Low entropy and the mean curvature flow with surgery

On self shrinkers of medium entropy in $\mathbb{R}^4$

“…This extends to the noncompact case joint work of the author and S. Wang [31] on compact self shrinkers M n ⊂ R n+1 when n = 3, where they showed for each n ≥ 3 closed self shrinkers M n with Λ(M ) < Λ n−2 are diffeomorophic to S n , which in turn extends a result of Colding, Ilmanen, Minicozzi, and White [13] which says closed self shrinkers with entropy less than Λ n−1 Λ n−2 are diffeomorphic to S n , hence weakening the assumed entropy bound. In a similar manner, the result above extends (in a weaker sense than the compact case) a result of Bernstein and L. Wang [5] for noncompact shrinkers in R 4 , where they showed (amongst other results, see corollary 1.4 therein) asymptotically conical self shrinkers M 3 ⊂ R 4 satisfying λ(M ) ≤ Λ 2 the stronger conclusion that they are diffeomorphic to R 3 ; our argument does at least recover their statement though as discussed at the end of the proof.…”

“…With this in mind, the author showed in his previous article [30] that one can then construct a flow with surgery using the RMCF on suitable perturbations of noncompact self shrinkers, and that as one lets the surgery parameters degenerate indeed the surgery converges to the level set flow when n = 2. This can be readily combined with the aforementioned joint work with S. Wang [31] to show the following:…”

“…The next natural entropy condition to consider then is a bound by Λ n−2 , which in this paper we refer to naturally as a medium entropy bound; one might expect studying surfaces with this entropy bound to be tractable because morally it implies that mean convex singularities encountered will be 2-convex, which as implied above in the discussion on surgery are the easiest to consider/flow through (after convex ones). Indeed this philosophy was carried out in the compact case in the the joint work with S. Wang [31] ("low" in its title refers to what we define as medium). An important observation for the argument of this paper is that this philosophy can be extended to the noncompact setting, but there are some important new issues to consider.…”

“…The main result of the paper is the following, in part inspired by arguments in [4,23,26,30,31,42]. The basic idea is by considering renormalized mean curvature flows out of (appropriate perturbations of) asymptotically conical self shrinkers below we may use the entropy assumption to constrain which types of singularities may occur.…”

Precise asymptotics near a pinched disk singularity formed by mean curvature flow

“…In [7,8], Gerhard and Sinestrari, Haslhofer and Kleiner developed the existence of mean curvature flow with surgery starting from arbitrary 2-convex hypersurfaces, with a detailed description of a neighborhood of the singularities and surgery procedure with controlled geometry and topology. For the results on non 2-convex flows, see [14,2,15].…”