A functional-analytic proof of the conformal welding theorem

Transcription

1 A functional-analytic proof of the conformal welding theorem Eric Schippers 1 Wolfgang Staubach 2 1 Department of Mathematics University of Manitoba Winnipeg, Canada 2 Department of Mathematics Uppsala Universitet Uppsala, Sweden CMS Winter Meeting 2012 Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

2 Statement of the theorem Conformal welding theorem Definition A quasiconformal map φ : A B between open connected domains A and B in C is a homeomorphism such that 1 φ is absolutely continuous on almost every vertical and horizontal line in every closed rectangle [a, b] [c, d] A 2 for some fixed k < 1. f f k Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

3 Statement of the theorem Conformal welding theorem Definition A quasiconformal map φ : A B between open connected domains A and B in C is a homeomorphism such that 1 φ is absolutely continuous on almost every vertical and horizontal line in every closed rectangle [a, b] [c, d] A 2 for some fixed k < 1. Definition f f k A quasisymmetric map φ : S 1 S 1 is a homeomorphism which is the boundary values of some quasiconformal map H : D D. Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

4 Statement of the theorem Conformal welding theorem Theorem (Conformal welding theorem) Let φ : S 1 S 1 be a quasisymmetric map and let α > 0. There is a pair of maps f : D C and g : D C such that 1 f is one-to-one and holomorphic, and has a quasiconformal extension to C 2 g is one-to-one and holomorphic except for a simple pole at, and has a quasiconformal extension to C 3 f (0) = 0, g( ) = and g ( ) = α. 4 φ = g 1 f on S 1. Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

5 Statement of the theorem Conformal welding theorem Theorem (Conformal welding theorem) Let φ : S 1 S 1 be a quasisymmetric map and let α > 0. There is a pair of maps f : D C and g : D C such that 1 f is one-to-one and holomorphic, and has a quasiconformal extension to C 2 g is one-to-one and holomorphic except for a simple pole at, and has a quasiconformal extension to C 3 f (0) = 0, g( ) = and g ( ) = α. 4 φ = g 1 f on S 1. standard proof uses existence and uniqueness to solutions of the Beltrami equation. We will give another proof using symplectic geometry and Grunsky inequalities. Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

6 The function spaces and composition operator The function spaces H and H Let H denote the space of L 2 functions h on S 1 such that n ĥ(n) 2 <. n= Define h 2 = ĥ(0) 2 + n ĥ(n) 2. n= Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

7 The function spaces and composition operator The function spaces H and H Let H denote the space of L 2 functions h on S 1 such that n ĥ(n) 2 <. n= Define h 2 = ĥ(0) 2 + n ĥ(n) 2. n= We will also consider with norm H = {h H : ĥ(0) = 0} h 2 = n ĥ(n) 2. n= Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

8 The function spaces and composition operator Decomposition of H H + = {h H : h = H = {h H : h = h n e inθ } n=1 1 n= h n e inθ }. Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

9 The function spaces and composition operator Decomposition of H H + = {h H : h = H = {h H : h = h n e inθ } n=1 1 n= h n e inθ }. It is well-known that we have the following isometries H + = D(D) = {h : D C : h 2 da < h(0) = 0} D H = D(D ) = {h : D C : h 2 da < h( ) = 0} D Summarized in Nag and Sullivan. Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

10 The function spaces and composition operator Composition operators on H and H We consider two composition operators C φ : H H C φ h = h φ Ĉ φ : H H C φ h = h φ 1 h φ(e iθ ) dθ. 2π S 1 Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

11 The function spaces and composition operator Composition operators on H and H We consider two composition operators C φ : H H C φ h = h φ Ĉ φ : H H C φ h = h φ 1 h φ(e iθ ) dθ. 2π S 1 Theorem (Nag and Sullivan, quoting notes of Zinsmeister) Ĉ φ is bounded if φ is a quasisymmetry. Theorem (S and Staubach) If φ is a quasisymmetry then C φ is bounded. Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

12 Sketch of a new proof The proof Sketch of proof Treat φ as a composition operator C φ on H: we want to solve for unknown functions f and g in equation f φ 1 = g, g 1 = α. Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

13 Sketch of a new proof The proof Sketch of proof Treat φ as a composition operator C φ on H: we want to solve for unknown functions f and g in equation f φ 1 = g, g 1 = α. Using the decomposition H = [H + ] [C H ], the welding equation can be written ( ) ( ) ( ) M++ M C φ f = + f g+ = M + M 0 g so Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

14 Sketch of a new proof The proof Sketch of proof Treat φ as a composition operator C φ on H: we want to solve for unknown functions f and g in equation f φ 1 = g, g 1 = α. Using the decomposition H = [H + ] [C H ], the welding equation can be written ( ) ( ) ( ) M++ M C φ f = + f g+ = M + M 0 g so M ++ f = g + and M + f = g. where g + = g 1 z = αz and g = g 0 + g 1 /z + g 2 /z 2 +. Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

15 Sketch of a new proof The proof Sketch of proof Treat φ as a composition operator C φ on H: we want to solve for unknown functions f and g in equation f φ 1 = g, g 1 = α. Using the decomposition H = [H + ] [C H ], the welding equation can be written ( ) ( ) ( ) M++ M C φ f = + f g+ = M + M 0 g so M ++ f = g + and M + f = g. where g + = g 1 z = αz and g = g 0 + g 1 /z + g 2 /z 2 +. which leads to the solution f = M 1 ++ g + g = M + f. Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

16 The proof Sketch of proof What are the gaps? We need to show that M ++ is invertible: will use symplectic geometry and results of Nag and Sullivan, Takhtajan and Teo. The solutions in H so obtained have the desired properties: conformal with quasiconformal extensions: will use Grunsky inequalities. Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

17 The proof Sketch of proof What are the gaps? We need to show that M ++ is invertible: will use symplectic geometry and results of Nag and Sullivan, Takhtajan and Teo. The solutions in H so obtained have the desired properties: conformal with quasiconformal extensions: will use Grunsky inequalities. Here we go! Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

18 The proof Symplectic geometry of H Symplectic structure on H For f, g H let ω(f, g) = i f n g n. n= If one restricts to the real subspace (such ( that ˆf ( n) = ˆf (n)) this is a non-degenerate anti-symmetric form 2Im ˆf ) n=1 (n)ĝ(n). Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

19 The proof Symplectic geometry of H Symplectic structure on H For f, g H let ω(f, g) = i f n g n. n= If one restricts to the real subspace (such ( that ˆf ( n) = ˆf (n)) this is a non-degenerate anti-symmetric form 2Im ˆf ) n=1 (n)ĝ(n). Theorem (Nag and Sullivan) If φ : S 1 S 1 is quasisymmetric then Ĉφ is a symplectomorphism (that is, ω(ĉφf, Ĉφg) = ω(f, g)). Note that Ĉφ has the form ( A B B A ). Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

20 The proof Symplectic geometry of H The infinite Siegel disc (Nag and Sullivan) Definition The infinite Siegel disc S is the set of maps Z : H H + such that Z T = Z and I Z Z is positive definite. Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

21 The proof Symplectic geometry of H The infinite Siegel disc (Nag and Sullivan) Definition The infinite Siegel disc S is the set of maps Z : H H + such that Z T = Z and I Z Z is positive definite. Context: the graph of each Z is a Lagrangian subspaces of H symplectomorphisms Ĉφ act on them. Definition Let L be the set of bounded linear maps of the form (P, Q) : H H where P : H H + and Q : H H are bounded operators satisfying P T P Q T Q > 0 and Q T P = P T Q. Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

22 Two facts The proof Symplectic geometry of H Q invertible PQ 1 S (P, Q) L. (P, Q)Q 1 = (PQ 1, I) has the same image as (P, Q) Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

23 Invariance of L The proof Symplectic geometry of H L is invariant under bounded symplectomorphisms. Proposition If Ψ is a bounded symplectomorphism which preserves H R then ( ) P Ψ L. Q Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

24 Invertibility The proof Symplectic geometry of H Proposition If (P, Q) L then Q has a left inverse. Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

25 Invertibility The proof Symplectic geometry of H Proposition If (P, Q) L then Q has a left inverse. Proof. If Qv = 0 then by the positive-definiteness of Q T Q P T P ( ) 0 v T Q T Q P T P v = v T P T Pv = Pv 2. ( ) Thus Pv = 0. This implies that v T Q T Q P T P v = 0 so v = 0. Thus Q is injective, or equivalently Q has a left inverse. Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

26 Invertibility of A The proof Symplectic geometry of H Note that A = M ++, the matrix we needed to show was invertible. Theorem (S, Staubach) Let φ : S 1 S 1 be a quasisymmetry, with ( A B Ĉ φ 1 = B A Then A is invertible and Z = BA 1 S. ). Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

27 Invertibility of A The proof Symplectic geometry of H Note that A = M ++, the matrix we needed to show was invertible. Theorem (S, Staubach) Let φ : S 1 S 1 be a quasisymmetry, with ( A B Ĉ φ 1 = B A Then A is invertible and Z = BA 1 S. Note: This theorem was proven originally by Takhtajan and Teo. However their proof uses the conformal welding theorem, so we must provide a new one. ). Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

28 Proof The proof Symplectic geometry of H Proof: Invertibility of A: ( A B B A So A has a left inverse. ) ( 0 I ) = ( B A ) L. Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

29 Proof The proof Symplectic geometry of H Proof: Invertibility of A: ( A B B A ) ( 0 I ) = ( B A ) L. So A has a left inverse. Apply to φ 1 (also a quasisymmetry) ( Ĉ φ = A T B T B T A T ). So A T has a left inverse; thus A is a bounded bijection so it is invertible. Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

30 Proof continued The proof Symplectic geometry of H Let Z = BA 1. Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

31 Proof continued The proof Symplectic geometry of H Let Z = BA 1. Recall: (B, A) L BA 1 S. Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

32 The proof Definition of Grunsky matrix Grunsky matrices and proof Let g(z) = g 1 z + g 0 + g 1 z + g 2 z 2 +. The Grunsky matrix b mn of g is defined by log g(z) g(w) z w = b mn z m w n. m,n=1 Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

33 The proof Grunsky matrices and proof Grunsky matrix and welding maps Theorem (Takhtajan and Teo) Let f (z) = f 1 z + f 2 z 2 + D(D) and g = g 1 z + g where g D(D ), and let φ : S 1 S 1 be a quasisymmetry. Assume that g φ = f on S 1. Let ( ) ( ) A B A B Ĉ φ = and Ĉ B A φ 1 = B A (1) 1 If g 1 0, then the Grunsky matrix of g is BA 1. 2 If f 1 0, then the Grunsky matrix of f is BA 1. Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

34 The proof Grunsky matrices and proof Grunsky matrix and welding maps Theorem (Takhtajan and Teo) Let f (z) = f 1 z + f 2 z 2 + D(D) and g = g 1 z + g where g D(D ), and let φ : S 1 S 1 be a quasisymmetry. Assume that g φ = f on S 1. Let ( ) ( ) A B A B Ĉ φ = and Ĉ B A φ 1 = B A (1) 1 If g 1 0, then the Grunsky matrix of g is BA 1. 2 If f 1 0, then the Grunsky matrix of f is BA 1. Note: Their statement assumes that f and g are the maps in the conformal welding theorem. However their proof only uses the assumptions above and invertibility of g. Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

35 The proof Grunsky matrices and proof Recap: proof of conformal welding theorem Proof: (1) For a quasisymmetry φ. ( A B Ĉ φ 1 = B A ) A is invertible. Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

36 The proof Grunsky matrices and proof Recap: proof of conformal welding theorem Proof: (1) For a quasisymmetry φ. ( A B Ĉ φ 1 = B A ) A is invertible. (2) We may find f, g H such that f φ 1 = g using ( ) ( ) M++ M C φ 1f = + f = 0 M + M ( g+ g ) where g + = g 1 z = αz and g = g 0 + g 1 /z + g 2 /z 2 + which has the solution f = M 1 ++ g + g = M + f. Note that M ++ = A. Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

37 Proof continued The proof Grunsky matrices and proof (3) BA 1 is the Grunsky matrix of g under these assumptions, by the theorem of Takhtajan and Teo. Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

38 Proof continued The proof Grunsky matrices and proof (3) BA 1 is the Grunsky matrix of g under these assumptions, by the theorem of Takhtajan and Teo. (4) Z = BA 1 satsifies I Z Z is positive definite since Z S. Thus Z k < 1 some k. Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

39 Proof continued The proof Grunsky matrices and proof (3) BA 1 is the Grunsky matrix of g under these assumptions, by the theorem of Takhtajan and Teo. (4) Z = BA 1 satsifies I Z Z is positive definite since Z S. Thus Z k < 1 some k. (5) By a classical theorem of Pommerenke if Z k < 1 then g is univalent and quasiconformally extendible. A bit of work shows the same for f. Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21

40 References References 1 Lehto, Olli. Univalent functions and Teichmüller spaces. Graduate Texts in Mathematics 109 Springer-Verlag, New York, Nag, Subhashis; Sullivan, Dennis. Teichmüller theory and the universal period mapping via quantum calculus and the H 1/2 space on the circle. Osaka Journal of Mathematics 32 (1995), no. 1, Pommerenke, Christian Univalent functions. Vandenhoeck & Ruprecht, Göttingen (1975). 4 Tatkhajan, Leon A. and Teo, Lee-Peng. Weil-Petersson metric on the universal Teichmüller space. Memoirs of the American Mathematical Society 183 (2006), no Vaisman, Izu. Symplectic geometry and secondary characteristic classes. Progress in Mathematics, 72. Birkhäuser Boston, Inc., Boston, MA, Eric Schippers (Manitoba) Conformal welding theorem CMS Winter Meeting / 21