Convexity estimates for a nonhomogeneous mean curvature flow

Abstract

We study the evolution of a closed immersed hypersurface whose speed is given by a function φ(H ) of the mean curvature asymptotic to H/ ln H for large H . Compared with other nonlinear functions of the curvatures, this speed has some good properties which allow for an easier study of the formation of singularities in the nonconvex case. We prove apriori estimates showing that any surface with positive mean curvature at the initial time becomes asymptotically convex near a singularity. Similar estimates also hold for the mean curvature flow; for the flow considered here they admit a simpler proof based only on the maximum principle.

Convexity estimates for hypersurfaces moving by convex curvature functions

“…However, a slight modification reveals that it is still possible to compare compact solutions with spheres. curvature having a certain asymptotic behaviour [Alessandroni and Sinestrari 2010]. In a companion paper [Andrews et al 2012], we were able to exploit the simplified structure of the evolution equation for the second fundamental form in two dimensions (see also [Schulze 2006;Andrews 2007;McCoy 2011]) to prove that an asymptotic convexity estimate holds in surprising generality for flows of surfaces, namely for any surface flow (1-1)-(1-2) satisfying conditions (i)-(iv).…”

Convexity estimates for surfaces moving by curvature functions

“…Second, there is in general no Harnack inequality available sufficient to classify type-II singularities, although the latter is known for quite a wide sub-class of flows [4]. And finally, until recently, there was no analogue of the Huisken-Sinestrari asymptotic convexity estimate for most other flows, with the notable exception of the result of Alessandroni and Sinestrari [1], which applies to a special class of flows by functions of the mean curvature having a certain asymptotic behaviour. In a companion paper [11], the authors prove that an asymptotic convexity estimate holds for fully non-linear flows (1.1) satisfying Conditions 1.1 if, in addition, the speed f is a convex function.…”

“…We remark that the proof of Theorem 1.2 utilises a Stampacchia iteration procedure analogous to those of [24,26,27], whereas the result of [1] is proved more directly, using the maximum principle.…”

“…Unfortunately, the final two terms of the evolution equation (3.3) can be positive, and we cannot obtain the required estimate directly from the maximum principle, as in [1,35]. However, the Stampacchia iteration method of [24,26] is still available to us.…”

“…Note that we lose no generality by assuming further that Γ contains (1,1) and f is normalised such that f (1, 1) = 1. Furthermore, since f is symmetric, we may at each point (x, t) ∈ M × [0, T ) assume that κ 2 (x, t) ≥ κ 1 (x, t).…”