Matching univalent functions Pavel Gumenyuk joint research with - PowerPoint PPT Presentation

Matching univalent functions Pavel Gumenyuk joint research with Erlend Grong and Alexander Vasil’ev University of Bergen Saratov State University

2 Matching functions and conformal welding 1. Matching functions and conformal welding Definition 1. Suppose that: • f is a conformal mapping of D := { z : | z | < 1 } onto a Jordan domain D ; • ϕ is a conformal mapping of D ∗ := C \ D onto a Jordan domain D ∗ . Then the functions f and ϕ are said to be matching if D and D ∗ are complementary domains, i. e. D ∩ D ∗ = ∅ and Γ := ∂D = ∂D ∗ . University of Bergen Saratov State University

3 Matching functions and conformal welding Using fractional-linear change of variables, we can assume that: (i) 0 ∈ D and ∞ ∈ D ∗ ; (ii) f (0) = f ′ (1) − 1 = 0; (iii) ϕ ( ∞ ) = ∞ . S := { f : D → C : f is analytic, univalent, and subject to normalization (ii) } . Problem 1. Given f ∈ S s. t. f ( D ) is a Jordan domain, find a univalent meromorphic function ϕ which matches the function f . A pair of matching functions ( f, ϕ ) defines the homeomorphism of the unit circle S 1 , γ = f − 1 ◦ ϕ, γ : S 1 → S 1 . (1) Definition 2. Representation (1) of a homeomorphism γ : S 1 → S 1 by means of matching functions is called the conformal welding . University of Bergen Saratov State University

4 Matching functions and conformal welding Problem 2. Find the conformal welding for a given orientation preserv- ing homeomorphism γ : S 1 → S 1 , i. e. the pair of matching univalent functions ( f, ϕ ) such that γ = f − 1 ◦ ϕ . Problem 2 has a unique solution for all homeomorphisms γ that are quasisymmetric, i. e. satisfies � e i ( t + h ) � � e it � � � γ − γ � � � � � < C γ < + ∞ , for all t, h ∈ R , 0 < | h | < π. (2) � � � � � � e i ( t − h ) e it − γ γ � � � A. Pfluger, 1960; O. Lehto & K.I. Virtanen, 1960. Also follows from the Ahlfors – Beurling Extension Theorem, A. Beurling & L. Ahlfors, 1956 Existence and uniqueness of the conformal welding for the constant C γ replaced in right-hand side of (2) with ρ ( h ) = O (log h ), O. Lehto, 1970; G.L. Jones, 2000. University of Bergen Saratov State University

5 Matching functions and conformal welding Denote by: • Lip α ( X, Y ) the class of all functions h : X → Y which are H¨ older- continuous with exponent α ; • S the class of all analytic univalent functions f : D → C such that f (0) = f ′ (0) − 1 = 0; • S qc := { f ∈ S : f can be extended to q. c. homeomorphism of C } ; • S 1 ,α := { f ∈ S : ∂f ( D ) is a C 1 ,α -smooth Jordan curve } , α ∈ (0 , 1); • S ∞ := { f ∈ S : ∂f ( D ) is a C ∞ -smooth Jordan curve } ; • Homeo + qs ( S 1 ) the group of all orientation preserving q. s. homeo- morphisms γ : S 1 → S 1 . Remark 1. The conformal welding establishes one-to-one correspon- dence between S qc and Homeo + qs ( S 1 ) / Rot( S 1 ). • To calculate γ ∈ Homeo + qs ( S 1 ) for given f ∈ S qc one have to solve Problem 1, which is to find the function ϕ matching f . University of Bergen Saratov State University

6 Matching functions and conformal welding • To determine f ∈ S qc for given γ ∈ Homeo + qs ( S 1 ) one have to solve the Beltrami equation � ¯ if z ∈ D ∗ , ∂u ( z ) /∂u ( z ) , ∂ ˜ ¯ f ( z ) = µ ( z ) ∂ ˜ f ( z ) , µ ( z ) := (3) 0 , otherwise, � � � � ∂ := 1 ∂x − i ∂ ∂ ∂ := 1 ∂x + i ∂ ∂ ¯ , , 2 ∂y 2 ∂y with the normalization f ′ (0) − 1 = 0 , f ( ∞ ) = ∞ and ˜ ˜ f (0) = ˜ (4) where u is any q. c. automorphism of D ∗ such that u | S 1 = γ − 1 . u ( ∞ ) = ∞ and Then f | D ∗ ◦ u − 1 f := ˜ ϕ := ˜ f | D , (5) are matching functions, f ∈ S qc , and γ = f − 1 ◦ ϕ . University of Bergen Saratov State University

7 Main results 2. Main results The following theorem establishes more explicit relation between f , ϕ , For fixed f ∈ S 1 ,α define the operator and γ for the smooth case. I f : Lip α ( S 1 , R ) → Hol( D ) by the formula � 2 sf ′ ( s ) � I f [ v ]( z ) := − 1 v ( s ) ds � s , z ∈ D . (6) 2 πi S 1 f ( s ) f ( s ) − f ( z ) Suppose f ∈ S 1 ,α and ϕ , ϕ ( ∞ ) = ∞ , are matching Theorem 1. Then the kernel of the operator I f : Lip α ( S 1 , R ) → Hol( D ) functions. is the one-dimensional manifold ker I f = span { v 0 } , where ( ψ ◦ f )( z ) v 0 ( z ) := 1 ψ := ϕ − 1 , z ∈ S 1 . (7) f ′ ( z )( ψ ′ ◦ f )( z ) , z Moreover, the function v 0 is positive on S 1 and satisfies the following condition � 2 π dt v 0 ( e it ) = 2 π. (8) 0 University of Bergen Saratov State University

8 Kirillov’s manifold Remark 2. Theorem 1 reduces Problem 1 to solution of the equation I f [ v ] = 0 . (9) Indeed, given f and v 0 , one can calculate ψ = ϕ − 1 on the boundary of D ∗ by solving the following differential equation ψ ′ ( u ) = H ( u ) ψ ( u ) , u ∈ ∂D ∗ , 1 H ◦ f − 1 , for z ∈ S 1 . H := ˜ ˜ H ( z ) := (10) zf ′ ( z ) v 0 ( z ) Complex Solutions to I f [ v ] = 0 . Suppose f ∈ S 1 ,α and ϕ , ϕ ( ∞ ) = ∞ , are matching Theorem 2. functions and γ := f − 1 ◦ ϕ . Then the kernel of the operator I f : Lip α ( S 1 , C ) → Hol( D ) is the set of all functions v of the form v ( z ) = v 0 ( z ) · ( h ◦ γ − 1 )( z ) , z ∈ S 1 , (11) where v 0 is defined by (7) and h is an arbitrary holomorphic function in D ∗ admitting Lip α ( S 1 , C ) -extension to S 1 . University of Bergen Saratov State University

9 Kirillov’s manifold 3. Operator I f and Kirillov’s manifold By Diff + ( S 1 ) denote the Lie – Fr´ echet group of all orientation preserv- ing C ∞ -smooth diffeomorphisms of S 1 . In 1987 A.A. Kirillov proposed to use the 1-to-1 correspondence be- tween S qc and Homeo + qs ( S 1 ) / Rot( S 1 ) established by conformal weld- ing to represent the homogeneous manifold M := Diff + ( S 1 ) / Rot( S 1 ) (Kirillov’s manifold) via univalent functions. The bijection K : M → S ∞ allows to bring the complex structure from S ∞ to M . A.A. Kirillov proved that the (left) action of Diff + ( S 1 ) on M is holomorphic w.r.t. this complex structure. The infinitesimal version of K : M → S ∞ is more explicit and expressed by means of I f [ v ]. Consider the variation of γ ∈ Diff + ( S 1 ) given by γ ε ( ζ ) := γ ( ζ ) δγ ( ζ ) , δγ := exp iε ( v ◦ γ ) , (12) University of Bergen Saratov State University

10 Kirillov’s manifold where v ∈ C ∞ ( S 1 , R ) is regarded as an element of T id Diff + ( S 1 ). Variation (12) of γ results in the following variation of f ∈ S ∞ f ε := K ( γ ε ) = f + δf, � 2 f 2 ( z ) v ( s ) sf ′ ( s ) � δf ( z ) = ε ds � s = iεf 2 ( z ) I f [ v ]( z ) . (13) 2 π S 1 f ( s ) f ( z ) − f ( s ) Remark 3. A natural consequence of this is that I f [ v ]( z ) = 0 for all z ∈ D if and only if the variation of γ produces no variation of [ γ ] ∈ M (up to higher order terms), which can be reformulated as follows: the element of T γ Diff + ( S 1 ) represented by v ◦ γ is tangent to the one- dimensional manifold γ ◦ Rot( S 1 ) = [ γ ] ⊂ Diff + ( S 1 ) . The latter is equivalent to � � T id Rot( S 1 ) � constant functions on S 1 � v ∈ Ad γ = Ad γ , (14) University of Bergen Saratov State University

11 Kirillov’s manifold where Ad γ stands for the differential of β �→ γ ◦ β ◦ γ − 1 at β = id. Elementary calculations show that Ad γ u = u ◦ γ − 1 � ′ , β # := π − 1 ◦ β ◦ π � � # , � γ − 1 where π : R → S 1 is the universal covering, π ( x ) = e ix . As a conclusion we get The kernel of I f : C ∞ ( S 1 , R ) → Hol( D ) is a one- Proposition 1. dimensional manifold and coincides with span { 1 / ( γ − 1 ) # } . Remark 4. This Proposition is the special case of Theorem 1 for C ∞ - smooth case. It shows that Problem 2 (of finding conformal welding) is reduced by Theorem 1 to finding solution to I f [1 / ( γ − 1 ) # ] = 0 , (15) regarded as equation w.r.t. f ∈ S ∞ . University of Bergen Saratov State University

12 An Example 4. An Example of matching functions Given an integer n > 1, let us consider quadratic differentials Ψ( ζ ) dζ 2 := − dζ 2 ζ 2 ; n − 1 W ( w ) dw 2 := − w n − 2 dw 2 w k := e 2 πik/n ; � , P ( w ) := ( w − w k ) , P ( w ) k =0 Z ( z ) dz 2 := − z n − 2 dz 2 , Q ( z ) n − 1 | z k | z k := re 2 πik/n , � Q ( z ) := κ ( z k − z )( z − 1 /z k ) , r ∈ (0 , 1) . z k k =0 where κ > 0 is such that � � Z ( z ) dz = 2 π (16) S 1 for the appropriately chosen branch of the square root. University of Bergen Saratov State University

13 An Example n = 5 1 1 0.5 0.5 -1 -0.5 0.5 1 -1 -0.5 0.5 1 -0.5 -0.5 -1 -1 The structure of trajectories W ( w ) dw 2 > 0 and Z ( z ) dz 2 > 0. University of Bergen Saratov State University