Some Remarks on Huisken’s Monotonicity Formula for Mean Curvature Flow

We discuss a monotone quantity related to Huisken’s monotonicity formula and some technical consequences for mean curvature flow. CONTENTS 1. Maximizing Huisken’s Monotonicity Formula 1 2. Applications 6 2.1. A No–Breathers Result 6 2.2. Singularities 7 References 9 1. MAXIMIZING HUISKEN’S MONOTONICITY FORMULA For an immersed hypersurface M ⊂ R, we call A and H respectively its second fundamental form and its mean curvature. Let Mt = φ(M, t) be the mean curvature flow (MCF) of an n–dimensional compact hypersurface in R, defined by the smooth family of immersions φ : M × [0, T )→ R which satisfies ∂tφ = Hν where ν is the “inner” unit normal vector field to the hypersurface. Huisken in [7] found his fundamental monotonicity formula (1.1) d dt ∫ M e |x−p|2 4(C−t) [4π(C − t)]n/2 dμt(x) = − ∫ M e |x−p|2 4(C−t) [4π(C − t)])n/2 ∣∣∣∣H + 〈x− p | ν〉 2(C − t) ∣∣∣∣2 dμt(x) ≤ 0 , for every p ∈ R, in the time interval [0,min{C, T}). Here dμt is the canonical measure on M associated to the metric induced by the immersion at time t. We call the quantity ∫ M e − |x−p| 2 4(C−t) [4π(C−t)]n/2 dμt(x), the Huisken’s functional. Such formula was generalized by Hamilton in [5, 6] as follows, suppose that we have a positive smooth solution of ut = −∆u in R × [0, C) then, in the time interval [0,min{C, T}), there holds