# Volume preserving mean curvature flows near strictly stable sets in flat torus

## TL;DR

This paper proves that in low-dimensional flat tori, volume-preserving mean curvature flows starting near strictly stable sets exist forever and converge exponentially to a translated stable set, under certain smoothness conditions.

## Contribution

It establishes a new stability theorem for volume-preserving mean curvature flows near strictly stable sets in flat tori for dimensions 3 and 4.

## Key findings

- Flow has infinite lifetime when starting near stable sets.
- Flow converges exponentially fast to a translate of the stable set.
- Convergence occurs in the $W^{2,5}$-sense.

## Abstract

In this paper we establish a new stability result for the smooth volume preserving mean curvature flow in flat torus $\mathbb T^n$ in low dimensions $n=3,4$. The result says roughly that if the initial set is near to a strictly stable set in $\mathbb T^n$ in $H^3$-sense, then the corresponding flow has infinite lifetime and converges exponentially fast to a translate of the strictly stable (critical) set in $W^{2,5}$-sense.