# Mean curvature flow in asymptotically flat product spacetimes

## TL;DR

This paper studies the long-term evolution of spacelike hypersurfaces under mean curvature flow in asymptotically flat Lorentzian product spacetimes, proving convergence to a flat slice for suitable initial conditions.

## Contribution

It establishes long-time existence and convergence of mean curvature flow in asymptotically flat Lorentzian manifolds, extending understanding of geometric flows in such spacetimes.

## Key findings

- Flow exists for all time
- Flow converges to a flat slice
- Initial conditions ensure uniform convergence

## Abstract

We consider the long-time behaviour of the mean curvature flow of spacelike hypersurfaces in the Lorentzian product manifold $M\times\mathbb{R}$, where $M$ is asymptotically flat. If the initial hypersurface $F_0\subset M\times\mathbb{R}$ is uniformly spacelike and asymptotic to $M\times\left\{s\right\}$ for some $s\in\mathbb{R}$ at infinity, we show that a mean curvature flow starting at $F_0$ exists for all times and converges uniformly to $M\times\left\{s\right\}$ as $t\to \infty$.