Mean Curvature Flow in de Sitter space

TL;DR

This paper investigates the long-term behavior of mean convex spacelike graphs evolving under mean curvature flow in de Sitter space, showing convergence to the flat slicing under certain conditions, with implications for the cosmic no hair conjecture.

Contribution

It establishes convergence results for mean curvature flow of spacelike graphs in de Sitter space, connecting geometric flow behavior to cosmological conjectures.

Findings

Flow surfaces become graphical over expanding balls with gradient approaching 1

Flow converges smoothly to the flat slicing of de Sitter space

Results relate to the cosmic no hair conjecture

Abstract

We study mean convex mean curvature flow $M_s$ of local spacelike graphs in the flat slicing of de Sitter space. We show that if the initial slice is of non-negative time and is graphical over a large enough ball, and if $M_s$ is of bounded mean curvature, then as $s$ goes to infinity, $M_s$ becomes graphical in expanding balls, over which the gradient function converges to $1$. In particular, if $p_s$ is the point lying over the center of the domain ball in $M_s$, then $(M_s,p_s)$ converges smoothly to the flat slicing of de Sitter space. This has some relation to the mean curvature flow approach to the cosmic no hair conjecture.