By G Lefort
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Additional info for Algebra & Analysis, Problems & Solutions
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).  G. Huisken, C. Sinestrari, Mean curvature ﬂow singularities for mean convex surfaces, Calc. Variations 8 (1999), 1–14.  G. Huisken, C. Sinestrari, Convexity estimates for mean curvature ﬂow and singularities of mean convex surfaces, Acta Math. 183 (1999), 45–70. 42 Bibliography  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  for the Ricci ﬂow. However, the method of proof is quite diﬀerent. In fact, the result of  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 .