Authors’ abstract: In this paper we study a spacially compact space-time

$(M,g)$ evolved through a conformal Killing vector (CKV) field

$\xi $ such that: (a) the normal component of

$\xi $ is constant on each space-like slice

${\Sigma}$ and each

${\Sigma}$ has constant mean curvature; (b) the stress energy tensor obeys the mixed energy condition; (c) the conformal scalar function is non-decreasing along the evolution CKV field

$\xi $. We prove that: (i)

$\xi $ is homothetic and orthogonal to

${\Sigma}$; (ii)

${\Sigma}$ is hyperbolic and totally umbilical in

$M$; and (iii)

$M$ is a vacuum space-time. We also discuss a physically important case of Killing horizon when

$\xi $ is a null Killing vector field and

${\Sigma}$ degenerates to a null hypersurface.