Basins of attraction in Loewner equations

Abstract

We prove that any Loewner PDE on the unit ball $\B^q$ whose driving term $h(z,t)$ vanishes at the origin and satisfies the bunching condition $\ell m(Dh(0,t))\geq k(Dh(0,t))$ for some $\ell\in \mathbb{R}^+$, admits a solution given by univalent mappings $(f_t\colon \B^q\to\C^q)_{t\geq 0}$. This is done by discretizing time and considering the abstract basin of attraction. If $\ell<2$, then the range $\cup_{t\geq 0} f_t(\B^q)$ of any such solution is biholomorphic to $\C^q$.