Gabriela Kohr

2. Univalent subordination chains in several complex variables
J. Pfaltzgraff (1974); T. Poreda (1990); M. Elin, S. Reich, D. Shoikhet
(2000); I. Graham, H. Hamada, G.K, M.K. (2008); L. Arosio; M. Voda
(2011).
Theorem
Let A : [0, ∞) → L(C n , C n ) be such that (a)-(c) hold. Also let
h : B n × [0, ∞) → C n satisfy the following conditions:
(i) h(·, t) ∈ N and Dh(0, t) = A(t) for t ≥ 0;
(ii) h(z, ·) is measurable on [0, ∞) for each z ∈ B n .
Then for each z ∈ B n and s ≥ 0, the initial value problem
(2.1)
∂v
∂t
= −h(v, t), a.e. t ≥ s, v(z, s, s) = z,
has a unique solution v = v(z, s, t) such that v(·, s, t) is a univalent
Schwarz mapping, v(z, s, ·) is Lipschitz continuous on [s, ∞) locally
uniformly with respect to z ∈ B n and Dv(0, s, t) = exp(− t
s A(τ)dτ).
h(z, t)-generating vector field (Herglotz vector field).
Gabriela Kohr (UBB Cluj) Geometric and analytic aspects of Loewner chains 22 / 62