MCF-stack: Special Solutions of Mean Curvature Flow

§ 3.30 Invariant translators in Sol geometry2BC769F3A4AD

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. Pipoli — 2019-06-19 (first arXiv submission).
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For $\mathrm{Sol}_3$ with metric $e^{2z}dx^2+e^{-2z}dy^2+dz^2$, Pipoli classifies surfaces invariant under a one-parameter isometry group that translate under another Killing field. Write $F_1=\partial _ x$, $F_2=\partial _ y$, $F_3=-x\partial _ x+y\partial _ y+\partial _ z$. For an $F_1$-invariant surface and driving field $V=\eta F_1+\lambda F_2+\mu F_3$, the case $\mu =0$, $\lambda \ne 0$ gives complete graphs over slabs or half-planes, with the asymptotic alternatives in Theorem 1.1. For $\mu \ne 0$ the two components off a distinguished orbit approach a special vertical plane or a logarithmic minimal surface. Mixed horizontal invariance is covered by Theorem 1.2; invariance with a nonzero $F_3$ component forces the stationary minimal alternatives in Theorem 1.3. These graphs include behavior impossible for complete Euclidean graphical translators. Their construction is by profile ODE, not by merely inserting a logarithm into Euclidean space. The eternal motion is the flow of the specified Killing field; for $F_3$ it is $(x,y,z)\mapsto (e^{-t}x,e^ ty,z+t)$. The diagram illustrates this anisotropic isometry on a schematic invariant graph, with the metric and stationary exceptions stated. No common entropy or genus conclusion is assigned beyond the graph cases.

Sources: [Pipoli2019].

Invariant translators in Sol geometry. Fixed Sol metric and anisotropic Killing motion. A coordinate graph is schematic; the different scaling of x and y is an isometry, not parabolic homothety.
Source-informed schematic. Fixed Sol metric and anisotropic Killing motion. A coordinate graph is schematic; the different scaling of x and y is an isometry, not parabolic homothety. 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 graphical cases of Theorems 1.1–1.2 · Riemannian ambient manifold · Embedded
Ambient space
Riemannian ambient manifold
Topology
Graph over the specified slab or half-plane
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.
Other invariant profile cases · Riemannian ambient manifold · Not recorded
Ambient space
Riemannian ambient manifold
Topology
Determined by the profile and invariant orbit
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. [Pipoli2019] Giuseppe Pipoli. Invariant translators of the Solvable group. arXiv preprint, 2019. URL: https://arxiv.org/abs/1906.08077, arXiv:1906.08077.All uses in this project ↩ 1
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