MCF-stack: Special Solutions of Mean Curvature Flow

§ 3.36 Invariant translators in the special linear group9C3485CD7693

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. López–Munteanu — 2024-05-24 (first arXiv submission).
9C3485CD7693
3535C925421E

López–Munteanu classify invariant translating surfaces in $SL(2,\mathbb R)$ with its standard left-invariant metric. In Iwasawa coordinates $(x,y,\theta )$, $y>0$ and $\theta \in S^1$, the metric is

\[ g=\frac{dx^2+dy^2}{4y^2}+\left(d\theta +\frac{dx}{2y}\right)^2. \]

The invariant subgroups are the parabolic, hyperbolic and elliptic factors $N,A,K$. The driving field is selected from the Killing algebra, and is separate from the field tangent to the invariant orbits. The resulting classification contains explicit special cases and ODE-defined families, as well as minimal stationary exceptions. The source's scalar mean curvature is the average of the principal curvatures: a profile satisfying $H=\langle N,X\rangle $ evolves under trace MCF by the isometries generated by $2X$. Complete Killing motion is eternal, but does not make an incomplete profile complete. Global topology, properness and embeddedness must therefore be checked for the chosen subgroup, profile domain and circle identifications; no blanket assertion is recorded here. The visualization is an orbit-space and fiber diagram in the $\mathbb H^2(-4)$ base with its circle fiber. It does not depict the universal cover as identical to $SL(2,\mathbb R)$ or assign Euclidean metric geometry to the coordinate plot.

Sources: [LopezMunteanu2024].

Invariant translators in the special linear group. Iwasawa base and circle-fiber orbit schematic. A selected Killing motion is shown through time; fiber identifications and the curved metric are retained in the labels.
Source-informed schematic. Iwasawa base and circle-fiber orbit schematic. A selected Killing motion is shown through time; fiber identifications and the curved metric are retained in the labels. 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.

Invariant surfaces in SL(2,ℝ) · Riemannian ambient manifold · Depends on the construction
Ambient space
Riemannian ambient manifold
Topology
Profile × invariant orbit, with the stated circle identifications
Embeddedness
Depends on the construction: The status depends on the chosen input and closing or quotient conditions.
Compactness
Not specified for this subfamily
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. [LopezMunteanu2024] Rafael López and Marian Ioan Munteanu. Translators of the mean curvature flow in the special linear group $SL(2,\mathbb R)$. arXiv preprint, 2024. URL: https://arxiv.org/abs/2405.15957, arXiv:2405.15957.All uses in this project ↩ 1
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