#### Vol. 13, No. 4, 2020

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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
##### Abstract

We prove, in all dimensions $n\ge 2$, that there exists a convex translator lying in a slab of width $\pi sec𝜃$ in ${ℝ}^{n+1}$ (and in no smaller slab) if and only if $𝜃\in \left[0,\frac{\pi }{2}\right]$. 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
Primary: 53A10
##### Milestones
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