Global existence and geometry of constant mass aspect function foliation in perturbed Schwarzschild spacetime

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.