A Bernstein problem for translating solutions to the mean curvature flow
Abstract
We study entire graphical translating solutions of the mean curvature flow, div ((∇ G)/(√(1+|∇ G|2))) =1/(√(1+|∇ G|2)) in ℝN. Every such graph is mean-convex, since its mean curvature is the vertical component of its unit normal. In dimension two, mean-convex translating solitons are convex, and an entire graphical translator is therefore the rotationally symmetric bowl soliton. In higher dimensions Wang constructed non-rotational entire convex translating graphs. We prove that a further loss of rigidity occurs at the Bernstein dimension: for every N≥ 8 there exists a one-parameter family of entire graphical translators that are mean-convex but not convex. The construction starts from the Bombieri--De Giorgi--Giusti (BDG) entire minimal graph in ℝ8 and develops a singular perturbation theory for the translator equation around it. The main new feature is a transition layer near Simons' cone: the translating term breaks the odd symmetry of the minimal graph, and after a suitable recentering the matching problem is governed by a parabolic inner equation. A detailed analysis of this layer, together with weighted Jacobi theory on the BDG graph and global barriers, yields the desired entire solutions.
Related
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BibTeX
@misc{davila2026bernstein,
title = {{A Bernstein problem for translating solutions to the mean curvature flow}},
author = {Juan Davila and Manuel del Pino and Juncheng Wei},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04675},
eprint = {2609.04675},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04675},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - A Bernstein problem for translating solutions to the mean curvature flow
AU - Juan Davila
AU - Manuel del Pino
AU - Juncheng Wei
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.04675
UR - https://arxiv.org/abs/2609.04675
AB - We study entire graphical translating solutions of the mean curvature flow, div ((∇ G)/(√(1+|∇ G|^2))) =1/(√(1+|∇ G|^2)) in ℝ^N. Every such graph is mean-convex, since its mean curvature is the vertical component of its unit normal. In dimension two, mean-convex translating solitons are convex, and an entire graphical translator is therefore the rotationally symmetric bowl soliton. In higher dimensions Wang constructed non-rotational entire convex translating graphs. We prove that a further loss of rigidity occurs at the Bernstein dimension: for every N≥ 8 there exists a one-parameter family of entire graphical translators that are mean-convex but not convex. The construction starts from the Bombieri--De Giorgi--Giusti (BDG) entire minimal graph in ℝ^8 and develops a singular perturbation theory for the translator equation around it. The main new feature is a transition layer near Simons' cone: the translating term breaks the odd symmetry of the minimal graph, and after a suitable recentering the matching problem is governed by a parabolic inner equation. A detailed analysis of this layer, together with weighted Jacobi theory on the BDG graph and global barriers, yields the desired entire solutions.
ER -