Rolling CR-manifolds Setsuo TANIGUCHI Faculty of Arts and Science, - PDF document

Rolling CR-manifolds Setsuo TANIGUCHI Faculty of Arts and Science, Kyushu University November 9, 2015 S. Taniguchi (Kyushu Univ) Rolling CR-manifolds November 9, 2015 1 / 16 Introduction Introcuction (The Eells-Elworthy-Malliavin Approach) To construct Bm { X x t } t ≧ 0 on an n -dim Rie mfd M starting at x ∈ M ; Rolling M along the Brownian motion { B t } t ≥ 0 on R n ; ( O ( M ) , π ) : Orthon frame bndle / M { L 1 , . . . , L n } : fundamental v files on O ( M ) n � { r r t } t ≧ 0 : dr t = L α ( r t ) ◦ dB α t , r 0 = r ∈ O ( M ) α = 1 X x t = π ( r r t ) ( π ( r ) = x ) ⇒ Bm on M Do the same thing on a strictly pseudoconvex CR-mfd CR-manifold 1 CR-Brownian motion 2 Heat kernel 3 Dirichlet problem 4 Shot time asymptotics 5 Joint work with Hiroki Kondo S. Taniguchi (Kyushu Univ) Rolling CR-manifolds November 9, 2015 2 / 16

CR-manifold Strongly pseudoconvex CR-mfd M is said to be a (2 n + 1) -dim CR-mfd iff real (2 n + 1) -dimensional oriented C ∞ -mfd 1 ∃ complex n -dim subbundle T 1 , 0 of the complexified tangent 2 bundle C TM s.t. T 1 , 0 ∩ T 0 , 1 = { 0 } , where T 0 , 1 = T 1 , 0 Frobenius cond is fulfilled: [ T 1 , 0 , T 1 , 0 ] ⊂ T 1 , 0 3 ∃ 1-form θ � 0 on M s.t. θ ( H ) = { 0 } ( H : = Re( T 1 , 0 ⊕ T 0 , 1 ) ) The Levi form L θ is defined by L θ ( Z , W ) = − i d θ ( Z , W ) ( Z , W ∈ Γ ∞ ( T 1 , 0 ⊕ T 0 , 1 )) , where Γ ∞ ( T 1 , 0 ) is the totality of C ∞ -sections to T 1 , 0 . M is strongly pseudoconvex iff the Levi form is strictly positive. (Assumed here after) S. Taniguchi (Kyushu Univ) Rolling CR-manifolds November 9, 2015 3 / 16 CR-manifold Examples ⋄ Real submfd of complex mfd N : complex mfd of (complex) dim n + 1 M : real 2 n + 1 -dim submfd of N T 1 , 0 = T 1 , 0 N ∩ C TM , where T 1 , 0 N is the hol tangent bdl/N ⋄ H n = C n × R : Heisenberg gr ( z , t ) · ( w , s ) = ( z + w , s + t + 2 Im � z , w � ) α ∂ Z α = ∂ z α + i z ∂ ∂ t T 1 , 0 = span C { Z 1 , . . . , Z n } n � � z α dz α − z α dz α � θ = dt + i α = 1 n � α α = 1 dz α ∧ dz d θ = 2i L θ ( Z α , Z β ) = δ αβ S. Taniguchi (Kyushu Univ) Rolling CR-manifolds November 9, 2015 4 / 16

CR-manifold Kohn-Rossi Laplacian ψ : = θ ∧ ( d θ ) n ; vol form θ : = the dual on H ∗ of L θ on H = Re ( T 1 , 0 ⊕ T 0 , 1 ) . L ∗ � � u , v � θ = uv ψ , � M L ∗ θ ( ω, η ) ψ ( u , v ∈ C ∞ 0 ( M ) , ω, η ∈ Γ ( H ∗ ) ). � ω, η � θ = M d b : = r 0 ◦ d , ∂ b : = r 1 ◦ d , where r 0 : T ∗ M → H ∗ , r 1 : T ∗ M → T ∗ 0 , 1 : projections ∗ ∂ b : Sublaplacian ∆ b : = d b ∗ d b , K-R Laplacian � b : = ∂ b � ∆ b u , v � θ = � d b u , d b v � θ , � � b u , v � θ = � ∂ b u , ∂ b v � θ . ∃ 1 T : v field transversal to H with T ⌋ d θ = 0 and T ⌋ θ = 1 . Then � b = ∆ b + i nT . S. Taniguchi (Kyushu Univ) Rolling CR-manifolds November 9, 2015 5 / 16 CR-manifold Tanaka-Webster connection J : H → H ; the complex structure; J = i on T 1 , 0 , = − i on T 0 , 1 g θ : the Webster metric; for X , Y ∈ H g θ ( X , Y ) = d θ ( X , JY ) , g θ ( X , T ) = 0 , & g θ ( T , T ) = 1 Tanaka-Webster connection: ∃ ! lin conn ∇ ∇ X ( Γ ∞ ( H )) ⊂ Γ ∞ ( H ) ( ∀ X ∈ Γ ∞ ( TM ) ), ∇ J = 0 , ∇ g θ = 0 T ∇ ( Z , W ) = 0 , T ∇ ( Z , V ) = 2i L θ ( Z , V ) T for Z , W ∈ Γ ∞ ( T 1 , 0 ) , V ∈ Γ ∞ ( T 0 , 1 ) where T ∇ ( Z , W ) = ∇ Z W − ∇ W Z − [ Z , W ] T ∇ ( T , JX ) + J ( T ∇ ( T , X )) = 0 for X ∈ Γ ∞ ( TM ) � n � : = { 1 , . . . , n } & � � n � � : = { 0 , 1 , . . . , n , 1 , . . . , n } , where α ; Z α : = Z α For a local orthon frame { Z α } α ∈� n � ( Z α ∈ Γ ∞ ( U ; T 1 , 0 ) ), ∇ Z A Z B = � � Γ C AB Z C , A , B ∈ � � n � ( Z 0 = T ) � C ∈� � n � Γ C AB = 0 if ( B , C ) � { ( β, γ ) , ( β, γ ) | β, γ ∈ � n �} S. Taniguchi (Kyushu Univ) Rolling CR-manifolds November 9, 2015 6 / 16

CR-manifold Fundamental vector fields p : [ a , b ] → T 1 , 0 is parallel along p : [ a , b ] → M � p ( t ) ∈ ( T 1 , 0 ) p ( t ) and ∇ ˙ p = 0 iff � p � U ( T 1 , 0 ) : = ∐ x ∈ M { r : C n → ( T 1 , 0 ) x : isometric } , π ( r ) = x U ( n ) -principal bundle p : [ a , b ] → U ( T 1 , 0 ) is a horizontal lift of p : [ a , b ] → M � p ( t ) ξ is pararel along p ( ∀ ξ ∈ C n ) iff π ( � p ) = p , � For v ∈ T x M , η ∈ T r ( U ( T 1 , 0 )) ( π ( r ) = x ) is a holizontal lift of v p (0) = r , ˙ p , a holizontal lift of p s.t. � p (0) = η, π ∗ η = v . if ∃ � � For v ∈ T x M and r ∈ U ( T 1 , 0 ) with π ( r ) = x , ∃ ! η r ( v ) ∈ T r ( U ( T 1 , 0 )) ; holizontal lift of v ( L α ) r : = η r ( re α ) , α ∈ � n � ; v fields on U ( T 1 , 0 ) where { e α } α ∈� n � is the standard basis of C n S. Taniguchi (Kyushu Univ) Rolling CR-manifolds November 9, 2015 7 / 16 CR-manifold Local expression For a local orhon frame { Z α } α ∈� n � of T 1 , 0 , α ( r ) } ∈ C n × n by r ( e α ) = � set { e β β ∈� n � e β α ( r )( Z β ) π ( r ) . � � � ∂ ∂ e β Γ γ ε e β Γ γ ε e β L α = α Z β − βδ e δ βδ e δ − α α ∂ e γ ∂ e γ β ∈� n � β,γ,δ,ε ∈� n � β,γ,δ,ε ∈� n � ε ε S. Taniguchi (Kyushu Univ) Rolling CR-manifolds November 9, 2015 8 / 16

CR-Brownian motion CR-Brownian motion { B t = ( B 1 t , . . . , B n t ) } t ≧ 0 : C n -valud continuous martingale with � B α , B β � t = 0 & � B α , B β � t = δ αβ t . { r r t } t ≧ 0 : the unique sol to the SDE on U ( T 1 , 0 ) : n � dr t = α = 1 { L α ( r t ) ◦ dB α t + L α ( r t ) ◦ dB α r 0 = r ∈ U ( T 1 , 0 ) t } , Q r :=the distribution of { r r t } t ≧ 0 on C ([0 , ∞ ); U ( T 1 , 0 )) . P x : = Q r ◦ π − 1 ( r ∈ π − 1 ( x ) ) ( π : U ( T 1 , 0 ) → M : proj) Rem: Q r ◦ π − 1 = Q r ′ ◦ π − 1 if π ( r ) = π ( r ′ ) , since r ( t , ur , uB ) = r ( t , r , B ) ( u ∈ U ( n ) ). Let X t : C ([0 , ∞ ); M ) → M be the coord pr. { ( { X t } t ≧ 0 , P x ) , x ∈ M } is the diffusion generated by − 1 2 ∆ b (CR-Brownian motion) � L α L α + L α L α �� n � � ∵ − 1 2 ∆ b = 1 � � C ∞ ( M ) 2 α = 1 S. Taniguchi (Kyushu Univ) Rolling CR-manifolds November 9, 2015 9 / 16 Heat kernel Partial hyperellipticity Let P , N be C ∞ -mfds, φ : P → N be C ∞ , A 0 , . . . , A n be C ∞ -v fields on P . X t be the sol to the SDE on P : n � dX t = α = 1 A α ( X t ) ◦ dB α t + A 0 ( X t ) dt Y t : = φ ( X t ) Theorem 0 (T83) ( φ ∗ ) p ( L p ) = T φ ( p ) N ( ∀ p ∈ P ), Assume that where � � � � � [ A i 1 , . . . , [ A i k , A j ] . . . ] � 1 ≤ j ≤ n , 0 ≤ i ℓ ≤ n , k ∈ Z ≥ 0 . L = Then Y t admits a C ∞ -density function. P = O ( M ) , N = M ⇒ ∃ heat kernel for 1 2 ∆ M S. Taniguchi (Kyushu Univ) Rolling CR-manifolds November 9, 2015 10 / 16

Heat kernel Heat kernel Coming back to the CR-mfd; Let { Z α } be a local orthonormal frame for T 1 , 0 . Then n � e β ( π ∗ ) L α = α Z β . β = 1 ( π ∗ )[ L α , L α ] = − i T mod { Z β , Z β ; β = 1 , . . . , n } . Hence � ( π ∗ ) r Re L α , ( π ∗ ) r Im L α , ( π ∗ ) r [ Re L α , Im L α ] : 1 ≤ α ≤ n � span R = T π ( r ) M ( ∀ r ∈ U ( T 1 , 0 ) ). Under a suitable non-explosion assumption (assumed hereafter), by Theorem 0, Theorem 1 ∃ p ∈ C ∞ ((0 , ∞ ) × M × M ) s.t. P x ( X t ∈ dy ) = p ( t , x , y ) ψ ( dy ) . S. Taniguchi (Kyushu Univ) Rolling CR-manifolds November 9, 2015 11 / 16 Dirichlet problem Dirichlet problem Let G be a rel cpt conn open set in M with C 3 -bdry τ ′ = inf { t ≥ 0 : X t � G } . For f ∈ C ( ∂ G ) , define u f ( x ) : = E x [ f ( X τ ′ )] . Theorem 2 (Probabilistically) ⋄ u f ∈ C ( G ) ⋄ � u f , ∆ b v � θ = 0 ( ∀ v ∈ C 0 ( G ) ) & u f | ∂ G = f . Under local orthon frame { Z α } , n � α = 1 { Z 2 α + Z 2 α } + b , − ∆ b = [ Re Z α , Im Z α ] = 1 2 T mod { Re Z β , Im Z β ; 1 ≤ β ≤ n } Corollary 3 u f ∈ C ∞ ( G ) ; u f is a classical sol to the Diriclet problem for ∆ b S. Taniguchi (Kyushu Univ) Rolling CR-manifolds November 9, 2015 12 / 16