Loewner Theory in the Unit Disk P avel G umenyuk COLLOQUIUM at the - PowerPoint PPT Presentation

Universita’ di Roma TOR VERGATA Loewner Theory in the Unit Disk P avel G umenyuk COLLOQUIUM at the Faculty of Mathematics — Universidad de Sevilla — ESPAÑA, February 13, 2013 1/41

Outline Universita’ di Roma TOR VERGATA Classical Loewner Theory Origin of Loewner Theory Loewner’s construction Chordal Loewner Equation Representation of the whole class S Some interesting results and applications Applications to Extremal Problems Criteria for univalence SLE Conditions for slit dynamics More Topics to mention New approach Semigroups of Conformal Mappings Evolution Families Herglotz Vector Fields 2/41

Classical Loewner Theory Universita’ di Roma TOR VERGATA The starting point of Loewner Theory is the seminal paper by Czech – German mathematician Karel Löwner (1893 – 1968) known also as Charles Loewner Untersuchungen über schlichte konforme Abbildungen des Einheitskreises , Math. Ann. 89 (1923), 103–121. In this paper Loewner introduced a new method to study the famous Bieberbach Conjecture concerning the so-called class S . 3/41 Classical Loewner Theory

Bieberbach’s Conjecture Universita’ di Roma TOR VERGATA Ludwig Bieberbach, 1916: analytic properties of conformal mappings f : D into D := { z : | z | < 1 } , f ( 0 ) = 0 , f ′ ( 0 ) = 1 . − − − → C , Class S By S we denote the class of all holomorphic univalent functions + ∞ � a n z n , (1) f ( z ) = z + z ∈ D . n = 2 the famous Bieberbach Conjecture (1916) ∀ f ∈ S ∀ n = 2 , 3 , . . . (2) | a n | � n Bieberbach (1916): n = 2; Loewner (1923): n = 3; . . . de Branges (1984): all n � 2 — using Loewner’s method 4/41 Classical Loewner Theory

Elementary properties of class S . Universita’ di Roma TOR VERGATA there is no natural linear structure in the class S ; – the class S is not even a convex set in Hol ( D , C ) ; – the class S is compact w.r.t. local uniform convergence in D ; + + Uni 0 ( D , D ) := � � ϕ ∈ Hol ( D , D ) : ϕ is univalent and ϕ ( 0 ) = 0 , ϕ ′ ( 0 ) > 0 is a topological semigroup w.r.t. the composition operation ( ϕ, ψ ) �→ ψ ◦ ϕ and the topology of locally uniform convergence. 5/41 Classical Loewner Theory

Loewner’s construction 1 Universita’ di Roma TOR VERGATA Loewner considered the dense subclass S ′ ⊂ S of all slit mappings , � S ′ := f ∈ S : f ( D ) = C \ Γ , where Γ is Figure 1 � a Jordan arc extending to ∞ . Loewner’s construction 2 ◮ Consider f ∈ S ′ and let Γ := C \ f ( D ) . ◮ Choose a parametrization γ : [ 0 , + ∞ ] → Γ , γ (+ ∞ ) = ∞ . � � ◮ Consider the domains Ω t := C \ γ , t � 0. [ t , + ∞ ] ◮ By Riem. Mapping Th’m ∀ t � 0 ∃ ! conformal mapping f t : D onto f t ( 0 ) = 0 , f ′ t ( 0 ) > 0 . − − − → Ω t , ◮ Note that f 0 = f . t ( 0 ) = e t . ◮ Reparameterizing Γ : ∀ t � 0 f ′ 6/41 Classical Loewner Theory

Loewner’s construction 2 Universita’ di Roma TOR VERGATA Loewner’s Theorem The family ( f t ) is of class C 1 w.r.t. t (even if Γ is NOT smooth!) ◮ Moreover, ∃ ! continuous function ξ : [ 0 , + ∞ ) → T := ∂ D ◮ t ( z ) 1 + ξ ( t ) z ∂ f t ( z ) = zf ′ z ∈ D , t � 0 . (3) (the Loewner PDE) , ∂ t 1 − ξ ( t ) z The following IVP (for the classical Loewner ODE) ◮ = − w ( t ) 1 + ξ ( t ) w ( t ) dw ( t ) (4) dt 1 − ξ ( t ) w ( t ) ∀ s � 0 ∀ z ∈ D has a unique solution w = w z , s : [ s , + ∞ ) → D . For all s � 0, f s ( z ) = lim (5) t → + ∞ e t w z , s ( t ) . ◮ 7/41 Classical Loewner Theory

Loewner’s construction 3 Universita’ di Roma TOR VERGATA As a corollary Every f ∈ S ′ is generated by some (uniquely defined) continuous function ξ : [ 0 , + ∞ ) → T . Namely f ( z ) = lim t → + ∞ e t w z , 0 ( t ) , (6) where w = w z , 0 is the solution to the IVP = − w ( t ) 1 + ξ ( t ) w ( t ) dw ( t ) t � 0 , w ( 0 ) = z . (7) , dt 1 − ξ ( t ) w ( t ) Answer (the converse Loewner Theorem) Yes: for any continuous ξ : [ 0 , + ∞ ) → T relations (6) (7) define a function f ∈ S . But: f ∈ S ′ ? — NOT necessarily! [Kufarev 1947] 8/41 Classical Loewner Theory

Loewner’s construction 4 Universita’ di Roma TOR VERGATA Conclusion A dense subclass of S is represented by a linear space: � � f ∈ S 0 ⊃ S ′ Loewner [ 0 , + ∞ ) , R ξ ( t ) := e iu ( t ) C ∋ u �→ �− − − − − − − → equations Remark For any simply connected domain 0 ∈ B � C , a dense subclass U 0 B ⊃ U ′ B of � � f ∈ Hol ( D , B ) : f is univalent in D , f ( 0 ) = 0 , f ′ ( 0 ) > 1 U B := can be represented in a similar way. def f ∈ U ′ f ∈ U B , f ( D ) = B \ [a slit] . ⇐ = = = = ⇒ B 9/41 Classical Loewner Theory

Loewner’s construction 5 Universita’ di Roma TOR VERGATA Representation of U B A dense subclass U 0 B ⊂ U B is represented by the formula � � (8) f ( z ) = F w z , 0 ( T ) where: F : D onto → B conformally with F ( 0 ) = 0, F ′ ( 0 ) > 0; − − − ◮ � � T := log F ′ ( 0 ) / f ′ ( 0 ) ; ◮ w z , 0 is the solution to ◮ = − w ( t ) 1 + ξ ( t ) w ( t ) dw ( t ) t ∈ [ 0 , T ] , w ( 0 ) = z , (9) , dt 1 − ξ ( t ) w ( t ) and ξ : [ 0 , T ] → T is continuous. 10/41 Classical Loewner Theory

Chordal Loewner Equation 1 Universita’ di Roma TOR VERGATA Previously we considered the conformal mappings normalized at the internal point z = 0. For applications it is important to consider also normalization at a boundary point . H := { ζ : Im ζ > 0 } P . P . Kufarev, V. V. Sobolev, and L. V. Sporysheva, 1968, considered the following class � � f ∈ Hol ( H , H ) : f is univalent in H , and satisfies (10) R := . � � Hydrodynamic normalization: lim = 0 . (10) f ( z ) − z H ∋ z →∞ If H \ f ( H ) is bounded, then f extends meromorphically to O ( ∞ ) and the hydrodynamic normalization is equivalent to f ( z ) = z − ℓ ( f ) / z + c 2 / z 2 + c 3 / z 3 + . . . (11) Note that ℓ ( f ) � 0, with ℓ ( f ) = 0 f = id H . ⇐⇒ 11/41 Classical Loewner Theory

Chordal Loewner Equation 2 Universita’ di Roma TOR VERGATA f ( ζ ) = ζ − ℓ ( f ) + o ( 1 /ζ ) (12) ζ as H ∋ ζ → ∞ ; f t ( ζ ) = ζ − ℓ ( f ) − 2 t ζ + o ( 1 /ζ ) (13) as H ∋ ζ → ∞ . ℓ ( f t ) = 2 ( T − t ) , T := ℓ ( f ) / 2 12/41 Classical Loewner Theory

Chordal Loewner equation 3 Universita’ di Roma TOR VERGATA The analogue of classical Loewner ODE — aka radial Loewner equation = − w ( t ) 1 + ξ ( t ) w ( t ) dw ( t ) w ( 0 ) = z ∈ D , , dt 1 − ξ ( t ) w ( t ) in the case of the class R considered by Kufarev et al is Kufarev’s ODE — aka chordal Loewner equation 2 dw ( t ) w ( 0 ) = ζ ∈ H , = λ ( t ) − w ( t ) , dt where λ : [ 0 , T ] → R is a continuous function. 13/41 Classical Loewner Theory

General form of radial Loewner equation 1 Universita’ di Roma TOR VERGATA Pavel Parfen’evich Kufarev Christian Pommerenke Tomsk (1909 – 1968) (Copenhagen, 17 December 1933) 14/41 Classical Loewner Theory

General form of radial Loewner equation 2 Universita’ di Roma TOR VERGATA The radial Loewner equation can be thought as a special case of a more general equation. = − w ( t ) 1 + ξ ( t ) w ( t ) dw ( t ) dt 1 − ξ ( t ) w ( t ) � ������������ �� ������������ � � � p w ( t ) , t Note that: � � CHF1. p ( · , t ) ∈ Hol and Re p ( · , t ) > 0 for a.e. t � 0; D , C CHF2. p ( 0 , t ) = 1 for a.e. t � 0; CHF3. p ( z , · ) is measurable on [ 0 , + ∞ ) for all z ∈ D . Definition A function p : D × [ 0 , + ∞ ) → C is said to be a classical Herglotz function if it satisfies CHF1 – CHF3. 15/41 Classical Loewner Theory

General form of radial Loewner equation 3 Universita’ di Roma TOR VERGATA Loewner – Kufarev equation dw ( t ) � � t � 0 , w ( 0 ) = z ∈ D , (14) = − wp w ( t ) , t , dt where p is a classical Herglotz function, i.e. � � CHF1. p ( · , t ) ∈ Hol and Re p ( · , t ) > 0 for a.e. t � 0; D , C CHF2. p ( 0 , t ) = 1 for a.e. t � 0; CHF3. p ( z , · ) is measurable on [ 0 , + ∞ ) for all z ∈ D . � � f ∈ Hol ( D , C ) : f is univalent , f ( 0 ) = f ′ ( 0 ) − 1 = 0 S := . Generates the whole class S t → + ∞ e t w z , 0 ( t ) , f ( z ) = lim (15) z ∈ D . 16/41 Classical Loewner Theory

Applications to univalent functions Universita’ di Roma TOR VERGATA Here we mention some important applications of the classical Loewner Theory to the problems for univalent functions. The class S : + ∞ � f : D → C univalent holomorphic normalized by f ( z ) = z + a n z n . n = 2 This class is compact, so for any continuous map (16) J : S → R there exists J max := max f ∈S J ( f ) . Extremal Problem: is the problem to find J max and all the functions f ∗ ∈ S such that J ( f ∗ ) = J max ( extremal functions ). Coefficient functionals: J ( f ) := J ( a 2 , . . . , a n ) . 17/41 Some interesting results and applications