Date: 15 May 2011

Surface diffusion flow near spheres

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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 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.

Communicated by G. Huisken.