Holomorphic evolution: metamorphosis of the Loewner equation

This is a survey on recent results on the Loewner theory in one and several complex manifoldsComment: 26 page

Simultaneous models for commuting holomorphic self-maps of the ball

We prove that a finite family of commuting holomorphic self-maps of the unit ball $\mathbb{B}^q\subset \mathbb{C}^q$ admits a simultaneous holomorphic conjugacy to a family of commuting automorphisms of a possibly lower dimensional ball, and that such conjugacy satisfies a universal property. As an application we describe when a hyperbolic and a parabolic holomorphic self-map of $\mathbb{B}^q$ can commute.Comment: Final version, to appear on Adv. Mat

Support points and the Bieberbach conjecture in higher dimension

We describe some open questions related to support points in the class $S^0$ and introduce some useful techniques toward a higher dimensional Bieberbach conjecture