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On the mean curvature evolution of two-convex hypersurfaces. (English) Zbl 1275.53059

Summary: We study the mean curvature evolution of smooth, closed, two-convex hypersurfaces in \(\mathbb {R}^{n+1}\) for \(n \geq 3\). Within this framework we effect a reconciliation between the flow with surgeries – recently constructed by G. Huisken and C. Sinestrari in [Invent. Math. 175, No. 1, 137–221 (2009; Zbl 1170.53042)] – and the well-known weak solution of the level-set flow: we prove that the two solutions agree in an appropriate limit of the surgery parameters and in a precise quantitative sense. Our proof relies on geometric estimates for certain \(L^p\)-norms of the mean curvature which are of independent interest even in the setting of classical mean curvature flow. We additionally show how our construction can be used to pass these estimates to limits and produce regularity results for the weak solution.

MSC:

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces

Citations:

Zbl 1170.53042
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