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Dynamical stability of various convex graphical translators

Junyoung Park

TL;DR

The paper addresses the stability of graphical translators for mean curvature flow by proving the long-time existence of graphical solutions from continuous slab data and establishing dynamical stability for a class of translators, including the grim reaper, 2D translators in $\\ extbf{R}^3$, and asymptotically cylindrical translators. The approach combines barrier methods (ancient pancakes), Hamilton's differential Harnack inequality, and a recent classification framework for asymptotically cylindrical translators to show that perturbed flows converge to translating solutions modulo rigid motions. Key contributions include weakening closeness assumptions, providing convergence results without full $C^{2+α}$ perturbation requirements, and connecting stability to the broader translator classification program. The results advance understanding of type II singularity models in mean curvature flow and offer a robust, barrier-driven methodology applicable to convex graphical translators in higher dimensions.

Abstract

In the first part of the paper, we prove the existence of longtime solution to mean curvature flow starting from a graph of a continuous function defined over a slab. Then, we establish dynamical stability results for various types of graphical translators to mean curvature flow, namely the grim reaper, two dimensional graphical translators, and asymptotically cylindrical translators.

Dynamical stability of various convex graphical translators

TL;DR

The paper addresses the stability of graphical translators for mean curvature flow by proving the long-time existence of graphical solutions from continuous slab data and establishing dynamical stability for a class of translators, including the grim reaper, 2D translators in , and asymptotically cylindrical translators. The approach combines barrier methods (ancient pancakes), Hamilton's differential Harnack inequality, and a recent classification framework for asymptotically cylindrical translators to show that perturbed flows converge to translating solutions modulo rigid motions. Key contributions include weakening closeness assumptions, providing convergence results without full perturbation requirements, and connecting stability to the broader translator classification program. The results advance understanding of type II singularity models in mean curvature flow and offer a robust, barrier-driven methodology applicable to convex graphical translators in higher dimensions.

Abstract

In the first part of the paper, we prove the existence of longtime solution to mean curvature flow starting from a graph of a continuous function defined over a slab. Then, we establish dynamical stability results for various types of graphical translators to mean curvature flow, namely the grim reaper, two dimensional graphical translators, and asymptotically cylindrical translators.
Paper Structure (6 sections, 20 theorems, 242 equations)

This paper contains 6 sections, 20 theorems, 242 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Existence of graphical solution
  4. Dynamical stability of grim reaper
  5. Dynamical stability of graphical translators in $\mathbf{R}^3$
  6. Dynamical stability of asymptotically cylindrical translators

Key Result

Theorem 1.1

For each $b > 0$, $n \geq 1$, let $\Omega^n_b = \mathbf{R}^{n-1} \times (-b,b)$. For given $u_0 \in C^0(\Omega^n_b)$, there exists $u \in C^{\infty}(\Omega^n_b \times (0, \infty)) \cap C^0(\Omega^n_b \times [0, \infty))$ so that $u$ solves the initial value problem If we further assume that for each $J > 0$, one can find $\delta > 0$ so that $|u_0(x)| > J$ for all $x \in \Omega^n_b$ with $b - \de

Theorems & Definitions (50)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Definition 1.1
  • Theorem 1.9
  • ...and 40 more