Gradient flow for the Willmore functional

Abstract

We consider two-dimensional, compact immersed surfaces in R n moving by the gradient of the L 2 integral of their curvature. It is not known whether solutions to this fourth-order, geometric evolution equation may develop singularities in finite time. We give a lower bound on the lifespan of a smooth solution, which depends only on how much the curvature of the initial surface is concentrated in space.