Finite entropy translating solitons in slabs

TL;DR

This paper investigates finite entropy translating solitons within slabs, introduces asymptotic invariants called wing numbers, and establishes a uniqueness theorem for certain simply connected solitons with entropy 3.

Contribution

It introduces wing numbers as asymptotic invariants for translating solitons in slabs and provides a formula for computing entropy based on these invariants, demonstrating entropy quantization.

Findings

Entropy is quantized into integer steps for this class of solitons.

Wing numbers can be used to compute the entropy of translating solitons.

Uniqueness of certain translating pitchforks is established under specified conditions.

Abstract

We study translating solitons for the mean curvature flow, $\Sigma^2\subseteq\mathbb{R}^3$ which are contained in slabs, and are of finite genus and finite entropy. As a first consequence of our results, we can enumerate connected components of slices to define asymptotic invariants $\omega^\pm(\Sigma)\in\mathbb{N}$, which count the numbers of "wings''. Analyzing these, we give a method for computing the entropies $\lambda(\Sigma)$ via a simple formula involving the wing numbers, which in particular shows that for this class of solitons the entropy is quantized into integer steps. Finally, combining the concept of wing numbers with Morse theory for minimal surfaces, we prove the uniqueness theorem that if $\Sigma$ is a complete embedded simply connected translating soliton contained in a slab with entropy $\lambda(\Sigma)=3$ and containing a vertical line, then $\Sigma$ is one of the translating pitchforks of Hoffman-Mart\'in-White