MCF-stack: Special Solutions of Mean Curvature Flow

§ 3.32 Hyperbolic Killing translators and catenoids (de Lima–Ramos–dos Santos, 2023; de Lima–Ramos, 2026)EA9BCA366717

Earliest recorded reference: (first arXiv submission).

Paper credits and chronology

Paper credits are chronological and include every author of the cited papers. They identify sources, not exclusive discovery credit. Background-only citations do not determine the date of this example. A classical example may predate the references recorded here.

  1. de Lima–Ramos–dos Santos — 2023-07-26 (first arXiv submission).
  2. de Lima–Ramos — 2026-02-02 (first arXiv submission).
EA9BCA366717
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In upper-half-space $\mathbb H^{n+1}=\{ z>0\} $ with metric $z^{-2}(dx^2+dz^2)$, the maps $\Phi _\tau (x,z)=(e^\tau x,e^\tau z)$ are hyperbolic isometries. de Lima–Ramos–dos Santos construct rotational translating catenoids and translation-invariant grim-reaper analogues in $\mathbb H^3$, as well as helicoidal rotators-translators. The 2026 work of de Lima–Ramos supplies rotational catenoid barriers in all dimensions $n+1\ge 3$. These should not be called Euclidean self-expanders: the ambient metric is fixed and the motion is isometric. The rotational examples are properly embedded annuli, with topology $S^{n-1}\times \mathbb R$ and two ends. Their ODE construction and the speed of the generating Killing field are part of the data. The uniqueness statements for the surface catenoids and grim reapers impose the symmetry/asymptotic hypotheses of the cited theorems. Horosphere stability in the later paper concerns radial graphical perturbations with decay; it is not a stability theorem for every catenoid. The figure depicts the rotational and invariant profiles in the half-space model and compares isometric dilation times. Its curves are source-informed schematics, not a computation of the nonlinear profile ODE.

Sources: [LimaRamosSantos2026, LimaRamos2026].

Hyperbolic Killing translators and catenoids. Upper-half-space profile and rotational orbit schematic under hyperbolic dilation. The boundary plane remains fixed; dilation is an ambient hyperbolic isometry.
Source-informed schematic. Upper-half-space profile and rotational orbit schematic under hyperbolic dilation. The boundary plane remains fixed; dilation is an ambient hyperbolic isometry. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Rotational annuli in ℍ³ · Riemannian ambient manifold · Embedded
Ambient space
Riemannian ambient manifold
Topology
S¹ × ℝ
Embeddedness
Embedded: The cited construction gives an embedding for this case.
Compactness
Noncompact
Surface genus
Genus zero.
Topological ends
Two topological ends.
Entropy
Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Gaussian area
Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Rotational catenoids in ℍⁿ⁺¹, n ≥ 3 · Riemannian ambient manifold · Embedded
Ambient space
Riemannian ambient manifold
Topology
Sⁿ⁻¹ × ℝ
Embeddedness
Embedded: The cited construction gives an embedding for this case.
Compactness
Noncompact
Surface genus
Not a smooth surface; surface genus is inapplicable.
Topological ends
Two topological ends.
Entropy
Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Gaussian area
Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Grim and helicoidal surface constructions · Riemannian ambient manifold · Not recorded
Ambient space
Riemannian ambient manifold
Topology
Depends on the specified invariant profile
Embeddedness
Not recorded: The reviewed statement gives an immersion; no general embedding assertion is recorded.
Compactness
Noncompact
Surface genus
Not recorded.
Topological ends
Depends on the input; asymptotic wings are not automatically topological ends.
Entropy
Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Gaussian area
Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.

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References on this page

  1. [LimaRamosSantos2026] R. F. de Lima, A. K. Ramos, and J. P. dos Santos. Solitons to Mean Curvature Flow in the hyperbolic 3-space. Revista Matemática Iberoamericana, 42:769–802, 2026. URL: https://arxiv.org/abs/2307.14136, arXiv:2307.14136, doi:10.4171/rmi/1593.All uses in this project ↩ 1
  2. [LimaRamos2026] Ronaldo F. de Lima and Álvaro K. Ramos. Dynamical Stability of Translating Solitons to Mean Curvature Flow in Hyperbolic Space. arXiv preprint, 2026. URL: https://arxiv.org/abs/2602.02424, arXiv:2602.02424.All uses in this project ↩ 1
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