MCF-stack: Special Solutions of Mean Curvature Flow

§ 3.11 Scherkenoid translators (Hoffman–Ilmanen–Martín–White, 2019; Hoffman–Martín–White, 2019)9E847D647A0C

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. Hoffman–Ilmanen–Martín–White — 2019-01-25 (first arXiv submission).
  2. Hoffman–Martín–White — 2019-03-11 (first arXiv submission).
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Hoffman–Martín–White construct Scherkenoids for $0<\alpha <\pi $ and $w\geq \pi $ as complete, properly embedded, singly periodic translator surfaces in $\mathbb R^3$; see [HMW20], Theorems 2.3 and 9.2. In the catalogue's upward convention, the fundamental piece is the graph of $U$ over

\[ \Omega (\alpha ,w)=\{ (x,y):0<y<w,\ x>y\cot \alpha \} , \]

with $U\to +\infty $ on the two horizontal rays and $U\to -\infty $ on the finite slanted edge. The graph closes along the two vertical lines above the corners. Repeated half-turns about those lines produce the complete surface, with period $2(w\cot \alpha ,w,0)$. This is a boundary-value construction, not a closed formula for $U$.

The connected illustration fixes $\alpha =\pi /2$ and allows $w$ to vary. It shows how a single central sheet opens into alternating wings. Far along a wing, after recentering height, the limiting graph has the explicit profile

\[ G_ w(x,y)=-\left(\frac{w}{\pi }\right)^2\log \sin \left(\frac{\pi y}{w}\right) +x\sqrt{\left(\frac{w}{\pi }\right)^2-1},\qquad x\in \mathbb R,\quad 0<y<w. \]

This formula describes the tilted grim-reaper limit, not the whole Scherkenoid. For $w=\pi $ the tilt vanishes. In the upward convention, the height limits are the plane $x=y\cot \alpha $ as $z\to -\infty $ and the planes $y=nw$, $n\in \mathbb Z$, as $z\to +\infty $.

The drawing follows Figures 5 and 11 and Section 9 of [HMW20], arXiv version 4, pages 7 and 26–30. Its joining profile is schematic, not a numerical solution of the translator equation. The paper uses downward velocity; reflecting height gives the upward flow $M_ t=M+t e_3$ for all $t\in \mathbb R$. See also the earlier survey [HIMWNotes19], Section 5. No stability or entropy conclusion is inferred from the drawing.

One connected Scherkenoid schematic with a central sheet and alternating wings joined along vertical reflection lines. Upward convention, alpha pi over two, adjustable strip width at least pi. Not a solved translator.
Connected construction schematic. A central sheet opens into alternating grim-reaper-shaped wings. Rotate the model, isolate its fundamental half-strip piece, or change the width w ≥ π to see the wings tilt. The angle is fixed at α = π/2. Based on Hoffman–Martín–White, Figures 5 and 11; upward translation convention. The joining shape is schematic, not an exact or numerically solved Scherkenoid. Outer edges are display cuts. Open drawing · Numerical references and credits

Half-strip x greater than zero, y between zero and w, with positive infinite height on the horizontal rays and negative infinite height on the finite vertical edge. The corner points mark vertical reflection lines.
Exact construction domain. The half-strip for α = π/2, with the boundary heights reflected to the upward convention. The two corners lift to vertical lines on the surface; half-turns about these lines continue the fundamental graph. This diagram shows the domain and boundary data, not an explicit formula for the graph. Open drawing · Numerical references and credits
Ambient space, topology, embeddedness and entropy

The source family has 0 < α < π and w ≥ π at unit translation speed. The illustration fixes α = π/2 and varies w. Asymptotic planes in a height direction are not automatically topological ends; the displayed joining shape is schematic.

Periodic surface; global topology not recorded · Euclidean space ℝ³ · Embedded
Ambient space
Euclidean space ℝ³
Topology
Periodic surface; global topology not recorded
Embeddedness
Embedded: The complete Scherkenoid translators in the cited construction are properly embedded.
Compactness
Noncompact
Surface genus
Not recorded
Topological ends
Not recorded
Entropy
Not recorded: Not recorded in this entry.
Gaussian area
Not recorded: Not recorded in this entry.

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References on this page

  1. [HMW20] David Hoffman, Francisco Martín, and Brian White. Scherk-like translators for mean curvature flow. 2020. Later detailed source for the constructions surveyed in the supplied 2019 notes. Theorems 2.3, 3.1, 6.1 and 11.1 and the infinite-entropy discussion checked in arXiv v4; proofs remain external to this catalogue. URL: https://arxiv.org/abs/1903.04617v4.All uses in this project ↩ 1 ↩ 2
  2. [HIMWNotes19] David Hoffman, Tom Ilmanen, Francisco Martín, and Brian White. Notes on translating solitons for mean curvature flow. 2019. Supplied 20-page manuscript dated January 29, 2019; catalogue citations use its PDF page numbers, not the later published edition. URL: https://arxiv.org/abs/1901.09101.All uses in this project ↩ 1
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