Ancient asymptotically cylindrical flows and applications

  title={Ancient asymptotically cylindrical flows and applications},
  author={Kyeongsu Choi and Robert Haslhofer and Or Hershkovits and Brian White},
  journal={Inventiones mathematicae},
In this paper, we prove the mean-convex neighborhood conjecture for neck singularities of the mean curvature flow in $\mathbb{R}^{n+1}$ for all $n\geq 3$: we show that if a mean curvature flow $\{M_t\}$ in $\mathbb{R}^{n+1}$ has an $S^{n-1}\times \mathbb{R}$ singularity at $(x_0,t_0)$, then there exists an $\varepsilon=\varepsilon(x_0,t_0)>0$ such that $M_t\cap B(x_0,\varepsilon)$ is mean-convex for all $t\in(t_0-\varepsilon^2,t_0+\varepsilon^2)$. As in the case $n=2$, which was resolved by the… 
Nonlocal estimates for the Volume Preserving Mean Curvature Flow and applications
. We obtain estimates on nonlocal quantities appearing in the Volume Preserving Mean Curvature Flow (VPMCF) in the closed, Euclidean setting. As a result we demonstrate that blowups of finite time
Ancient solutions and translators of Lagrangian mean curvature flow
. Suppose that M is an almost calibrated, exact, ancient solution of Lagrangian mean curvature flow in C n . We show that if M has a blow-down given by the static union of two Lagrangian subspaces
On the dynamics of formation of generic singularities of mean curvature flow
  • Z. Gang
  • Mathematics
    Journal of Functional Analysis
  • 2022
For Γn−1 ⊂ ∂B1 ⊂ R, consider Σ ⊂ B1 a hypersurface with ∂Σ = Γ, with least area among all such surface. (This is known as the Plateau problem). It might happen that Σ is singular. For example,
On the Stability of Cylindrical Singularities of the Mean Curvature Flow
We study the rescaled mean curvature flow (MCF) of hypersurfaces that are global graphs over a fixed cylinder of arbitrary dimensions. We construct an explicit stable manifold for the rescaled MCF of
Uniqueness and stability of singular Ricci flows in higher dimensions
In this short note, we observe that the Bamler-Kleiner proof of uniqueness and stability for 3-dimensional Ricci flow through singularities generalizes to singular Ricci flows in higher dimensions
A note on blowup limits in 3d Ricci flow
. We prove that Perelman’s ancient ovals occur as blowup limit in 3d Ricci flow through singu- larities if and only if there is an accumulation of spherical singularities.
We consider ancient noncollapsed mean curvature flows in R whose tangent flow at −∞ is a bubble-sheet. We carry out a fine spectral analysis for the bubble-sheet function u that measures the
Hearing the shape of ancient noncollapsed flows in $\mathbb{R}^{4}$
We consider ancient noncollapsed mean curvature flows in R 4 whose tangent flow at −∞ is a bubble-sheet. We carry out a fine spectral analysis for the bubble-sheet function u that measures the


A Topological Property of Asymptotically Conical Self-Shrinkers of Small Entropy
For any asymptotically conical self-shrinker with entropy less than or equal to that of a cylinder we show that the link of the asymptotic cone must separate the unit sphere into exactly two
Embedded self-similar shrinkers of genus 0
We confirm a well-known conjecture that the round sphere is the only compact, embedded self-similar shrinking solution to the mean curvature flow with genus $0$. More generally, we show that the only
A strong maximum principle for varifolds that are stationary with respect to even parametric elliptic functionals
  • Indiana Univ. Math. J., 38(3):683–691
  • 1989
Complexity of parabolic systems
This work proves that in any dimension and codimension any ancient flow that is cylindrical at − ∞ $-\infty $ must be a flow of hypersurfaces in a Euclidean subspace and shows rigidity of cylinders as shrinkers in all dimension and allcodimension in a very strong sense.
Topology of Closed Hypersurfaces of Small Entropy
We use a weak mean curvature flow together with a surgery procedure to show that all closed hypersurfaces in $\mathbb{R}^4$ with entropy less than or equal to that of $\mathbb{S}^2\times \mathbb{R}$,
Uniqueness of blowups and Łojasiewicz inequalities
Once one knows that singularities occur, one naturally wonders what the singu- larities are like. For minimal varieties the first answer, already known to Federer-Fleming in 1959, is that they weakly
Generic mean curvature flow I; generic singularities
It has long been conjectured that starting at a generic smooth closed embedded surface in R^3, the mean curvature flow remains smooth until it arrives at a singularity in a neighborhood of which the
Ancient low entropy flows
  • mean convex neighborhoods, and uniqueness. arXiv:1810.08467
  • 2018
Nonfattening of Mean Curvature Flow at Singularities of Mean Convex Type
We show that a mean curvature flow starting from a compact, smoothly embedded hypersurface M ⊆ ℝn + 1 remains unique past singularities, provided the singularities are of mean convex type, i.e., if