On complete embedded translating solitons of the mean curvature flow that are of finite genus

Abstract

We desingularise the union of $3$ Grim paraboloids along Costa-Hoffman-Meeks surfaces in order to obtain what we believe to be the first examples in $\Bbb{R}^3$ of complete embedded translating solitons of the mean curvature flow of arbitrary non-trivial genus. This solves a problem posed by Mart\'in, Savas-Halilaj and Smoczyk.Comment: The presentation has been moderately redrafted and numerous typos have been corrected. Hopefully the paper is now considerably more accessible. Notation, terminology and conventions are described in detail in Appendix