Evolution families and the Loewner equation I: the unit disc

By B. Filippo, Manuel D. Contreras, S. Díaz-Madrigal. 2008

98 citations recorded by the source corpus.

Abstract In this paper we introduce a general version of the Loewner differential equation which allows us to present a new and unified treatment of both the radial equation introduced in 1923 by K. Loewner and the chordal equation introduced in 2000 by O. Schramm. In particular, we prove that evolution families in the unit disc are in one to one correspondence with solutions to this new type of Loewner equations. Also, we give a Berkson–Porta type formula for non-autonomous weak holomorphic vector fields which generate such Loewner differential equations and study in detail geometric and dynamical properties of evolution families.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3eb8665210c9a34a16e4827b3c0b00b4a8ee539a

How researchers see this paper

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In Applied Mathematics

1. Filippo Bracci
2. Manuel D. Contreras
3. S. Díaz-Madrigal
4. B. Filippo
5. Leandro Arosio
6. H. Hamada
7. G. Kohr

In Geometry and Topology

1. Filippo Bracci
2. S. Díaz-Madrigal
3. Manuel D. Contreras

In Mathematical Physics

1. Filippo Bracci
2. Manuel D. Contreras
3. S. Díaz-Madrigal
4. Leandro Arosio

Citations in the corpus

• Evolution families and the Loewner equation II: complex hyperbolic manifolds (2008, methodology): “If M = D the unit disc in C, in [4] we proved that Proposition 4.”