On the existence of translating solutions of mean curvature flow in slab regions

Bourni, Theodora; Langford, Mat; Tinaglia, Giuseppe

doi:10.2140/apde.2020.13.1051

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On the existence of translating solutions of mean curvature flow in slab regions

Theodora Bourni, Mat Langford and Giuseppe Tinaglia

Vol. 13 (2020), No. 4, 1051–1072

DOI: 10.2140/apde.2020.13.1051

Abstract

We prove, in all dimensions $n \geq 2$ , that there exists a convex translator lying in a slab of width $π sec 𝜃$ in $ℝ^{n + 1}$ (and in no smaller slab) if and only if $𝜃 \in [0, \frac{π}{2}]$ . We also obtain convexity and regularity results for translators which admit appropriate symmetries and study the asymptotics and reflection symmetry of translators lying in slab regions.

Keywords

mean curvature flow, translators, ancient solutions

Mathematical Subject Classification 2010

Primary: 53A10

Milestones

Received: 12 June 2018

Revised: 3 April 2019

Accepted: 18 April 2019

Published: 13 June 2020

Authors

Theodora Bourni
Department of Mathematics University of Tennessee Knoxville, TN United States

Mat Langford
Department of Mathematics University of Tennessee Knoxville, TN United States

Giuseppe Tinaglia
Department of Mathematics King’s College London London United Kingdom

Vol. 13, No. 4, 2020