https://neckpinch.com/authors/manuel-del-pino/New arXiv papers by Manuel del Pino, from Neckpinch Tue, 03 Nov 2026 08:41:12 GMT https://neckpinch.com/papers/2609.04675/arxiv:2609.04675Fri, 04 Sep 2026 00:00:00 GMTJuan Davila, Manuel del Pino, Juncheng Wei — We study entire graphical translating solutions of the mean curvature flow, div (frac∇G√(1+|∇G|²)) =frac1√(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 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.Mean curvature flowSelf-shrinkers & translatorsStability & uniquenessRotational symmetry & warped products