MCF-stack: Special Solutions of Mean Curvature Flow

§ 3.33 Helicoidal rotators-translators in the hyperbolic-plane product65C371A89575

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 — 2024-02-07 (first arXiv submission).
65C371A89575
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For each pitch $h>0$, de Lima–Ramos–dos Santos construct a one-parameter family of complete helicoidal solitons in $\mathbb H^2\times \mathbb R$. In the hyperboloid model a profile $\sigma (u)=(\alpha (u),\sqrt{1+|\alpha (u)|^2})$ generates

\[ X(u,v)=(e^{vJ}\alpha (u),\sqrt{1+|\alpha (u)|^2},hv),\qquad u,v\in \mathbb R, \]

where $J(x,y)=(-y,x)$. The profile is determined by the curvature ODE in Section 2, not by an Archimedean spiral. Its trace has two unbounded properly embedded arms spiraling around the center; this does not by itself establish embeddedness of their union or of the full immersion. The surface is a rotator exactly when it is a translator for $-h\partial _ z$ in the source convention. Rotation and vertical translation describe the same evolving sets because of the helicoidal symmetry; they must not be added to double the speed. These are eternal isometric flows. The illustration uses the two-arm profile and helical orbits in disk × height coordinates, labeled schematic. No Euclidean Gaussian area, entropy, or unrestricted uniqueness claim is assigned.

Sources: [LimaRamosSantos2025].

Helicoidal rotators-translators in the hyperbolic-plane product. Source-informed two-arm spiral and helicoidal orbit diagram in disk × height coordinates. Rotation and vertical translation are alternative descriptions of the same set evolution.
Source-informed schematic. Source-informed two-arm spiral and helicoidal orbit diagram in disk × height coordinates. Rotation and vertical translation are alternative descriptions of the same set evolution. 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.

Complete helicoidal surface in ℍ² × ℝ · Riemannian ambient manifold · Not recorded
Ambient space
Riemannian ambient manifold
Topology
Immersed parameter plane ℝ²
Embeddedness
Not recorded: The reviewed statement gives an immersion; no general embedding assertion is recorded.
Compactness
Noncompact
Surface genus
Genus zero.
Topological ends
One topological end.
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. [LimaRamosSantos2025] Ronaldo F. de Lima, Álvaro K. Ramos, and João Paulo dos Santos. Rotators-Translators to Mean Curvature Flow in $\mathbb H^2\times \mathbb R$. Archiv der Mathematik, 124:343–353, 2025. URL: https://arxiv.org/abs/2402.04757, arXiv:2402.04757, doi:10.1007/s00013-024-02076-9.All uses in this project ↩ 1
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