Asymptotically flat 3-manifolds arise as space-like slices of Lorentzian 4-manifolds modelling isolated gravitating systems in General Relativity. The lecture discusses how radial families of 2-dimensional surfaces governed by mean curvature flow can serve to describe physical concepts such as mass, center of mass and quasi-local mass of the isolated system in terms of geometric quantities such as an isoperimetric deficit. Particular emphasis will be given to the interplay between analytical techniques and geometric and physical concepts.