Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A new family of translating solitons in hyperbolic space

Antonio Bueno, Rafael López

Abstract

If $ξ$ is a Killing vector field of the hyperbolic space $\h^3$ whose flow are parabolic isometries, a surface $Σ\subset\h^3$ is a $ξ$-translator if its mean curvature $H$ satisfies $H=\langle N,ξ\rangle$, where $N$ is the unit normal of $Σ$. We classify all $ξ$-translators invariant by a one-parameter group of rotations of $\h^3$, exhibiting the existence of a new family of grim reapers. We use these grim reapers to prove the non-existence of closed $ξ$-translators.

A new family of translating solitons in hyperbolic space

Abstract

If is a Killing vector field of the hyperbolic space whose flow are parabolic isometries, a surface is a -translator if its mean curvature satisfies , where is the unit normal of . We classify all -translators invariant by a one-parameter group of rotations of , exhibiting the existence of a new family of grim reapers. We use these grim reapers to prove the non-existence of closed -translators.
Paper Structure (7 sections, 8 theorems, 21 equations, 2 figures)

This paper contains 7 sections, 8 theorems, 21 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['t1']}
  3. Classification of the $\xi$-grim reapers
  4. Proof of Theorem \ref{['t2']}: case $a=0$
  5. Proof of Theorem \ref{['t2']}: case $a\neq0$
  6. Non-existence of closed $\xi$-translators
  7. Appendix: grim reapers in $\mathbb H^3$

Key Result

Theorem 1.2

Figures (2)

  • Figure 1: The two types of generating curves of grim reapers when $\xi=\partial_y$.
  • Figure 2: Left: the phase plane and the orbit passing through $(2,0)$. Right: the corresponding curve $\alpha$ of maximum height $2$.

Theorems & Definitions (15)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Remark 3.1
  • Proposition 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 5 more