Mean curvature flow with generic initial data

Abstract

We show that the mean curvature flow of generic closed surfaces in R 3 avoids asymptotically conical and non-spherical compact singularities. We also show that the mean curvature flow of generic closed low-entropy hypersurfaces in R 4 is smooth until it disappears in a round point. The main technical ingredient is a longtime existence and uniqueness result for ancient mean curvature flows that lie on one side of asymptotically conical or compact shrinking solitons.

Initial Perturbation of the Mean Curvature Flow for closed limit shrinker

“…We also want to compare our work with Chodosh-Choi-Mantoulidis-Schulze's work [CCMS1]. In [CCMS1], Chodosh-Choi-Mantoulidis-Schulze have another interpretation of positive perturbations.…”

“…We also want to compare our work with Chodosh-Choi-Mantoulidis-Schulze's work [CCMS1]. In [CCMS1], Chodosh-Choi-Mantoulidis-Schulze have another interpretation of positive perturbations. After a positive perturbation, the perturbed MCF will be staying on one side of the original MCF by the avoidance principle.…”

“…Compared with the previous work on generic perturbations of MCF like [CM1], [BS], [Su], our perturbation is applied to the initial data, while the previous perturbations are applied to a timing very closed to the singular time. Recently, Chodosh-Choi-Mantoulidis-Schulze also study the perturbation of the initial data of a MCF in [CCMS1] and [CCMS2]. In particular, they proved that after a generic initial positive perturbation, the perturbed MCF will avoid any singularities modeled by non-generic closed and conical self-shrinkers.…”

“…For instance, for any orbit with an initial condition on the stable manifold, its forward orbit approaches Σ exponentially under the RMCF, while for any orbit with an initial condition on the unstable manifold, its backward orbit approaches Σ exponentially under the RMCF. The latter gives ancient solutions which were constructed by [ChM,CCMS1]. For a generic initial point, its orbit is shadowed by an orbit on the center-unstable manifold, which has a finite dimension.…”

Translating surfaces under flows by sub-affine-critical powers of Gauss curvature

“…An emphasis should be given to Section 2.3 which accounts for a perspective that a solution is a one-parameter family of functions parametrized by a height variable s = log l. Then the translator can be seen as a solution to a first-order ODE in (2.31) and (2.32). Indeed, this formulation allows us to apply the Merle-Zaag's ODE Lemma A.1 in [49] which plays the crucial role in recent researches of the classification of ancient flows; [9,10,11,12,13,14,15,33,17,26,34,35,37].…”