Surface diffusion flow near spheres

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 \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.

19 pages

References in corpus (1)

• Lifespan theorem for constrained surface diffusion flows

Cited by in corpus (13)

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• A family of parameter-dependent diffeomorphisms acting on function spaces over a Riemannian manifold and applications to geometric flows
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• Lifespan theorem for simple constrained surface diffusion flows
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• On well-posedness, stability, and bifurcation for the axisymmetric surface diffusion flow
• On the isoperimetric inequality and surface diffusion flow for multiply winding curves
• The Willmore flow with prescribed isoperimetric ratio
• Stability of the surface diffusion flow and volume-preserving mean curvature flow in the flat torus
• On the flow of non-axisymmetric perturbations of cylinders via surface diffusion
• Willmore Flow of Complete Surfaces
• Gradient Flows of Higher Order Yang-Mills-Higgs Functionals
• Higher order geometric flow of hypersurfaces in a Riemannian manifold