  Turing patterns emerge from a spatially uniform state following a linear
instability driven by diffusion. Features of the eventual pattern (stabilized
by non-linearities) are already present in the initial unstable modes. On a
uniform flat surface or perfect sphere, the unstable modes and final patterns
are degenerate, reflecting translational/rotational symmetry. This symmetry can
be broken, e.g. by a bump on a flat substrate or by deforming a sphere. As the
diffusion operator on a two dimensional manifold depends on the underlying
curvature, the degeneracy of the initial unstable mode is similarly reduced.
Different shapes can pin different modes. We adapt methods of conformal mapping
and perturbation theory to analytically examine how bumps and ripples entrain
modes of the diffusion operator on cylinders and spheres. We confirm these
results numerically, and provide closed form expressions that describe how
non-uniformities in curvature pin diffusion-driven instabilities and the
resulting patterns.
