# Pinning of Diffusional Patterns by Non-Uniform Curvature

## TL;DR

This paper investigates how non-uniform curvature on surfaces influences the pinning and selection of diffusion-driven patterns, using conformal mapping and perturbation theory, with implications for biological and physical systems.

## Contribution

It introduces a novel theoretical framework combining conformal mapping and perturbation theory to analyze pattern pinning by curvature inhomogeneities.

## Key findings

- Curvature inhomogeneities can pin diffusion patterns.
- The theory draws an analogy to quantum mechanics in a geometry-dependent potential.
- Numerical confirmation supports the theoretical predictions.

## Abstract

Diffusion-driven patterns appear on curved surfaces in many settings, initiated by unstable modes of an underlying Laplacian operator. On a flat surface or perfect sphere, the patterns are degenerate, reflecting translational/rotational symmetry. Deformations, e.g. by a bulge or indentation, break symmetry and can pin a pattern. We adapt methods of conformal mapping and perturbation theory to examine how curvature inhomogeneities select and pin patterns, and confirm the results numerically. The theory provides an analogy to quantum mechanics in a geometry-dependent potential and yields intuitive implications for cell membranes, tissues, thin films, and noise-induced quasipatterns.