Formal Definition The Witt algebra is a Lie algebra of Killing (metric preserving) vector fields defined on C \ { 0 } The Witt basis is given by the holomorphic vector fields L n = − z n +1 ∂ ∂z, n ∈ Z . The Lie-Poisson bracket of two Killing fields is [ L m , L n ] = ( m − n ) L m + n . owner Chains, September 2008 – p.16/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Formal Definition The Virasoro Algebra is a central extension of the Witt algebra by C ( Witt ⊕ C ): owner Chains, September 2008 – p.17/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Formal Definition The Virasoro Algebra is a central extension of the Witt algebra by C ( Witt ⊕ C ): [ L m , L n ] V ir = ( m − n ) L m + n + c 12 n ( n 2 − 1) δ n, − m . owner Chains, September 2008 – p.17/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Formal Definition The Virasoro Algebra is a central extension of the Witt algebra by C ( Witt ⊕ C ): [ L m , L n ] V ir = ( m − n ) L m + n + c 12 n ( n 2 − 1) δ n, − m . The constant c is the central charge and it is a constant of the theory. owner Chains, September 2008 – p.17/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Conformal Field Theory In Conformal Field Theory Virasoro Algebra Vir appears as an infinite dimensional algebra generated by the coefficients of the Laurent expansion of the analytic component T of the momentum-energy tensor, Virasoro generators. owner Chains, September 2008 – p.18/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Conformal Field Theory In Conformal Field Theory Virasoro Algebra Vir appears as an infinite dimensional algebra generated by the coefficients of the Laurent expansion of the analytic component T of the momentum-energy tensor, Virasoro generators. The corresponding Virasoro-Bott group vir appears as the space of reparametrization of a closed string. owner Chains, September 2008 – p.18/54 Workshop on Holomorphic Iteration, Semigroups and L¨
KdV hierarchy Phase space (field variables) u ( e ix , t ) so that S 1 × R is our space-time. Simplifying u → u ( x, t ) , where the new u is a 2 π periodic smooth function. owner Chains, September 2008 – p.19/54 Workshop on Holomorphic Iteration, Semigroups and L¨
KdV hierarchy KdV equation u t = 6 uu x + u xxx has an infinite number of conserved quantities (first integrals) I k [ u ] , e.g., (1 � � � u 2 dx, I 1 = 2( u ′ 2 ) + u 3 ) dx, . . . I − 1 = udx, I 0 = � polynomial ( d . . . , I = dx , · u ) dx. which are all in involution ( I − 1 - mass, I 0 - energy). owner Chains, September 2008 – p.19/54 Workshop on Holomorphic Iteration, Semigroups and L¨
KdV hierarchy Hierarchy is constructed as u = [ u, I n ] ≡ d δI n ˙ δu . dx owner Chains, September 2008 – p.19/54 Workshop on Holomorphic Iteration, Semigroups and L¨
KdV hierarchy Hierarchy is constructed as u = [ u, I n ] ≡ d δI n ˙ δu . dx Lax reformulation L = − ∂ 2 + u, 2 n +1 ˙ 2 ) ≥ 0 . L = [ L, A n ] , A n = 4 i ( L owner Chains, September 2008 – p.19/54 Workshop on Holomorphic Iteration, Semigroups and L¨
KdV via Virasoro Virasoro generators [ L m , L n ] V ir = ( m − n ) L m + n + c 12 n ( n 2 − 1) δ n, − m . owner Chains, September 2008 – p.20/54 Workshop on Holomorphic Iteration, Semigroups and L¨
KdV via Virasoro Virasoro generators [ L m , L n ] V ir = ( m − n ) L m + n + c 12 n ( n 2 − 1) δ n, − m . Define u = 6 L n e − inx − 1 � c 4 n ∈ Z owner Chains, September 2008 – p.20/54 Workshop on Holomorphic Iteration, Semigroups and L¨
KdV via Virasoro Virasoro generators [ L m , L n ] V ir = ( m − n ) L m + n + c 12 n ( n 2 − 1) δ n, − m . Then, using δ ( x ) = 1 n ∈ Z e inx , we obtain � 2 π [ u ( x ) , u ( y )] = 6 π c ( − δ ′′′ ( x − y )+4 u ( x ) δ ′ ( x − y )+2 u ′ δ ( x − y )) . owner Chains, September 2008 – p.20/54 Workshop on Holomorphic Iteration, Semigroups and L¨
KdV via Virasoro Virasoro generators [ L m , L n ] V ir = ( m − n ) L m + n + c 12 n ( n 2 − 1) δ n, − m . Then, using δ ( x ) = 1 n ∈ Z e inx , we obtain � 2 π [ u ( x ) , u ( y )] = 6 π c ( − δ ′′′ ( x − y )+4 u ( x ) δ ′ ( x − y )+2 u ′ δ ( x − y )) . � 2 π Taking I 0 = 1 u 2 dx , we obtain 2 0 u = c 6 π [ u, I 0 ] = − u ′′′ + 6 uu ′ . ˙ owner Chains, September 2008 – p.20/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Realization on the Unit Circle The Lie-Fréchet group of the sense preserving diffeos Diff S 1 ; owner Chains, September 2008 – p.21/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Realization on the Unit Circle The Lie-Fréchet group of the sense preserving diffeos Diff S 1 ; The Lie algebra Vect S 1 of real vector fields φ ( θ ) d dθ ; owner Chains, September 2008 – p.21/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Realization on the Unit Circle The Lie-Fréchet group of the sense preserving diffeos Diff S 1 ; The Lie algebra Vect S 1 of real vector fields φ ( θ ) d dθ ; φ ( θ + 2 π ) = φ ( θ ) ; owner Chains, September 2008 – p.21/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Realization on the Unit Circle The Lie-Fréchet group of the sense preserving diffeos Diff S 1 ; The Lie algebra Vect S 1 of real vector fields φ ( θ ) d dθ ; φ ( θ + 2 π ) = φ ( θ ) ; The commutator [ φ 1 , φ 2 ] = φ 1 φ ′ 1 φ 2 . 2 − φ ′ owner Chains, September 2008 – p.21/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Some obstacles Finite dimension: Lie algebra-Lie group correspondence. Infinite dimension: Lie-Banach, Lie-Fréchet. The Lie algebra Vect S 1 can not be lifted to the Lie-Fréchet group Diff S 1 . What are coordinated on Vect S 1 ? owner Chains, September 2008 – p.22/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Kirillov’s construction Kirillov proposed to consider the homogeneous space Diff S 1 /S 1 , and the Lie algebra Vect S 1 /const = Vect 0 S 1 � 2 π (i.e., φdθ = 0 ). 0 owner Chains, September 2008 – p.23/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Complex Structure A complex structure for Vect 0 S 1 ∞ � φ ( θ ) = a n cos nθ + b n sin nθ, n =1 ∞ � J [ φ ]( θ ) = − a n sin nθ + b n cos nθ, n =1 Complexification Vect 0 S 1 ⊗ C = Vect + 0 S 1 ⊕ Vect − 0 S 1 ; projections: ∞ φ → v := 1 ( a n ∓ ib n ) e inθ ∈ Vect ± � 0 S 1 . 2( φ ∓ iJ [ φ ]) = n =1 owner Chains, September 2008 – p.24/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Gelfand-Fuchs cocycle Virasoro algebra is a unique (modulo isomorphisms) non-trivial central extension of Vect 0 S 1 ⊗ C ; owner Chains, September 2008 – p.25/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Gelfand-Fuchs cocycle Virasoro algebra is a unique (modulo isomorphisms) non-trivial central extension of Vect 0 S 1 ⊗ C ; dθ – Fourier basis; v n = − ie inθ d owner Chains, September 2008 – p.25/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Gelfand-Fuchs cocycle Virasoro algebra is a unique (modulo isomorphisms) non-trivial central extension of Vect 0 S 1 ⊗ C ; dθ – Fourier basis; v n = − ie inθ d [ v n , v m ] = ( m − n ) v n + m ; owner Chains, September 2008 – p.25/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Gelfand-Fuchs cocycle Virasoro algebra is a unique (modulo isomorphisms) non-trivial central extension of Vect 0 S 1 ⊗ C ; dθ – Fourier basis; v n = − ie inθ d [ v n , v m ] = ( m − n ) v n + m ; 12 n ( n 2 − 1) δ n, − m , c ∈ R ; Gelfand-Fuchs cocycle: ω ( v n , v m ) = c owner Chains, September 2008 – p.25/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Gelfand-Fuchs cocycle Virasoro algebra is a unique (modulo isomorphisms) non-trivial central extension of Vect 0 S 1 ⊗ C ; dθ – Fourier basis; v n = − ie inθ d [ v n , v m ] = ( m − n ) v n + m ; 12 n ( n 2 − 1) δ n, − m , c ∈ R ; Gelfand-Fuchs cocycle: ω ( v n , v m ) = c [ v n , v m ] V ir = ( m − n ) v n + m + ω ( v n , v m ) . owner Chains, September 2008 – p.25/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Conformal welding Realization Diff S 1 /S 1 : owner Chains, September 2008 – p.26/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Conformal welding Realization Diff S 1 /S 1 : η y z = f ( ζ ) = ζ + c 1 ζ 2 + . . . S 1 Γ ξ x 0 0 1 U Ω 1 z = g ( ζ ) = a 1 ζ + a 0 + a − 1 ζ + . . . owner Chains, September 2008 – p.26/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Conformal welding Realization Diff S 1 /S 1 : η y z = f ( ζ ) = ζ + c 1 ζ 2 + . . . S 1 Γ ξ x 0 0 1 U Ω 1 z = g ( ζ ) = a 1 ζ + a 0 + a − 1 ζ + . . . γ = f − 1 ◦ g | S 1 ∈ Diff S 1 /S 1 , f ∈ S ⇆ γ ∈ Diff S 1 /S 1 . We identify the space of smooth Jordan curves and Diff S 1 /S 1 . owner Chains, September 2008 – p.26/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Schaeffer and Spencer Variation � 2 δ φ f ( z ) = f 2 ( ζ ) � wf ′ ( w ) φ ( w ) dw � w ( f ( w ) − f ( z )) , 2 π f ( w ) S 1 owner Chains, September 2008 – p.27/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Schaeffer and Spencer Variation � 2 δ φ f ( z ) = f 2 ( ζ ) � wf ′ ( w ) φ ( w ) dw � w ( f ( w ) − f ( z )) , 2 π f ( w ) S 1 exponential map φ ∈ Vect S 1 → Diff S 1 /S 1 . owner Chains, September 2008 – p.27/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Schaeffer and Spencer Variation � 2 δ φ f ( z ) = f 2 ( ζ ) � wf ′ ( w ) φ ( w ) dw � w ( f ( w ) − f ( z )) , 2 π f ( w ) S 1 exponential map φ ∈ Vect S 1 → Diff S 1 /S 1 . δ φ transfers the complex structure J from Vect 0 S 1 to T S : J ( δ φ ) := δ J ( φ ) . owner Chains, September 2008 – p.27/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Schaeffer and Spencer Variation � 2 δ φ f ( z ) = f 2 ( ζ ) � wf ′ ( w ) φ ( w ) dw � w ( f ( w ) − f ( z )) , 2 π f ( w ) S 1 exponential map φ ∈ Vect S 1 → Diff S 1 /S 1 . δ φ transfers the complex structure J from Vect 0 S 1 to T S : J ( δ φ ) := δ J ( φ ) . Complexification T S = T S + ⊕ T S − : δ φ ∓ iJ ( δ φ ) ∈ T S ± ∞ � ( a n ∓ ib n ) e inθ . v = φ ∓ iJ ( φ ) = n =1 owner Chains, September 2008 – p.27/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Kirillov’s vector fields Observe that v = δ φ ∓ iJ ( δ φ ) = δ φ ∓ iJ ( φ ) . owner Chains, September 2008 – p.28/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Kirillov’s vector fields Observe that v = δ φ ∓ iJ ( δ φ ) = δ φ ∓ iJ ( φ ) . Taking v k = − iζ k , k = 1 , 2 , . . . for T S + , we obtain δ v k ( f ) = L k ( f )( ζ ) = ζ k +1 f ′ ( ζ ) . owner Chains, September 2008 – p.28/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Kirillov’s vector fields Observe that v = δ φ ∓ iJ ( δ φ ) = δ φ ∓ iJ ( φ ) . Taking v k = − iζ k , k = 1 , 2 , . . . for T S + , we obtain δ v k ( f ) = L k ( f )( ζ ) = ζ k +1 f ′ ( ζ ) . Taking v − k = − iζ − k , k = 1 , 2 , . . . for T S − , we obtain δ v − k ( f ) = L − k ( f )( ζ ) = very difficult expressions . owner Chains, September 2008 – p.28/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Kirillov’s vector fields Observe that v = δ φ ∓ iJ ( δ φ ) = δ φ ∓ iJ ( φ ) . Taking v k = − iζ k , k = 1 , 2 , . . . for T S + , we obtain δ v k ( f ) = L k ( f )( ζ ) = ζ k +1 f ′ ( ζ ) . Taking v − k = − iζ − k , k = 1 , 2 , . . . for T S − , we obtain δ v − k ( f ) = L − k ( f )( ζ ) = very difficult expressions . Commutators: [ L m , L n ] = ( n − m ) L m + n , [ L − n , L n ] = 2 nL 0 , where L 0 ( f )( z ) = zf ′ ( z ) − f ( z ) . L 0 corresponds to rotation. owner Chains, September 2008 – p.28/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Kirillov’s vector fields Observe that v = δ φ ∓ iJ ( δ φ ) = δ φ ∓ iJ ( φ ) . Taking v k = − iζ k , k = 1 , 2 , . . . for T S + , we obtain δ v k ( f ) = L k ( f )( ζ ) = ζ k +1 f ′ ( ζ ) . Taking v − k = − iζ − k , k = 1 , 2 , . . . for T S − , we obtain δ v − k ( f ) = L − k ( f )( ζ ) = very difficult expressions . It is easily seen from f ε ( z ) = e − iε f ( e iε z ) = f ( z ) + iε ( zf ′ ( z ) − f ( z )) + o ( ε ) . owner Chains, September 2008 – p.28/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Virasoro algebra Virasoro algebra (complex) is a central extension [ L m , L n ] V ir = ( m − n ) L m + n + c 12 n ( n 2 − 1) δ n, − m , c ∈ C . We concentrate our attention on T S + . In affine coordinates we get Kirillov’s operators: ∞ � L j = ∂ j + ( k + 1) c k ∂ j + k ∂ j = ∂/∂c j , k =1 j = 1 , 2 , . . . owner Chains, September 2008 – p.29/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Algebraic structure of L-K η y z = w ( ζ, t ) = e t ζ + . . . S 1 Ω( s ) ξ x 0 0 1 Ω( t ) U Ω( t ) ⊂ Ω( s ) as t < s . The Löwner-Kufarev equation w ( ζ, t ) = ζw ′ ( ζ, t ) p ( ζ, t ) , Re p ( ζ, t ) > 0 , ˙ | ζ | < 1 . owner Chains, September 2008 – p.30/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Curve in coefficient body f ( z, t ) = e − t w ( z, t ) = z (1 + � ∞ n =1 c n z n ) ; smooth curve: ( c 1 ( t ) , . . . , c n ( t ) , . . . ) ; tangent vector: ˙ c n ∂ n + . . . , ∂ n = ∂c n ; ∂ c 1 ∂ 1 + · · · + ˙ recalculation in a new basis { L 1 , . . . , L n , . . . } c 1 ∂ 1 + · · · + ˙ ˙ c n ∂ n + · · · = u 1 L 1 + . . . u n L n + . . . ; owner Chains, September 2008 – p.31/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Curve in coefficient body The Löwner-Kufarev equation w ( ζ, t ) = ζw ′ ( ζ, t ) p ( ζ, t ) , Re p ( ζ, t ) > 0 , ˙ | ζ | < 1 . f ( z, t ) = e − t w ( z, t ) = z (1 + � ∞ n =1 c n z n ) ; compare with the Löwner equation ˙ c n ∂ n + . . . ) f = zf ′ p ( z, t ) − f = , f = (˙ c 1 ∂ 1 + · · · + ˙ = ( L 0 + u 1 L 1 + . . . u n L n + . . . ) f, where p ( z, t ) = 1 + u 1 z + · · · + u n z n + . . . . ∞ ( k + 1) c k ∂ n + k , L n f ( z ) = z n +1 f ′ ( z ) . � L n = ∂ n + k =1 owner Chains, September 2008 – p.32/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Curve in coefficient body compare with the Löwner equation ˙ c n ∂ n + . . . ) f = zf ′ p ( z, t ) − f = , f = (˙ c 1 ∂ 1 + · · · + ˙ = ( L 0 + u 1 L 1 + . . . u n L n + . . . ) f, where p ( z, t ) = 1 + u 1 z + · · · + u n z n + . . . . ∞ ( k + 1) c k ∂ n + k , L n f ( z ) = z n +1 f ′ ( z ) . � L n = ∂ n + k =1 What is L 0 ? owner Chains, September 2008 – p.33/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Curve in coefficient body compare with the Löwner equation ˙ c n ∂ n + . . . ) f = zf ′ p ( z, t ) − f = , f = (˙ c 1 ∂ 1 + · · · + ˙ = ( L 0 + u 1 L 1 + . . . u n L n + . . . ) f, where p ( z, t ) = 1 + u 1 z + · · · + u n z n + . . . . ∞ ( k + 1) c k ∂ n + k , L n f ( z ) = z n +1 f ′ ( z ) . � L n = ∂ n + k =1 The answer is in the normalization f ( z, t ) = z (1 + � ∞ n =1 c n ( t ) z n ) owner Chains, September 2008 – p.34/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Dynamics The dynamics without normalization and subordination; f ( z, t ) = a 0 ( t ) z + a 1 ( t ) z 2 + . . . ; tangent vector: ˙ a n ∂ n + . . . ; a 0 ∂ 0 + · · · + ˙ recalculation in the new basis a 0 ∂ 0 + · · · + ˙ ˙ a n ∂ n + · · · = u 0 L 0 + . . . u n L n + . . . ; L n = a 0 ∂ n + 2 a 1 ∂ n +1 + . . . , ∂ n = ∂a n , n = 0 , 1 , . . . ; ∂ ∂ n f = z n +1 , L n ( f ) = z n +1 f ′ , ˙ c n ∂ n + · · · = zf ′ p ( z, t ) = u 0 L 0 + u 1 L 1 + . . . u n L n + . . . f = ˙ c 1 ∂ 1 + · · · + ˙ where p ( z, t ) = u 0 + u 1 z + · · · + u n z n + . . . . owner Chains, September 2008 – p.35/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Projections The dynamics is performed in the space of co-dimension 1: ∂ 0 L 0 ∂ n , L n ∂ 1 , L 1 We consider two projections: w.r.t. ∂ 0 and w.r.t. L 0 . owner Chains, September 2008 – p.36/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Analytic form of projections ∂ 0 a 0 z 2 + . . . a 0 f ( z, t ) = z + a 1 1 F 1 ( z, t ) = 1 p ( z, t ) − ˙ a 0 ˙ F 1 = zF ′ F 1 , a 0 where u 0 = ˙ a 0 a 0 . owner Chains, September 2008 – p.37/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Analytic form of projections ∂ 0 a 0 z 2 + . . . a 0 f ( z, t ) = z + a 1 1 F 1 ( z, t ) = 1 p ( z, t ) − ˙ a 0 ˙ F 1 = zF ′ F 1 , a 0 where u 0 = ˙ a 0 a 0 . c n ∂ n + · · · = ˆ L 0 + u 1 ˆ L 1 + · · · + u n ˆ c 1 ∂ 1 + · · · + ˙ ˙ L n + . . . where ˆ 1 − F 1 ) , ˆ 1 (in particular, L k F 1 = z k +1 F ′ L 0 F 1 = u 0 ( zF ′ a 0 = e t ⇒ Löwner-Kufarev), c k = a k a 0 , ∂ k = ∂c k . ∂ The Löwner PDE is an analytic form of the recalculation of the tangent vector from the basis ∂ n to the basis L n and the projection w.r.t. a 0 = e t . owner Chains, September 2008 – p.37/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Analytic form of projections L 0 0 z 2 + . . . F 2 ( z, t ) = f ( 1 a 0 z, t ) = z + a 1 a 2 2 p ( z , t ) − ˙ a 0 ˙ F 2 = zF ′ zF ′ 2 , a 0 a 0 where u 0 = ˙ a 0 . a 0 owner Chains, September 2008 – p.38/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Analytic form of projections L 0 0 z 2 + . . . F 2 ( z, t ) = f ( 1 a 0 z, t ) = z + a 1 a 2 2 p ( z , t ) − ˙ a 0 ˙ F 2 = zF ′ zF ′ 2 , a 0 a 0 where u 0 = ˙ a 0 . a 0 c n ∂ n + · · · = u 1 ˜ L 1 + · · · + u n ˜ c 1 ∂ 1 + · · · + ˙ ˙ L n + . . . where ˜ a k L k F 1 = z k +1 F ′ 1 , c k = , ∂ k = ∂c k . ∂ a k +1 0 owner Chains, September 2008 – p.38/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Commutators In all cases: L n ( f ) = z n +1 f ′ for n = 1 , 2 , 3 , . . . L n ( f ) = z n +1 f ′ for n = 0 , 1 , 2 , 3 , . . . L n ( f ) = z n +1 f ′ for n = 1 , 2 , 3 , . . . , L 0 = zf ′ − f , owner Chains, September 2008 – p.39/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Commutators In all cases: L n ( f ) = z n +1 f ′ for n = 1 , 2 , 3 , . . . L n ( f ) = z n +1 f ′ for n = 0 , 1 , 2 , 3 , . . . L n ( f ) = z n +1 f ′ for n = 1 , 2 , 3 , . . . , L 0 = zf ′ − f , TheWitt commutator relation is { L n , L m } = ( m − n ) L n + m . owner Chains, September 2008 – p.39/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Coefficient bodies Curves within the coefficient bodies M n = ( c 1 , c 2 , . . . , c n ) , f ( z, t ) = z (1 + � ∞ n =1 c n ( t ) z n ) , f ∈ S . For n = 1 , M 1 = {| c 1 | ≤ 2 } ; For n = 2 non-trivial A. C. Schaeffer, D. C. Spencer, Coefficient Regions for Schlicht Functions , American Mathematical Society Colloquium Publications, Vol. 35. American Mathematical Society, New York, 1950. owner Chains, September 2008 – p.40/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Body M 2 Crossections ( c 1 , Re c 2 , Im c 2 ) ( Re c 1 , Im c 1 , c 2 ) owner Chains, September 2008 – p.41/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Curves in M n The operators L j restricted onto M n give truncated vector fields n − j � L j = ∂ j + ( k + 1) c k ∂ j + k ; k =1 Let c ( t ) = be a smooth trajectory in M n ; � � c 1 ( t ) , . . . , c n ( t ) c ( t ) = ˙ ˙ c 1 ( t ) ∂ 1 + . . . + ˙ c n ( t ) ∂ n = u 1 L 1 + u 2 L 3 + . . . + u n L n . L j are differential 1-st order operators. The operator L = � | L k | 2 is elliptic. owner Chains, September 2008 – p.42/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Hamiltonian Turning to co-vectors L k → l k we write the Hamiltonian defined on the co-tangent bundle n c, ψ, ¯ � | l k | 2 , H ( c, ¯ ψ ) = k =1 where n − k l k = ¯ � ( j + 1) c j ¯ ψ k + ψ k + j . j =1 The co-vectors ¯ ψ k correspond to ∂ k owner Chains, September 2008 – p.43/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Hamiltonian system ∂ H = ¯ c 1 ˙ = l 1 ∂ ¯ ψ 1 . . . = . . . . . . . . . . . . n − 1 ∂ H = ¯ � ( j + 1) c j ¯ c n ˙ = l n + l n − j ∂ ¯ ψ n j =1 n − p − ∂ H ˙ ¯ � l k ¯ ψ p = = − ( p + 1) ψ k + p ∂ c p k =1 . . . = . . . . . . . . . . . . − ∂ H ˙ ¯ ψ n = = 0 . ∂ c n owner Chains, September 2008 – p.44/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Geodesics, Lagrangian c 1 ( t ) ∂ 1 + . . . + ˙ ˙ c n ( t ) ∂ n = u 1 L 1 + u 2 L 3 + . . . + u n L n Results: u k , k = 1 , 2 . . . , n ; l k = ¯ u k = � n − k ˙ j =1 ( j − k )¯ u j u j + k , k =1 | l k | 2 = const, along geodesics, H = � n c, ¯ � n Lagrangian L = (˙ ψ ) − H = 1 k =1 | u k | 2 , 2 turning to ∞ dimension and to the Löwner-Kufarev equation, H 2 . L = � p � 2 owner Chains, September 2008 – p.45/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Some conclusions The Löwner-Kufarev PDE can be considered as an algebraic recalculation of basis. From this point of view, the driving term p ( z, t ) does need to be of Re p > 0 . Alternate Löwner equation. One of the reasons: Brownian motion on Jordan curves. owner Chains, September 2008 – p.46/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Brownian motion on Jordan curves We consider the canonical Brownian motion on the group of diffeomorphisms of the unit circle and on the space of Jordan curves. owner Chains, September 2008 – p.47/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Brownian motion on Jordan curves The regularized canonical Brownian motion on Diff S 1 is a stochastic flow on S 1 associated to the Itô stochastic differential equation dg r x,t = dζ r x,t ( g r x,t ) , ∞ r n � ζ r √ x,t ( θ ) = n 3 − n ( x 2 n ( t ) cos nθ − x 2 n − 1 ( t ) sin nθ ) , n =1 where { x k } is a sequence of independent real-valued Brownian motions and r ∈ (0 , 1) and the series for ζ r x,t ( θ ) is a Gaussian trigonometric series. owner Chains, September 2008 – p.48/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Brownian motion on Jordan curves Kunita’s theory of stochastic flows asserts that the mapping x,t ( θ ) is a C ∞ diffeomorphism and the limit θ → g r x,t = g x,t exists uniformly in θ . The random r → 1 − g r lim homeomorphism g x,t is called canonical Brownian motion on Diff S 1 . (Airault, Fang, Malliavin, Ren, Zhang). ; owner Chains, September 2008 – p.48/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Brownian motion on Jordan curves The canonical Brownian motion given on Diff S 1 can be defined also on the space of C ∞ -smooth Jordan curves by conformal welding. No subordination, alternate behavior. owner Chains, September 2008 – p.48/54 Workshop on Holomorphic Iteration, Semigroups and L¨
L-K representation again Any univalent function f : U → Ω , f ( z ) = z + c 1 z 2 + . . . ( f ∈ S ) can be represented as a limit t →∞ e t w ( z, t ) . f ( z ) = lim owner Chains, September 2008 – p.49/54 Workshop on Holomorphic Iteration, Semigroups and L¨
L-K representation again Any univalent function f : U → Ω , f ( z ) = z + c 1 z 2 + . . . ( f ∈ S ) can be represented as a limit t →∞ e t w ( z, t ) . f ( z ) = lim The function ζ = w ( z, t ) , � � ∞ � w ( z, t ) = e − t z c n ( t ) z n 1 + , n =1 satisfies dw dt = − wp ( w, t ) , with the initial condition w ( z, 0) = z . owner Chains, September 2008 – p.49/54 Workshop on Holomorphic Iteration, Semigroups and L¨
L-K representation again Any univalent function f : U → Ω , f ( z ) = z + c 1 z 2 + . . . ( f ∈ S ) can be represented as a limit t →∞ e t w ( z, t ) . f ( z ) = lim If p ( z, t ) is analytic in z ∈ U and smooth in ˆ U = U ∪ S 1 , then w ( z, t ) analytic, univalent in z ∈ U and smooth in ˆ U = U ∪ S 1 . owner Chains, September 2008 – p.49/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Hamiltonian system for ODE The Hamiltonian system d ( e t w ( z, t )) = e t w (1 − p ( w, t )) = δH δψ = { H, ψ } . dt owner Chains, September 2008 – p.50/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Hamiltonian system for ODE The Hamiltonian system d ( e t w ( z, t )) = e t w (1 − p ( w, t )) = δH δψ = { H, ψ } . dt Hamiltonian is ∞ ψ ( z, t ) dz � e t w ( z, t )(1 − p ( w ( z, t ) , t )) ¯ � ψ n z n , H = iz , ψ ( z ) = 1 z ∈ S 1 ψ ( z, t ) is holomorphic in ˆ U , owner Chains, September 2008 – p.50/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Hamiltonian system for ODE The Hamiltonian system d ( e t w ( z, t )) = e t w (1 − p ( w, t )) = δH δψ = { H, ψ } . dt Hamiltonian is ∞ ψ ( z, t ) dz � e t w ( z, t )(1 − p ( w ( z, t ) , t )) ¯ � ψ n z n , H = iz , ψ ( z ) = 1 z ∈ S 1 ψ ( z, t ) is holomorphic in ˆ U , d ¯ ψ ψ = − δH dt = − (1 − p ( w, t ) − wp ′ ( w, t )) ¯ δ ( e t w ) = { H, e t w } . owner Chains, September 2008 – p.50/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Hamiltonian system for ODE The Hamiltonian system d ( e t w ( z, t )) = e t w (1 − p ( w, t )) = δH δψ = { H, ψ } . dt Hamiltonian is ∞ ψ ( z, t ) dz � e t w ( z, t )(1 − p ( w ( z, t ) , t )) ¯ � ψ n z n , H = iz , ψ ( z ) = 1 z ∈ S 1 ψ ( z, t ) is holomorphic in ˆ U , The conservative quantity is L ( z ) = e t w ′ ( z, t ) ¯ ψ ( z, t ) . All equations are given on the unit circle | z | = 1 . owner Chains, September 2008 – p.50/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Conservative Quantities Considering L ( z ) < 0 = [ e t w ′ ( z, t ) ¯ ψ ( z, t )] < 0 we get the Kirillov’s fields ∞ � L j = ∂ j + ( k + 1) c k ∂ j + k , k =1 where ¯ ψ k is the co-vector for ∂ k . owner Chains, September 2008 – p.51/54 Workshop on Holomorphic Iteration, Semigroups and L¨
Conservative Quantities Considering L ( z ) < 0 = [ e t w ′ ( z, t ) ¯ ψ ( z, t )] < 0 we get the Kirillov’s fields ∞ � L j = ∂ j + ( k + 1) c k ∂ j + k , k =1 where ¯ ψ k is the co-vector for ∂ k . The operators L n are defined on the co-tangent space to the class ˜ S (smooth on S 1 ). owner Chains, September 2008 – p.51/54 Workshop on Holomorphic Iteration, Semigroups and L¨
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