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Convexity estimates for mean curvature flow and singularities of mean convex surfaces. (English) Zbl 0992.53051
In this nice paper, the authors obtain the following classification of singularities which form in hypersurfaces with nonnegative mean curvature moving under mean curvature flow:
Consider a smooth immersion of a closed, \(n\)-dimensional hypersurface with nonnegative mean curvature, which evolves by mean curvature flow on a finite maximal time interval \([0,T).\) Then any rescaled limit of a singularity that forms as \(t \rightarrow T\) is weakly convex.
This implies that for a hypersurface of nonnegative mean curvature, the limiting flow of a Type-II singularity has convex surfaces \(\widetilde M_{\tau},\) \(\tau \in \mathbb R\). Moreover, \(\widetilde M_{\tau}\) is either a strictly convex translating soliton, or it splits as a product of \(\mathbb R^{n-m}\) with a lower-dimensional strictly convex translating soliton in \(\mathbb R^{m+1}.\) This classification complements that of Type-I singularities [G. Huisken, J. Differ. Geom. 31, 285-299 (1990; Zbl 0694.53005); Proc. Sympos. Pure Math. 54, Part 1, 175-191 (1993; Zbl 0791.58090)].
To prove the result, the authors derive a priori bounds on the symmetric functions of the principal curvatures of the evolving hypersurfaces. More precisely, let \((\lambda_{1}, \ldots,\lambda_{n})\) be the principle curvatures, and let \(S_{k}(\lambda)=\sum_{1\leq i_{1}<i_{2}\ldots<i_{k}\leq n} \lambda_{i_{1}}\ldots\lambda_{i_{k}}\). Then for each \(k,\) \( 2\leq k \leq n\), and any \(\eta>0\), there exists a constant \(C\) depending only on \(\eta, k, n,\) and the initial data, so that \[ S_{k}(\lambda) \geq -\eta H^{k} - C_{\eta,k}. \] The idea of the derivation is to proceed inductively on the degree \(k\) of the symmetric functions, with the \(k=2\) case contained in a previous paper of the authors [Calc. Var. Partial Differ. Equ. 8, 1-14 (1999; Zbl 0992.53052)].
A crucial step is to perturb the second fundamental form in such a way as to be able to work with the quotient of the new consecutive symmetric functions as a test function.

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
35K55 Nonlinear parabolic equations
Full Text: DOI
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