# Ancient solutions for Andrews' hypersurface flow

## TL;DR

This paper constructs ancient solutions for Andrews' hypersurface flow, showing their asymptotic behavior as they collapse to a point and become oval, with rescaled limits resembling round cylinders, extending known results from Ricci and mean curvature flows.

## Contribution

It introduces explicit ancient solutions for Andrews' hypersurface flow and analyzes their asymptotic geometric limits, expanding understanding of singularity models in geometric flows.

## Key findings

- Solutions collapse to a round point at singular time
- Solutions become more oval as time goes to negative infinity
- Rescaled limits are round cylinders of the form S^J x R^{n-J}

## Abstract

We construct the ancient solutions of the hypersurface flows in Euclidean spaces studied by B. Andrews in 1994. As time $t \rightarrow 0^-$ the solutions collapse to a round point where $0$ is the singular time. But as $t\rightarrow-\infty$ the solutions become more and more oval. Near the center the appropriately-rescaled pointed Cheeger-Gromov limits are round cylinder solutions $S^J \times \mathbb{R}^{n-J}$, $1 \leq J \leq n-1$. These results are the analog of the corresponding results in Ricci flow ($J=n-1$) and mean curvature flow.