MCF-stack: Special Solutions of Mean Curvature Flow

§ 2.26 Shrinking ends asymptotic to isoparametric cones33FBBB16A0B1

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. Chang–Spruck — 2015-10-24 (first arXiv submission).
33FBBB16A0B1
EDCE90129831

Chang–Spruck construct self-shrinking ends asymptotic to cones over isoparametric hypersurfaces of the sphere. The invariant ansatz reduces the shrinker equation to a profile ODE; the admissible foliation and its multiplicities are part of the data. A nonminimal leaf is used for the nontrivial end; the minimal cone is the stationary exceptional case. These are exterior ends with an inner boundary, not a claimed global complete smoothing of every such cone. Their dimension is that of the cone hypersurface. The figure uses a labeled orbit-space profile and distinguishes the inner boundary from the asymptotic cone. Under the normalization $\mathbf H=-X^\perp /2$, the end evolves as $\sqrt{-t}\, \Sigma $ for $t<0$, including motion of its inner boundary. No boundary condition for a separate fixed-barrier MCF problem is imposed. No global entropy or uniqueness conclusion beyond the cited end problem is recorded.

Sources: [ChangSpruck2017].

Shrinking ends asymptotic to isoparametric cones. A numerical local profile or a labeled orbit-space schematic, with an explicit inner boundary and dashed asymptotic direction. Homothetic time does not remove that boundary.
Source-informed orbit-space schematic. A numerical local profile or a labeled orbit-space schematic, with an explicit inner boundary and dashed asymptotic direction. Homothetic time does not remove that boundary. 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.

Surface end in ℝ³ · Euclidean space ℝ³ · Not recorded
Ambient space
Euclidean space ℝ³
Topology
Exterior end or orbit-space disk as specified; boundary retained
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 recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area
Not recorded: No quantitative conclusion is recorded from the cited construction.
Higher-dimensional ends · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Not recorded
Ambient space
Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology
Exterior end or orbit-space disk as specified; boundary retained
Embeddedness
Not recorded: The reviewed statement gives an immersion; no general embedding assertion is recorded.
Compactness
Noncompact
Surface genus
Not a surface.
Topological ends
Not recorded for the full parameter manifold.
Entropy
Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area
Not recorded: No quantitative conclusion is recorded from the cited construction.

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References on this page

  1. [ChangSpruck2017] Po-Yao Chang and Joel Spruck. Self-shrinkers to the mean curvature flow asymptotic to isoparametric cones. Transactions of the American Mathematical Society, 369:6565–6582, 2017. URL: https://arxiv.org/abs/1510.07183, arXiv:1510.07183, doi:10.1090/tran/6957.All uses in this project ↩ 1
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