Lectures on mean curvature flow and related equations by Ilmanen T.
By Ilmanen T.
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Formally, this justifies the slogan “developing singularities are modeled by self-shrinkiers”, which we used as motivation for the computer searches. We call this limit surface, if it exists, a blowup of Mt at (x0 , t0 ). Example. The blowup of a shrinking convex surface at its unique point of disappearance is a self-shrinking sphere [Hu1]. Blowups always exist in the weak sense of Brakke (moving varifolds); see [I6] or White [W2]. Our aim in the next section is to prove that the blowups exist classically for surfaces in R3 .
H. Poincare Non-Lineare 7 (1990) 335-344. [C] Chopp, D. “Numerical Computation of Self-Similar Solutions for Mean Curvature Flow,” Exper. Math. 3 (1994) 1-16. [CIL] Crandall, M. , “User’s Guide to Viscosity Solutions of Second Order Partial Differential Equations,” Bull. Amer. Math. Soc. 27 (1992) 1-67. , “Some Conjectures on Flow by Mean Curvature,” white paper, 1990. , “New Problems on Minimizing Movements,” Boundary Value Problems for PDE and Applications, C. L. , Masson, 1992, 81-88. , “On Regularity for Mean Curvature Flow of Hypersurfaces,” preprint.
It is not know whether the full blowup sequence converges. Sketch of Proof. The proof mostly involves known techniques in the compactness theory of 2-dimensional surfaces, with a few twists. For general background, see Sacks-Uhlenback [SaU], Schoen-Yau [SY], Choi-Schoen [CS], Gulliver [Gu1, Gu2], and many others; see also Ecker [E]. λ . We will show that Let Mtλ be the rescaled surface. Write M λ = M−1 a subsequence of {M λ }λ>0 passes to a smooth limit. Let us fix R > 0 and prove this in BR . It may later be extended to R3 by diagonalization.