Meromorphic quadratic differentials with half-plane structures

We prove the existence of "half-plane differentials" with prescribed local data on any Riemann surface. These are meromorphic quadratic differentials with higher-order poles which have an associated singular flat metric isometric to a collection of euclidean half-planes glued by an interval-exchange map on their boundaries. The local data is associated with the poles and consists of the integer order, a non-negative real residue, and a positive real leading order term. This generalizes a result of Strebel for differentials with double-order poles, and associates metric spines with the Riemann surface.Comment: 46 pages, 23 figures. Some minor corrections in v2, and a clarification added in section 1

Conformal mapping methods for interfacial dynamics

The article provides a pedagogical review aimed at graduate students in materials science, physics, and applied mathematics, focusing on recent developments in the subject. Following a brief summary of concepts from complex analysis, the article begins with an overview of continuous conformal-map dynamics. This includes problems of interfacial motion driven by harmonic fields (such as viscous fingering and void electromigration), bi-harmonic fields (such as viscous sintering and elastic pore evolution), and non-harmonic, conformally invariant fields (such as growth by advection-diffusion and electro-deposition). The second part of the article is devoted to iterated conformal maps for analogous problems in stochastic interfacial dynamics (such as diffusion-limited aggregation, dielectric breakdown, brittle fracture, and advection-diffusion-limited aggregation). The third part notes that all of these models can be extended to curved surfaces by an auxilliary conformal mapping from the complex plane, such as stereographic projection to a sphere. The article concludes with an outlook for further research.Comment: 37 pages, 12 (mostly color) figure