Local space time constant mean curvature and constant expansion foliations

TL;DR

This paper develops a method to construct and analyze local foliations of surfaces with prescribed mean curvature in a manifold, with applications to general relativity, including conditions for their existence and uniqueness.

Contribution

It introduces a Lyapunov Schmidt reduction approach to construct and analyze local constant mean curvature and constant expansion foliations in a geometric setting.

Findings

Constructed local foliations around a point with prescribed mean curvature.

Proved uniqueness of these foliations under certain conditions.

Identified nonexistence conditions for specific foliations.

Abstract

Inspired by the small sphere-limit for quasi-local energy we study local foliations of surfaces with prescribed mean curvature. Following the strategy used by Ye in 1991 to study local constant mean curvature foliations, we use a Lyapunov Schmidt reduction in an n+1 dimensional manifold equipped with a symmetric 2-tensor to construct the foliations around a point, prove their uniqueness and show their nonexistence conditions. To be specific, we study two foliation conditions. First we consider constant space-time mean curvature surfaces. These foliations were used by Cederbaum and Sakovich to characterize the center of mass in general relativity. Second, we study local foliations of constant expansion surfaces.