# Strong spherical rigidity of ancient solutions of expansive curvature flows

## TL;DR

This paper proves that for certain expanding geometric flows, the only ancient solutions converging to a point are spheres, demonstrating a strong spherical rigidity property without restrictive assumptions.

## Contribution

It establishes a uniqueness result for ancient solutions of expanding curvature flows, showing that the homothetic sphere is the only such solution under broad conditions.

## Key findings

- Homothetic sphere is the unique ancient solution converging to a point.
- The result holds without assumptions on the speed other than positivity and monotonicity.
- Expanding flows have stronger spherical rigidity compared to contractive cases.

## Abstract

We consider geometric flows of hypersurfaces expanding by a function of the extrinsic curvature and we show that the homothethic sphere is the unique solution of the flow which converges to a point at the initial time. The result does not require assumptions on the speed other than positivity and monotonicity and it is proved using a reflection argument. Our theorem shows that expanding flows exhibit stronger spherical rigidity, if compared with the classification results of ancient solutions in the contractive case.