MCF-stack: Special Solutions of Mean Curvature Flow

§ 3.12 The pitchfork translator (Hoffman–Ilmanen–Martín–White, 2019; Hoffman–Martín–White, 2019; Gama–Martín–Møller, 2022; Martín–Sáez–Tsiamis–White, 2024)64055D697644

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).
  3. Gama–Martín–Møller — 2022-09-04 (first arXiv submission).
  4. Martín–Sáez–Tsiamis–White — 2024-11-25 (first arXiv submission).
64055D697644
B5E7AB44607B

The pitchfork is a complete, properly embedded, simply connected, nongraphical translator in $\mathbb R^3$. The supplied notes describe it as a limit of Scherk-type examples, without an explicit formula.

  • It has genus zero. This distinguishes it from Nguyen's periodic infinite-genus tridents despite their similar asymptotic plane arrangements.

  • After reflection to upward translation, it is asymptotic to one plane as $z\to -\infty $ and to three parallel planes as $z\to +\infty $. These are height-direction asymptotics, not a claim that the surface has four topological ends.

  • Its translation yields an eternal noncompact flow. The later construction paper records finite topology and quadratic area growth for its pitchfork subfamily.

See [HIMWNotes19], Section 5 and Figure 7, PDF pages 11–12, and [HMW20]. The existence proof is external. The full construction supplies a pitchfork for each half-width $w\geq \pi $, lying in $|y|<w$. Theorem 1 of [MSTW26] proves uniqueness up to isometry for fixed width in this semigraphical family. Moreover, [GMM25], Theorem 1.5, characterizes pitchforks among complete embedded simply connected translators of finite width, entropy $3$, and containing a vertical line. These hypotheses are essential to the stated classification.

To read the picture, remove the vertical $z$-axis. One component is a graph $z=v(x,y)$ over $\mathbb R\times (0,w)$; the other is its image under $(x,y,z)\mapsto (-x,-y,z)$. In the upward convention the boundary values are

\[ v(x,0)=-\infty \quad (x<0),\qquad v(x,0)=+\infty \quad (x>0),\qquad v(x,w)=+\infty . \]

Theorem 12.1 and Corollary 12.2 of [HMW20] give the smooth extension along the axis, negative Gauss curvature, and the planar and tilted-grim-reaper asymptotics. For $w=\pi $, the grim-reaper profile, after vertical recentering at $y=\pi /2$, is $-\log \sin y$ for $0<y<\pi $. This is an asymptotic profile, not a formula for the pitchfork. The interactive drawing illustrates this width using two colored halves and horizontal sections, following Figures 6 and 14 of that paper. Its connecting geometry is schematic, not a computed solution of the translator equation.

A connected pitchfork surface: teal and ochre graphical halves joined along a vertical line, approaching one plane below and three parallel planes above. Original schematic, not a solved translator.
Source-informed schematic. The half-width w = π case, in the upward convention. Two graphical halves join along the vertical z-axis; the second is obtained by a half-turn. Rotate the surface, isolate one half, or move through horizontal sections. Based on Hoffman–Martín–White, Figures 6 and 14 and Theorem 12.1. This original drawing is not a numerical solution of the translator equation; all outer edges are display cuts. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy

The one-plane/three-plane asymptotics do not give four topological ends.

Plane ℝ² (simply connected noncompact surface) · Euclidean space ℝ³ · Embedded
Ambient space
Euclidean space ℝ³
Topology
Plane ℝ² (simply connected noncompact surface)
Embeddedness
Embedded: The complete pitchfork translator is properly embedded.
Compactness
Noncompact
Surface genus
0
Topological ends
1
Entropy
Exact formula: λ = 3; Gama–Martín–Møller, Theorem 1.4 and the pitchfork asymptotics.
Gaussian area
Not recorded: Not recorded in this entry.

Compare examples by these properties →

References on this page

  1. [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
  2. [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
  3. [MSTW26] Francisco Martín, Mariel Sáez, Raphael Tsiamis, and Brian White. Classification of semigraphical translators. 2026. arXiv:2411.16889v2. Selected theorem statements and construction passages checked November 6, 2026; proofs are cited, not independently verified. URL: https://arxiv.org/abs/2411.16889v2.All uses in this project ↩ 1
  4. [GMM25] Eddygledson Souza Gama, Francisco Martín, and Niels Martin Møller. Finite entropy translating solitons in slabs. 2025. arXiv:2209.01640v3. Selected theorem statements and construction passages checked November 6, 2026; proofs are cited, not independently verified. URL: https://arxiv.org/abs/2209.01640v3.All uses in this project ↩ 1
Cite this page

Comments

Comments are not enabled yet.