Gabriela Kohr

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2. Univalent subordination chains in several complex variables Remark The first part of the talk refers to univalent subordination chains with the normalization Df (0, t) = e t 0 A(τ)dτ for t ≥ 0. f (·, s) ≺ f (·, t) is equivalent to the existence of a Schwarz mapping v = v(z, s, t)-transition mapping associated with f (z, t), such that f (z, s) = f (v(z, s, t), t), z ∈ B n , 0 ≤ s ≤ t < ∞. (vs,t)-evolution family associated to f (z, t). Remark Let f (z, t) be a univalent subordination chain and let v(z, s, t) be the transition mapping associated with f (z, t). Then (i) v(·, s, t) is biholomorphic on B n and v(z, s, s) = z, z ∈ B n , s ≥ 0. (ii) v(z, s, ·) is decreasing on [s, ∞), for all z ∈ B n and s ≥ 0. (iii) Semigroup property: v(z, s, τ) = v(v(z, s, t), t, τ), z ∈ B n , 0 ≤ s ≤ t ≤ τ < ∞. Gabriela Kohr (UBB Cluj) Geometric and analytic aspects of Loewner chains 10 / 62
2. Univalent subordination chains in several complex variables S(B n ) = {h ∈ H(B n ) : h biholomorphic , h(0) = 0, Dh(0) = In}; S(B 1 ) = S • Every function f ∈ S can be embedded in a Loewner chain. • In C n , n ≥ 2, there exist mappings f ∈ S(B n ) that cannot be embedded in Loewner chains f (z, t) such that {e −t f (·, t)}t≥0 is a normal family on B n . • Every function f ∈ S has parametric representation, i.e. f (z) = limt→∞ e t v(z, t), where v = v(z, t) is the unique Lipschitz continuous solution on [0, ∞) of the initial value problem ∂v = −vp(v, t) a.e. t ≥ 0, v(z, 0) = z, ∂t for some choice of p = p(z, t) such that p(·, t) ∈ P for almost all t ∈ [0, ∞) and p(z, ·) is measurable on [0, ∞) for z ∈ U. P = {p ∈ H(U) : p(0) = 1, Re p(z) > 0, z ∈ U}. • S 0 (B 1 ) = S; S 0 (B n ) S(B n ) for n ≥ 2, where S 0 (B n ) is the family of mappings which have parametric representation. Gabriela Kohr (UBB Cluj) Geometric and analytic aspects of Loewner chains 11 / 62
Page 1 and 2: Geometric and analytic aspects of L
Page 3 and 4: 1. Main contributions 1. Contributi
Page 5 and 6: 1. Main contributions • I. Graham
Page 7 and 8: 1. Main contributions • F. Bracci
Page 9: 2. Univalent subordination chains i
Page 13 and 14: 2. Univalent subordination chains i
Page 37 and 38: 3. Parametric representation on the
Page 43 and 44: Theorem 3. Parametric representatio
Page 45 and 46: Remark 3. Parametric representation
Page 47 and 48: 4. Subordination chains in complex
Page 49 and 50: Theorem 4. Subordination chains in
Page 51 and 52: Theorem 4. Subordination chains in
Page 53 and 54: Remark 4. Subordination chains in c
Page 55 and 56: 4. Subordination chains in complex
Page 57 and 58: 5. Open problems Conjecture 5. Open
Page 59 and 60: Definition 5. Open problems Let Ω
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5. Open problems References 1. P. D
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Gabriela Kohr

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