Mean curvature flow of star-shaped hypersurfaces

Low Entropy and the Mean Curvature Flow with Surgery

“…Above β is a lower bound on H which for our case follows easily from the diameter bounds on the initial surface. The proof of this goes exactly as in the proof of theorem 3.1 in [36], where Lin proves it for starshaped mean curvature flow. The only point of that proof that might require some clarification is how to check his claim 3.8 (a one-sided minimization result that allows one to upgrade Hausdorff convergnece of a certain sequence in the proof to smooth convergence on compact sets).…”

“…Before moving on to describing the construction of the surgery flow we briefly discuss the aforementioned local curvature estimates for F -noncollapsing flows-these were used in the above proposition. Lin showed these for starshaped mean curvature flow (theorem 3.1 in [36]), although he remarks (specifically see remark 2.4 in [36]) that these estimates are true for a wide class of flows including ours: Theorem 4.5. (Local curvature estimate).…”

“…There are other convexity assumptions under which a surgery theory are almost certainly possible; as noted in [36] F α-noncollapsing is preserved for any F of the form F = a 1 x, ν + (a 2 + 2a 1 t)H and the necessary estimates for surgery should go through (the case above is a 1 = −1, a 2 = 2). Note also that when n = 2, the flow with surgery is possible (after deforming) for any self shrinker, even of "high" entropy; topologically this seems to give no information not already known by the classical theory of surfaces.…”

“…One can easily then check using this weighted area that the proof of claim 3.8 in [36] goes through. Now we are ready to describe how the flow with surgery is constructed.…”

“…Perturbing a low entropy self shrinker as in [19] we obtain a low entropy surface with H − x,ν 2 > 0. From [43,36] a related convexity assumption is preserved under the flow and in fact adapting some estimates of Lin [36] we may then run the surgery essentially using that high curvature regions will still be mean convex. Combining it with a recent result of Hershkovits and White [27] yields: Theorem 1.4.…”