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Classification of Semigraphical Translators

Francisco Martín, Mariel Sáez, Raphael Tsiamis, Brian White

Abstract

We complete the classification of semigraphical translators for mean curvature flow in $\mathbb{R}^3$ that was initiated by Hoffman-Martín-White. Specifically, we show that there is no solution to the translator equation on the upper half-plane with alternating positive and negative infinite boundary values, and we prove the uniqueness of pitchfork and helicoid translators. The proofs use Morse-Radó theory for translators and an angular maximum principle.

Classification of Semigraphical Translators

Abstract

We complete the classification of semigraphical translators for mean curvature flow in that was initiated by Hoffman-Martín-White. Specifically, we show that there is no solution to the translator equation on the upper half-plane with alternating positive and negative infinite boundary values, and we prove the uniqueness of pitchfork and helicoid translators. The proofs use Morse-Radó theory for translators and an angular maximum principle.

Paper Structure

This paper contains 6 sections, 8 theorems, 87 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. An Angular Maximum Principle
  4. Uniqueness for Pitchforks
  5. Uniqueness for Translating Helicoids
  6. Nonexistence of The Yeti

Key Result

Theorem 1

A semigraphical translator $M$ in $\mathbf{R}^3$ is one of the following: Up to isometries of $\mathbf{R}^3$,

Figures (2)

  • Figure 1: Left: A fundamental piece of the pitchfork of width $\pi$. Right: The whole surface, obtained from the fundamental piece by a $180^{\circ}$ rotation around the $z$-axis.
  • Figure 2: Left: A fundamental piece of the helicoid of width $\pi/2$. Right: Part of the surface, obtained by successive reflections along the vertical boundary lines.

Theorems & Definitions (25)

  • Theorem 1
  • Theorem 2
  • proof
  • Theorem 3
  • Remark 4
  • proof
  • Claim 1
  • proof
  • Theorem 5
  • Remark 6
  • ...and 15 more