MCF-stack: Special Solutions of Mean Curvature Flow

§ 5.35 Hypersurface singularities with bounded mean curvature88709443EE99

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. Stolarski — 2020-03-13 (first arXiv submission).
88709443EE99
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Stolarski constructs smooth properly embedded noncompact hypersurface flows in $\mathbb R^ N$, $N\ge 8$, that develop a finite-time singularity while the scalar mean curvature remains uniformly bounded. The second fundamental form becomes unbounded, so cancellation between principal curvatures is essential. The construction develops the Velázquez singularity model; higher-dimensional examples can be obtained by the stated product extensions. The theorem does not assert a compact example merely because the introduction discusses possible compactification. At the parabolic scale the model is a minimal cone; at a finer curvature scale one sees a smooth minimal desingularization. These are different rescalings of the same degenerating region. They are distinct from the rotational bowl-type degenerate neck pinch and the Lagrangian Stolarski–Su examples. No exact time-dependent parametrization or numeric entropy is provided. The radial-block diagram shows a narrowing neighborhood of the Simons cone and identifies the two zoom scales. It is not a shrinking family of minimal surfaces claimed to solve MCF.

Sources: [Stolarski2023].

Hypersurface singularities with bounded mean curvature. Radial-block schematic of a Simons-cone singularity. The ordinary view, parabolic magnification, and finer neck scale are distinguished; interpolated curves do not solve MCF.
Source-informed schematic. Radial-block schematic of a Simons-cone singularity. The ordinary view, parabolic magnification, and finer neck scale are distinguished; interpolated curves do not solve MCF. 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.

Ambient ℝᴺ, N ≥ 8 · Euclidean dimension ≥ 5 · Embedded
Ambient space
Euclidean dimension ≥ 5
Topology
Noncompact hypersurface; minimal-cone desingularization geometry
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
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. [Stolarski2023] Maxwell Stolarski. Existence of Mean Curvature Flow Singularities with Bounded Mean Curvature. Duke Mathematical Journal, 172:1235–1292, 2023. URL: https://arxiv.org/abs/2003.06383, arXiv:2003.06383, doi:10.1215/00127094-2023-0005.All uses in this project ↩ 1
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