MCF-stack: Special Solutions of Mean Curvature Flow

§ 3.8 Tilted grim-reaper cylinders9A8E5E528EEB

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 — 2018-05-28 (first arXiv submission).
9A8E5E528EEB
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Let $a\in \mathbb R$ and $A=1+a^2$. An exact unit-speed upward translator in $\mathbb R^3$ is

\[ X(x,y,t)=\left(x,y,-A\log \cos (x/\sqrt A)+ay+t\right),\quad |x|<\frac\pi 2\sqrt A,\quad y,t\in \mathbb R. \]

  • This is a complete, properly embedded noncompact surface diffeomorphic to $\mathbb R^2$. It has zero Gauss curvature and is invariant under translation in the direction $(0,1,a)$.

  • Its horizontal graph domain is a strip of width $\pi \sqrt{1+a^2}$. The case $a=0$ is the ordinary grim-reaper cylinder. The underlying surface is a rotation and dilation of that cylinder.

  • At a fixed width greater than $\pi $, the two signs of $a$ give the two opposite tilts. Translation in a tangent invariant direction does not change the evolving image.

  • Products with Euclidean factors give higher-dimensional translators. Graphical weighted stability holds, as in the preceding example.

See [HIMWGraphical18], equation (1.4), PDF page 3, with exchanged horizontal coordinates and reflected vertical direction. These cylinders must be distinguished from the strictly convex $\Delta $-wings on strips of the same width.

Tilted grim-reaper cylinders. Exact tilted grim-reaper formula, with the infinite ends clipped.
Exact formula. Exact tilted grim-reaper formula, with the infinite ends clipped. The displayed tilt parameter is a = 0.65. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy
Plane ℝ² · Euclidean space ℝ³ · Embedded
Ambient space
Euclidean space ℝ³
Topology
Plane ℝ²
Embeddedness
Embedded: The tilted graph and its stated Euclidean products are properly embedded.
Compactness
Noncompact
Surface genus
0
Topological ends
1
Entropy
Not recorded: Not recorded in this entry.
Gaussian area
Not recorded: Not recorded in this entry.
Products in higher dimensions · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded
Ambient space
Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology
Euclidean products, topology ℝⁿ
Embeddedness
Embedded: The tilted graph and its stated Euclidean products are properly embedded.
Compactness
Noncompact
Surface genus
Not a surface
Topological ends
1
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. [HIMWGraphical18] David Hoffman, Tom Ilmanen, Francisco Martín, and Brian White. Graphical translators for mean curvature flow. 2018. Supplied 28-page preprint dated July 5, 2018. Catalogue locators refer to this edition. The inverted ratio in Theorem 2.4 is corrected using equation (1.4). URL: https://arxiv.org/abs/1805.10860v2.All uses in this project ↩ 1
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