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Rigidity and non-existence results for collapsed translators

Debora Impera, Niels Martin Møller, Michele Rimoldi

TL;DR

The article addresses the problem of characterizing collapsed self-translating solitons in $\mathbb{R}^3$ with finite entropy. It introduces a weighted parabolic framework using the drift Laplacian $L=\Delta-e_3\cdot\nabla$ and a universal $L$-superharmonic function to prove a sharp rigidity result: among slab-confined translators of width $\pi$, the grim reaper cylinder is the unique surface that is bounded from below. The authors also derive nonexistence results for slab widths $\omega<\pi$ and for certain slanted halfspaces when $\omega=\pi$, and show a boundary-preserving rigidity to vertical planes under prescribed boundary data. Overall, the work provides a direct, parabolicity-based approach to rigidity and nonexistence results for collapsed translators, complementing existing mean-convex classifications and highlighting the role of finite entropy and the Ilmanen conformal metric in translating solitons.

Abstract

We prove a rigidity result for mean curvature self-translating solitons, characterizing the grim reaper cylinder as the only finite entropy self-translating 2-surface in $\mathbb{R}^3$ of width $π$ and bounded from below. The proof makes use of parabolicity in a weighted setting applied to a suitable universally $L$-superharmonic function defined on translaters in such slabs.

Rigidity and non-existence results for collapsed translators

TL;DR

The article addresses the problem of characterizing collapsed self-translating solitons in with finite entropy. It introduces a weighted parabolic framework using the drift Laplacian and a universal -superharmonic function to prove a sharp rigidity result: among slab-confined translators of width , the grim reaper cylinder is the unique surface that is bounded from below. The authors also derive nonexistence results for slab widths and for certain slanted halfspaces when , and show a boundary-preserving rigidity to vertical planes under prescribed boundary data. Overall, the work provides a direct, parabolicity-based approach to rigidity and nonexistence results for collapsed translators, complementing existing mean-convex classifications and highlighting the role of finite entropy and the Ilmanen conformal metric in translating solitons.

Abstract

We prove a rigidity result for mean curvature self-translating solitons, characterizing the grim reaper cylinder as the only finite entropy self-translating 2-surface in of width and bounded from below. The proof makes use of parabolicity in a weighted setting applied to a suitable universally -superharmonic function defined on translaters in such slabs.
Paper Structure (11 sections, 12 theorems, 49 equations)

This paper contains 11 sections, 12 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Examples
  3. Some preliminary material
  4. Background on weighted manifolds and parabolicity.
  5. Useful facts about translators
  6. Stability for translators
  7. Rigidity in slabs
  8. Rigidity of collapsed translators with prescribed boundary
  9. Equidistant surfaces to translators with respect to the conformal metric
  10. Geodesics of $(\mathbb{R}^{3}, \widetilde{\langle\,,\,\rangle}$)
  11. $\widetilde{\langle\,,\,\rangle}$-equidistant surfaces to translators

Key Result

Proposition 1.1

Let $\Sigma^2\subseteq\mathbb{R}^{3}$ be a translator, which is a graph $\{(x_1,x_2,u(x_1,x_2))\}$ and contained in a vertical slab of width $\pi$. Then, up to rigid motions, $\Sigma$ is a standard grim reaper cylinder $\Gamma\times\mathbb{R}$.

Theorems & Definitions (22)

  • Proposition 1.1: HIMW, BLT20
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Remark 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Definition 3.1
  • Proposition 3.2: IPS, ILPS
  • Lemma 3.3
  • ...and 12 more