# Convergence result and blow-up examples for the Guan--Li mean curvature flow on warped product spaces

## TL;DR

This paper studies the convergence and blow-up behavior of a geometric flow on warped product spaces, establishing optimal conditions for convergence and providing examples to demonstrate their sharpness.

## Contribution

It extends the convergence analysis of the Guan--Li mean curvature flow to warped product spaces and identifies optimal initial data conditions for convergence.

## Key findings

- Convergence under a specific modulus of continuity condition.
- Examples showing the optimality of the convergence condition.
- Extension of flow analysis from space forms to warped product spaces.

## Abstract

We examine the question of convergence of solutions to a geometric flow which was introduced by Guan and Li for starshaped hypersurfaces in space forms and generalized by Guan, Li, and Wang to the case of warped product spaces. We obtain a convergence result under a condition on the optimal modulus of continuity of the initial data. Moreover we show by examples that this condition is optimal at least in the one-dimensional case.