Mean curvature flow singularities for mean convex surfaces. (English) Zbl 0992.53052

Huisken, Gerhard; Sinestrari, Carlo

Calc. Var. Partial Differ. Equ. 8, No. 1, 1-14 (1999).

Summary: We study the evolution by mean curvature of a smooth \(n\)-dimensional surface \({\mathcal M}\subset \mathbb{R}^{n+1}\), compact and with positive mean curvature. We first prove an estimate on the negative part of the scalar curvature of the surface. Then we apply this result to study the formation of singularities by rescaling techniques, showing that there exists a sequence of rescaled flows converging to a smooth limit flow of surfaces with nonnegative scalar curvature. This gives a classification of the possible singular behaviour for mean convex surfaces in the case \(n=2\).

MSC:

53C44	Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
35K55	Nonlinear parabolic equations
53A10	Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
58E12	Variational problems concerning minimal surfaces (problems in two independent variables)

Keywords:

evolution by mean curvature; scalar curvature; singularities; mean convex surfaces