Surface diffusion flow near spheres

TL;DR

This paper analyzes the behavior of surfaces evolving under surface diffusion flow, proving conditions for convergence to spheres and characterizing singularity formation using integral and pointwise curvature estimates.

Contribution

It establishes new criteria for convergence to spheres and describes the nature of singularities in surface diffusion flow near spheres.

Findings

Flow starting near a sphere converges exponentially to a round sphere.

Singularities involve curvature concentration and converge to nonumbilic stationary surfaces.

Stationary solutions with certain curvature conditions are unions of umbilic hypersurfaces.

Abstract

We consider closed immersed hypersurfaces evolving by surface diffusion flow, and perform an analysis based on local and global integral estimates. First we show that a properly immersed stationary (\Delta H \equiv 0) hypersurface in \R^3 or \R^4 with restricted growth of the curvature at infinity and small total tracefree curvature must be an embedded union of umbilic hypersurfaces. Then we prove for surfaces that if the L^2 norm of the tracefree curvature is globally initially small it is monotonic nonincreasing along the flow. We also derive pointwise estimates for all derivatives of the curvature assuming that its L^2 norm is locally small. Using these results we show that if a singularity develops the curvature must concentrate in a definite manner, and prove that a blowup under suitable conditions converges to a nonumbilic embedded stationary surface. We obtain our main result as a consequence: the surface diffusion flow of a surface initially close to a sphere in L^2 is a family of embeddings, exists for all time, and exponentially converges to a round sphere.