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Rotators-Translators to Mean Curvature Flow in $\mathbb H^2\times\mathbb R$

Ronaldo F. de Lima, Álvaro K. Ramos, João Paulo dos Santos

Abstract

We establish the existence of one-parameter families of helicoidal surfaces of $\mathbb H^2\times\mathbb R$ which, under mean curvature flow, simultaneously rotate about a vertical axis and translate vertically.

Rotators-Translators to Mean Curvature Flow in $\mathbb H^2\times\mathbb R$

Abstract

We establish the existence of one-parameter families of helicoidal surfaces of which, under mean curvature flow, simultaneously rotate about a vertical axis and translate vertically.
Paper Structure (2 sections, 4 theorems, 52 equations, 4 figures)

This paper contains 2 sections, 4 theorems, 52 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorems \ref{['th-main']} and \ref{['th-translator']}

Key Result

Theorem 1

For any $h>0,$ there exists a one-parameter family of complete rotators to MCF in $\mathbb{H} ^2\times\mathbb{R}$ which are helicoidal surfaces of pitch $h.$ For each such surface, the trace of the generating curve in $\mathbb{H} ^2$ consists of two unbounded properly embedded arms centered at a poi

Figures (4)

  • Figure 1: A $1$-pitched helicoidal rotator-translator in $\mathbb{H} ^2\times\mathbb{R}$ (right) and its generating curve in the Poincaré disk $\mathbb{H} ^2$ (left).
  • Figure 2: Phase portrait of system \ref{['eq-system007']} for $h=2.$
  • Figure 3: Graph of $\uptau.$
  • Figure 4: Generating curve of a $1$-pitched helicoidal rotator-translator of $\mathbb{H} ^2\times\mathbb{R}$ in the hyperboloid model of $\mathbb{H} ^2$.

Theorems & Definitions (24)

  • Theorem 1
  • Theorem 2
  • Definition 3
  • Proposition 4
  • proof
  • Lemma 5
  • proof : Proof of Theorem \ref{['th-main']}
  • Claim 6
  • proof : Proof of Claim \ref{['claim-noconstantsolutionsH2xR']}
  • Claim 7
  • ...and 14 more