MCF-stack: Special Solutions of Mean Curvature Flow

§ 3.35 Parabolic Killing grim reapers in hyperbolic space2C2BD4493223

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–López — 2024-02-08 (first arXiv submission).
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Bueno–López construct a family of grim-reaper analogues in $\mathbb H^3$ driven by the parabolic Killing field $\xi =a\partial _ x+b\partial _ y$ in the upper-half-space model. The metric is $z^{-2}(dx^2+dy^2+dz^2)$ and the isometries are $(x,y,z)\mapsto (x+a\tau ,y+b\tau ,z)$. The construction classifies the relevant invariant surfaces via a planar ODE. It differs from the hyperbolic-axis translation family, whose half-space coordinates dilate. With the average-curvature convention in this paper, the corresponding trace-MCF time is $\tau =2t$. The ambient isometry group exists for all real time. The nontrivial profiles and their domains are those of Section 3; no uniform embedding or completeness assertion is inferred for arbitrary initial data of the ODE. The rotational no-go results of Section 2 eliminate other proposed invariant cases rather than constructing additional solitons. The figure draws the asymptotic shape and horizontal Killing direction in a fixed half-space chart; it is schematic and is not a Euclidean grim curve asserted to solve the hyperbolic equation.

Sources: [BuenoLopez2024a].

Parabolic Killing grim reapers in hyperbolic space. Parabolic horizontal motion in the upper-half-space model, with the ideal boundary marked. Schematic profile; the horizontal isometry is distinct from hyperbolic dilation.
Source-informed schematic. Parabolic horizontal motion in the upper-half-space model, with the ideal boundary marked. Schematic profile; the horizontal isometry is distinct from hyperbolic dilation. 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.

Parabolic invariant surfaces in ℍ³ · Riemannian ambient manifold · Not recorded
Ambient space
Riemannian ambient manifold
Topology
Invariant profile construction; case-dependent domain
Embeddedness
Not recorded: The reviewed statement gives an immersion; no general embedding assertion is recorded.
Compactness
Noncompact
Surface genus
Not recorded.
Topological ends
Not recorded for the full parameter manifold.
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. [BuenoLopez2024a] Antonio Bueno and Rafael López. A new family of translating solitons in hyperbolic space. arXiv preprint, 2024. URL: https://arxiv.org/abs/2402.05533, arXiv:2402.05533.All uses in this project ↩ 1
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