A Finiteness Theorem Via the Mean Curvature Flow with Surgery

Abstract

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 .

Regularity and stability results for the level set flow via the mean curvature flow with surgery

“…The first subsection introducing the mean curvature flow we borrow quite liberally from the author's previous paper [28], although a couple additional comments are made concerning the flow of noncompact hypersurfaces. The second subsection concerns the mean curvature flow with surgery as defined by Haslhofer and Kleiner in [17] which differs from the original formulation of the flow with surgery by Huisken and Sinestrari in [17] (see also [6]).…”

“…One such approach is the mean curvature flow with surgery developed by Huisken, Sinestrari [22] (and Brendle and Huisken [6] for the surface case) and later Haslhofer and Kleiner in [17]). The mean curvature flow with surgery "cuts" the manifold into pieces with very well understood geometry and topology and for this and the explicit nature of the flow with surgery is particularly easy to understand (and makes it a useful tool to understand the topology of the space of applicable hypersurfaces; see [7] or [28]). To be able to do this however unfortunately boils down eventually to understanding the nature of the singularities very well and establishing certain quite strong estimates, and all this has only been carried out (in R n+1 at least) for 2-convex compact hypersurfaces.…”

Low Entropy and the Mean Curvature Flow with Surgery

“…There are several corollaries of the surgery; the first consequence was first noted in the original mean curvature flow with surgery paper by Huisken and Sinestrari [30]: Corollary 1.2. If M ∈ M, then M ∼ = S n or a finite connect sum of S n−1 × S 1 The second corollary, an extension of the first corollary, was observed in the 2convex case by the first named author in [40]; only set monotonicity of the MCF was required so the proof immediately adapts to the low entropy case: We turn next to applications of the flow to self shrinkers of low entropy. Perturbing a low entropy self shrinker as in [19] we obtain a low entropy surface with H − x,ν 2 > 0.…”

“…The first subsection introducing the mean curvature flow we borrow quite liberally from the first named author's previous paper [40] although a small introduction to self shrinkers has also been included. The second subsection concerns the mean curvature flow with surgery as constructed by Haslhofer and Kleiner in [23].…”

“…To gain the isotopy one wishes to argue as in the inductive proof of theorem 1.3 in [40] by "extending the necks" found by the surgery. In the mean convex case what one does is consider the future paths, which we'll refer to below as strings, traced out by the tips of the surgery caps under the flow with surgery and "thicken" (take a small tubular neighborhood) the paths to gain an isotopy to a tubular neighborhood of an embedded tree (i.e.…”