  We analyze the competitive evolution of codimension one and two morphologies
within the $H^{-1}$ gradient flow of the strong Functionalized Cahn-Hilliard
equation. On a slow time scale a sharp hypersurface reduction yields a
degenerate Mullins-Sekerka evolution for both codimension one and two
hypersurfaces, leading to a geometric flow that depends locally on curvatures
couples to the dynamic value of the spatially constant far-field chemical
potential. Both codimension one and two morphologies admit two classes of
bifurcations, one leads to pearling, a short-wavelength in-plane modulation of
interfacial width, the other flips motion by curvature to the locally-ill posed
motion against curvature, which leads to fingering instabilities. We present a
bifurcation diagram for the morphological competition, and compare our results
quantitatively to simulations of the full system and qualitatively to
simulations of self-consistent mean field models and laboratory experiments;
illuminating the role of the pearling bifurcation in the development of complex
network morphologies.
