§ 3.1 The grim reaper and its cylinders (Hoffman–Ilmanen–Martín–White, 2018; Hoffman–Ilmanen–Martín–White, 2019)505486401C44

For unit upward translation, an exact parametrization in $\mathbb R^{n+1}$ is

For $n=1$ omit $y$. More generally the planar curve of speed $c>0$ has height $-c^{-1}\log \cos (cx)+ct$, on $|x|<\pi /(2c)$.

• The curve and all its Euclidean products are complete, properly embedded, noncompact, and diffeomorphic to $\mathbb R^ n$. Their motion is eternal.

• At unit speed the curve has curvature magnitude $\cos x$ and total curvature $\pi $. Its two ends have the vertical asymptotes $x=\pm \pi /2$.

• The product is invariant under translations in its $\mathbb R^{n-1}$ factor. For $n\geq 2$ its second fundamental form has rank one; in $\mathbb R^3$ its Gauss curvature is zero, so it is not strictly convex.

• As a translating graph it is stable for compactly supported variations of the translator weighted area. This is weighted stability, not a claim about stability for unweighted Euclidean area.

The supplied notes [HIMWNotes19], Example 1.6, PDF page 4, use downward translation; the displayed surface is its reflection in a horizontal plane. Graphical weighted minimality is discussed in [HIMWGraphical18], Section 2, PDF page 5.