Local space time constant mean curvature and constant expansion foliations

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