Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions. (English) Zbl 1482.53117
In [Invent. Math. 217, 35–76 (2019; Zbl 1414.53059)], the authors of the paper under review prove that the rotationally symmetric bowl soliton is the only noncompact ancient solution of mean curvature flow in \(\mathbb{R}^3\) which is strictly convex and non-collapsed. In the paper under review, using similar techniques, the authors classify all noncompact ancient solutions to mean curvature flow in \(\mathbb{R}^{n+1}\), \(n\geq 3\), which are convex, uniformly two-convex, and non-collapsed, proving that such an ancient solution is a rotationally symmetric translating soliton. As a direct consequence, the following interesting result is derived: Let us consider an arbitrary closed, embedded, two-convex hypersurface in \(\mathbb{R}^{n+1}\), and evolve it by mean curvature flow. Then it follows that, at the first singular time, the only possible blow-up limits are shrinking round spheres, shrinking round cylinders, and the unique rotationally symmetric translating soliton.
Reviewer: Gabriel Eduard Vilcu (Ploieşti)
MSC:
53E10 | Flows related to mean curvature |
53A07 | Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces |
Citations:
Zbl 1414.53059References:
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