  The constant mass function foliation has been shown useful for studying the
null Penrose inequality on a null hypersurface, because of the monotonicity
formula of Hawking mass along such a foliation. In this paper, we show the
global existence of the constant mass aspect function foliation on a nearly
spherically symmetric incoming null hypersurface, emanating from a spacelike
surface near the apparent horizon to the past null infinity in a vacuum
perturbed Schwarzschild spacetime. Moreover, we study the geometry of the
constant mass aspect function foliation, by comparing with the spherically
symmetric foliation in the Schwarzschild spacetime. The knowledge about the
geometry of the foliation is essential for investigating the perturbation of
the constant mass aspect function foliation, which is the core in the
application to the null Penrose inequality for a vacuum perturbed Schwarzschild
spacetime.
