MCF-stack: Special Solutions of Mean Curvature Flow

§ 2.29 Lens-shaped shrinking surface clusters82C42A1C42B4

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. Baldi–Haus–Mantegazza — 2018-11-19 (first arXiv submission).
82C42A1C42B4
521D2D744A5F

Baldi–Haus–Mantegazza construct a rotationally symmetric self-shrinking lens-shaped cluster in $\mathbb R^3$. Two bounded disk-type caps and an unbounded planar exterior meet along a common circle at $120$ degrees. The upper and lower caps are exchanged by reflection in the horizontal plane. Each sheet satisfies $\mathbf H=-X^\perp /2$ away from the triple circle; the equal-angle condition supplies the junction balance. The flow is homothetic for $t<0$. This is a cluster with a junction curve, not a smooth closed sphere or an immersed surface without boundary. The exterior sheet gives a noncompact cluster; the caps themselves are compact disks. Surface genus for the whole singular union is inapplicable. The source proves existence by shooting a rotational profile; it does not identify the caps with spherical caps. The picture therefore uses a numerical rotational shooting profile and keeps the exterior planar sheet, triple circle and $120$-degree contact distinct. Gaussian integrals use the sum of sheet areas, with no numerical entropy value assigned.

Sources: [BaldiHausMantegazza2019].

Lens-shaped shrinking surface clusters. Numerically shot rotational caps meeting a planar exterior at 120 degrees. The triple circle is highlighted and the three homothetic times have a common scale.
Numerical ODE cluster. Numerically shot rotational caps meeting a planar exterior at 120 degrees. The triple circle is highlighted and the three homothetic times have a common scale. 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.

Two-dimensional cluster in ℝ³ · Euclidean space ℝ³ · Embedded
Ambient space
Euclidean space ℝ³
Topology
Two disks and a planar exterior joined along a triple circle
Embeddedness
Embedded: Embedded sheets meeting at the prescribed triple circle; the union is not smooth there.
Compactness
Noncompact
Surface genus
Not a smooth surface; surface genus is inapplicable.
Topological ends
One topological end.
Entropy
Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area
Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

Compare examples by these properties →

References on this page

  1. [BaldiHausMantegazza2019] Pietro Baldi, Emanuele Haus, and Carlo Mantegazza. Existence of a lens-shaped cluster of surfaces self-shrinking by mean curvature. Mathematische Annalen, 375:1857–1881, 2019. URL: https://arxiv.org/abs/1811.07822, arXiv:1811.07822, doi:10.1007/s00208-019-01890-9.All uses in this project ↩ 1
Cite this page

Comments

Comments are not enabled yet.