Loewner chains and parametric representation in several complex variables

By Ian D. Graham, G. Kohr, Mirela Kohr. 2003

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Abstract Let B be the unit ball of C n with respect to an arbitrary norm. We study certain properties of Loewner chains and their transition mappings on the unit ball B. We show that any Loewner chain f(z,t) and the transition mapping v(z,s,t) associated to f(z,t) satisfy locally Lipschitz conditions in t locally uniformly with respect to z∈B. Moreover, we prove that a mapping f∈H(B) has parametric representation if and only if there exists a Loewner chain f(z,t) such that the family {e−tf(z,t)}t⩾0 is a normal family on B and f(z)=f(z,0) for z∈B. Also we show that univalent solutions f(z,t) of the generalized Loewner differential equation in higher dimensions are unique when {e−tf(z,t)}t⩾0 is a normal family on B. Finally we show that the set S0(B) of mappings which have parametric representation on B is compact.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d2a63b0dda52be4b872860352311c822356d2552

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Evolution families and the Loewner equation II: complex hyperbolic manifolds (2008, background): “We refer the reader to [7], [8], [9] and references therein.”
Variation of Loewner Chains, Extreme and Support Points in the Class $$S^0$$S0 in Higher Dimensions (2014, background): “(ii) It is known that S0 is compact in the topology of uniform convergence on compacta, and that S0 = S for n ≥ 2 (see [19]; see also [12] and [18]).”
Asymptotically spirallike mappings in several complex variables (2008, background): “For A = In, this result was obtained in [9] and [ 16 ] (see also [14]; cf. [33, Theorem 3] in the case of maximum norm).”
Solutions for the generalized Loewner differential equation in several complex variables (2010, background): “valent mappings in several complex variables (see [14,21, 22 ,41,42,51,52]):”