The extension problem of the mean curvature flow (I)

Abstract

We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in R 3 .

Self-shrinkers with bounded HA

“…} is a mean curvature flow, i.e., ∂ t x = −Hn. We obtain some kind of ǫ−regularity from Corollary 3.11 and Theorem 3.7 in Li-Wang [15] and the interior estimates of Ecker and Huisken in [17].…”

“…By virtue of the energy estimate above and the ǫ−regularity from Li-Wang [15] we find that the space of properly embedded self-shrinkers with uniformly bounded entropy, uniformly bounded |HA| and uniformly bounded local energy is compact. The organization of this paper is as follows.…”

“…Then Σ has finite weighted volume Volwhere C is a positive constant depending only on Σ e −f dv.Provided bounded mean curvature we can also get volume ratio lower bound. See Lemma 3.5 in Li-Wang [15].Lemma 2.4. (Lemma 3.5 of [15]) Let Σ n ֒→ R n+1 be a properly immersed hypersurface in B(x 0 , r 0 ) with x 0 ∈ Σ and sup Σ |H| ≤ Λ.…”

“…Provided bounded mean curvature we can also get volume ratio lower bound. See Lemma 3.5 in Li-Wang [15].…”

Type II smoothing in mean curvature flow

“…Dropping the assumption of mean convexity, it was shown in [7,8,10] by Lin-Sesum and Le-Sesum, and in [13] by Xu-Ye-Zhao that for mean curvature flow of closed hypersurfaces the mean curvature needs to blow up at the first singular time, given some extra assumptions, such as having only Type I singularities or being close to a sphere in the L 2 sense. More recently, in [9], Li and Wang showed, using a quite involved argument that in the case of closed surfaces in R 3 the mean curvature always blows up at the first singular time. The question of boundedness of the mean curvature on a singular mean curvature flow is therefore completely settled in the case of compact surfaces in R 3 , and a variety of extra assumptions for hypersurfaces in higher dimensions.…”