Scherk-like Translators for Mean Curvature Flow

2019/03/11 by David Hoffman, Hoffman, David, Francisco Martín +3 · 2 citations
Mathematics · #53C21 #53C42 #53C44 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1903.04617

openalex publication_date 2019/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove existence and uniqueness for a two-parameter family of translators for mean curvature flow. We get additional examples by taking limits at the boundary of the parameter space. Some of the translators resemble well-known minimal surfaces (Scherk's doubly periodic minimal surfaces, helicoids), but others have no minimal surface analogs. A one-parameter subfamily of the examples (the pitchforks) have finite topology and quadratic area growth, and thus might arise as blowups at singularities of initially smooth, closed surfaces flowing by mean curvature flow.

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• The mean curvature flow of subgroups on Lie groups of dimension three
• Existence proofs for rotationally symmetric translating solutions to mean curvature flow

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