  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.
