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Rigidity of Solitons to the Mean Curvature Flow in $\mathbb{H}^3$ as Translation Surfaces

Tarcios Andrey Ferreira, João Paulo dos Santos

TL;DR

The paper classifies translation surfaces in $\mathbb{H}^3$ generated by a horospherical and a planar curve under mean curvature flow into rigid families: minimal surfaces are either totally geodesic planes or minimal translation cylinders; translators are restricted to horospheres, geodesic planes, or grim-reaper-type translators; and conformal solitons are confined to totally geodesic planes or Grim-reaper cylinders, with explicit parameterizations and associated ODEs. The authors derive these results by modeling $\mathbb{H}^3$ as a Lie group, computing the mean curvature via conformal changes, and reducing the MCF soliton conditions to one-dimensional ODEs for the generating curves, supported by auxiliary lemmas on ODE behavior. The work extends rigidity phenomena from Euclidean settings to hyperbolic geometry, providing sharp classifications and explicit profiles that illuminate the structure of self-similar solutions in $\mathbb{H}^3$. Overall, the paper offers a unified framework for understanding how translation-surface geometry interacts with MCF solitons in a non-Euclidean ambient, with potential implications for singularity analysis and geometric flow comparisons.

Abstract

In the half-space model of the hyperbolic three space with the hyperbolic metric, this same space can be seen as the Lie group, hence, a translation surface is a surface that is given by the product of two curves $α$ and $β$ in this group. Here we present the rigidity of this kind of surfaces for some particular products in the context of minimal surfaces and solitons to the Mean Curvature Flow, also known as self-similar solutions.

Rigidity of Solitons to the Mean Curvature Flow in $\mathbb{H}^3$ as Translation Surfaces

TL;DR

The paper classifies translation surfaces in generated by a horospherical and a planar curve under mean curvature flow into rigid families: minimal surfaces are either totally geodesic planes or minimal translation cylinders; translators are restricted to horospheres, geodesic planes, or grim-reaper-type translators; and conformal solitons are confined to totally geodesic planes or Grim-reaper cylinders, with explicit parameterizations and associated ODEs. The authors derive these results by modeling as a Lie group, computing the mean curvature via conformal changes, and reducing the MCF soliton conditions to one-dimensional ODEs for the generating curves, supported by auxiliary lemmas on ODE behavior. The work extends rigidity phenomena from Euclidean settings to hyperbolic geometry, providing sharp classifications and explicit profiles that illuminate the structure of self-similar solutions in . Overall, the paper offers a unified framework for understanding how translation-surface geometry interacts with MCF solitons in a non-Euclidean ambient, with potential implications for singularity analysis and geometric flow comparisons.

Abstract

In the half-space model of the hyperbolic three space with the hyperbolic metric, this same space can be seen as the Lie group, hence, a translation surface is a surface that is given by the product of two curves and in this group. Here we present the rigidity of this kind of surfaces for some particular products in the context of minimal surfaces and solitons to the Mean Curvature Flow, also known as self-similar solutions.

Paper Structure

This paper contains 18 sections, 8 theorems, 125 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conformal metrics and the corresponding mean curvatures
  4. Translators to the MCF in TEXT
  5. Conformal solitons to the MCF in TEXT
  6. Translation surfaces in TEXT
  7. Main results
  8. Geometry of translation surfaces in TEXT
  9. Auxiliary results
  10. Proof or results
  11. Proof of theorem \ref{['theominfirst']}
  12. Proof of theorem \ref{['theoSolitonfirst']}
  13. Proof of theorem \ref{['theconfSolitonfirst']}
  14. Proof of theorem \ref{['Theo_general']}
  15. Declarations
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\alpha : I \subset \mathbb{R} \to \mathbb{H}^3$ a curve contained in a horosphere $\mathcal{H}_a = \{ (x,y,a) \in \mathbb{H}^3 : a \textnormal{ is constant} \}$ and $\beta: J \subset \mathbb{R} \to \mathbb{H}^3$ is a plane curve (contained in a totally geodesic hyperbolic plane). Then the tra

Figures (3)

  • Figure 1: ODE solution and surface of Theorem \ref{['theominfirst']}
  • Figure 2: ODE numerical solution and Surface of Theorem \ref{['theoSolitonfirst']}.
  • Figure 3: Ode solution and Surface of Theorem \ref{['theconfSolitonfirst']} .

Theorems & Definitions (19)

  • Theorem 1.1
  • Remark 2.1
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.2
  • Remark 2.4
  • Theorem 2.3
  • Remark 2.5
  • Remark 2.6
  • ...and 9 more