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 $\C^q$.

Arosio, L. (2012). Basins of attraction in Loewner equations. ANNALES ACADEMIAE SCIENTIARUM FENNICAE. MATHEMATICA, 37(1), 563-570 [10.5186/aasfm.2012.3742].

Basins of attraction in Loewner equations

AROSIO, LEANDRO
2012-01-01

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 $\C^q$.
2012
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/03 - GEOMETRIA
English
Arosio, L. (2012). Basins of attraction in Loewner equations. ANNALES ACADEMIAE SCIENTIARUM FENNICAE. MATHEMATICA, 37(1), 563-570 [10.5186/aasfm.2012.3742].
Arosio, L
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/136208
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