Surface diffusion flow near spheres

摘要: We consider closed immersed hypersurfaces evolving by surface diffusion flow, and perform an analysis based on local global integral estimates. First we show that a properly stationary (ΔH ≡ 0) hypersurface in \({\mathbb{R}^3}\) or \({\mathbb{R}^4}\) with restricted growth of the curvature at infinity small total tracefree must be embedded union umbilic hypersurfaces. Then prove for surfaces if L2 norm is globally initially it monotonic nonincreasing along flow. also derive pointwise estimates all derivatives assuming its locally small. Using these results singularity develops concentrate definite manner, blowup under suitable conditions converges to nonumbilic surface. obtain our main result as consequence: flow close sphere family embeddings, exists time, exponentially round sphere.