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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.

MSC:
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
35K55 Nonlinear parabolic equations
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[1] Andrews, B., Contraction of convex hypersurfaces in Euclidean space.Calc. Var. Partial Differential Equations, 2 (1994), 151–171. · Zbl 0805.35048 · doi:10.1007/BF01191340
[2] Caffarelli, L., Nirenberg, L. &Spruck, J., The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian.Acta Math., 155 (1985), 261–301. · Zbl 0654.35031 · doi:10.1007/BF02392544
[3] Gårding, L., An inequality for hyperbolic polynomials.J. Math. Mech., 8 (1959), 957–965. · Zbl 0090.01603
[4] Hamilton, R. S., Four-manifolds with positive curvature operator.J. Differential Geom., 24 (1986), 153–179. · Zbl 0628.53042
[5] –, The formation of singularities in the Ricci flow, inSurveys in Differential Geometry, Vol. II (Cambridge, MA, 1993), pp. 7–136. Internat. Press, Cambridge, MA, 1993.
[6] – Harnack estimate for the mean curvature flow.J. Differential Geom., 41 (1995), 215–226. · Zbl 0827.53006
[7] Hardy, G. H., Littlewood, J. E. &Pólya, G.,Inequalities. Cambridge Univ. Press, Cambridge, 1934.
[8] Huisken, G., Flow by mean curvature of convex surfaces into spheres.J. Differential Geom., 20 (1984), 237–266. · Zbl 0556.53001
[9] – Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature.Invent. Math., 84 (1986), 463–480. · Zbl 0589.53058 · doi:10.1007/BF01388742
[10] – Asymptotic behaviour for singularities of the mean curvature flow.J. Differential Geom., 31 (1990), 285–299. · Zbl 0694.53005
[11] – Local and global behaviour of hypersurfaces moving by mean curvature.Proc. Sympos. Pure Math., 54 (1993), 175–191. · Zbl 0791.58090
[12] Huisken, G. &Sinestrari, C., Mean curvature flow singularities for mean convex surfaces.Calc. Var. Partial Differential Equations, 8 (1999), 1–14. · Zbl 0992.53052 · doi:10.1007/s005260050113
[13] Marcus, M. &Lopes, L., Inequalities for symmetric functions and Hermitian matrices.Canad. J. Math., 9 (1957), 305–312. · Zbl 0079.02103 · doi:10.4153/CJM-1957-037-9
[14] Smoczyk, K., Starshaped hypersurfaces and the mean curvature flow.Manuscripta Math., 95 (1998), 225–236. · Zbl 0903.53039
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