Calculus of Variations and Partial Differential Equations

, Volume 44, Issue 1, pp 131–151

Surface diffusion flow near spheres

Article

DOI: 10.1007/s00526-011-0429-4

Cite this article as:
Wheeler, G. Calc. Var. (2012) 44: 131. doi:10.1007/s00526-011-0429-4

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 (ΔH ≡ 0) hypersurface in \({\mathbb{R}^3}\) or \({\mathbb{R}^4}\) 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 L2 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 L2 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 L2 is a family of embeddings, exists for all time, and exponentially converges to a round sphere.

Mathematics Subject Classification (2000)

53C4458J35

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.School of Mathematics and Applied StatisticsUniversity of WollongongWollongongAustralia