Paper · Sep 4, 2026
A Bernstein problem for translating solutions to the mean curvature flow
As on arXiv
Abstract
We study entire graphical translating solutions of the mean curvature flow, 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 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 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.
Comments: 61 pages; comments welcome
Its neighbourhood
Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
-
Rigidity results for mean curvature flow graphical translators moving in non-graphical direction
Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “graphical translators”, “entire graphical”, “graphical translating”
-
Rotationally symmetric translating solitons of fully nonlinear extrinsic geometric flows: Classification and Applications
Topics Rotational symmetry & warped products, Self-shrinkers & translators, Stability & uniqueness · Shared terms “graphical translators”, “entire graphical”, “translating solitons”
-
Existence proofs for rotationally symmetric translating solutions to mean curvature flow
Topics Rotational symmetry & warped products, Self-shrinkers & translators · Shared terms “translating solution”, “rotationally symmetric”
-
Dynamical Stability of Translating Solitons to Mean Curvature Flow in Hyperbolic Space
Topics Rotational symmetry & warped products, Self-shrinkers & translators, Stability & uniqueness · Shared terms “translating solitons”, “graphical”, “rotationally”
-
Rotationally symmetric translating solutions to extrinsic geometric flows
Topics Rotational symmetry & warped products, Self-shrinkers & translators · Shared terms “translating solution”, “rotationally symmetric”, “bowl soliton”
-
Translating Solutions of a Generalized Mean Curvature Flow in a Cylinder: I. Constant Boundary Angles
Topics Rotational symmetry & warped products, Self-shrinkers & translators, Stability & uniqueness · Shared terms “translating solution”
-
Translating solitons to higher order mean curvature flows in Riemannian products
Topics Rotational symmetry & warped products, Self-shrinkers & translators · Shared terms “translating solitons”, “rotationally symmetric”
-
Translators of the Mean Curvature Flow in Hyperbolic Einstein's Static Universe
Topics Rotational symmetry & warped products, Self-shrinkers & translators, Stability & uniqueness · Shared terms “rotationally”
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 6A.10Curve Shortening and the First Geometric Flows
Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.