By A. T. Fomenko

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**Example text**

Convexity Invariance 47 Finally, further invariance properties for the mean curvature ﬂow can be obtained again by means of Hamilton’s maximum principle for tensors [56, Sections 4 and 8] (whose proof is a generalization of the argument above), see Appendix C. Let us ﬁrst recall a deﬁnition; we say that an immersed hypersurface is k-convex, for some 1 ≤ k ≤ n, if the sum of the k smallest principal curvatures is nonnegative at every point. In particular, one-convexity coincides with convexity, while n-convexity means nonnegativity of the mean curvature H, that is, mean convexity.

11). • Almgren, Taylor and Wang discretization-minimization procedure in [2]. • Ilmanen’s approximation in [77, 78]. 3. One can show that the mean curvature ﬂow shares a kind of the usual regularizing property of parabolic equations, for instance, any C 2 initial hypersurface becomes analytic at every positive time, in the sense that it is not the map ϕt which becomes analytic, but the image hypersurface ϕt (M ) ⊂ Rn+1 , that is, it admits an analytic reparametrization. Moreover, with the right deﬁnition, one can let evolve a hypersurface with corners or other singularities and these latter immediately vanish, see for instance [14, 16] and [38, 118].

P+q+r=k | p,q,r∈N Proof. We work by induction on k ∈ N. 1); we then suppose that the formula holds for k − 1. We have, by the previous lemma, ∂ k ∂ ∇ hij = ∇ ∇k−1 hij + ∇k−1 A ∗ ∇A ∗ A ∂t ∂t = ∇ ∆∇k−1 hij + ∇p A ∗ ∇q A ∗ ∇r A p+q+r=k−1 | p,q,r∈N +∇ k−1 A ∗ ∇A ∗ A = ∇∆∇k−1 hij + ∇p A ∗ ∇q A ∗ ∇r A. p+q+r=k | p,q,r∈N Interchanging now the Laplacian and the covariant derivative and recalling that Riem = A ∗ A, we have the conclusion, as all the extra terms we get are of the form A ∗ A ∗ ∇k A and A ∗ ∇A ∗ ∇k−1 A.