Gabriela Kohr

More documents

Recommendations

Info

2. Univalent subordination chains in several complex variables • S 0 (B n ) is compact, but S(B n ) is not compact for n ≥ 2. • There exist mappings f ∈ S 0 (B n ) such that 1 2 D2 f (0) > 2. • In dimension n ≥ 2, there exist Loewner chains f (z, t) such that {e −t f (·, t)}t≥0 is not a normal family. • Let p : U × [0, ∞) → C be a function such that p(·, t) ∈ P, t ≥ 0, and p(z, ·) is measurable on [0, ∞), z ∈ U. Then there exists a unique Loewner chain f (z, t) which satisfies the Loewner differential equation ∂f ∂t (z, t) = zf ′ (z, t)p(z, t), a.e. t ≥ 0, ∀z ∈ U. • In dimension n ≥ 2, the Loewner differential equation ∂f (z, t) = Df (z, t)h(z, t), a.e. t ≥ 0, ∀z ∈ U, ∂t does not have a unique normalized univalent solution f (z, t). Gabriela Kohr (UBB Cluj) Geometric and analytic aspects of Loewner chains 12 / 62
2. Univalent subordination chains in several complex variables • Many classical results in the theory of holomorphic mappings in C n do not hold in infinite dimensional complex Banach spaces. • Montel’s theorem does not hold in the infinite dimensional setting (but surprisingly, Vitali’s theorem does). • On a domain in Cn , any univalent (holomorphic and injective) mapping into Cn is also biholomorphic. However, this result is no longer true in infinite dimensional complex Banach spaces. For example, if f : ℓ2 → ℓ2 is given by f (x) = (x 2 1 , x 3 1 , x 2 2 , x 3 2 , . . .), then f is univalent on the unit ball of ℓ2, but is not biholomorphic. • On a domain in C n any univalent mapping is open. Heath and Suffridge (1980): an example of a univalent mapping on the unit ball B of a complex Banach space which is not biholomorphic, f (B) contains an open set, but f (B) is not open. Gabriela Kohr (UBB Cluj) Geometric and analytic aspects of Loewner chains 13 / 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 and 10: 2. Univalent subordination chains i
Page 11: 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
Page 61 and 62: 5. Open problems References 1. P. D

Gabriela Kohr

Create successful ePaper yourself

Delete template?

Save as template?