metric on R . If ˜ f satisfies (4) and is injective on ∂ D then d ( R ( ˜ f ( ζ )) , ˜ f ( ζ )) → as | ζ | → . Proof.
We divide the proof into the cases when U ˜ f has one critical point and when it hasnone. We work in the spherical metric because, first, ˜ f has a spherically continuous extensionto ∂ D (by [6]), and second, when U ˜ f has no critical points we have to allow for shifting ˜ f by a M¨obius transformation.Suppose U ˜ f has a unique critical point, which, by (11), we can take to be at 0. The proofof Lemma 3 shows that there is an a > , e iθ )(1 − r ) ∂∂r U ˜ f ( re iθ ) ≥ a for all r ≥ r >
0. (This corresponds to dW/ds ≥ a in the proof of Lemma 3, where s is thehyperbolic arclength parameter.) It follows that(39) (1 − | ζ | ) (cid:107)∇U ˜ f ( ζ ) (cid:107) ≥ a > , for all | ζ | ≥ r > |R ( ˜ f ( ζ )) − ˜ f ( ζ ) | = e σ ( ζ ) (cid:107)∇ log U ˜ f ( ζ ) (cid:107) = U ˜ f ( ζ ) e σ ( ζ ) (cid:107)∇U ˜ f ( ζ ) (cid:107) = 1 U ˜ f ( ζ ) 1(1 − | ζ | ) (cid:107)∇U ˜ f ( ζ ) (cid:107) . This tends to 0 as | ζ | → U ˜ f becomes infinite (Lemma 3) and (1 − | ζ | ) (cid:107)∇U ˜ f ( ζ ) (cid:107) stays bounded below.Next, supposing that U ˜ f has no critical point, we produce one. That is, let T be a M¨obiustransformation so that U ( T ◦ ˜ f ) has a critical point at 0. The preceding argument can berepeated verbatim to conclude that(40) (cid:107)R (cid:48) ( T ( ˜ f ( ζ ))) − T ( ˜ f ( ζ )) (cid:107) → | ζ | → , where R (cid:48) is the reflection for the surface Σ (cid:48) = T (Σ). If the reflections were conformallynatural, if we knew that R (cid:48) ◦ T = T ◦ R , then we would be done. Instead, we argue asfollows.Let ζ ∈ D , ζ (cid:54) = 0. The number (cid:107)R (cid:48) ( T ( ˜ f ( ζ ))) − T ( ˜ f ( ζ )) (cid:107) is the length of the diameterof the circle C T ( ˜ f ( ζ )) based at T ( ˜ f ( ζ )) that defines the reflection R (cid:48) , and it tends to 0by (40). But now, if C ˜ f ( ζ ) is the circle based at ˜ f ( ζ ), for the surface Σ then R ( ˜ f ( ζ ))is on C ˜ f ( ζ ) (diametrically opposite to ˜ f ( ζ )), and then T ( R ( ˜ f ( ζ ))) ∈ C T ( ˜ f ( ζ )) . Therefore (cid:107) T ( R ( ˜ f ( ζ ))) − T ( ˜ f ( ζ )) (cid:107) → | ζ | →
1, whence in the spherical metric d ( R ( ˜ f ( ζ )) , ˜ f ( ζ ))tends to 0 as well and the proof is complete. (cid:3) Remark.
Theorem 5 shows that R is indeed a reflection across ∂ Σ, and that the extension E ˜ f is continuous at ∂ D . In Section 5.4 we will show that R is quasiconformal, a propertyneeded for the proof of Theorem 3 on the quasiconformal extension of the planar harmonicmapping f = h + ¯ g . This separate fact is not necessary for the proof of Theorem 2, but itfollows from limiting cases of the estimates in Section 5.3. Let Σ ∗ = R (Σ). Then Σ ∪ ∂ Σ ∪ Σ ∗ is a topological sphere that is the image of C by thequasiconformal mapping E ˜ f of R , in other words it is a quasisphere. Or, one might alsoregard Σ as a (nonplanar) quasidisk, and given the many analytic and geometric character-izations of planar quasidisks (see for example Gehring’s survey [13]) one might ask if anyhave analogues for Σ. Some results in this direction are due to W. Sierra, who has shown in[18] that if ˜ f satisfies (4) then Σ is a John domain (a John surface) in its metric geometry,and if ˜ f is also bounded then Σ is linearly connected. Both these notions come from thegeometry of planar quasidisks and we will not define therm here, see [9]. Although Σ ∗ willmost likely not be a minimal surface, one might expect it also to have these properties.3.2. The Ahlfors-Weill Extension.
It is possible to express the reflection R in termsintrinsic to the surface Σ. Recall the function λ Σ from (15), with λ Σ ◦ ˜ f = ( U ˜ f ) . Using (36)and (37) it is easy to verify that(41) R ( ˜ w ) = ˜ w + 2 J ( ∇ log λ Σ ( ˜ w )) , where, following Ahlfors, J is the M¨obius inversion centered at the origin,(42) J ( p ) = p | p | . The formula (41) will be important in Section 6. Here, we make contact with the classicalAhlfors-Weill extension, which, when ˜ f is analytic in D , can be written as F ( z ) = ˜ f ( z ) , z ∈ D , ˜ f ( ζ ) + (1 − | ζ | ) ˜ f (cid:48) ( ζ )¯ ζ − (1 − | ζ | ) ˜ f (cid:48)(cid:48) ( ζ )˜ f (cid:48) ( ζ ) = M ˜ f (1 / ¯ ζ, ζ ) ζ = 1 / ¯ z, z ∈ C \ D . Ahlfors and Weill did not express their extension in this form; see [8].Alternatively, if λ Ω | dw | is the Poincar´e metric on Ω = f ( D ) then F ( z ) = ˜ f ( z ) , z ∈ D , ˜ f ( ζ ) + 1 ∂ w log λ Ω ( ˜ f ( ζ )) , ζ = 1 / ¯ z, z ∈ C \ D . The equation (41) for the reflection gives exactly(43) R ( ˜ f ( ζ )) = ˜ f ( ζ ) + 1 ∂ w log λ Ω ( ˜ f ( ζ ))when ˜ f is analytic.The reflection defining the Ahlfors-Weill extension was expressed in a form like (43) alsoby Epstein [12]. Still another interesting geometric construction, using Euclidean circles ofcurvature, was given by D. Minda [15].The Ahlfors-Weill reflection is conformally natural, meaning in this case that if M is aM¨obius transformation of C and Ω (cid:48) = M (Ω) with corresponding reflection R (cid:48) , then R (cid:48) ◦ M = M ◦ R . From the perspective of the present paper this is so because all the tangent planes T z (Ω) to Ω can be identified with C , which is preserved by the extensions to R of the M¨obiustransformations. In the more general setting, if M is a M¨obius transformation of R then UASICONFORMAL EXTENSIONS TO SPACE OF WEIERSTRASS-ENNEPER LIFTS 15 Σ (cid:48) = M (Σ) also supports a circle bundle C (Σ (cid:48) ) and hence an associated reflection R (cid:48) . Butwhile it is true that C ( M (Σ)) = M ( C (Σ)) it is not true that R (cid:48) ◦ M = M ◦ R , i.e., the reflection is not conformally natural. The reason is that the reflections R and R (cid:48) use tangent planes for Σ and Σ (cid:48) , while M may map tangent planes for Σ to tangent planesor tangent spheres for Σ (cid:48) . We do not know how to define a conformally natural reflection,at least one that is suited to our analysis.3.3. A Bound on (cid:107)∇ log U ˜ f (cid:107) and a Classical Distortion Theorem. In the proof ofTheorem 5 we needed the lower bound (39) on (cid:107)∇U ˜ f (cid:107) . In Section 5.3, where we bound thedilatation of the extension E ˜ f to prove its quasiconformality, we will need a correspondingupper bound (to be used again in connection to (38)). We state the result as Lemma 4. If ˜ f satisfies (4) then (44) (cid:107)∇ log U ˜ f ( ζ ) (cid:107) ≤ √ − | ζ | . Proof.
From (37), we want to show(45) (cid:12)(cid:12)(cid:12)(cid:12) ∂ z σ ( ζ ) − ¯ ζ − | ζ | (cid:12)(cid:12)(cid:12)(cid:12) ≤ √ − | ζ | . For this we first derive a lower bound for ∇| ∂ z σ | . Let τ = | ∂ z σ | , so that τ = ∂ z σ∂ ¯ z σ and2 τ ∂ z τ = ∂ zz σ∂ z σ + ∂ z, ¯ z σ∂ z σ = ( ∂ z ¯ z σ + | ∂ z σ | ) ∂ z σ + ( ∂ zz σ − ( ∂ z σ ) ) ∂ z σ. Then 2 τ | ∂ z τ | ≥ ( ∂ z ¯ z σ + | ∂ z σ | ) τ − | ∂ z ¯ z σ − ( ∂ z σ ) | τ ≥ τ − τ (1 − | ζ | ) , because in the first term σ z ¯ z ≥ | ∂ z τ | ≥ τ − − | ζ | ) . The desired estimate at ζ = 0 is(46) τ (0) = | ∂ z σ (0) | ≤ √ , and this will follow by showing that an initial condition a = τ (0) > √ τ becomes infinite in D . To this end, consider v ( t ) = τ ( ζ ( t )) along arc lengthparametrized integral curves t (cid:55)→ ζ ( t ) to ∇ τ . There exists such an integral curve starting atthe origin because (cid:107)∇ τ (0) (cid:107) = 2 | ∂ z τ (0) | ≥ τ (0) − > . The function v ( t ) satisfies v (cid:48) ( t ) = 2 | ∂ z τ ( z ( t )) | ≥ v ( t ) − − | ζ ( t ) | ) ≥ v ( t ) − − t ) , since | ζ ( t ) | ≤ t .We compare v ( t ) with the solution y ( t ) of y (cid:48) = y − − t ) , y (0) = a, which is given by y = 12 n (cid:48)(cid:48) n (cid:48) , where n ( t ) = n ( t )1 − an ( t ) , and n ( t ) = 1 √ t ) √ − (1 − t ) √ (1 + t ) √ + (1 − t ) √ . Because a > √ < t < an ( t ) = 1. The function y ( t ) is increasingfor 0 ≤ t < t and becomes unbounded there. There are then two possibilities. Either v ( t )becomes infinite before or at t , or the integral curve ceases to exist before that time. Butwhile v ( t ) is finite it is bounded below by y ( t ) ≥ a , hence |∇ τ | does not vanish, as shownabove, so the integral curve can be continued. We conclude that v ( t ) must become infinitebefore or at t , and this contradiction shows that (46) must hold.To deduce (45) at an arbitrary point ζ ∈ D we consider˜ f ( z ) = ˜ f ( M ( z )) , M ( z ) = z + ζ ζ z . Then ˜ f satisfies (4) and its conformal factor is e σ ( z ) = e σ ( M ( z )) | M (cid:48) ( z ) | . From this ∂ z σ (0) = (1 − | ζ | ) ∂ z σ ( ζ ) − ¯ ζ , and (45) at ζ is obtained from | ∂ z σ (0) | ≤ √ (cid:3) Remarks.
Suppose equality holds in (44), so in (45), at some ζ ∈ D . By composing ˜ f witha M¨obius transformation of D onto itself we may suppose ζ = 0. The argument shows that ∂ z ¯ z σ must then vanish along the integral curve ζ ( t ) from the origin. Hence the curvature ofthe minimal surface Σ vanishes on a continuum and Σ must therefore be a planar. In turnthis means that ˜ f = h + α ˜ h for some constant α < h for which(4) holds. Because ∂ z σ = (1 / h (cid:48)(cid:48) /h (cid:48) ) we see, from the case of equality in the analytic case,that h must be an affine transformation of a rotation of the function n .Lemma 4, on the one hand expressed as in (45), is reminiscent of the classical distortiontheorem for univalent functions, see, e.g., [10]. Namely, if f is analytic and injective in D then(47) (cid:12)(cid:12)(cid:12)(cid:12) f (cid:48)(cid:48) ( ζ ) f (cid:48) ( ζ ) − ¯ ζ − | ζ | (cid:12)(cid:12)(cid:12)(cid:12) ≤ − | ζ | , with equality holding at a point exactly when f is a rotation of the Koebe function k ( ζ ) = ζ/ (1 − ζ ) . On the other hand, it was observed in [17] that (47) can be written in terms ofthe Poincar´e metric λ Ω | dw | on Ω = f ( D ) as (cid:107)∇ log λ Ω (cid:107) ≤ λ Ω . For the harmonic case we recall (14) and (15) where we had U ˜ f = ( λ Σ ◦ f ) / with λ Σ playingthe role of the Poincar´e metric on Σ. The bound (44) becomes (cid:107)∇ log λ Σ (cid:107) ≤ √ λ Σ . UASICONFORMAL EXTENSIONS TO SPACE OF WEIERSTRASS-ENNEPER LIFTS 17 Best M¨obius Approximations, II: Dependence on the base point
We return to properties of best M¨obius approximations and examine how M ˜ f ( z, ζ ) varieswith the base point ζ . First, when ˜ f is analytic and M ˜ f ( z, ζ ) is given by (32) and (33) wefind(48) ∂ ζ M ˜ f ( z, ζ ) = 12 ˜ f (cid:48) ( ζ ) S ˜ f ( ζ ) m ( z, ζ ) . In particular(49) ∂ ζ M ˜ f ( z, ζ ) (cid:12)(cid:12)(cid:12) ζ = z = 0 . An additional such result is how the conformal factor | ( M ˜ f ) (cid:48) ( z, ζ ) | = | ∂ z M ˜ f ( z, ζ ) | dependson ζ , for which we obtain(50) ∂ ζ log | ∂ z M ˜ f ( z, ζ ) | = 12 S ˜ f ( ζ ) m ( z, ζ ) . Again in particular(51) ∂ ζ | ( M ˜ f ) (cid:48) ( z, ζ ) | (cid:12)(cid:12)(cid:12) ζ = z = 0 . Equations (49) and (51) have counterparts in the harmonic case.Starting with (28), ∂ ξ M ˜ f ( z, ζ ) = ∂ ξ ˜ f ( ζ ) + Re { ∂ ξ m ( z, ζ ) } ∂ ξ ˜ f ( ζ ) + Re { m ( z, ζ ) } ∂ ξξ ˜ f ( ζ )+ Im { ∂ ξ m ( z, ζ ) } ∂ η ˜ f ( ζ ) + Im { m ( z, ζ ) } ∂ ξη ˜ f ( ζ ) , (52)and we calculate that(53) ∂ ξ m ( z, ζ ) = − ∂ ξ ∂ ζ σ ( ζ )( z − ζ ) (1 − ∂ ζ σ ( ζ )( z − ζ )) . Similarly,(54) ∂ η m ( z, ζ ) = − i + ∂ η ∂ ζ σ ( ζ )( z − ζ ) (1 − ∂ ζ σ ( ζ )( z − ζ )) . To do more with (52) we need to work with the second derivatives of ˜ f . Recalling that N isthe unit normal to Σ, we can write ∂ ξξ ˜ f = α N + β ∂ ξ ˜ f + γ ∂ η ˜ f ,∂ ξη ˜ f = α N + β ∂ ξ ˜ f + γ ∂ η ˜ f ,∂ ηη ˜ f = α N + β ∂ ξ ˜ f + γ ∂ η ˜ f . The α ij are the components of the second fundamental form.We find the other coefficients in terms of the derivatives of σ . For example, starting with (cid:104) ∂ ξξ ˜ f , ∂ ξ ˜ f (cid:105) = β e σ , from the first equation we have also β e σ = (cid:104) ∂ ξξ ˜ f , ∂ ξ ˜ f (cid:105) = 12 ∂ ξ (cid:104) ∂ ξ ˜ f , ∂ ξ ˜ f (cid:105) = 12 ∂ ξ ( e σ ) = ( ∂ ξ σ ) e σ . Thus β = ∂ ξ σ. Similar arguments apply to finding the other coefficients, and the final equations are:(55) ∂ ξξ ˜ f = α N + ∂ ξ σ ∂ ξ ˜ f − ∂ η σ ∂ η ˜ f ,∂ ξη ˜ f = α N + ∂ η σ ∂ ξ ˜ f + ∂ ξ σ ∂ η ˜ f ,∂ ηη ˜ f = α N − ∂ ξ σ ∂ ξ ˜ f + ∂ η σ ∂ η ˜ f . The derivatives and the α ij are to be evaluated at ζ , and N = N ˜ w .Substituting this into (52),(56) ∂ ξ M ˜ f ( z, ζ ) = [Re { ∂ ξ m ( z, ζ ) + ∂ ξ σ ( ζ ) m ( z, ζ ) } + Im { ∂ η σ ( ζ ) m ( z, ζ ) } ] ∂ ξ ˜ f ( ζ )+ [Im { ∂ ξ m ( z, ζ ) + ∂ ξ σ ( ζ ) m ( z, ζ ) } − Re { ∂ η σ ( ζ ) m ( z, ζ ) } ] ∂ η ˜ f ( ζ )+ [ α ( ζ ) Re { m ( z, ζ ) } + α ( ζ ) Im { m ( z, ζ ) } ] N ˜ w Now let C ( z, ζ ) = 1 + ∂ ξ m ( z, ζ ) + 2 ∂ ζ σ ( ζ ) m ( z, ζ ) , so that(57) ∂ ξ M ˜ f ( z, ζ ) = Re { C ( z, ζ ) } ∂ ξ ˜ f ( ζ ) + Im { C ( z, ζ ) } ∂ η ˜ f ( ζ )+ [ α ( ζ ) Re { m ( z, ζ ) } + α ( ζ ) Im { m ( z, ζ ) } ] N ˜ w . The terms in the expression for C ( z, ζ ) combine to result in C ( z, ζ ) = ( ∂ ξ ∂ ζ σ ( ζ ) − ∂ ζ σ ( ζ ) )( z − ζ ) (1 − ∂ ζ σ ( ζ )( z − ζ )) = m ( z, ζ ) ( ∂ ξ ∂ ζ σ ( ζ ) − ∂ ζ σ ( ζ ) ) . We can take this further, for with ∂ ξ = ∂ ζ + ∂ ¯ ζ , and ∂ ξ ∂ ζ σ ( ζ ) − ∂ ζ σ ( ζ ) = ∂ ζζ σ ( ζ ) − ∂ ζ σ ( ζ ) + ∂ ζ ¯ ζ σ ( ζ ) we obtain the final form(58) C ( z, ζ ) = m ( z, ζ ) (cid:18) S ˜ f ( ζ ) − e σ ( ζ ) K ( ˜ f ( ζ )) (cid:19) . For later use, let(59) v ( z, ζ ) = Re { C ( z, ζ ) } ∂ ξ ˜ f ( ζ ) + Im { C ( z, ζ ) } ∂ η ˜ f ( ζ ) ,v n ( z, ζ ) = α ( ζ ) Re { m ( z, ζ ) } + α ( ζ ) Im { m ( z, ζ ) } , exhibiting(60) ∂ ξ M ˜ f ( z, ζ ) = v ( z, ζ ) + v n ( z, ζ ) N ˜ w as resolved into velocities tangential to and normal to T ˜ w (Σ). If Σ were planar the normalcomponent would not be present. For the tangential component,(61) (cid:107) v ( z, ζ ) (cid:107) ≤ e σ ( ζ ) | m ( z, ζ ) | | ( |S ˜ f ( ζ ) | + 12 e σ ( ζ ) | K ( ˜ f ( ζ )) | ) , and for the normal component,(62) | v n ( z, ζ ) | ≤ e σ ( ζ ) | m ( z, ζ ) | (cid:113) | K ( ˜ f ( ζ )) | by the Cauchy-Schwarz inequality and using that for minimal surfaces(63) α + α = e σ | K | . UASICONFORMAL EXTENSIONS TO SPACE OF WEIERSTRASS-ENNEPER LIFTS 19
We record some corresponding equations for ∂ η M ˜ f ( z, ζ ). The calculations are very similar,but the end result is a little different, namely(64) ∂ η M ˜ f ( z, ζ ) = − Im { C (cid:48) ( z, ζ ) } ∂ ξ ˜ f ( ζ ) + Re { C (cid:48) ( z, ζ ) } ∂ η ˜ f ( ζ )+ [ α ( ζ ) Re { m ( z, ζ ) } + α ( ζ ) Im { m ( z, ζ ) } ] N ˜ w , where(65) C (cid:48) ( z, ζ ) = m ( z, ζ ) (cid:18) S ˜ f ( ζ ) + 14 e σ ( ζ ) K ( ˜ f ( ζ )) (cid:19) . Note that the curvature enters with a plus sign.As consequences of these expressions we have(66) ∂ ξ M ˜ f ( z, ζ ) (cid:12)(cid:12)(cid:12) z = ζ = 0 , ∂ η M ˜ f ( z, ζ ) (cid:12)(cid:12)(cid:12) z = ζ = 0 . This is the analog to (49).The result we seek on the conformal factor (cid:107) D M ˜ f ( z, ζ ) (cid:107) analogous to (51) is(67) ∂ ξ (cid:107) D M ˜ f ( z, ζ ) (cid:107) (cid:12)(cid:12)(cid:12) ζ = z = 0 , ∂ η (cid:107) D M ˜ f ( z, ζ ) (cid:107) (cid:12)(cid:12)(cid:12) ζ = z = 0 . Because z (cid:55)→ M ˜ f ( z, ζ ) is conformal it suffices to show that(68) ∂ ξ (cid:107) ∂ x M ˜ f ( z, ζ ) (cid:107) (cid:12)(cid:12)(cid:12) ζ = z = 0 = ∂ η (cid:107) ∂ x M ˜ f ( z, ζ ) (cid:107) (cid:12)(cid:12)(cid:12) ζ = z , z = x + iy. For this ∂ x M ˜ f ( z.ζ ) = Re { ∂ x m ( z, ζ ) } ∂ ξ ˜ f ( ζ ) + Im { ∂ x m ( z, ζ ) } ∂ η ˜ f ( ζ ) , whence from (31) (cid:107) ∂ x M ˜ f ( z, ζ ) (cid:107) = | ∂ x m ( z, ζ ) | e σ ( ζ ) = e σ ( ζ ) | − ∂ ζ σ ( ζ )( z − ζ ) | . Next, we considerlog (cid:107) ∂ x M ˜ f ( z, ζ ) (cid:107) = σ ( ζ ) − log(1 − ∂ ζ σ ( ζ )( z − ζ )) − log(1 − ∂ ζ σ ( ζ ) ( z − ζ ))= σ ( ζ ) − log(1 − ∂ ζ σ ( ζ )( z − ζ )) − log(1 − ∂ ¯ ζ σ ( ζ ) ( z − ζ )) . To establish (68) we can work with ∂ ζ : ∂ ζ log (cid:107) ∂ x M ˜ f ( z, ζ ) (cid:107) = ( ∂ ζζ σ ( ζ ) − ∂ ζ σ ( ζ ) )( z − ζ )1 − ∂ ζ σ ( ζ )( z − ζ ) + ∂ ζ ¯ ζ σ ( ζ )( z − ζ )1 − ∂ ¯ ζ σ ( ζ ) ( z − ζ )= S ˜ f ( ζ )( z − ζ )1 − ∂ ζ σ ( ζ )( z − ζ ) − e σ ( ζ ) K ( ˜ f ( ζ ))( z − ζ )1 − ∂ ¯ ζ σ ( ζ ) ( z − ζ )= 12 S ˜ f ( ζ ) m ( z, ζ ) − e σ ( ζ ) K ( ˜ f ( ζ )) m ( z, ζ ) . Equations (67) follow from m ( ζ, ζ ) = 0. An Application to Horospheres.
In the next section we will use the equations abovein the proof of the quasiconformality of the extension E ˜ f . One aspect of this is the geometryof horospheres in H and their images under the mappings M ˜ f ( p, ζ ) for varying ζ .It is a result from hyperbolic geometry that if H ζ is a horosphere in H of Euclidean radius a and with base point ζ , and if M : H → H is a M¨obius transformation, then the Euclideanradius a (cid:48) of the image horosphere M ( H ζ ), M ( ζ ) (cid:54) = ∞ , is(69) a (cid:48) = (cid:107) DM ( ζ ) (cid:107) a. Certainly the analogous formula holds for M ˜ f ( p, ζ ) mapping a horosphere H ζ ⊂ H to ahorosphere H ˜ w ⊂ H w (Σ), ˜ w = M ˜ f ( ζ, ζ ), but more can be said. Fix a horosphere H ζ ⊂ H of Euclidean radius a . For a different base point ζ the mapping p (cid:55)→ M ˜ f ( p, ζ ) takes C tothe tangent plane T ˜ w (Σ), ˜ w = M ˜ f ( ζ, ζ ) and takes H ζ to a horosphere H ˜ w (cid:48) ⊂ H w (Σ) basedat ˜ w (cid:48) = M ˜ f ( ζ , ζ ) and of Euclidean radius, say, a (cid:48) . Then ˜ w (cid:48) and a (cid:48) are both functions of ζ = ξ + iη . From (66) we conclude(70) ∂ ξ ˜ w (cid:48) | ζ = ζ = 0 , ∂ η ˜ w (cid:48) | ζ = ζ , while from (68) and (69) we also have(71) ∂ ξ a (cid:48) | ζ = ζ = 0 , ∂ η a (cid:48) | ζ = ζ = 0 . Put another way, to first order at ζ , H ˜ w = M ˜ f ( H ζ , ζ ) = M ˜ f ( H ζ , ζ ) = H ˜ w (cid:48) . Quasiconformality of the Extension E ˜ f Recall how the extension is defined, in (9), for points in space: E ˜ f ( p ) = (cid:40) M ˜ f ( p, ζ ) , p ∈ C ζ , ˜ f ( p ) , p ∈ ∂ D . We will establish the existence of a constant k ( ρ ) such that(72) 1 k ( ρ ) ≤ max (cid:107) X (cid:107) =1 (cid:107) D E ˜ f ( p ) X (cid:107) min (cid:107) X (cid:107) =1 (cid:107) D E ˜ f ( p ) X (cid:107) ≤ k ( ρ ) , for p in the upper half-space; the arguments and estimates are identical if p is in the lowerhalf-space. Since E ˜ f is a homeomorphism of R it then follows that E ˜ f is k ( ρ )-quasiconformaleverywhere.A point p ∈ H is the intersection of a circle C ζ with a horosphere H ζ in H that is tangentto D at ζ , and to assess the distortion one can regard E ˜ f as acting in directions tangent toand normal to the circles C ζ . As C ζ is orthogonal to H ζ at p the objective is thus to estimate (cid:107) D E ˜ f ( p )( X ) (cid:107) when a unit vector X is tangent to C ζ at p and when it is tangent to H ζ at p .For this, we add a parameter t ∈ R to the circle-horosphere configuration, aiming to adaptto minimal surfaces the parallel flow in hyperbolic space introduced by Epstein [12] in hisstudy of the classical Ahlfors-Weill extension.We need a number of notions and formulas from the hyperbolic geometry of the upper half-space H . To begin with, the upper hemisphere over D in H , denoted S (0) with parameter UASICONFORMAL EXTENSIONS TO SPACE OF WEIERSTRASS-ENNEPER LIFTS 21 t = 0, is the envelope of the family of horospheres H ζ (0), ζ ∈ D , of Euclidean radius a ( ζ,
0) = 12 (1 − | ζ | ) . Starting at the point p ( ζ,
0) = C ζ ∩ H ζ (0), follow the hyperbolic geodesic C ζ at unit speedfor a time t to the point p ( ζ, t ) = C ζ ∩ H ζ ( t ), where the horosphere H ζ ( t ) (still based at ζ )has radius(73) a ( ζ, t ) = e t a ( ζ,
0) = 12 e t (1 − | ζ | ) . Here t > p ( ζ, t ) upward from S (0) along C ζ and t < p ( ζ, t ) downwardfrom S (0) along C ζ . Fixing t and varying ζ defines a surface S ( t ) that is simply a portionof a sphere that intersects the complex plane along ∂ D . It is the envelope of the family ofhorospheres H ζ ( t ) and p ( ζ, t ) is the point of tangency between S ( t ) and H ζ ( t ). Varying t as well then gives a family of hyperbolically parallel surfaces in H . For t < S ( t ) lies inside S (0) and for t > S (0). The limiting cases as t → ∓∞ are,respectively, D and its exterior.The mapping ζ (cid:55)→ p ( ζ, t ) is a parametrization of S ( t ), and one obtains(74) p ( ζ, t ) = (cid:18) e t e t | ζ | ξ, e t e t | ζ | η, e t (1 − | ζ | )1 + e t | ζ | (cid:19) , ζ = ξ + iη. It is an important fact that this is a conformal mapping. The corresponding conformalmetric on D is(75) 1 + e t e t | ζ | | dζ | . Consider now the configuration in the image on applying E ˜ f . The circles C ζ in the bundle C ( D ) map to corresponding circles C ˜ w , ˜ w = M ˜ f ( ζ, ζ ), in the bundle C (Σ), and for each ζ we have the one-parameter family of horospheres H ˜ w ( t ) = M ˜ f ( H ζ ( t ) , ζ )in H w (Σ). The circle C ˜ w intersects each of the horospheres H ˜ w ( t ) orthogonally and we write˜ p ( ζ, t ) = C ˜ w ∩ H ˜ w ( t ), so that for each t the surfaceΣ( t ) = E ˜ f ( S ( t ))is parametrized by(76) ζ (cid:55)→ ˜ p ( ζ, t ) = M ˜ f ( p ( ζ, t ) , ζ ) , ζ ∈ D . However , due to the curvature of Σ the surface Σ( t ) need not be the envelope of the horo-spheres H ˜ w ( t ); they need not be tangent to Σ( t ) at ˜ p ( ζ, t ) and the circle C ˜ w need not beorthogonal to Σ( t ) there. Put another way, while the derivative of E ˜ f in the direction ofa circle C ζ will be tangent to the circle C ˜ w , the derivative of E ˜ f in directions tangent to S ( t ) need not necessarily be tangent to the corresponding horospheres H ˜ w . This is the keydifference in geometry between our considerations and the case when ˜ f is analytic and Σ isplanar as considered by Epstein. It is also the reason that the dilatation of the extensiondoes not turn out to be a clean (1 + ρ ) / (1 − ρ ).Fix a point p = p ( ζ , t ) . In the direction of C ζ the extension acts as the fixed M¨obius transformation M ˜ f ( p, ζ ) andwe can express the derivative of E ˜ f in that direction using the hyperbolic geometry of H and of H w (Σ). The calculation of the derivatives of E ˜ f at p in directions tangent to S ( t )is equivalent to finding the derivatives of the mapping (76) in the ζ -variable. This is whywe need the results in Section 4, and we are aided further by the fact that ζ (cid:55)→ p ( ζ, t )is a conformal mapping with a known conformal factor. Calculating the derivative in the ξ -direction, the quantity we want is(77) D ( M ˜ f )( p ( ζ, t ) , ζ ) ∂ ξ p ( ζ, t ) + ∂ ξ M ˜ f ( p ( ζ, t ) , ζ )evaluated at ζ = ζ . The first term contains the contribution from the differential of themapping M ˜ f while the second considers the variation of the M¨obius approximations frompoint to point. There is no essential difference in estimating the derivative in the η -directionso we consider only (77). This is a consequence of the formulas (57) – (65) in Section 4 and(74), or also because we can as well work with ˜ f ◦ M for any M¨obius transformation M ofthe disk.The expression (77) represents a vector in the image of E ˜ f and, to make use of thegeometry, when ζ = ζ we want to resolve it into components tangent to and normal to H ˜ w ( t ) – this is central to the argument. First note that since M ˜ f ( p, ζ ) maps H ζ ( t )to H ˜ w ( t ), the first term is tangent to H ˜ w ( t ) at E ˜ f ( p ( ζ, t )) = M ˜ f ( p ( ζ, t ) , ζ ). Moreover,because M ˜ f ( p, ζ ) is a hyperbolic isometry between H and H w (Σ), the size of this first term isdetermined by | ∂ ξ p ( ζ, t ) | together with the heights of p ( ζ, t ) above C and of M ˜ f ( p ( ζ, t ) , ζ )above T ˜ w (Σ). We will show:(A) The first term in (77) is the dominant one when considering the contributions to thecomponent tangent to H ˜ w ( t ).(B) Except for the factor | ∂ ξ p ( ζ, t ) | , this first term equals in size the derivative of E ˜ f inthe direction of the circle C ζ .(C) The size of the terms in (77) normal to H ˜ w ( t ) are comparable to the derivative of E ˜ f in the direction of the circle C ζ .5.1. Horospheres and Hyperbolic Stereographic Projection.
While the first term in(77) is relatively straightforward to analyze, the second is not. To do so we will use a variantof stereographic projection based on horospheres and hyperbolic geodesics that is well suitedto our geometric arrangements and enables us to express a point p ∈ H using planar data.For the model case, suppose p ∈ H lies on a horosphere H having Euclidean radius a thatis based at 0 ∈ C . Let C be the hyperbolic geodesic in H passing through p with endpoints0 and a point q ∈ C . To relate p to q we introduce the angle of elevation φ of the point p assited from the origin. Let let e r be the unit vector in the radial direction in C and let N bethe upward unit normal to C in H . See Figure 1 (in profile). Then p = (cid:107) q (cid:107) cos φ ((cos φ ) e r + (sin φ ) N ) , or alternatively(78) p = (cos φ ) q + 2 a (sin φ ) N . Note that φ depends on a . Letting 2 r = (cid:107) q (cid:107) , note also that(79) r cos φ = a sin φ, UASICONFORMAL EXTENSIONS TO SPACE OF WEIERSTRASS-ENNEPER LIFTS 23
H C qp Φ a Φ N e r Figure 1.
Hyperbolic stereographic projection of q to p .and that the height of p above the plane is(80) h ( p ) = 2 a sin φ = r sin 2 φ. Just as for classical stereographic projection, rays in C from the origin correspond tomeridians on H and circles in C concentric to the origin correspond to parallels on H . Itfollows that the mapping q (cid:55)→ p is conformal, and it is not difficult to show that the conformalmetric on the plane is cos φ | dq | .We want to use this to compute ∂ ξ M ˜ f ( p ( ζ, t ) , ζ ) at ζ . Again, p = p ( ζ , t ) = C ζ ∩ H ζ ( t ). As in Section 4.1, consider a point ζ ∈ D different from ζ and the M¨obius transfor-mation M ˜ f ( p, ζ ). The image C = M ˜ f ( C ζ , ζ ) is a circle orthogonal to T ˜ w (Σ), ˜ w = M ˜ f ( ζ, ζ ),passing through M ˜ f ( ζ , ζ ) and M ˜ f ( ζ ∗ , ζ ). The image H = M ˜ f ( H ζ ( r ) , ζ ) is a horospherein H w (Σ) tangent to T ˜ w (Σ) at M ˜ f ( ζ , ζ ); say its radius is a . We can suppose that M ˜ f ( ζ , ζ ) isthe origin of coordinates in T ˜ w (Σ) and apply (78) with C and H as above and q = M ˜ f ( ζ ∗ , ζ )to write(81) M ˜ f ( p , ζ ) = (cos φ ) M ˜ f ( ζ ∗ , ζ ) + 2 a (sin φ ) N . Here N is the normal to T ˜ w (Σ), and the quantities φ , a and N depend on ζ .Now recall also from Section 4.1 that to first order at ζ we have H ˜ w = M ˜ f ( H ζ , ζ ) = M ˜ f ( H ζ , ζ ). Thus for the purposes of computing the derivative ∂ ξ M ˜ f ( p ( ζ, t ) , ζ ) at ζ wecan regard M ˜ f ( ζ ∗ , ζ ) as varying in the fixed plane T ˜ w (Σ) and as being projected to thefixed horosphere H ˜ w along a geodesic C whose one endpoint stays fixed at ˜ w and whoseother endpoint is varying in T ˜ w (Σ).5.2. Components of ∂ ξ M ˜ f ( p, ζ ) . Directly from (81) we compute ∂ ξ M ˜ f ( p , ζ ) = − φ sin φ ( ∂ ξ φ ) M ˜ f ( ζ ∗ , ζ ) + (cos φ ) ∂ ξ M ˜ f ( ζ ∗ , ζ )+ 2( ∂ ξ a )(sin φ ) N + 4 a sin φ cos φ ( ∂ ξ φ ) N + 2 a (sin φ ) ∂ ξ N . Evaluate this at ζ = ζ using ∂ ξ a | ζ = ζ = 0 from (71), and to ease notation write φ , a , N for these quantities at ζ : ∂ ξ M ˜ f ( p , ζ ) = ( − φ sin φ ( ∂ ξ φ ) )( M ˜ f ( ζ ∗ , ζ ) − a N )+ cos φ ∂ ξ M ˜ f ( ζ ∗ , ζ ) + 2 a sin φ ( ∂ ξ N ) We invoke (60) for z = ζ ∗ to substitute for ∂ ξ M ˜ f ( ζ ∗ , ζ ):(82) ∂ ξ M ˜ f ( p , ζ ) = ( − φ sin φ ( ∂ ξ φ ) )( M ˜ f ( ζ ∗ , ζ ) − a N )+ cos φ v ( ζ ∗ , ζ ) + cos φ v n ( ζ ∗ , ζ ) N + 2 a (sin φ )( ∂ ξ N ) . Now isolate the terms(83) V = V ( ζ ) = ( − φ sin φ ( ∂ ξ φ ) )( M ˜ f ( ζ ∗ , ζ ) − a N ) + cos φ v ( ζ ∗ , ζ ) . V is formed by omitting the terms that are present because Σ has curvature. Because ofthe first-order congruence, as above, if there is no curvature the expression for V ( ζ ), for ζ = ξ + iη varying in the ξ -direction from ζ , is exactly the velocity of a point moving onthe fixed horosphere H ˜ w under the hyperbolic stereographic projection of a point movingwith velocity v ( ζ ∗ , ζ ) in the fixed plane T ˜ w (Σ). Thus in general when Σ has curvature, V ( ζ ), having no curvature terms, is tangent to H ˜ w at E ˜ f ( p ) = M ˜ f ( p , ζ ). Moreover thefact that hyperbolic stereographic projection is conformal with velocities scaling by cos φ allows us to say, with (61) and (cid:107) ∂ ξ ˜ f (cid:107) = (cid:107) ∂ η ˜ f (cid:107) = e σ , that(84) (cid:107) V ( ζ ) (cid:107) = cos φ (cid:107) v ( ζ ∗ , ζ ) (cid:107)≤
12 cos φ e σ ( ζ ) | m ( ζ ∗ , ζ ) | | ( |S ˜ f ( ζ ) | + 12 e σ ( ζ ) | K ( ˜ f ( ζ )) | ) . We turn to the remaining two terms in (82),(85) w = cos φ v n ( ζ ∗ , ζ ) N + 2 a (sin φ )( ∂ ξ N ) , and seek to write this in the form W + W ⊥ , for W = W ( ζ ) tangent to and W ⊥ = W ⊥ ( ζ )normal to H ˜ w at E ˜ f ( p ) = M ˜ f ( p , ζ ), respectively.For this we use polar coordinates ( r, θ ) in the plane T ˜ w (Σ) with ˜ w = M ˜ f ( ζ , ζ ) as theorigin and we let e r and e θ be the orthonormal vectors in the radial and angular directions,respectively. From the formula (28) for M ˜ f we have(86) M ˜ f ( ζ ∗ , ζ ) − ˜ w = Re { m ( ζ ∗ , ζ ) } ∂ ξ ˜ f ( ζ ) + Im { m ( ζ ∗ , ζ ) } ∂ η ( ζ ) , and because (as a general fact)( ∂ ξ N ) = − ( α ( ζ ) ∂ ξ ˜ f ( ζ ) + α ( ζ ) ∂ η ˜ f ( ζ )) , we get (cid:104) ( ∂ ξ N ) , M ˜ f ( ζ ∗ , ζ ) − ˜ w (cid:105) = − ( α ( ζ ) Re { m ( ζ ∗ , ζ ) } + α ( ζ ) Im { m ( ζ ∗ , ζ ) } )= − v n ( ζ ∗ , ζ ) . Let(87) 2 r = (cid:107)M ˜ f ( ζ ∗ , ζ ) − ˜ w (cid:107) . UASICONFORMAL EXTENSIONS TO SPACE OF WEIERSTRASS-ENNEPER LIFTS 25
Recalling that r cos φ = a sin φ from (79), we can write w as w = cos φ v n ( ζ ∗ , ζ ) N − a sin φ v n ( ζ ∗ , ζ ) 12 r e r + (cid:104) w , e θ (cid:105) e θ = cos φ v n ( ζ ∗ , ζ ) N − sin φ cos φ v n ( ζ ∗ , ζ ) e r + (cid:104) w , e θ (cid:105) e θ . The vector (cid:104) w , e θ (cid:105) e θ is tangent to H ˜ w at E ˜ f ( p ) and points along the latitude through E ˜ f ( p ). We can find its magnitude. From (86),2 r e r = Re { m ( ζ ∗ , ζ ) } ∂ ξ ˜ f ( ζ ) + Im { m ( ζ ∗ , ζ ) } ∂ η ( ζ ) , and so 2 r e θ = − Im { m ( ζ ∗ , ζ ) } ∂ ξ ˜ f ( ζ ) + Re { m ( ζ ∗ , ζ ) } ∂ η ˜ f ( ζ ) . Thus referring back to (85), (cid:104) w , e θ (cid:105) = 2 a sin φ (cid:104) ( ∂ ξ N ) , e θ (cid:105) = 2 a r sin φ e − σ ( ζ ) ( α ( ζ ) Im { m ( ζ ∗ , ζ ) } − α ( ζ ) Re { m ( ζ ∗ , ζ ) } )= sin φ cos φ e − σ ( ζ ) ( α ( ζ ) Im { m ( ζ ∗ , ζ ) } − α ( ζ ) Re { m ( ζ ∗ , ζ ) } ) . The remaining terms in w , u = cos φ v n ( ζ ∗ , ζ ) N − sin φ cos φ v n ( ζ ∗ , ζ ) e r , must be further resolved to u = u + W ⊥ , where u is tangent to the longitude through E ˜ f ( p ) and W ⊥ is normal to H ˜ w at E ˜ f ( p ).It is easy to check that the vector u makes an angle π/ − φ with the tangent plane tothe sphere H ˜ w at E ˜ f ( p ), whence (cid:107) u (cid:107) = (cid:107) u (cid:107) cos (cid:16) π − φ (cid:17) = sin φ cos φ | v n ( ζ ∗ , ζ ) | , while (cid:107) W ⊥ (cid:107) = (cid:107) u (cid:107) sin (cid:16) π − φ (cid:17) = cos φ | v n ( ζ ∗ , ζ ) | . Therefore we can write w = W + W ⊥ , with W = u + (cid:104) w , e θ (cid:105) e θ . We compute (cid:107) W (cid:107) = (cid:107) u (cid:107) + (cid:104) w , e θ (cid:105) = (cid:107) u (cid:107) + sin φ cos φ [Im { α ( ζ ) m ( ζ ∗ , ζ ) } − Re { α ( ζ ) m ( ζ ∗ , ζ ) } ] = sin φ cos φ [(Re { α ( ζ ) m ( ζ ∗ , ζ ) + Im α ( ζ ) m ( ζ ∗ , ζ )) + (Im { α ( ζ ) m ( ζ ∗ , ζ ) } − Re { α ( ζ ) m ( ζ ∗ , ζ ) } ) ]= sin φ cos φ ( α ( ζ ) + α ( ζ ) )(Re { m ( ζ ∗ , ζ ) } + Im { m ( z ∗ , ζ ) } )= sin φ cos φ | m ( ζ ∗ , ζ ) | e σ ( ζ ) | K ( ˜ f ( ζ ) | , where in the last line we have used α + α = e σ | K | , (63). Finally, from the bound on | v n | in (62), (cid:107) W ⊥ (cid:107) = cos φ | v n | ≤ cos φ | m ( ζ ∗ , ζ ) | e σ ( ζ ) (cid:113) | K ( ˜ f ( ζ ) | . Taken together,(88) (cid:107) V (cid:107) ≤
12 cos φ | m ( ζ ∗ , ζ ) | e σ ( ζ ) ( |S ˜ f ( ζ ) | + 12 e σ ( ζ ) | K ( ˜ f ( ζ )) | ) , (cid:107) W (cid:107) ≤ sin φ cos φ | m ( ζ ∗ , ζ ) | e σ ( ζ ) (cid:113) | K ( ˜ f ( ζ )) | , (cid:107) W ⊥ (cid:107) ≤ cos φ | m ( ζ ∗ , ζ ) | e σ ( ζ ) (cid:113) | K ( ˜ f ( ζ ) | . For the calculations in the next section it will be convenient to write these inequalities alittle differently. First,(89) | m ( ζ ∗ , ζ ) | = 2 e − σ ( ζ ) r from (86) and (87). Then the bounds for (cid:107) V (cid:107) and (cid:107) W ⊥ (cid:107) become(90) (cid:107) V (cid:107) ≤ r cos φ e − σ ( ζ ) ( |S ˜ f ( ζ ) | + 12 e σ ( ζ ) | K ( ˜ f ( ζ )) | ) , (cid:107) W ⊥ (cid:107) ≤ r cos φ e σ ( ζ ) (cid:113) | K ( ˜ f ( ζ ) | . We also bring in the radius a of the horosphere H ˜ w ,(91) a = 12 e σ ( ζ ) e t (1 − | ζ | ) , using (69) and (73), and the equation r cos φ = a sin φ from (79). Then the bound for (cid:107) W (cid:107) is(92) (cid:107) W (cid:107) ≤ r cos φ e − t (cid:113) | K ( ˜ f ( ζ )) | − | ζ | . We summarize the principal results of this section in a lemma.
Lemma 5.
Let p = p ( ζ , t ) ∈ H ζ ( t ) and let φ be the angle of elevation of E ˜ f ( p ) ∈ H ˜ w = M ˜ f ( H ζ , ζ ) measured from ˜ w . Let r = (cid:107)M ˜ f ( ζ ∗ , ζ ) − ˜ w (cid:107) .Then ∂ ξ M ˜ f ( p , ζ ) = V + W + W ⊥ , where V and W are tangent to and W ⊥ is normal to H ˜ w at E ˜ f ( p ) , with (93) (cid:107) V (cid:107) ≤ r cos φ e − σ ( ζ ) ( |S ˜ f ( ζ ) | + 12 e σ ( ζ ) | K ( ˜ f ( ζ )) | ) , (cid:107) W (cid:107) ≤ r cos φ e − t (cid:113) | K ( ˜ f ( ζ ) | − | ζ | , (cid:107) W ⊥ (cid:107) ≤ r cos φ e σ ( ζ ) (cid:113) | K ( ˜ f ( ζ ) | . UASICONFORMAL EXTENSIONS TO SPACE OF WEIERSTRASS-ENNEPER LIFTS 27
Estimating the Dilatation of E ˜ f . The proof of Theorem 2 is completed by derivingthe bounds(94) 1 k ( ρ ) ≤ max (cid:107) X (cid:107) =1 (cid:107) D E ˜ f ( p ) X (cid:107) min (cid:107) X (cid:107) =1 (cid:107) D E ˜ f ( p ) X (cid:107) ≤ k ( ρ ) , We continue with the notation as above, in particular working at the fixed point p = p ( ζ , t ) = C ζ ∩ H ζ ( t ) and ˜ w = ˜ f ( ζ ). We estimate (cid:107) D E ˜ f ( p )( X ) (cid:107) , (cid:107) X (cid:107) = 1 for X tangent to and normal to C ζ at p . Case I : X tangent to C ζ . Since E ˜ f restricted to C ζ coincides with M ˜ f ( p, ζ ) we have D E ˜ f ( p )( X ) = D M ˜ f ( p , ζ )( X ), which is tangent to C ˜ w at E ˜ f ( p ). The magnitude of D M ˜ f ( p , ζ )( X ) can in turn be expressed as the ratio of the heights of p ∈ H above C and of E ˜ f ( p ) = M ˜ f ( p , ζ ) ∈ H w (Σ) above T ˜ w (Σ), say(95) (cid:107) D E ˜ f ( p ) X (cid:107) = h ( E ˜ f ( p )) h ( p ) . Case II : X normal to C ζ . In this case X is a unit vector tangent to the surface S ( t ) at p and we take it to be in the ∂ ξ -direction; recall (77) and the accompanying discussion.Letting δ = (cid:107) ∂ ξ p ( ζ , t ) (cid:107) , we obtain D E ˜ f ( p )( X ) = D M ˜ f ( p , ζ )( X ) + 1 δ ∂ ξ M ˜ f ( p , ζ )= D M ˜ f ( p , ζ )( X ) + 1 δ ( V + W + W ⊥ ) . Hence (cid:107) D E ˜ f ( p )( X ) (cid:107) ≤ (cid:107) D M ˜ f ( p , ζ )( X ) (cid:107) + 1 δ ( (cid:107) V (cid:107) + (cid:107) W (cid:107) + (cid:107) W ⊥ (cid:107) )= h ( E ˜ f ( p )) h ( p ) + 1 δ ( (cid:107) V (cid:107) + (cid:107) W (cid:107) + (cid:107) W ⊥ (cid:107) ) . On the other hand, since D M ˜ f ( p , ζ )( X ) is tangent to the image horosphere H ˜ w = M ˜ f ( H ζ , ζ ) we also have(96) (cid:107) D E ˜ f ( p )( X ) (cid:107) ≥ (cid:107) D M ˜ f ( p , ζ )( X ) (cid:107) − δ ( (cid:107) V (cid:107) + (cid:107) W (cid:107) )= h ( E ˜ f ( p )) h ( p ) − δ ( (cid:107) V (cid:107) + (cid:107) W (cid:107) ) . To deduce (94) we thus want to show two things:(1) There exists a constant κ = κ ( ρ ) < δ ( (cid:107) V (cid:107) + (cid:107) W (cid:107) ) ≤ κ h ( E ˜ f ( p )) h ( p ) . (2) There exists a constant κ = κ ( ρ ) < ∞ such that(98) 1 δ (cid:107) W ⊥ (cid:107) ≤ κ h ( E ˜ f ( p )) h ( p ) . From the formula (74) for p ( ζ, t ) we calculate δ = 1 + e t e t | ζ | , and for the height of p above C , h ( p ) = e t − | ζ | e t | ζ | . In the image, the height of E ˜ f ( p ) above T ˜ w (Σ) is h ( E ˜ f ( p )) = 2 r sin φ cos φ , from (80), where 2 r = (cid:107)M ˜ f ( ζ ∗ , ζ ) − ˜ w (cid:107) , as in (87). We also bring in the radius a of thehorosphere H ˜ w , a = 12 e σ ( ζ ) e t (1 − | ζ | ) , using (69) and (73), and the equation r cos φ = a sin φ from (79). We then obtain(99) δ h ( E ˜ f ( p ) h ( p ) = 4 r cos φ e − σ ( ζ ) e − t (1 − | ζ | ) . The estimates in (93) allow us to express the sought for upper bound (97) as(100) |S ˜ f ( ζ ) | + 12 e σ ( ζ ) | K ( ˜ f ( ζ )) | + 2 e − t e σ ( ζ ) (cid:113) | K ( ˜ f ( ζ )) | − | ζ | ≤ κ e − t )(1 − | ζ | ) . Let A denote the left-hand side of (100). Since ˜ f satisfies (4),(101) A ≤ ρ (1 − | ζ | ) − e σ ( ζ ) | K ( ˜ f ( ζ ) | + 2 e − t e σ ( ζ ) (cid:113) | K ( ˜ f ( ζ ) | − | ζ | = 2 ρ (1 − | ζ | ) − B + 2 B C = 2 ρ (1 − | ζ | ) + 2 C −
12 ( B − C ) , where we have put B = e σ ( ζ ) (cid:113) | K ( ˜ f ( ζ )) | , C = e − t (1 − | ζ | ) . The curvature bound (5) implies(102) B ≤ √ ρ − | ζ | = e t (cid:112) ρ C . Let (cid:15) ∈ (0 , τ by(103) e τ = √ (cid:15) √ ρ . Suppose first that t ≤ τ . Then (102) implies B ≤ (cid:15)C . Hence from (101),(104) A ≤ ρ (1 − | ζ | ) + 2(1 − (1 − (cid:15) ) ) C = 2 ρ (1 − | ζ | ) + 2 ρ e − t (1 − | ζ | ) , UASICONFORMAL EXTENSIONS TO SPACE OF WEIERSTRASS-ENNEPER LIFTS 29 setting(105) ρ = 1 − (1 − (cid:15) ) . The number ρ increases with (cid:15) ∈ (0 ,
1) and 0 < ρ <
1. Hence (100) holds in this case with κ = max { ρ, ρ } .Suppose next that τ ≤ t . Begin with A ≤ ρ (1 − | ζ | ) + 2 C = 2 ρ (1 − | ζ | ) + 2 e − t (1 − | ζ | ) ≤ ρ (1 + e − t )(1 − | ζ | ) provided ρ can be chosen so that ρ + e − t ≤ ρ (1 + e − t ), that is so that e − t ≤ ρ − ρ − ρ . Using (103) and τ ≤ t we are led to the optimal value ρ = ρ (cid:15) ρ + 2 (cid:15) . Here ρ is decreasing for (cid:15) ∈ (0 ,
1) and 3 ρ ρ < ρ < . Finally, we choose (cid:15) as the unique solution in (0 ,
1) for which ρ = ρ . This common valuedefines a constant κ for which (100) holds, and thus proves the estimate (97). Remark.
Before continuing, we note that a simple approximation for this common valuewhen ρ ∼ ρ = ρ ( (cid:15) ), ρ = ρ ( (cid:15) ) by straightlines, giving κ = 2 + ρ − ρ . It is of some interest to be more precise. The equation1 − (1 − (cid:15) ) = ρ (cid:15) ρ + 2 (cid:15) leads to ρ = h ( (cid:15) ) = 2 (cid:15) (2 − (cid:15) )3 (cid:15) − (cid:15) + 1 . The function h ( (cid:15) ) is monotonically increasing for (cid:15) ∈ (0 ,
1) with h (0) = 0 and h (1) = 1 andhas an inverse (cid:15) = k ( ρ ) with the same properties. Then the constant κ in (100) and (97) is κ = 1 − (1 − g ( ρ )) . Moreover, using h (cid:48) (1) = 0, h (cid:48)(cid:48) (1) = − (cid:15) ∼ h ( (cid:15) ) ∼ −
32 (1 − (cid:15) ) , hence κ ∼ −
23 (1 − ρ ) . This agrees to first order with the approximation (2 + ρ ) / (4 − ρ ) for (cid:15) ∼ Finally, we prove (98), which, with (93) and (99) amounts to e σ ( ζ ) (cid:113) | K ( ˜ f ( ζ ) | ≤ κ r e − t )(1 − | ζ | ) , for an appropriate κ . But now from (38), in Section 3.1 on the reflection across ∂ Σ, we have2 r = e σ ( ζ ) (cid:107)∇ log U ˜ f ( ζ ) (cid:107) , and so the inequality we must show reduces to e σ ( ζ ) (cid:113) | K ( ˜ f ( ζ ) | (cid:107)∇ log U ˜ f ( ζ ) (cid:107) ≤ κ (1 + e − t )(1 − | ζ | ) Lemma 4 provides the bound (cid:107)∇ log U ˜ f ( ζ ) (cid:107) ≤ √ − | ζ | , so it suffices to find κ with √ e σ ( ζ ) (cid:113) | K ( ˜ f ( ζ ) | ≤ κ − | ζ | , which by (5) will hold for κ = 2 √ ρ. The estimates (97) and (98) together show that1 − κ ≤ max (cid:107) X (cid:107) =1 (cid:107) D E ˜ f ( p ) X (cid:107) min (cid:107) X (cid:107) =1 (cid:107) D E ˜ f ( p ) X (cid:107) ≤ κ + κ , which proves that E ˜ f is quasiconformal in R with constant k ( ρ ) = 1 + κ + κ − κ . The proof of Theorem 2 is complete.In the classical case, when ˜ f is analytic and Σ is planar, the result generalizes the classicalAhlfors-Weill theorem to provide an extension to space, with the classical dilatation as well. Corollary 1. If f is analytic in D and |S f ( ζ ) | ≤ ρ (1 − | z | ) , ρ < , then E f is (1 + ρ ) / (1 − ρ ) -quasiconformal.Proof. In this case the curvature is zero. We see from (100) that we may take κ = ρ andfrom (93) and (98) that we may take κ = 0. (cid:3) UASICONFORMAL EXTENSIONS TO SPACE OF WEIERSTRASS-ENNEPER LIFTS 31
Quasiconformality of the Reflection R . Recall from Section 3.1 the reflection R : Σ → Σ ∗ , defined via the circle bundle C (Σ) on Σ by setting R ( ˜ w ) = ˜ w ∗ , where forthe circle C ˜ w we have C ˜ w ∩ T ˜ w (Σ) = { ˜ w, ˜ w ∗ } . Under R the unique critical point of U ˜ f is mapped to the point at infinity while the rest of Σ is mapped to the surface Σ ∗ , andΣ ∪ Σ ∗ ∪ {∞} is a topological sphere. A limiting case of the preceding estimates allows usto deduce that R is quasiconformal.We must find bounds for (cid:107) D E ˜ f ( p ) X (cid:107) when p ∈ Σ and X is tangent to Σ at p . This is alimit of the estimates in Case II, above, as t → −∞ . For κ , following the analysis when t ≤ τ starting from (103), we may let τ → −∞ as well, whence (cid:15) → ρ → κ = ρ. The value of κ is as before, namely κ = 2 √ ρ . Thus Corollary 2. If ˜ f satisfies (4) with ρ < then the reflection R is k ( ρ ) -quasiconformal with k ( ρ ) = 1 + ρ + 2 √ ρ − ρ = (1 + √ ρ ) − ρ . As before, when ˜ f is analytic and Σ is planar the estimates involving the curvature andthe second fundamental form do not enter. In that case the bound reduces to the classical(1 + ρ ) / (1 − ρ ). Remark.
It is possible to show directly that R is quasiconformal using the formula R ( ˜ w ) = ˜ w + 2 J ( ∇ log λ Σ ( ˜ w )) , J ( p ) = p/ (cid:107) p (cid:107) . Very briefly, we can regard ˜ w (cid:55)→ R ( ˜ w ) as a vector field along Σ (not tangent to Σ) and thencompute its covariant derivative ∇ X R in the direction of a vector X , (cid:107) X (cid:107) = 1, tangent toΣ. Here ∇ X R is the Euclidean covariant derivative on R .In terms of the function λ Σ on Σ, (14), and the gradient Λ = (cid:107)∇ log λ Σ (cid:107) one can show4 λ Λ (1 − ρ ) ≤ (cid:107)∇ X R(cid:107) ≤ λ Λ (1 + √ ρ ) , and the corollary follows. The derivation is interesting, but it requires more preparation.6. Quasiconformal Extension of Planar Harmonic mappings
In this section we consider the problem of injectivity and quasiconformal extension forthe planar harmonic mapping f = h + ¯ g under the assumption that its lift ˜ f satisfies (4).Our method is simply to project from Σ ∪ Σ ∗ to the plane, and the reward is the similarityof the resulting extension of the planar map to the classical Ahlfors-Weill formula appliedseparately to h and ¯ g .However, (4) alone is not enough. In fact, we are in a situation reminiscent of the originalAhlfors-Weill proof, where we need to know first that the projection is locally injective –geometrically that Σ ∪ Σ ∗ is locally a graph. If we assume that f is locally injective, sense-preserving, and that its dilatation ω is the square of an analytic function, so | ω ( ζ ) | <
1, thenat least the surface Σ is locally a graph; see [11]. It may exhibit several sheets if f is notinjective, and the analysis in Lemma 6 below suggests that without a stronger assumptionon the dilatation the reflected surface Σ ∗ need not be locally a graph. To address the latterwe have the following result. Lemma 6.
Suppose that f = h + ¯ g is locally injective with dilatation ω the square of ananalytic function, and that ˜ f satisfies (4) for a ρ < . If ω satisfies (107) sup ζ ∈ D (cid:112) | ω ( ζ ) | < − √ ρ √ ρ , ζ ∈ D , then Σ ∗ is locally a graph.Proof. Fix a point ˜ w = ˜ f ( ζ ) on Σ. Let ϑ be the angle of inclination with respect to thevertical of the tangent plane T ˜ w (Σ). From the formulas for the components of ˜ f , i.e., theformulas for the Weierstrass-Enneper lift, see [11], one can show that(108) tan ϑ = 2 (cid:112) | ω ( ζ ) | − | ω ( ζ ) | . Now let X be a unit tangent vector to Σ at ˜ w and let ( D E ˜ f ( ζ )( X )) (cid:62) and ( D E ˜ f ( ζ )( X )) ⊥ berespectively the tangential and normal components of D E ˜ f ( ζ )( X ) Then the angle of incli-nation of the tangent plane T ˜ w ∗ (Σ ∗ ) to Σ ∗ at ˜ w ∗ = R ( ˜ w ) is ϑ + tan − (cid:107) ( D E ˜ f ( ζ )( X )) ⊥ (cid:107)(cid:107) ( D E ˜ f ( ζ )( X )) (cid:62) (cid:107) . The surface Σ ∗ will be locally a graph at ˜ w ∗ if this angle is < π/
2, and using (108) thiscondition can be written(109) 2 (cid:112) | ω ( ζ ) | − | ω ( ζ ) | (cid:107) ( D E ˜ f ( ζ )( X )) ⊥ (cid:107)(cid:107) ( D E ˜ f ( ζ )( X )) (cid:62) (cid:107) < . We can use limiting cases of previous estimates for (cid:107) D E ˜ f (cid:107) to bound the ratio, namely(96), (97) and (98), with κ = ρ , from (106), and κ = 2 √ ρ . This results in2 (cid:112) | ω | − | ω | (cid:107) ( D E ˜ f ( ζ )( X )) ⊥ (cid:107)(cid:107) ( D E ˜ f ( ζ )( X )) (cid:62) (cid:107) ≤ (cid:112) | ω ( ζ ) | − ω ( ζ ) √ ρ − ρ . The right-hand side will be < (cid:112) | ω ( ζ ) | < − √ ρ √ ρ . (cid:3) We restate Theorem 3 from the introduction, and proceed with the proof.
Theorem. If f = h + ¯ g is a locally injective harmonic mapping of D whose lift ˜ f satis-fies (4) for a ρ < and whose dilatation ω satisfies (107) , then f is injective and has aquasiconformal extension to C given by F ( ζ ) = f ( ζ ) , ζ ∈ D f ( ζ ∗ ) + (1 − | ζ ∗ | ) h (cid:48) ( ζ ∗ )¯ ζ ∗ − (1 − | ζ ∗ | ) ∂ z σ ( ζ ∗ ) + (1 − | ζ ∗ | ) g (cid:48) ( ζ ∗ ) ζ ∗ − (1 − | ζ ∗ | ) ∂ ¯ z σ ( ζ ∗ ) , ζ ∗ = 1¯ ζ , , ζ / ∈ D . UASICONFORMAL EXTENSIONS TO SPACE OF WEIERSTRASS-ENNEPER LIFTS 33
Proof.
Without loss of generality we can assume that the unique critical point of U ˜ f is theorigin. Let Π : R → C be the projection Π( x , x , x ) = x + ix . We know that Σ ∪ Σ ∗ islocally a graph over C , and hence the mapping F ( ζ ) = (cid:40) f ( ζ ) , ζ ∈ D , (Π ◦ R )( ˜ f ( ζ ∗ )) , ζ (cid:54)∈ D is locally injective. Note that F = Π ◦ E ˜ f restricted to C .Locating the critical point of U ˜ f at the origin implies that F ( z ) → ∞ as | z | → ∞ . Bythe monodromy theorem we conclude that F is a homeomorphism of C . In particular, theunderlying harmonic mapping f is injective. Moreover, the assumption on ω implies thatthe inclinations of both Σ and Σ ∗ are bounded away from π/
2, making the projection Πquasiconformal. Since by Corollary 2 the reflection R is quasiconformal, so is F .Let us verify that F has the stated form. From the Weierstrass-Enneper formulas (seeagain [11]), ∂ ξ ˜ f = (Re { h (cid:48) + g (cid:48) } , Im { h (cid:48) − g (cid:48) } , { h (cid:48) √ ω } ) ,∂ η ˜ f = ( − Im { h (cid:48) + g (cid:48) } , Re { h (cid:48) − g (cid:48) } , − { h (cid:48) √ ω } ) , from which Π( ∂ ξ ˜ f ) = h (cid:48) + ¯ g (cid:48) , Π( ∂ η ˜ f ) = i ( h (cid:48) − ¯ g (cid:48) ) . Now recall that the reflection R is given in terms of the best M¨obius approximation, from(35), and when ζ is outside D we want the projection of˜ f ( ζ ∗ ) + Re { m ( ζ ∗ , ζ ) } ∂ ξ ( ζ ∗ ) + Im { m ( ζ ∗ , ζ ) } ∂ η ˜ f ( ζ ∗ ) . This is f ( ζ ∗ ) + Re { m ( ζ ∗ , ζ )( h (cid:48) ( ζ ∗ )+¯ g (cid:48) ( ζ ∗ )) + i Im { m ( ζ ∗ , ζ ) } ( h (cid:48) ( ζ ∗ ) − ¯ g (cid:48) ( ζ ∗ ))= f ( ζ ∗ ) + m ( ζ ∗ , ζ ) h (cid:48) + m ( ζ ∗ , ζ ) ¯ g (cid:48) ( ζ ∗ ) , which is exactly the formula for F ( ζ ). (cid:3) We can also conclude that Σ ∪ Σ ∗ is a graph over C . References
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