Local Prescribed Mean Curvature foliations in cosmological spacetimes

TL;DR

This paper proves a local in time existence theorem for spacelike foliations with prescribed mean curvature in cosmological spacetimes, addressing topological restrictions and fixing the diffeomorphism invariance.

Contribution

It introduces a geometrically defined time function for foliations with prescribed mean curvature, overcoming limitations of constant mean curvature foliations.

Findings

Establishes local existence of prescribed mean curvature foliations in cosmological spacetimes.

Provides a method to fix diffeomorphism invariance in spacetime foliations.

Overcomes topological restrictions associated with constant mean curvature surfaces.

Abstract

A theorem about local in time existence of spacelike foliations with prescribed mean curvature in cosmological spacetimes will be proved. The time function of the foliation is geometrically defined and fixes the diffeomorphism invariance inherent in general foliations of spacetimes. Moreover, in contrast to the situation of the more special constant mean curvature foliations, which play an important role in the global analysis of spacetimes, this theorem overcomes the existence problem arising from topological restrictions for surfaces of constant mean curvature.