Doubly periodic self-translating surfaces for the mean curvature flow

TL;DR

This paper constructs new doubly periodic self-translating surfaces for the mean curvature flow using configurations of grim reaper cylinders, revealing that such surfaces can have non-quadratic volume growth, unlike self-shrinkers.

Contribution

It introduces a novel construction of doubly periodic self-translating surfaces with specific geometric properties, extending previous work on grim reaper cylinders.

Findings

Self-translating surfaces can have non-quadratic volume growth.

Periodic configurations with grim reaper cylinders are flexible for construction.

The work extends the understanding of mean curvature flow self-translators.

Abstract

We construct new examples of self-translating surfaces for the mean curvature flow from a periodic configuration with finitely many grim reaper cylinders in each period. Because this work is an extension of the author's article on the desingularization of a finite family of grim reaper cylinders, we simply discuss the ideas of the construction and here prove only that the periodic configuration has the necessary flexibility. These examples show that self-translating surfaces do not necessarily have quadratic volume growth rate in contrast to self-shrinking surfaces.