Translating solitons in arbitrary codimension

We study the translating solitons of the mean curvature flow. Although many authors study translating solitons in codimension one, there are few references and examples for higher codimensional cases except for Lagrangian translating solitons. First we observe non-trivial examples of translating solitons in arbitrary codimension. We will see that they have the property called parallel principal normal (PPN). Inspired by this fact and the work of Smoczyk for selfshrinkers in 2005, we then characterize the complete translating solitons with PPN.

Keywords

mean curvature flow, translating soliton, parallel principal normal

2010 Mathematics Subject Classification

Primary 53A07. Secondary 53C44.

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The author was partly supported by the COLABS program (Tohoku Univ.) and the Grant-in-Aid for JSPS Fellows.

Received 16 November 2015

Accepted 1 April 2016

Published 9 February 2018