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Huisken, A. Polden, Geometric evolution equations for hypersurfaces, in: S. Hidebrandt, M. ), Calculus of Variations and Geometric Evolution Problems (Cetraro, 1996), Springer-Verlag, Berlin, Heidelberg (1999). [46] G. Huisken, C. Sinestrari, Mean curvature flow singularities for mean convex surfaces, Calc. Variations 8 (1999), 1–14. [47] G. Huisken, C. Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math. 183 (1999), 45–70. 42 Bibliography [48] G.

Then, for any η > 0, there exists a constant Cη > 0 such that |A|2 − H2 ≤ ηH 2 + Cη n−1 on Mt for any t ∈ [0, T ). Sketch of the proof. Let us consider, for η ∈ IR and σ ∈ [0, 2] , the function fσ,η = 1 + η)H 2 |A|2 − ( n−1 H 2−σ . 1). 6). In fact, Z can be negative on nonconvex surfaces. A typical example is when λ1 < 0 and λ2 = · · · = λn > 0; then Z < 0, even if |λ1 | is small compared to the other curvatures. 3) on Mt for any t > 0. 1, we aim at estimating the Lp norms of the positive part of fσ,η .

Proof. Let us first remark that a similar result holds for the analogous function considered in [33] for the Ricci flow. However, the method of proof is quite different. In fact, the result of [33] follows from an application of the maximum principle. In our case, instead, the additional factor H σ induces the presence of a positive zeroorder term in the evolution equation for fσ that cannot be directly compensated by the other terms. More precisely, one finds ∂fσ 2(1 − σ) 2 ≤ Δfσ + ∇H, ∇fσ − 4−σ |H∇i hkl − hkl ∇i H| + σ|A|2 fσ .

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