Loewner Chains Tiziano Casavecchia Mathematics Department L. - PowerPoint PPT Presentation

Loewner Chains Loewner Chains Tiziano Casavecchia Mathematics Department L. Tonelli of Pisa University-IMUS 25-May-2010

Loewner Chains outline 1 Classical Loewner Chains 2 Chordal Loewner Equation 3 Modern General Setting 4 The Problem

Loewner Chains Classical Loewner Chains Classical Loewner Chains in unit disc In 1923 Charles Loewner [4] developed a tool to embed a univalent map from the unit disc to a slit-domain in a ‘continuous’ family of univalent maps. He proved that such a family of univalent maps f t : D �→ C satisfied the following partial differential equation ∂ f t ∂ t = z k ( t ) + z ∂ f t ∂ z , k ( t ) − z where k : [0 , + ∞ ) �→ ∂ D is a continuous map; the map k ( t ) encodes information about the slit.

Loewner Chains Classical Loewner Chains Applications of Classical Loewner Theory Applications of Loewner Theory were: extremal problems for univalent maps; univalence criteria for holomorphic maps; geometric function theoretic properties (spirallike and starlike maps); estimates for coefficients of univalent maps. It was one of the tool of De Branges proof of Bieberach Conjecture.

Loewner Chains Classical Loewner Chains Pommerenke setting for Loewner Theory Definition [5] Let f : D × [0 , + ∞ ) �→ C be a map holomorphic in D . Then f t is a Loewner Chain if f t is univalent; for each t ≥ s ≥ 0 we have f s ( D ) � f t ( D ); t (0) = e t ; f t (0) = 0 and f ′

Loewner Chains Classical Loewner Chains Characterization of Loewner Chains Theorem [5, Theorem 6.3] Let f : D (0 , R ) × [0 , + ∞ ) �→ C , with R < 1 , be a map holomorphic in D (0 , R ) and absolutely continuous in [0 , + ∞ ) . Then f t extends to a Loewner Chain in D if and only if there exists a K := K ( R ) such that | f t ( z ) | ≤ e t K, there exists p : D × [0 , + ∞ ) �→ H , holomorphic in D , measurable in [0 , + ∞ ) , with p (0 , t ) = 1 and such that ∂ f t ∂ t = zp ( z , t ) ∂ f t ∂ z .

Loewner Chains Chordal Loewner Equation Chordal Loewner Equation Starting from the forties several Russian mathematicians studied evolution equations quite similar to classical Loewmer Chains. Basically they are a family of maps f t : H �→ C such that f t is a holomorphic univalent map, absolutely continuous in t ; f t satisfies the following partial equation ∂ f t 2 ∂ f t ∂ t = ∂ w , w − ik ( t ) where k : [0 , + ∞ ) �→ R is piece-wise continuous.

Loewner Chains Chordal Loewner Equation usage of Chordal Loewner equation was in extremal problems for univalent maps; fluido-dynamics; Schramm [6] introduced a stochastic Loewner equation. This tool was used to obtain several results in statistical mechanics and to solve the Mandelbrot Conjecture in 2000. Werner working with Schramm won the Fields Medal.

Loewner Chains Modern General Setting Modern general Definition Quite recently in [1–3], a new framework was developed to study Loewner chains. Definition A family ( f t ) 0 ≤ t ≤ + ∞ of holomorphic maps of the unit disc D will be called a Loewner Chain if each map f t : D �→ C is univalent; f s ( D ) ⊆ f t ( D ) for 0 ≤ s ≤ t ; for any compact K ⊆ D and any T ≥ 0 there exists a positive constant c K , T such that | f s ( z ) − f t ( z ) | ≤ c K , T | s − t | .

Loewner Chains Modern General Setting Loewner Chains and Differential Equations Theorem Let ( f t ) be a Loewner chain. Then there exists a set N ⊆ (0 , + ∞ ) of zero Lebesgue measure, such that for every s ∈ (0 , + ∞ ) \ N the map z ∈ D �→ ∂ f s ∂ s ( s ) is well defined and N is independent of z; there exists a function p : D × [0 , + ∞ ) → H such that p is holomorphic in D , furthermore p is in L ∞ loc ((0 , + ∞ ) , C ) , and a measurable function τ : [0 , + ∞ ) → D such that ∂ f t ∂ t = − ( z − τ ( t ))( τ ( t ) z − 1) p ( z , t ) ∂ f t ∂ z .

Loewner Chains Modern General Setting When τ = 0, we get the classical Loewner Chains; when τ = 1 we get the chordal Loewner Chains. Also notice that for bpth p and τ time-independent, we recover the theory of one-parameter semigroups in the unit disc. the map τ is essentially unique (that is up to a null set); the Theorem can be inverted taking care of adding conditions in order to ensure that the Loewner chain is unique; the unions of images of a Loewner Chains can be the whole plane C (in which case it is unique) or, up to biholomorphic maps, a disc. when the maps f − 1 ◦ f s share a common Denjoy-Wolff point, t τ is that point and viceversa.

Loewner Chains The Problem Generalize Pommerenke Theorem The problem I was concerned with was to generalize Pommerenke Theorem to the case where τ is on the boundary. the natural conjecture in this case is that f t is ‘controlled’ by the angular derivative at τ at least on horocycles centered at τ ; my idea was first to generalize it to the case when τ ∈ D \ { 0 } . then or using a technique that allows to cover horocycles with pseudo-hyperbolic discs or a time-dependent family of automorphisms that conjugate the given Loewner chain to a classical one.

Loewner Chains The Problem general inner case Theorem Let ( f t ) 0 ≤ t ≤ + ∞ be a family of holomorphic maps defined on the pseudo-hyperbolic disc D ( τ, R ) , with f t ( τ ) = τ ∈ D and R < 1 . Then f t extends to a Loewner Chain in D if and only if f t is absolutely continuous with respect to t; there exists p : D × [0 , + ∞ ) → H , holomorphic in D , such that p ( · , t ) ∈ L ∞ loc ((0 , + ∞ ) , C ) and f t satisfies ∂ f t τ z − 1) p ( z , t ) ∂ f t ∂ t = − ( z − τ )(¯ ∂ z ; there exists a constant K := K ( R ) such that | f t ( z ) − τ | ≤ (1 − | τ | 2 ) exp λ ( t ) K, where � t 0 (1 − | τ | 2 ) p ( τ, ξ ) d ξ , and lim t → + ∞ ℜ λ ( t ) = + ∞ . λ ( t ) :=

Loewner Chains The Problem difficulties After some (weeks of) work I realized that the situation is more involved than I thought. Consider the following examples. � 1 + z � f t ( z ) = e t 1 − z − t ; f t ( z ) = 1 + z 1 − z − t . One of the problems is that the map � t λ ( t ) := 0 ∠ lim z → τ (1 − ¯ τ z ) p ( z , ξ ) d ξ , which is the opposite of the logarithm of the angular derivative, does not control the dynamics of f t neither the geometry of the domains f t ( D ).

Loewner Chains The Problem Thank’s for attention

Loewner Chains The Problem [1] F. Bracci, M. D. Contreras, and S. D´ ıaz-Madrigal, Evolution families and the Loewner equation I: the unit disc , J. Reine Angew. Math. (2008). to appear. [2] , Evolution families and the Loewner equation II: complex hyperbolic manifolds , Math. Ann 344 (2009), 947–962. [3] M. D. Contreras, S. D´ ıaz-Madrigal, and P. Gumenyuk, Loewner chains in the unit disc , Rev. Mat. Iberoamericana (2008). to appear. [4] Loewner K. Untersuchungen uber schlichte konforme abbildungen des einheitskreises , Math. Ann. 89 (1923), 103–21. [5] C. Pommerenke, Univalent functions , Vandenhoeck and Ruprecht, 1975. G¨ ottingen. [6] O. Schramm, Scaling limits of loop-erased random walks and uniform spanning trees , Israel J. Math 118 (2000), 221–288.