Classification of Semigraphical Translators

We complete the classification of semigraphical translators for mean curvature flow in $\mathbb{R}^3$ that was initiated by Hoffman-Martín-White. Specifically, we show that there is no solution to the translator equation on the upper half-plane with alternating positive and negative infinite boundary values, and we prove the uniqueness of pitchfork and helicoid translators. The proofs use Morse-Radó theory for translators and an angular maximum principle.

💡 Research Summary

The paper completes the classification of semigraphical translators for mean curvature flow (MCF) in ℝ³, a program initiated by Hoffman‑Martín‑White. A translator is a surface that moves by translation under MCF; after a rotation and scaling one may assume the velocity vector is –e₃. In the translator metric g_{ij}=e^{−z}δ_{ij} such surfaces are minimal. A semigraphical translator is a properly embedded translator that, away from a discrete collection of vertical lines, is the graph of a function over a domain in the (x,y)‑plane.

The authors first prove a new angular maximum principle (Theorem 2). If two solutions u₁, u₂ of the translator equation agree on the boundary of a half‑strip and their upward unit normals have a uniform positive component in the e₂‑direction for large x, then u₂−u₁ is constant. The proof introduces a function ω(p) measuring the angular rotation needed to map a point on the graph of u₁ to the graph of u₂; ω cannot attain an interior maximum by the strong maximum principle, forcing it to be identically zero.

Using this principle, the paper establishes the long‑standing uniqueness conjectures for pitchfork and translating helicoid translators.

Pitchfork uniqueness (Theorem 3). For any width w ≥ π there exists a translator f defined on ℝ×(0,w) with boundary values –∞ on the top edge, and alternating ±∞ on the bottom edge at x=0. The existence of such a surface was known, but uniqueness was open. Assuming two distinct solutions f₁, f₂, the authors translate one of them by a vector v so that the difference g=f₁−f₂(·+v) has a critical point. The zero set of g forms a tree with at least four ends. By analyzing the asymptotic behavior of the normal vectors (using known facts (F1)–(F3) about the Gauss map, convergence to e₂ at +∞, and a strictly decreasing function ψ describing the y‑derivative at –∞) they show that two of the ends must head to +∞. The angular maximum principle then forces g to vanish on the region bounded by these two ends, contradicting the assumption that g is non‑constant. Hence the pitchfork is uniquely determined by its width.

Helicoid uniqueness (Theorem 5). For any width w < π there exists a translator f on ℝ×(0,w) with boundary data that switch sign across a vertical line x=a on the lower edge and across x=0 on the upper edge, together with a vertical line L through (a,w). This surface is a fundamental piece of a translating helicoid; again existence was known, uniqueness was conjectural. The proof follows the same scheme: after a suitable horizontal translation the difference of two candidates has a critical point, its zero set is a tree, and the angular maximum principle together with the asymptotic normal behavior forces the difference to be constant. Consequently the helicoid is uniquely determined by its width.

Non‑existence of the “y‑eti”. The paper finally rules out a hypothetical semigraphical translator with alternating infinite boundary values on the upper half‑plane (the so‑called “y‑eti”). By sliding a family of grim‑reaper translators and employing a gradient bound derived from the angular maximum principle, the authors show that any such surface would give rise to a complete graphical translator over a half‑space, contradicting the known classification of graphical translators (HIMW19a). Hence the y‑eti does not exist.

The main result, Theorem 1, states that every semigraphical translator in ℝ³ is one of the following five types:

1. Doubly periodic Scherk translator (two parameters: width and angle).
2. Singly periodic Scherkenoid (two parameters: width and angle).
3. Singly periodic translating helicoid (unique up to the width).
4. Pitchfork (unique up to the width, which must be ≥ π).
5. Singly periodic trident (unique up to its period).

The previously conjectured sixth type, the y‑eti, is shown not to exist. The paper thus closes the classification program, providing a complete catalogue of semigraphical translators and establishing the rigidity of pitchforks and helicoids via a novel angular maximum principle. This contributes significantly to the understanding of singularity models in mean curvature flow and the geometry of translating solitons.