MCF-stack: Special Solutions of Mean Curvature Flow

§ 3.31 Graphical translators over compact symmetric spaces (Fujii, 2021; Fujii–Koike, 2023)1102596C432B

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. Fujii — 2021-09-30 (first arXiv submission).
  2. Fujii–Koike — 2023-08-30 (first arXiv submission).
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Fujii constructs vertical graphical translators in $S^ n\times \mathbb R$ whose height is constant on leaves of an isoparametric foliation. Fujii–Koike treat the corresponding invariant graphs in $N\times \mathbb R$ for compact symmetric spaces, including $\mathbb CP^ n$, $\mathbb HP^ n$ and the Cayley plane, and derive reduced equations for higher-rank cases. In the trace convention a vertical unit-speed graph $z=u(p)$ satisfies

\[ \operatorname {div}_ N\frac{\nabla u}{\sqrt{1+|\nabla u|^2}} =\frac1{\sqrt{1+|\nabla u|^2}}. \]

The invariant ansatz $u=V\circ r$ reduces this PDE to the source's ODE on a regular orbit interval. The allowed domain and behavior at singular orbits are essential: these are graphs over specified open subsets, not smooth entire graphs on a closed compact base. The latter would contradict integration of the positive right-hand side. Embeddedness follows on each single-valued graph domain; completeness is not uniform across the classified branches. The flow is $(p,u(p)+t)$ for all real $t$ on that domain. The figure shows a profile over an orbit interval, marking excluded singular leaves and graph blow-up boundaries; it is not a numerical computation of every symmetric-space case.

Sources: [Fujii2024, FujiiKoike2023].

Graphical translators over compact symmetric spaces. Orbit interval of a compact base and vertically moving graph profile. Open domain endpoints are marked; the drawing does not suggest an entire translator over a closed base.
Source-informed schematic. Orbit interval of a compact base and vertically moving graph profile. Open domain endpoints are marked; the drawing does not suggest an entire translator over a closed base. 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.

Graphs over domains in S² · Riemannian ambient manifold · Embedded
Ambient space
Riemannian ambient manifold
Topology
Graph over the specified spherical domain
Embeddedness
Embedded: The cited construction gives an embedding for this case.
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.
Higher-dimensional compact symmetric bases · Riemannian ambient manifold · Embedded
Ambient space
Riemannian ambient manifold
Topology
Graph over a specified invariant open subset of N
Embeddedness
Embedded: The cited construction gives an embedding for this case.
Compactness
Noncompact
Surface genus
Not a smooth surface; surface genus is inapplicable.
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. [Fujii2024] Tomoki Fujii. Graphical translating solitons for the mean curvature flow and isoparametric functions. Tohoku Mathematical Journal, 76:391–410, 2024. URL: https://arxiv.org/abs/2109.14809, arXiv:2109.14809, doi:10.2748/tmj.20230112.All uses in this project ↩ 1
  2. [FujiiKoike2023] Tomoki Fujii and Naoyuki Koike. Translators invariant under hyperpolar actions. arXiv preprint, 2023. URL: https://arxiv.org/abs/2308.15790, arXiv:2308.15790.All uses in this project ↩ 1
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