MCF-stack: Special Solutions of Mean Curvature Flow

§ 3.28 Rotational translators in the hyperbolic-plane productE33234341F7D

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. Bueno — 2018-03-07 (first arXiv submission).
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In $\mathbb H^2\times \mathbb R$, with $g=dr^2+\sinh ^2r\, d\varphi ^2+dz^2$, Bueno constructs an entire rotational bowl and a one-parameter family of properly embedded translating catenoids. The bowl is a complete disk; each catenoid is a complete annulus with two graphical ends outside a compact set, parametrized by its positive neck radius. These are intrinsic product-metric examples, not the Euclidean bowl and winglike solitons. The source uses the average of the two principal curvatures. Its unit-speed profile satisfies

\[ r'=\cos \theta ,\quad z'=\sin \theta ,\quad \theta '=2\cos \theta -\coth (r)\sin \theta . \]

Consequently, with the trace mean-curvature vector used by this catalogue, the same surface evolves by $z\mapsto z+2t$, for every $t\in \mathbb R$. The bowl has the smooth axis condition $\theta (r)=r+O(r^3)$; the annuli start at a positive minimum radius with vertical tangent. Uniqueness is within the rotational class and the additional asymptotic hypotheses of the source. Numerical meridians are revolved and displayed in the Poincaré-disk product coordinates $\rho =\tanh (r/2)$; Euclidean-looking distances in this drawing are not intrinsic distances.

Sources: [Bueno2018].

Rotational translators in the hyperbolic-plane product. Numerically integrated product-metric bowl and catenoid meridians in disk × height coordinates. Vertical speed is 2 in the catalogue trace convention.
Numerical ODE profile. Numerically integrated product-metric bowl and catenoid meridians in disk × height coordinates. Vertical speed is 2 in the catalogue trace convention. 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.

Bowl in ℍ² × ℝ · Riemannian ambient manifold · Embedded
Ambient space
Riemannian ambient manifold
Topology
ℝ²
Embeddedness
Embedded: The cited construction gives an embedding for this case.
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.
Catenoid in ℍ² × ℝ · 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.

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References on this page

  1. [Bueno2018] Antonio Bueno. Translating solitons of the mean curvature flow in the space $\mathbb H^2\times \mathbb R$. Journal of Geometry, 2018. URL: https://arxiv.org/abs/1803.02783, arXiv:1803.02783, doi:10.1007/s00022-018-0447-x.All uses in this project ↩ 1
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