Parametric Representation of Univalent Mappings in Several Complex Variables

By Ian D. Graham, H. Hamada, G. Kohr. Canadian Journal of Mathematics - Journal Canadien de Mathematiques, 2002

185 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7e28f418de0e7a38ff61941f9c8e80f4c40e331e

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Variation of Loewner Chains, Extreme and Support Points in the Class $$S^0$$S0 in Higher Dimensions (2014, methodology): “3) is related to that of a g-Loewner chain [11].”
An abstract approach to Loewner chains (2010, methodology): “Loewner-Kufarev PDE In this section we prove that L-Loewner chains on complete hyperbolic complex manifolds are the univalent solutions of the Loewner-Kufarev partial differential equation, as in the classical theory of Loewner chains on the unit ball B in C (see [13], [16]).”
Loewner chains and parametric representation in several complex variables (2003, methodology): “These results were generalized to the unit ball with respect to an arbitrary norm in C by Graham, Hamada, and Kohr [5].”
Sharp growth theorems and coefficient bounds for starlike mappings in several complex variables (2008, methodology): “For the set S ∗ g,k+1(B), we obtain the following growth result by an argument similar to that in the proof of [12, Theorem 10] (cf. [ 8, Theorem 2.2 ] and [22, Theorem 2.3]).”
Solving the Loewner PDE in complete hyperbolic starlike domains of $\C^N$ (2012, background): “The main remaining open problem, whether any Loewner PDE in a complete hyperbolic domain in C admits a univalent solution with values in C , has been given many partial positive answers, see [14, 18, 19, 3, 4, 5, 26].”
Asymptotically spirallike mappings in several complex variables (2008, background): “(ii) A mapping f 2 S(B n ) has parametric representation if and only if there exists a Loewner chain f(z,t) such that {e t f(·,t)}t�0 is a normal family on B n and f(z) = f(z,0), z 2 B n (see [ 9 ], [16], [14]; cf. [33]).”
Growth theorems and coefficient bounds for univalent holomorphic mappings which have parametric representation (2006, background): “These results were obtained by Graham, Ham and Kohr [5] in the casek = 1 (cf.”
Solutions for the generalized Loewner differential equation in several complex variables (2010, methodology): “This result is a generalization of [41, Theorem 2.1], [ 14, Theorem 1.4 ] and [22, Lemma 1.3].”