On the existence of solutions of the Helfrich flow and its center manifold near spheres

Abstract

The Helfrich variational problem is the minimizing problem of the bending energy among closed surface with prescribed area and enclosed volume. This is one of the models for shape transformation theory of human red blood cells. Here the associated gradient flow, called the Helfrich flow, is studied. The existence of this geometric flow is proved locally for arbitrary initial data, and globally near spheres. Furthermore its center manifold near spheres is investigated.