Relative volume comparison for asymptotically flat manifolds and rigidity of total mass

Asymptotically flat manifolds are being studied intensively for the positive mass theorem and the Riemannian Penrose inequality. We obtain an asymptotic relative volume comparison estimate. From the volume comparison, we show the rigidity of total mass with respect to some integral norm of Ricci curvature. Also we show that the integral norm of Ricci curvature has a positive lower bound for an asymptotically flat manifold whose boundary is a compact minimal hypersurface.