Uniqueness of ancient solutions to Gauss curvature flow asymptotic to a cylinder

TL;DR

This paper classifies ancient solutions to the Gauss curvature flow confined within a cylinder, showing only two types exist: a translating soliton and a compact oval solution formed by gluing two solitons.

Contribution

It provides a complete classification of ancient solutions asymptotic to a cylinder in the Gauss curvature flow, identifying the only two possible solutions.

Findings

Only two ancient solutions exist for each convex bounded cylinder.

The solutions are a non-compact translating soliton and a compact oval solution.

The oval solution is formed by gluing two translating solitons from opposite ends.

Abstract

We address the classification of ancient solutions to the Gauss curvature flow under the assumption that the solutions are contained in a cylinder of bounded cross section. For each cylinder of convex bounded cross-section, we show that there are only two ancient solutions which are asymptotic to this cylinder: the non-compact translating soliton and the compact oval solution obtained by gluing two translating solitons approaching each other from time $-\infty$ from two opposite ends.