§ 3.9 Strictly convex delta-wing translators9BBF93D61D7A

For every $b>\pi /2$, there is a complete strictly convex unit-speed translator that is a graph $z=u_ b(x,y)$ over

This is a nonlinear PDE characterization; no explicit height formula is supplied.

• The graph is embedded, noncompact, diffeomorphic to $\mathbb R^2$, and has positive Gauss curvature everywhere. After centering and a vertical translation, $u_ b(0,0)=0$ is its minimum and $u_ b$ is even in both coordinates.

• The width is exactly $2b>\pi $. Among complete translators on this strip that are not tilted grim reapers, it is unique up to translation. The centered solution depends smoothly on compact sets under limits of the width parameter.

• Along the two unbounded horizontal directions, translated graphs converge smoothly to the two oppositely tilted grim-reaper cylinders of the same strip width. Their angle is determined by

  \[ \cos \theta =\frac{\pi }{2b}. \]

  This relation follows from the explicit tilted-cylinder width; the supplied preprint has an inverted ratio in its printed statement of Theorem 2.4.

• Graphical weighted stability holds. This is a strictly convex surface, whereas the tilted grim-reaper cylinders on the same strip have zero Gauss curvature.

See [HIMWGraphical18], Theorems 1.1, 2.4, 4.1 and 5.7 and Corollary 5.5 (PDF pages 4, 6, 10–12). The construction, uniqueness, and asymptotic results are cited rather than reproduced here. The illustration numerically approximates the rectangular Dirichlet construction in Sections 3–4 at $b=\pi /\sqrt{2}$, then recenters at the minimum and displays a compact height-truncated patch. The paper uses downward translation; the picture reflects the vertical coordinate to match the upward convention here. Profile views compare the computed surface with the exact limiting profile $-2\log \cos (y/\sqrt{2})$ and longitudinal limiting slopes $\pm 1$. The computation has finite-domain and discretization error; it is not an explicit formula or a certified numerical enclosure.