Introduction Real analytic case C 2 -smooth case Lempert Theorem for C 2 -smooth strongly linearly convex domains Tomasz Warszawski Kraków-Vienna Workshop on Pluripotential Theory and Several Complex Variables 3–7 IX 2012
Introduction Real analytic case C 2 -smooth case This presentation is based on Ł. Kosiński, T. Warszawski, Lempert Theorem for strongly linearly convex domains , to appear in Ann. Pol. Math. and L. Lempert, Intrinsic distances and holomorphic retracts , in Complex analysis and applications ’81 ( Varna, 1981 ) , 341–364, Publ. House Bulgar. Acad. Sci., Sofia, 1984.
Introduction Real analytic case C 2 -smooth case Definition (Strong linear convexity) A domain D ⊂ C n is strongly linearly convex if D has C 2 -smooth boundary; there exists a defining function r of D such that � � � � n n � � ∂ 2 r � ∂ 2 r � � � ( a ) X j X k > ( a ) X j X k , � � ∂ z j ∂ z k ∂ z j ∂ z k � � j , k = 1 j , k = 1 a ∈ ∂ D , X ∈ T C D ( a ) ∗ .
Introduction Real analytic case C 2 -smooth case Lempert function � k D ( z , w ) = inf { p ( ζ, ξ ) : ζ, ξ ∈ D and ∃ f ∈ O ( D , D ) : f ( ζ ) = z , f ( ξ ) = w } , z , w ∈ D .
Introduction Real analytic case C 2 -smooth case Lempert function � k D ( z , w ) = inf { p ( ζ, ξ ) : ζ, ξ ∈ D and ∃ f ∈ O ( D , D ) : f ( ζ ) = z , f ( ξ ) = w } , z , w ∈ D . Kobayashi-Royden (pseudo)metric κ D ( z ; v ) = inf {| λ | − 1 / ( 1 − | ζ | 2 ) : λ ∈ C ∗ , ζ ∈ D and z ∈ D , v ∈ C n . ∃ f ∈ O ( D , D ) : f ( ζ ) = z , f ′ ( ζ ) = λ v } ,
Introduction Real analytic case C 2 -smooth case Lempert function � k D ( z , w ) = inf { p ( ζ, ξ ) : ζ, ξ ∈ D and ∃ f ∈ O ( D , D ) : f ( ζ ) = z , f ( ξ ) = w } , z , w ∈ D . Kobayashi-Royden (pseudo)metric κ D ( z ; v ) = inf {| λ | − 1 / ( 1 − | ζ | 2 ) : λ ∈ C ∗ , ζ ∈ D and z ∈ D , v ∈ C n . ∃ f ∈ O ( D , D ) : f ( ζ ) = z , f ′ ( ζ ) = λ v } , If z � = w (resp. v � = 0), a mapping for which the infimum is attained we call an extremal ( � k D -extremal or κ D -extremal)
Introduction Real analytic case C 2 -smooth case Carath´ eodory (pseudo)distance c D ( z , w ) = sup { p ( F ( z ) , F ( w )) : F ∈ O ( D , D ) } , z , w ∈ D .
Introduction Real analytic case C 2 -smooth case Carath´ eodory (pseudo)distance c D ( z , w ) = sup { p ( F ( z ) , F ( w )) : F ∈ O ( D , D ) } , z , w ∈ D . Carath´ eodory-Reiffen (pseudo)metric γ D ( z ; v ) = sup {| F ′ ( z ) v | : F ∈ O ( D , D ) , F ( z ) = 0 } , z ∈ D , v ∈ C n .
Introduction Real analytic case C 2 -smooth case Carath´ eodory (pseudo)distance c D ( z , w ) = sup { p ( F ( z ) , F ( w )) : F ∈ O ( D , D ) } , z , w ∈ D . Carath´ eodory-Reiffen (pseudo)metric γ D ( z ; v ) = sup {| F ′ ( z ) v | : F ∈ O ( D , D ) , F ( z ) = 0 } , z ∈ D , v ∈ C n . Theorem (Lempert Theorem for C 2 , Ł. Kosiński, T.W.) Let D ⊂ C n , n ≥ 2 , be a bounded strongly linearly convex domain. Then c D = � k D and γ D = κ D .
Introduction Real analytic case C 2 -smooth case Definition (Stationary mapping) Let D ⊂ C n be a domain with C 1 -smooth boundary. We call a holomorphic mapping f : D − → D a stationary mapping if
Introduction Real analytic case C 2 -smooth case Definition (Stationary mapping) Let D ⊂ C n be a domain with C 1 -smooth boundary. We call a holomorphic mapping f : D − → D a stationary mapping if (1) f extends to a holomorphic mapping in a neighborhood od D ( denoted by the same letter ) ;
Introduction Real analytic case C 2 -smooth case Definition (Stationary mapping) Let D ⊂ C n be a domain with C 1 -smooth boundary. We call a holomorphic mapping f : D − → D a stationary mapping if (1) f extends to a holomorphic mapping in a neighborhood od D ( denoted by the same letter ) ; (2) f ( T ) ⊂ ∂ D ;
Introduction Real analytic case C 2 -smooth case Definition (Stationary mapping) Let D ⊂ C n be a domain with C 1 -smooth boundary. We call a holomorphic mapping f : D − → D a stationary mapping if (1) f extends to a holomorphic mapping in a neighborhood od D ( denoted by the same letter ) ; (2) f ( T ) ⊂ ∂ D ; (3) there exists a real analytic function ρ : T − → R > 0 such that → ζρ ( ζ ) ν D ( f ( ζ )) ∈ C n extends to a the mapping T ∋ ζ �− mapping holomorphic in a neighborhood of D ( denoted by � f ) .
Introduction Real analytic case C 2 -smooth case Definition (Weak stationary mapping) Let D ⊂ C n be a domain with C 1 -smooth boundary. We call a holomorphic mapping f : D − → D a weak stationary mapping if
Introduction Real analytic case C 2 -smooth case Definition (Weak stationary mapping) Let D ⊂ C n be a domain with C 1 -smooth boundary. We call a holomorphic mapping f : D − → D a weak stationary mapping if (1’) f extends to a C 1 / 2 -smooth mapping on D ( denoted by the same letter ) ;
Introduction Real analytic case C 2 -smooth case Definition (Weak stationary mapping) Let D ⊂ C n be a domain with C 1 -smooth boundary. We call a holomorphic mapping f : D − → D a weak stationary mapping if (1’) f extends to a C 1 / 2 -smooth mapping on D ( denoted by the same letter ) ; (2’) f ( T ) ⊂ ∂ D ;
Introduction Real analytic case C 2 -smooth case Definition (Weak stationary mapping) Let D ⊂ C n be a domain with C 1 -smooth boundary. We call a holomorphic mapping f : D − → D a weak stationary mapping if (1’) f extends to a C 1 / 2 -smooth mapping on D ( denoted by the same letter ) ; (2’) f ( T ) ⊂ ∂ D ; (3’) there exists a C 1 / 2 -smooth function ρ : T − → R > 0 such that → ζρ ( ζ ) ν D ( f ( ζ )) ∈ C n extends to a the mapping T ∋ ζ �− mapping � f ∈ O ( D ) ∩ C 1 / 2 ( D ) .
Introduction Real analytic case C 2 -smooth case Definition ((Weak) E -mapping) Let D ⊂ C n , n ≥ 2, be a bounded strongly linearly convex domain.
Introduction Real analytic case C 2 -smooth case Definition ((Weak) E -mapping) Let D ⊂ C n , n ≥ 2, be a bounded strongly linearly convex domain. A holomorphic mapping f : D − → D is called a ( weak ) E - mapping if it is a (weak) stationary mapping and
Introduction Real analytic case C 2 -smooth case Definition ((Weak) E -mapping) Let D ⊂ C n , n ≥ 2, be a bounded strongly linearly convex domain. A holomorphic mapping f : D − → D is called a ( weak ) E - mapping if it is a (weak) stationary mapping and (4) setting ϕ z ( ζ ) := � z − f ( ζ ) , ν D ( f ( ζ )) � , ζ ∈ T , we have wind ϕ z = 0 for some z ∈ D .
Introduction Real analytic case C 2 -smooth case Definition ((Weak) E -mapping) Let D ⊂ C n , n ≥ 2, be a bounded strongly linearly convex domain. A holomorphic mapping f : D − → D is called a ( weak ) E - mapping if it is a (weak) stationary mapping and (4) setting ϕ z ( ζ ) := � z − f ( ζ ) , ν D ( f ( ζ )) � , ζ ∈ T , we have wind ϕ z = 0 for some z ∈ D . Theorem (Ł. Kosiński, T.W.) Let D ⊂ C n , n ≥ 2 , be a bounded strongly linearly convex domain. Then a holomorphic mapping f : D − → D is an extremal if and only if f is a weak E-mapping.
Introduction Real analytic case C 2 -smooth case Recall the most important facts when ∂ D is real analytic.
Introduction Real analytic case C 2 -smooth case Recall the most important facts when ∂ D is real analytic. Theorem (Lempert Theorem) Let D ⊂ C n , n ≥ 2 , be a bounded strongly linearly convex domain with real analytic boundary. Then c D = � k D and γ D = κ D .
Introduction Real analytic case C 2 -smooth case Recall the most important facts when ∂ D is real analytic. Theorem (Lempert Theorem) Let D ⊂ C n , n ≥ 2 , be a bounded strongly linearly convex domain with real analytic boundary. Then c D = � k D and γ D = κ D . Theorem (Lempert) Let D ⊂ C n , n ≥ 2 , be a bounded strongly linearly convex domain with real analytic boundary. Then a holomorphic mapping f : D − → D is an extremal if and only if f is an E-mapping.
Introduction Real analytic case C 2 -smooth case Proposition Let D ⊂ C n , n ≥ 2 , be a bounded strongly linearly convex domain with real analytic boundary. Then a weak stationary mapping f of D is a stationary mapping of D with the same associated mappings � f , ρ . Moreover, f is a complex geodesic, that is c D ( f ( ζ ) , f ( ξ )) = p ( ζ, ξ ) for any ζ, ξ ∈ D .
Introduction Real analytic case C 2 -smooth case Proposition Let D ⊂ C n , n ≥ 2 , be a bounded strongly linearly convex domain with real analytic boundary. Then a weak stationary mapping f of D is a stationary mapping of D with the same associated mappings � f , ρ . Moreover, f is a complex geodesic, that is c D ( f ( ζ ) , f ( ξ )) = p ( ζ, ξ ) for any ζ, ξ ∈ D . Proposition (Uniqueness of E -mappings) For any different z , w ∈ D ( resp. for any z ∈ D, v ∈ ( C n ) ∗ ) there exists a unique E-mapping f : D − → D such that f ( 0 ) = z, f ( ξ ) = w for some ξ ∈ ( 0 , 1 ) ( resp. f ( 0 ) = z, f ′ ( 0 ) = λ v for some λ > 0 ) ( unique = with exactness to Aut ( D )) .
Introduction Real analytic case C 2 -smooth case Definition (Family D ( c ) ) For a given c > 0 let the family D ( c ) consist of all pairs ( D , z ) , where D ⊂ C n , n ≥ 2, is a bounded strongly pseudoconvex domain with real analytic boundary and z ∈ D , satisfying
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