Conformal Growth in Laplacian Fields: A Loewner Equation Approach to Natural Branching Patterns

Branching structures in nature, from venation and vasculature to river networks, often emerge from unstable growth in Laplacian fields. We consider the conformal-mapping and Loewner framework as a general tool to study such processes, reducing the moving-boundary problem to the evolution of a conformal map. This approach allows the analysis of perturbation growth, branching, and tip-splitting across different geometries. At this stage, we present preliminary simulations and emphasize connections to both classical viscous fingering and biological pattern formation, aiming toward a unified description of complex growth patterns.