In this article we extend an unknottedness theorem for compact self shrinkers to the mean curvature flow to shrinkers with finite topology and one asymptotically conical end, which conjecturally comprises the entire set of self shrinkers with finite topology and one end. A partial result for asymptotically cylindrical such shrinkers is also given. The mean curvature flow itself is used in the argument presented.

Abstract. In this article, we use the recently developed mean curvature flow with surgery for 2-convex hypersurfaces to prove certain isotopy existence and finally extrinsic finiteness results (in the spirit of Cheeger's finiteness theorem) for the space of 2-convex closed embedded hypersurfaces in R n+1 .

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In this article we use the mean curvature flow with surgery to derive regularity estimates for the level set flow going past Brakke regularity in certain special conditions allowing for 2-convex regions of high density. We also show a stability result for the plane under the level set flow.

In this note we show that self shrinkers in R 3 are "topologically standard" in that any genus g compact self shrinker is ambiently isotopic to the standard genus g embedded surface in R 3 . As a consequence self shrinking tori are unknotted.