The Asymptotic Behaviour of the Riemann Mapping Function at Analytic - PowerPoint PPT Presentation

The Asymptotic Behaviour of the Riemann Mapping Function at Analytic Cusps Sabrina Lehner Faculty of Computer Science and Mathematics University of Passau 20. July 2015

Content 1. Motivation 2. Asymptotic Behaviour of the Riemann Mapping Function at Singular Boundary Points 2.1. Analytic Corners 2.2. Analytic Cusps 3. Conclusion

Content 1. Motivation 2. Asymptotic Behaviour of the Riemann Mapping Function at Singular Boundary Points 2.1. Analytic Corners 2.2. Analytic Cusps 3. Conclusion

Motivation Theorem (T. Kaiser 2009) Let Ω � C be bounded, simply connected, and semianalytic. Assume that the opening angle ∢ x is an irrational multiple of π for all singular boundary points x ∈ ∂ Ω . Then φ : Ω → E is definable in an o-minimal structure. x ∢ x Ω O-Minimality and Applications Sabrina Lehner 1 / 21

Motivation General Premises ◮ Let Ω � C be a simply connected domain with piecewise analytic boundary. ◮ Let 0 ∈ ∂ Ω be a singular boundary point. ◮ Let ϕ : Ω → H be a Riemann map with ϕ ( 0 ) = 0. O-Minimality and Applications Sabrina Lehner 2 / 21

Motivation General Premises ◮ Let Ω � C be a simply connected domain with piecewise analytic boundary. ◮ Let 0 ∈ ∂ Ω be a singular boundary point. ◮ Let ϕ : Ω → H be a Riemann map with ϕ ( 0 ) = 0. O-Minimality and Applications Sabrina Lehner 2 / 21

Motivation General Premises ◮ Let Ω � C be a simply connected domain with piecewise analytic boundary. ◮ Let 0 ∈ ∂ Ω be a singular boundary point. ◮ Let ϕ : Ω → H be a Riemann map with ϕ ( 0 ) = 0. O-Minimality and Applications Sabrina Lehner 2 / 21

Motivation Theorem (R. S. Lehman, 1957) Assume that the opening angle at 0 ∈ ∂ Ω is πα with 0 < α ≤ 2 . (a) If α / ∈ Q then ϕ has an asymptotic power series expansion at 0 of the following kind a k , l z k + l � α , k ≥ 0 , l ≥ 1 where a k , l ∈ C and a 0 , 1 � = 0 , i.e. a k , l z k + l � α = o ( z N ) ϕ ( z ) − k + l α ≤ N for all N ∈ N . O-Minimality and Applications Sabrina Lehner 3 / 21

Motivation (b) If α = p q , with p and q coprime, then ϕ has an asymptotic power series expansion at 0 of the following kind a k , l , m z k + l � α ( log ( z )) m k ≥ 0 , 1 ≤ l ≤ q , 0 ≤ m ≤ k p where a k , l , m ∈ C and a 0 , 1 , 0 � = 0 . O-Minimality and Applications Sabrina Lehner 4 / 21

Content 1. Motivation 2. Asymptotic Behaviour of the Riemann Mapping Function at Singular Boundary Points 2.1. Analytic Corners 2.2. Analytic Cusps 3. Conclusion

Content 1. Motivation 2. Asymptotic Behaviour of the Riemann Mapping Function at Singular Boundary Points 2.1. Analytic Corners 2.2. Analytic Cusps 3. Conclusion

Analytic Corners Definition (Analytic Corner) We say that Ω has an analytic corner at 0 if 0 is a singular boundary point and the boundary at 0 is locally given by two regular analytic arcs with opening angle πα where 0 < α ≤ 2 . 0 πα O-Minimality and Applications Sabrina Lehner 5 / 21

Analytic Corners Im Im ϕ Ω H Re Re O-Minimality and Applications Sabrina Lehner 6 / 21

Analytic Corners Theorem (L. Lichtenstein (1911), S. Warschawski (1955)) At an analytic corner at 0 with opening angle πα with 0 < α ≤ 2 we have at 0 on Ω 1 (a) ϕ ( z ) ∼ z α 1 α − 1 (b) ϕ ′ ( z ) ∼ z  1 α − n for α � = 1 ∼ z k , k ∈ N  (c) ϕ ( n ) ( z ) for n ≥ 2 1 α − n ) for α = 1 = O ( z k , k ∈ N  O-Minimality and Applications Sabrina Lehner 7 / 21

Contents 1. Motivation 2. Asymptotic Behaviour of the Riemann Mapping Function at Singular Boundary Points 2.1. Analytic Corners 2.2. Analytic Cusps 3. Conclusion

Analytic Cusps Definition (Analytic Cusp) We say that Ω has an analytic cusp at 0 if 0 is a singular boundary point and the boundary at 0 is locally given by two regular analytic arcs such that the opening angle vanishes. 0 O-Minimality and Applications Sabrina Lehner 8 / 21

Setting After applying a coordinate transformation we can assume that locally the boundary of Ω is given by the arcs Γ 1 and Γ 2 with the parameterisations γ 1 ( t ) = t and γ 2 ( t ) = t exp ( i ∢ Ω ( t )) , ∞ a k t k is a real power series with d ∈ N resp. Hereby, ∢ Ω ( t ) = � k = d and a d � = 0. O-Minimality and Applications Sabrina Lehner 9 / 21

Analytic Cusps Im ∢ Ω( t ) Ω Γ2 Re Γ1 t O-Minimality and Applications Sabrina Lehner 10 / 21

Analytic Cusps Im Im ϕ H Ω Re Re O-Minimality and Applications Sabrina Lehner 11 / 21

Analytic Cusps Theorem We have at 0 on Ω � d − 1 � b n z n − d + a log ( z ) � ϕ ( z ) ∼ exp n = 0 with π c n b n := n − d and a := π c d , where c k are the coefficients of the Laurent series ∞ 1 � ∢ Ω ( t ) = t − d c k t k . k = 0 O-Minimality and Applications Sabrina Lehner 12 / 21

Analytic Cusp Example Let � z ∈ C | 0 < | z | < 1 � 2 , 0 < arg ( z ) < | z | − | z | 2 Ω := then − π � � ϕ ( z ) ∼ exp z + π log ( z ) . O-Minimality and Applications Sabrina Lehner 13 / 21

Analytic Cusps Remark If a d + 1 = . . . = a 2 d = 0 we have � π � ϕ ( z ) ∼ exp − a d dz d at 0 on Ω . ∞ a k t k � Recall: ∢ Ω ( t ) = k = d O-Minimality and Applications Sabrina Lehner 14 / 21

Analytic Cusp Example Let � z ∈ C | 0 < | z | < 1 � 2 , 0 < arg ( z ) < a d | z | d Ω := then � π � ϕ ( z ) ∼ exp − . a d dz d O-Minimality and Applications Sabrina Lehner 15 / 21

Analytic Cusps Theorem We have for n ∈ N � d − 1 � b k z k − d + a log ( z ) ϕ ( n ) ( z ) ∼ exp � z − n ( d + 1 ) k = 0 at 0 on Ω . O-Minimality and Applications Sabrina Lehner 16 / 21

Analytic Cusps Inverse function ψ Im Im ψ H Ω Re Re O-Minimality and Applications Sabrina Lehner 17 / 21

Analytic Cusps Theorem Let ψ : H → Ω be a conformal map with ψ ( 0 ) = 0 . Then � 1 � π d ψ ( z ) ≃ − a d d log ( z ) at 0 on H . O-Minimality and Applications Sabrina Lehner 18 / 21

Analytic Cusps Theorem We have for n ∈ N � 1 d + 1 � 1 ψ ( n ) ( z ) ∼ z − n − log ( z ) at 0 on H . O-Minimality and Applications Sabrina Lehner 19 / 21

Content 1. Motivation 2. Asymptotic Behaviour of the Riemann Mapping Function at Singular Boundary Points 2.1. Analytic Corners 2.2. Analytic Cusps 3. Conclusion

Conclusion Contributions Asymptotic behaviour at analytic cusps of ◮ ϕ : Ω → H ◮ ϕ ( n ) ◮ ψ : H → Ω ◮ ψ ( n ) Open Questions ◮ Development of the mapping function in a generalized power series? ◮ O-minimality? O-Minimality and Applications Sabrina Lehner 20 / 21

Conclusion Contributions Asymptotic behaviour at analytic cusps of ◮ ϕ : Ω → H ◮ ϕ ( n ) ◮ ψ : H → Ω ◮ ψ ( n ) Open Questions ◮ Development of the mapping function in a generalized power series? ◮ O-minimality? O-Minimality and Applications Sabrina Lehner 20 / 21

Thank you! O-Minimality and Applications Sabrina Lehner 21 / 21