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Introduction Real analytic case C 2 -smooth case Lempert Theorem for C 2 -smooth strongly linearly convex domains Tomasz Warszawski Krakw-Vienna Workshop on Pluripotential Theory and Several Complex Variables 37 IX 2012 Introduction

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Introduction Real analytic case C2-smooth case

Lempert Theorem for C2-smooth strongly linearly convex domains

Tomasz Warszawski Kraków-Vienna Workshop on Pluripotential Theory and Several Complex Variables 3–7 IX 2012

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Introduction Real analytic case C2-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.

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Introduction Real analytic case C2-smooth case

Definition (Strong linear convexity) A domain D ⊂ Cn is strongly linearly convex if D has C2-smooth boundary; there exists a defining function r of D such that

n

j,k=1

∂2r ∂zj∂zk (a)XjX k >

n
j,k=1

∂2r ∂zj∂zk (a)XjXk

a ∈ ∂D, X ∈ T C

D(a)∗.

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Introduction Real analytic case C2-smooth case

Lempert function

kD(z, w) = inf{p(ζ, ξ) : ζ, ξ ∈ D and ∃f ∈ O(D, D) :

f (ζ) = z, f (ξ) = w}, z, w ∈ D.

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Introduction Real analytic case C2-smooth case

Lempert function

kD(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 κD(z; v) = inf{|λ|−1/(1 − |ζ|2) : λ ∈ C∗, ζ ∈ D and ∃f ∈ O(D, D) : f (ζ) = z, f ′(ζ) = λv}, z ∈ D, v ∈ Cn.

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Introduction Real analytic case C2-smooth case

Lempert function

kD(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 κD(z; v) = inf{|λ|−1/(1 − |ζ|2) : λ ∈ C∗, ζ ∈ D and ∃f ∈ O(D, D) : f (ζ) = z, f ′(ζ) = λv}, z ∈ D, v ∈ Cn. If z = w (resp. v = 0), a mapping for which the infimum is attained we call an extremal ( kD-extremal or κD-extremal)

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Introduction Real analytic case C2-smooth case

Carath´ eodory (pseudo)distance cD(z, w) = sup{p(F(z), F(w)) : F ∈ O(D, D)}, z, w ∈ D.

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Introduction Real analytic case C2-smooth case

Carath´ eodory (pseudo)distance cD(z, w) = sup{p(F(z), F(w)) : F ∈ O(D, D)}, z, w ∈ D. Carath´ eodory-Reiffen (pseudo)metric γD(z; v) = sup{|F ′(z)v| : F ∈ O(D, D), F(z) = 0}, z ∈ D, v ∈ Cn.

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Introduction Real analytic case C2-smooth case

Carath´ eodory (pseudo)distance cD(z, w) = sup{p(F(z), F(w)) : F ∈ O(D, D)}, z, w ∈ D. Carath´ eodory-Reiffen (pseudo)metric γD(z; v) = sup{|F ′(z)v| : F ∈ O(D, D), F(z) = 0}, z ∈ D, v ∈ Cn. Theorem (Lempert Theorem for C2, Ł. Kosiński, T.W.) Let D ⊂ Cn, n ≥ 2, be a bounded strongly linearly convex domain. Then cD = kD and γD = κD.

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Introduction Real analytic case C2-smooth case

Definition (Stationary mapping) Let D ⊂ Cn be a domain with C1-smooth boundary. We call a holomorphic mapping f : D − → D a stationary mapping if

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Introduction Real analytic case C2-smooth case

Definition (Stationary mapping) Let D ⊂ Cn be a domain with C1-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);

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Introduction Real analytic case C2-smooth case

Definition (Stationary mapping) Let D ⊂ Cn be a domain with C1-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;

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Introduction Real analytic case C2-smooth case

Definition (Stationary mapping) Let D ⊂ Cn be a domain with C1-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 the mapping T ∋ ζ − → ζρ(ζ)νD(f (ζ)) ∈ Cn extends to a mapping holomorphic in a neighborhood of D (denoted by f ).

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Introduction Real analytic case C2-smooth case

Definition (Weak stationary mapping) Let D ⊂ Cn be a domain with C1-smooth boundary. We call a holomorphic mapping f : D − → D a weak stationary mapping if

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Introduction Real analytic case C2-smooth case

Definition (Weak stationary mapping) Let D ⊂ Cn be a domain with C1-smooth boundary. We call a holomorphic mapping f : D − → D a weak stationary mapping if (1’) f extends to a C1/2-smooth mapping on D (denoted by the same letter);

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Introduction Real analytic case C2-smooth case

Definition (Weak stationary mapping) Let D ⊂ Cn be a domain with C1-smooth boundary. We call a holomorphic mapping f : D − → D a weak stationary mapping if (1’) f extends to a C1/2-smooth mapping on D (denoted by the same letter); (2’) f (T) ⊂ ∂D;

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Introduction Real analytic case C2-smooth case

Definition (Weak stationary mapping) Let D ⊂ Cn be a domain with C1-smooth boundary. We call a holomorphic mapping f : D − → D a weak stationary mapping if (1’) f extends to a C1/2-smooth mapping on D (denoted by the same letter); (2’) f (T) ⊂ ∂D; (3’) there exists a C1/2-smooth function ρ : T − → R>0 such that the mapping T ∋ ζ − → ζρ(ζ)νD(f (ζ)) ∈ Cn extends to a mapping f ∈ O(D) ∩ C1/2(D).

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Introduction Real analytic case C2-smooth case

Definition ((Weak) E-mapping) Let D ⊂ Cn, n ≥ 2, be a bounded strongly linearly convex domain.

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Introduction Real analytic case C2-smooth case

Definition ((Weak) E-mapping) Let D ⊂ Cn, 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

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Introduction Real analytic case C2-smooth case

Definition ((Weak) E-mapping) Let D ⊂ Cn, 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 ϕz(ζ) := z − f (ζ), νD(f (ζ)), ζ ∈ T, we have wind ϕz = 0 for some z ∈ D.

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Introduction Real analytic case C2-smooth case

Definition ((Weak) E-mapping) Let D ⊂ Cn, 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 ϕz(ζ) := z − f (ζ), νD(f (ζ)), ζ ∈ T, we have wind ϕz = 0 for some z ∈ D. Theorem (Ł. Kosiński, T.W.) Let D ⊂ Cn, n ≥ 2, be a bounded strongly linearly convex domain. Then a holomorphic mapping f : D − → D is an extremal if and

nly if f is a weak E-mapping.

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Introduction Real analytic case C2-smooth case

Recall the most important facts when ∂D is real analytic.

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Introduction Real analytic case C2-smooth case

Recall the most important facts when ∂D is real analytic. Theorem (Lempert Theorem) Let D ⊂ Cn, n ≥ 2, be a bounded strongly linearly convex domain with real analytic boundary. Then cD = kD and γD = κD.

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Introduction Real analytic case C2-smooth case

Recall the most important facts when ∂D is real analytic. Theorem (Lempert Theorem) Let D ⊂ Cn, n ≥ 2, be a bounded strongly linearly convex domain with real analytic boundary. Then cD = kD and γD = κD. Theorem (Lempert) Let D ⊂ Cn, 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.

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Introduction Real analytic case C2-smooth case

Proposition Let D ⊂ Cn, 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 cD(f (ζ), f (ξ)) = p(ζ, ξ) for any ζ, ξ ∈ D.

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Introduction Real analytic case C2-smooth case

Proposition Let D ⊂ Cn, 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 cD(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 ∈ (Cn)∗) 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) = λv for some λ > 0) (unique = with exactness to Aut(D)).

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Introduction Real analytic case C2-smooth case

Definition (Family D(c)) For a given c > 0 let the family D(c) consist of all pairs (D, z), where D ⊂ Cn, n ≥ 2, is a bounded strongly pseudoconvex domain with real analytic boundary and z ∈ D, satisfying

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Introduction Real analytic case C2-smooth case

Definition (Family D(c)) For a given c > 0 let the family D(c) consist of all pairs (D, z), where D ⊂ Cn, n ≥ 2, is a bounded strongly pseudoconvex domain with real analytic boundary and z ∈ D, satisfying (1) dist(z, ∂D) ≥ 1/c;

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Introduction Real analytic case C2-smooth case

Definition (Family D(c)) For a given c > 0 let the family D(c) consist of all pairs (D, z), where D ⊂ Cn, n ≥ 2, is a bounded strongly pseudoconvex domain with real analytic boundary and z ∈ D, satisfying (1) dist(z, ∂D) ≥ 1/c; (2) the diameter of D is not greater than c and D satisfies the interior ball condition with a radius 1/c;

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Introduction Real analytic case C2-smooth case

Definition (Family D(c)) For a given c > 0 let the family D(c) consist of all pairs (D, z), where D ⊂ Cn, n ≥ 2, is a bounded strongly pseudoconvex domain with real analytic boundary and z ∈ D, satisfying (1) dist(z, ∂D) ≥ 1/c; (2) the diameter of D is not greater than c and D satisfies the interior ball condition with a radius 1/c; (3) for any x, y ∈ D there exist m ≤ 8c2 and open balls B0, . . . , Bm ⊂ D of radii 1/(2c) such that x ∈ B0, y ∈ Bm and the distance between the centers of the balls Bj, Bj+1 is not greater than 1/(4c) for j = 0, . . . , m − 1;

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Introduction Real analytic case C2-smooth case

Definition (Family D(c)) (4) for any open ball B ⊂ Cn of a radius not greater than 1/c, intersecting non-emptily with ∂D, there exists a mapping Φ ∈ O(D, Cn) such that

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Introduction Real analytic case C2-smooth case

Definition (Family D(c)) (4) for any open ball B ⊂ Cn of a radius not greater than 1/c, intersecting non-emptily with ∂D, there exists a mapping Φ ∈ O(D, Cn) such that

(a) for any w ∈ Φ(B ∩ ∂D) there is a ball of a radius c containing Φ(D) and tangent to ∂Φ(D) at w (we call it the “exterior ball condition” with a radius c);

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Introduction Real analytic case C2-smooth case

Definition (Family D(c)) (4) for any open ball B ⊂ Cn of a radius not greater than 1/c, intersecting non-emptily with ∂D, there exists a mapping Φ ∈ O(D, Cn) such that

(a) for any w ∈ Φ(B ∩ ∂D) there is a ball of a radius c containing Φ(D) and tangent to ∂Φ(D) at w (we call it the “exterior ball condition” with a radius c); (b) Φ is biholomorphic in a neighborhood of B and Φ−1(Φ(B)) = B;

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Introduction Real analytic case C2-smooth case

Definition (Family D(c)) (4) for any open ball B ⊂ Cn of a radius not greater than 1/c, intersecting non-emptily with ∂D, there exists a mapping Φ ∈ O(D, Cn) such that

(a) for any w ∈ Φ(B ∩ ∂D) there is a ball of a radius c containing Φ(D) and tangent to ∂Φ(D) at w (we call it the “exterior ball condition” with a radius c); (b) Φ is biholomorphic in a neighborhood of B and Φ−1(Φ(B)) = B; (c) entries of all matrices Φ′ on B ∩ D and (Φ−1)′ on Φ(B ∩ D) are bounded in modulus by c;

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Introduction Real analytic case C2-smooth case

Definition (Family D(c)) (4) for any open ball B ⊂ Cn of a radius not greater than 1/c, intersecting non-emptily with ∂D, there exists a mapping Φ ∈ O(D, Cn) such that

(a) for any w ∈ Φ(B ∩ ∂D) there is a ball of a radius c containing Φ(D) and tangent to ∂Φ(D) at w (we call it the “exterior ball condition” with a radius c); (b) Φ is biholomorphic in a neighborhood of B and Φ−1(Φ(B)) = B; (c) entries of all matrices Φ′ on B ∩ D and (Φ−1)′ on Φ(B ∩ D) are bounded in modulus by c; (d) dist(Φ(z), ∂Φ(D)) ≥ 1/c;

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Introduction Real analytic case C2-smooth case

Definition (Family D(c)) (4) for any open ball B ⊂ Cn of a radius not greater than 1/c, intersecting non-emptily with ∂D, there exists a mapping Φ ∈ O(D, Cn) such that

(a) for any w ∈ Φ(B ∩ ∂D) there is a ball of a radius c containing Φ(D) and tangent to ∂Φ(D) at w (we call it the “exterior ball condition” with a radius c); (b) Φ is biholomorphic in a neighborhood of B and Φ−1(Φ(B)) = B; (c) entries of all matrices Φ′ on B ∩ D and (Φ−1)′ on Φ(B ∩ D) are bounded in modulus by c; (d) dist(Φ(z), ∂Φ(D)) ≥ 1/c;

(5) the normal vector νD is Lipschitz with a constant 2c;

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Introduction Real analytic case C2-smooth case

Definition (Family D(c)) (4) for any open ball B ⊂ Cn of a radius not greater than 1/c, intersecting non-emptily with ∂D, there exists a mapping Φ ∈ O(D, Cn) such that

(a) for any w ∈ Φ(B ∩ ∂D) there is a ball of a radius c containing Φ(D) and tangent to ∂Φ(D) at w (we call it the “exterior ball condition” with a radius c); (b) Φ is biholomorphic in a neighborhood of B and Φ−1(Φ(B)) = B; (c) entries of all matrices Φ′ on B ∩ D and (Φ−1)′ on Φ(B ∩ D) are bounded in modulus by c; (d) dist(Φ(z), ∂Φ(D)) ≥ 1/c;

(5) the normal vector νD is Lipschitz with a constant 2c; (6) an ε-hull of D, i.e. a domain Dε := {w ∈ Cn : dist(w, D) < ε}, is strongly pseudoconvex for any ε ∈ (0, 1/c].

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Introduction Real analytic case C2-smooth case

We go back to the situation when D ⊂ Cn, n ≥ 2, is a bounded strongly linearly convex domain with real analytic boundary.

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Introduction Real analytic case C2-smooth case

We go back to the situation when D ⊂ Cn, n ≥ 2, is a bounded strongly linearly convex domain with real analytic boundary. Proposition (Uniform estimates) Fix c > 0, let (D, z) ∈ D(c) and let f : (D, 0) − → (D, z) be an E-mapping. Then there is a uniform C > 0 (depending only on c) such that

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Introduction Real analytic case C2-smooth case

We go back to the situation when D ⊂ Cn, n ≥ 2, is a bounded strongly linearly convex domain with real analytic boundary. Proposition (Uniform estimates) Fix c > 0, let (D, z) ∈ D(c) and let f : (D, 0) − → (D, z) be an E-mapping. Then there is a uniform C > 0 (depending only on c) such that C −1 < ρ(ζ)−1 < C, ζ ∈ T;

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Introduction Real analytic case C2-smooth case

We go back to the situation when D ⊂ Cn, n ≥ 2, is a bounded strongly linearly convex domain with real analytic boundary. Proposition (Uniform estimates) Fix c > 0, let (D, z) ∈ D(c) and let f : (D, 0) − → (D, z) be an E-mapping. Then there is a uniform C > 0 (depending only on c) such that C −1 < ρ(ζ)−1 < C, ζ ∈ T; |f (ζ1) − f (ζ2)| ≤ C

|ζ1 − ζ2|,

ζ1, ζ2 ∈ D;

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Introduction Real analytic case C2-smooth case

We go back to the situation when D ⊂ Cn, n ≥ 2, is a bounded strongly linearly convex domain with real analytic boundary. Proposition (Uniform estimates) Fix c > 0, let (D, z) ∈ D(c) and let f : (D, 0) − → (D, z) be an E-mapping. Then there is a uniform C > 0 (depending only on c) such that C −1 < ρ(ζ)−1 < C, ζ ∈ T; |f (ζ1) − f (ζ2)| ≤ C

|ζ1 − ζ2|,

ζ1, ζ2 ∈ D; |ρ(ζ1) − ρ(ζ2)| ≤ C

|ζ1 − ζ2|,

ζ1, ζ2 ∈ T;

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Introduction Real analytic case C2-smooth case

We go back to the situation when D ⊂ Cn, n ≥ 2, is a bounded strongly linearly convex domain with real analytic boundary. Proposition (Uniform estimates) Fix c > 0, let (D, z) ∈ D(c) and let f : (D, 0) − → (D, z) be an E-mapping. Then there is a uniform C > 0 (depending only on c) such that C −1 < ρ(ζ)−1 < C, ζ ∈ T; |f (ζ1) − f (ζ2)| ≤ C

|ζ1 − ζ2|,

ζ1, ζ2 ∈ D; |ρ(ζ1) − ρ(ζ2)| ≤ C

|ζ1 − ζ2|,

ζ1, ζ2 ∈ T; | f (ζ1) − f (ζ2)| ≤ C

|ζ1 − ζ2|,

ζ1, ζ2 ∈ D.

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Introduction Real analytic case C2-smooth case

Proposition A weak E-mapping f : D − → D of a bounded strongly linearly convex domain D ⊂ Cn, n ≥ 2, is a unique kD-extremal for f (ζ), f (ξ) (resp. a unique κD-extremal for f (ζ), f ′(ζ)), where ζ, ξ ∈ D, ζ = ξ (unique = with exactness to Aut(D)). Moreover, f is a complex geodesic.

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Introduction Real analytic case C2-smooth case

Proposition A weak E-mapping f : D − → D of a bounded strongly linearly convex domain D ⊂ Cn, n ≥ 2, is a unique kD-extremal for f (ζ), f (ξ) (resp. a unique κD-extremal for f (ζ), f ′(ζ)), where ζ, ξ ∈ D, ζ = ξ (unique = with exactness to Aut(D)). Moreover, f is a complex geodesic. The proof is analogous as in the real analytic case.

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Introduction Real analytic case C2-smooth case

Lemma Let D ⊂⊂ Bn, n ≥ 2, be a strongly pseudoconvex domain with C2-smooth boundary. Take z ∈ D and let r be a defining function

f D such that

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Introduction Real analytic case C2-smooth case

Lemma Let D ⊂⊂ Bn, n ≥ 2, be a strongly pseudoconvex domain with C2-smooth boundary. Take z ∈ D and let r be a defining function

f D such that

r ∈ C2(Cn);

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Introduction Real analytic case C2-smooth case

Lemma Let D ⊂⊂ Bn, n ≥ 2, be a strongly pseudoconvex domain with C2-smooth boundary. Take z ∈ D and let r be a defining function

f D such that

r ∈ C2(Cn); D = {x ∈ Cn : r(x) < 0};

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Introduction Real analytic case C2-smooth case

Lemma Let D ⊂⊂ Bn, n ≥ 2, be a strongly pseudoconvex domain with C2-smooth boundary. Take z ∈ D and let r be a defining function

f D such that

r ∈ C2(Cn); D = {x ∈ Cn : r(x) < 0}; Cn \ D = {x ∈ Cn : r(x) > 0};

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Introduction Real analytic case C2-smooth case

Lemma Let D ⊂⊂ Bn, n ≥ 2, be a strongly pseudoconvex domain with C2-smooth boundary. Take z ∈ D and let r be a defining function

f D such that

r ∈ C2(Cn); D = {x ∈ Cn : r(x) < 0}; Cn \ D = {x ∈ Cn : r(x) > 0}; |∇r| = 1 on ∂D;

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Introduction Real analytic case C2-smooth case

Lemma Let D ⊂⊂ Bn, n ≥ 2, be a strongly pseudoconvex domain with C2-smooth boundary. Take z ∈ D and let r be a defining function

f D such that

r ∈ C2(Cn); D = {x ∈ Cn : r(x) < 0}; Cn \ D = {x ∈ Cn : r(x) > 0}; |∇r| = 1 on ∂D; n

j,k=1 ∂2r ∂zj∂zk (a)XjX k ≥ C|X|2 for any a ∈ ∂D and X ∈ Cn

with some constant C > 0.

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Introduction Real analytic case C2-smooth case

Lemma Let D ⊂⊂ Bn, n ≥ 2, be a strongly pseudoconvex domain with C2-smooth boundary. Take z ∈ D and let r be a defining function

f D such that

r ∈ C2(Cn); D = {x ∈ Cn : r(x) < 0}; Cn \ D = {x ∈ Cn : r(x) > 0}; |∇r| = 1 on ∂D; n

j,k=1 ∂2r ∂zj∂zk (a)XjX k ≥ C|X|2 for any a ∈ ∂D and X ∈ Cn

with some constant C > 0. Suppose that there exist C2-smooth functions rm : Cn − → R such that ∂|α|rm

∂xα

→ ∂|α|r

∂xα uniformly on Bn for α ∈ N2n 0 with |α| ≤ 2.

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Introduction Real analytic case C2-smooth case

Lemma Let D ⊂⊂ Bn, n ≥ 2, be a strongly pseudoconvex domain with C2-smooth boundary. Take z ∈ D and let r be a defining function

f D such that

r ∈ C2(Cn); D = {x ∈ Cn : r(x) < 0}; Cn \ D = {x ∈ Cn : r(x) > 0}; |∇r| = 1 on ∂D; n

j,k=1 ∂2r ∂zj∂zk (a)XjX k ≥ C|X|2 for any a ∈ ∂D and X ∈ Cn

with some constant C > 0. Suppose that there exist C2-smooth functions rm : Cn − → R such that ∂|α|rm

∂xα

→ ∂|α|r

∂xα uniformly on Bn for α ∈ N2n 0 with |α| ≤ 2. Let

Dm be a connected component of the set {x ∈ Cn : rm(x) < 0}, containing the point z.

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Introduction Real analytic case C2-smooth case

Then there is c > 0 such that (Dm, z) and (D, z) belong to D(c), m >> 1.

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Introduction Real analytic case C2-smooth case

Then there is c > 0 such that (Dm, z) and (D, z) belong to D(c), m >> 1. We omit the very technical proof. Generally, it relies on studying functions of the form Cn ∋ x − → rm(x) − t(|x − b|2 − R2) ∈ R, where t, R ∈ R and b ∈ Cn are fixed.

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Introduction Real analytic case C2-smooth case

Proofs of the main Theorems Losing no generality 0 ∈ D ⊂⊂ Bn, let r be defining for D as in the lemma

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Introduction Real analytic case C2-smooth case

Proofs of the main Theorems Losing no generality 0 ∈ D ⊂⊂ Bn, let r be defining for D as in the lemma Weierstrass Theorem = ⇒ ∃ real polynomials Pk on Cn ≃ R2n such that ∂|α|Pk

∂xα

→ ∂|α|r

∂xα uniformly on Bn for α ∈ N2n 0 with

|α| ≤ 2

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Introduction Real analytic case C2-smooth case

Proofs of the main Theorems Losing no generality 0 ∈ D ⊂⊂ Bn, let r be defining for D as in the lemma Weierstrass Theorem = ⇒ ∃ real polynomials Pk on Cn ≃ R2n such that ∂|α|Pk

∂xα

→ ∂|α|r

∂xα uniformly on Bn for α ∈ N2n 0 with

|α| ≤ 2 Dk,ε := {x ∈ Cn : Pk(x) + ε < 0}

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Introduction Real analytic case C2-smooth case

Proofs of the main Theorems Losing no generality 0 ∈ D ⊂⊂ Bn, let r be defining for D as in the lemma Weierstrass Theorem = ⇒ ∃ real polynomials Pk on Cn ≃ R2n such that ∂|α|Pk

∂xα

→ ∂|α|r

∂xα uniformly on Bn for α ∈ N2n 0 with

|α| ≤ 2 Dk,ε := {x ∈ Cn : Pk(x) + ε < 0} εm → 0, 0 < 3εm+1 < εm

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Introduction Real analytic case C2-smooth case

Proofs of the main Theorems Losing no generality 0 ∈ D ⊂⊂ Bn, let r be defining for D as in the lemma Weierstrass Theorem = ⇒ ∃ real polynomials Pk on Cn ≃ R2n such that ∂|α|Pk

∂xα

→ ∂|α|r

∂xα uniformly on Bn for α ∈ N2n 0 with

|α| ≤ 2 Dk,ε := {x ∈ Cn : Pk(x) + ε < 0} εm → 0, 0 < 3εm+1 < εm ∀m ∃km : Pkm − rBn < εm

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Introduction Real analytic case C2-smooth case

Proofs of the main Theorems Losing no generality 0 ∈ D ⊂⊂ Bn, let r be defining for D as in the lemma Weierstrass Theorem = ⇒ ∃ real polynomials Pk on Cn ≃ R2n such that ∂|α|Pk

∂xα

→ ∂|α|r

∂xα uniformly on Bn for α ∈ N2n 0 with

|α| ≤ 2 Dk,ε := {x ∈ Cn : Pk(x) + ε < 0} εm → 0, 0 < 3εm+1 < εm ∀m ∃km : Pkm − rBn < εm rm := Pkm + 2εm = ⇒ r < rm+1 < rm in Bn

SLIDE 62

Introduction Real analytic case C2-smooth case

Proofs of the main Theorems Losing no generality 0 ∈ D ⊂⊂ Bn, let r be defining for D as in the lemma Weierstrass Theorem = ⇒ ∃ real polynomials Pk on Cn ≃ R2n such that ∂|α|Pk

∂xα

→ ∂|α|r

∂xα uniformly on Bn for α ∈ N2n 0 with

|α| ≤ 2 Dk,ε := {x ∈ Cn : Pk(x) + ε < 0} εm → 0, 0 < 3εm+1 < εm ∀m ∃km : Pkm − rBn < εm rm := Pkm + 2εm = ⇒ r < rm+1 < rm in Bn Dm := a connected component of Dkm,2εm containing 0

SLIDE 63

Introduction Real analytic case C2-smooth case

Proofs of the main Theorems Losing no generality 0 ∈ D ⊂⊂ Bn, let r be defining for D as in the lemma Weierstrass Theorem = ⇒ ∃ real polynomials Pk on Cn ≃ R2n such that ∂|α|Pk

∂xα

→ ∂|α|r

∂xα uniformly on Bn for α ∈ N2n 0 with

|α| ≤ 2 Dk,ε := {x ∈ Cn : Pk(x) + ε < 0} εm → 0, 0 < 3εm+1 < εm ∀m ∃km : Pkm − rBn < εm rm := Pkm + 2εm = ⇒ r < rm+1 < rm in Bn Dm := a connected component of Dkm,2εm containing 0 Dm is a bounded strongly linearly convex domain with real analytic boundary and rm is its defining function for m >> 1

SLIDE 64

Introduction Real analytic case C2-smooth case

Proofs of the main Theorems Losing no generality 0 ∈ D ⊂⊂ Bn, let r be defining for D as in the lemma Weierstrass Theorem = ⇒ ∃ real polynomials Pk on Cn ≃ R2n such that ∂|α|Pk

∂xα

→ ∂|α|r

∂xα uniformly on Bn for α ∈ N2n 0 with

|α| ≤ 2 Dk,ε := {x ∈ Cn : Pk(x) + ε < 0} εm → 0, 0 < 3εm+1 < εm ∀m ∃km : Pkm − rBn < εm rm := Pkm + 2εm = ⇒ r < rm+1 < rm in Bn Dm := a connected component of Dkm,2εm containing 0 Dm is a bounded strongly linearly convex domain with real analytic boundary and rm is its defining function for m >> 1 Dm ⊂ Dm+1,

m Dm = D =

⇒ Lempert Theorem for C2.

SLIDE 65

Introduction Real analytic case C2-smooth case

Proofs of the main Theorems It suffices to show that for any different z, w ∈ D (resp. z ∈ D, v ∈ (Cn)∗) there is a weak E-mapping for z, w (resp. for z, v)

SLIDE 66

Introduction Real analytic case C2-smooth case

Proofs of the main Theorems It suffices to show that for any different z, w ∈ D (resp. z ∈ D, v ∈ (Cn)∗) there is a weak E-mapping for z, w (resp. for z, v) z, w ∈ Dm (resp. z ∈ Dm) for m >> 1 = ⇒ ∃ an E-mapping fm of Dm s.t. fm(0) = z, fm(ξm) = w, ξm ∈ (0, 1)

SLIDE 67

Introduction Real analytic case C2-smooth case

Proofs of the main Theorems It suffices to show that for any different z, w ∈ D (resp. z ∈ D, v ∈ (Cn)∗) there is a weak E-mapping for z, w (resp. for z, v) z, w ∈ Dm (resp. z ∈ Dm) for m >> 1 = ⇒ ∃ an E-mapping fm of Dm s.t. fm(0) = z, fm(ξm) = w, ξm ∈ (0, 1) (resp. fm(0) = z, f ′

m(0) = λmv, λm > 0)

SLIDE 68

Introduction Real analytic case C2-smooth case

Proofs of the main Theorems It suffices to show that for any different z, w ∈ D (resp. z ∈ D, v ∈ (Cn)∗) there is a weak E-mapping for z, w (resp. for z, v) z, w ∈ Dm (resp. z ∈ Dm) for m >> 1 = ⇒ ∃ an E-mapping fm of Dm s.t. fm(0) = z, fm(ξm) = w, ξm ∈ (0, 1) (resp. fm(0) = z, f ′

m(0) = λmv, λm > 0)

Dm ⊂ Dm+1 ⊂ Bn and fm are complex geodesics = ⇒ ∃ compact K ⊂ (0, 1) s.t. {ξm} ⊂ K

SLIDE 69

Introduction Real analytic case C2-smooth case

Proofs of the main Theorems It suffices to show that for any different z, w ∈ D (resp. z ∈ D, v ∈ (Cn)∗) there is a weak E-mapping for z, w (resp. for z, v) z, w ∈ Dm (resp. z ∈ Dm) for m >> 1 = ⇒ ∃ an E-mapping fm of Dm s.t. fm(0) = z, fm(ξm) = w, ξm ∈ (0, 1) (resp. fm(0) = z, f ′

m(0) = λmv, λm > 0)

Dm ⊂ Dm+1 ⊂ Bn and fm are complex geodesics = ⇒ ∃ compact K ⊂ (0, 1) s.t. {ξm} ⊂ K (resp. ∃ compact K ⊂ (0, ∞) s.t. {λm} ⊂ K)

SLIDE 70

Introduction Real analytic case C2-smooth case

Proofs of the main Theorems It suffices to show that for any different z, w ∈ D (resp. z ∈ D, v ∈ (Cn)∗) there is a weak E-mapping for z, w (resp. for z, v) z, w ∈ Dm (resp. z ∈ Dm) for m >> 1 = ⇒ ∃ an E-mapping fm of Dm s.t. fm(0) = z, fm(ξm) = w, ξm ∈ (0, 1) (resp. fm(0) = z, f ′

m(0) = λmv, λm > 0)

Dm ⊂ Dm+1 ⊂ Bn and fm are complex geodesics = ⇒ ∃ compact K ⊂ (0, 1) s.t. {ξm} ⊂ K (resp. ∃ compact K ⊂ (0, ∞) s.t. {λm} ⊂ K) ∃c > 0 : (Dm, z) ∈ D(c), m >> 1

SLIDE 71

Introduction Real analytic case C2-smooth case

Proofs of the main Theorems It suffices to show that for any different z, w ∈ D (resp. z ∈ D, v ∈ (Cn)∗) there is a weak E-mapping for z, w (resp. for z, v) z, w ∈ Dm (resp. z ∈ Dm) for m >> 1 = ⇒ ∃ an E-mapping fm of Dm s.t. fm(0) = z, fm(ξm) = w, ξm ∈ (0, 1) (resp. fm(0) = z, f ′

m(0) = λmv, λm > 0)

Dm ⊂ Dm+1 ⊂ Bn and fm are complex geodesics = ⇒ ∃ compact K ⊂ (0, 1) s.t. {ξm} ⊂ K (resp. ∃ compact K ⊂ (0, ∞) s.t. {λm} ⊂ K) ∃c > 0 : (Dm, z) ∈ D(c), m >> 1 fm, fm and ρm satisfy the uniform estimates

SLIDE 72

Introduction Real analytic case C2-smooth case

Proofs of the main Theorems Arzela-Ascoli Theorem + passing to subsequences = ⇒

fm → f uniformly on D = ⇒ f ∈ O(D) ∩ C1/2(D)

SLIDE 73

Introduction Real analytic case C2-smooth case

Proofs of the main Theorems Arzela-Ascoli Theorem + passing to subsequences = ⇒

fm → f uniformly on D = ⇒ f ∈ O(D) ∩ C1/2(D)

fm →

f uniformly on D = ⇒ f ∈ O(D) ∩ C1/2(D)

SLIDE 74

Introduction Real analytic case C2-smooth case

Proofs of the main Theorems Arzela-Ascoli Theorem + passing to subsequences = ⇒

fm → f uniformly on D = ⇒ f ∈ O(D) ∩ C1/2(D)

fm →

f uniformly on D = ⇒ f ∈ O(D) ∩ C1/2(D) ρm → ρ uniformly on T = ⇒ ρ ∈ C1/2(T), ρ > 0

SLIDE 75

Introduction Real analytic case C2-smooth case

Proofs of the main Theorems Arzela-Ascoli Theorem + passing to subsequences = ⇒

fm → f uniformly on D = ⇒ f ∈ O(D) ∩ C1/2(D)

fm →

f uniformly on D = ⇒ f ∈ O(D) ∩ C1/2(D) ρm → ρ uniformly on T = ⇒ ρ ∈ C1/2(T), ρ > 0 ξm → ξ ∈ (0, 1) (resp. λm → λ > 0)

SLIDE 76

Introduction Real analytic case C2-smooth case

Proofs of the main Theorems Arzela-Ascoli Theorem + passing to subsequences = ⇒

fm → f uniformly on D = ⇒ f ∈ O(D) ∩ C1/2(D)

fm →

f uniformly on D = ⇒ f ∈ O(D) ∩ C1/2(D) ρm → ρ uniformly on T = ⇒ ρ ∈ C1/2(T), ρ > 0 ξm → ξ ∈ (0, 1) (resp. λm → λ > 0)

f passes through z, w (resp. f (0) = z, f ′(0) = λv), f (D) ⊂ D, f (T) ⊂ ∂D

SLIDE 77

Introduction Real analytic case C2-smooth case

Proofs of the main Theorems Arzela-Ascoli Theorem + passing to subsequences = ⇒

fm → f uniformly on D = ⇒ f ∈ O(D) ∩ C1/2(D)

fm →

f uniformly on D = ⇒ f ∈ O(D) ∩ C1/2(D) ρm → ρ uniformly on T = ⇒ ρ ∈ C1/2(T), ρ > 0 ξm → ξ ∈ (0, 1) (resp. λm → λ > 0)

f passes through z, w (resp. f (0) = z, f ′(0) = λv), f (D) ⊂ D, f (T) ⊂ ∂D D is strongly pseudoconvex = ⇒ f (D) ⊂ D

SLIDE 78

Introduction Real analytic case C2-smooth case

Proofs of the main Theorems Arzela-Ascoli Theorem + passing to subsequences = ⇒

fm → f uniformly on D = ⇒ f ∈ O(D) ∩ C1/2(D)

fm →

f uniformly on D = ⇒ f ∈ O(D) ∩ C1/2(D) ρm → ρ uniformly on T = ⇒ ρ ∈ C1/2(T), ρ > 0 ξm → ξ ∈ (0, 1) (resp. λm → λ > 0)

f passes through z, w (resp. f (0) = z, f ′(0) = λv), f (D) ⊂ D, f (T) ⊂ ∂D D is strongly pseudoconvex = ⇒ f (D) ⊂ D The conditions (3’) and (4) from the definition of a weak E-mapping follow from the uniform convergence of suitable functions

SLIDE 79

Introduction Real analytic case C2-smooth case

Proofs of the main Theorems Arzela-Ascoli Theorem + passing to subsequences = ⇒

fm → f uniformly on D = ⇒ f ∈ O(D) ∩ C1/2(D)

fm →

f uniformly on D = ⇒ f ∈ O(D) ∩ C1/2(D) ρm → ρ uniformly on T = ⇒ ρ ∈ C1/2(T), ρ > 0 ξm → ξ ∈ (0, 1) (resp. λm → λ > 0)

f passes through z, w (resp. f (0) = z, f ′(0) = λv), f (D) ⊂ D, f (T) ⊂ ∂D D is strongly pseudoconvex = ⇒ f (D) ⊂ D The conditions (3’) and (4) from the definition of a weak E-mapping follow from the uniform convergence of suitable functions f is a weak E-mapping of D

SLIDE 80

Introduction Real analytic case C2-smooth case