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 (1)

• Normal forms and linearization of holomorphic dilation type semigroups in several variables

Cited by in corpus (1)

• Solving the Loewner PDE in complete hyperbolic starlike domains of $\C^N$