MCF-stack: Special Solutions of Mean Curvature Flow

§ 3.27 Nonconvex entire graphical translators2E699B408ADA

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. Davila–del Pino–Wei — 2026-09-04 (first arXiv submission).
2E699B408ADA
879611BCBD19

The 2026 preprint of Dávila–del Pino–Wei states that for every $N\ge 8$ there is a one-parameter family of entire smooth functions $G:\mathbb R^ N\to \mathbb R$ satisfying

\[ \operatorname {div}\frac{DG}{\sqrt{1+|DG|^2}}=\frac1{\sqrt{1+|DG|^2}}, \]

whose graphs are not convex. The graphs are embedded, complete and proper, with topology $\mathbb R^ N$, and translate with unit vertical velocity for $t\in \mathbb R$. The positive scalar quantity in the displayed graph equation expresses mean convexity with the source's normal convention; it does not imply nonnegative principal curvatures. The construction in $N=8$ perturbs Bombieri–De Giorgi–Giusti geometry; higher dimensions follow by flat products. It is separate from the convex Wang examples. This is a reported preprint construction, with no independent proof verification or stability conclusion recorded here. Its profile is not available as a closed formula. The figure is a labeled radial-coordinate schematic of the two four-dimensional blocks and the transition near the Simons cone; the plotted saddle is not an exact translator or a surface MCF in three-space.

Sources: [DavilaPinoWei2026].

Nonconvex entire graphical translators. Schematic section through the two four-dimensional radial blocks of the construction. The transition region is marked. Vertical shifts show the exact type of motion, but the drawn saddle is not the constructed graph.
Source-informed schematic. Schematic section through the two four-dimensional radial blocks of the construction. The transition region is marked. Vertical shifts show the exact type of motion, but the drawn saddle is not the constructed graph. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Entire graphs, N ≥ 8, ambient ℝ^(N+1) · Euclidean dimension ≥ 5 · Embedded
Ambient space
Euclidean dimension ≥ 5
Topology
ℝᴺ, N ≥ 8
Embeddedness
Embedded: The cited construction gives an embedding for this case.
Compactness
Noncompact
Surface genus
Not a smooth surface; surface genus is inapplicable.
Topological ends
One topological end.
Entropy
Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area
Not recorded: No quantitative conclusion is recorded from the cited construction.

Compare examples by these properties →

References on this page

  1. [DavilaPinoWei2026] Juan Davila, Manuel del Pino, and Juncheng Wei. A Bernstein problem for translating solutions to the mean curvature flow. arXiv preprint, 2026. URL: https://arxiv.org/abs/2609.04675, arXiv:2609.04675.All uses in this project ↩ 1
Cite this page

Comments

Comments are not enabled yet.