TITLE: Geometry of curve lengthening membranes.

ABSTRACT:   In this talk, I’ll present rigorous results of the transient evolution of bilayer interfaces evolving of the mass preserving L^2-gradient flow of the functionalized Cahn-Hilliard equation. The proof is based on energy modulated method, which requires a construction of slow manifold – bilayer manifold, composed of approximated solutions dressed around admissible interfaces. Subject to certain initial data, we first establish an orbital stability that the solution/orbits stay near the manifold as long as the projected interface is smooth enough.  We also identify open sets of initial data whose orbits converge to circular equilibrium after a significant transient, and derive a singularly perturbed interfacial evolution comprised of motion against curvature regularized by an asymptotically weak Willmore term.