The extension problem of the mean curvature flow (I)

On Ilmanen's multiplicity-one conjecture for mean curvature flow with type-I mean curvature

“…In general the multiplicityone conjecture is still wide open . In this paper, using the techniques from our previous work [33] we confirm the multiplicity-one conjecture under the assumption that the mean curvature is of type-I.…”

“…where H denotes the mean curvature of the hypersurface Σ t := x(t)(Σ) and n denotes the outward unit normal of Σ t . In the previous paper [33], we proved that the mean curvature of (1.1) must blow up at the first finite singular time for a closed smooth embedded mean curvature flow in R 3 . This paper can be viewed as a continuation of [33], and we will develop the techniques in [33] further to study the finite time singularities of mean curvature flow.…”

“…In [33], we showed Theorem 1.3 under the assumption that the mean curvature decays exponentially to zero. In this special case, the flow (1.4) converges smoothly to a plane passing through the origin with multiplicity one.…”

“…Furthermore, Le-Sesum [46] proved that the mean curvature is at least of type-I if (1.7) holds. Using Theorem 1.2, we can remove the type-I condition (1.7) of Le-Sesum's result as follows, which can also be viewed as an improvement of the extension theorem in [33].…”

“…We have to show that the flow (1.4) converges smoothly to a self-shrinker with multiplicity one. The strategy is similar to [33], we first show a weak-compactness theorem and obtain the flow convergence is smooth away from a singular set. Then we use stability argument to remove the singular set.…”