Interface growth in two dimensions: a Loewner-equation approach.

The problem of Laplacian growth in two dimensions is considered within the Loewner-equation framework. Initially the problem of fingered growth is revisited and an exact solution for a three-finger configuration is reported. Then a general class of growth models for an interface growing in the upper half-plane is introduced and the corresponding Loewner equation for the problem is derived. Several examples are given including interfaces with one or more tips as well as multiple growing interfaces. A generalization of our interface growth model in terms of "Loewner domains," where the growth rule is specified by a time evolving measure, is briefly discussed.