Resonances in Loewner equations

Abstract

We prove that given a Herglotz vector field on the unit ball of ${\mathbb{C}}^n$ of the form $H(z,t)=(a_1{z}_1,...,a_n{z}_n)+O(|z{|}^2)$ with $\Re a_j<0$ for all $j$, its evolution family admits an associated Loewner chain, which is normal if no real resonances occur. Hence the Loewner-Kufarev PDE admits a solution defined for all positive times.

23 pages, corrected typos, revised argument in Proposition 5.5, result unchanged

References in corpus (2)

• An abstract approach to Loewner chains
• Normal forms and linearization of holomorphic dilation type semigroups in several variables

Cited by in corpus (6)

• Solution of a Loewner Chain Equation in Several Complex Variables
• Solving the Loewner PDE in complete hyperbolic starlike domains of $\C^N$
• An abstract approach to Loewner chains
• Holomorphic evolution: metamorphosis of the Loewner equation
• Approximation properties of univalent mappings on the unit ball in ${\mathbb{C}}^n$
• Variation of Loewner chains, extreme and support points in the class $S^0$ in higher dimensions