Local prescribed mean curvature foliations in cosmological spacetimes

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 global analysis of spacetimes, this theorem overcomes the existence problem arising from topological restrictions for surfaces of constant mean curvature

The Global Existence Problem in General Relativity

“…The proof is based on writing a wave equation for the second fundamental form, a technique which was also used in the work of Choquet-Bruhat and Ruggeri [43] on harmonic time gauge. The prescribed mean curvature condition, has been used in the U(1) × U(1) symmetric case by Henkel [85] who proved global existence in this gauge. An important advantage of using PMC gauge is that one avoids the problem of proving existence of a CMC Cauchy surface.…”

“…These facts indicate that the CMC foliation condition is an interesting time gauge for the Einstein evolution equation. See also [79,65,22,23,85] for results relevant to existence of CMC hypersurfaces.…”

Theorems on Existence and Global Dynamics for the Einstein Equations

“…More recently, this restriction has been removed in certain cases by Henkel using a generalization of CMC foliations called prescribed mean curvature (PMC) foliations. A PMC foliation can be built that includes any given Cauchy surface [178] and global existence of PMC foliations can be proved in a way analogous to that previously done for CMC foliations [177, 176]. These global foliations provide barriers that imply the existence of a CMC hypersurface.…”

Theorems on Existence and Global Dynamics for the Einstein Equations

“…More recently, this restriction has been removed in certain cases by Henkel using a generalization of CMC foliations called prescribed mean curvature (PMC) foliations. A PMC foliation can be built that includes any given Cauchy surface [130] and global existence of PMC foliations can be proved in a way analogous to that previously done for CMC foliations [131, 132]. These global foliations provide barriers that imply the existence of a CMC hypersurface.…”