I will start my talk with an overview of recent results on canonical geometric foliations of asymptotically flat Riemannian manifolds that complete a program initiated by G. Huisken and S.-T. Yau in the early nineties. I will explain how these results are closely tied to what I call effective versions of the positive mass theorem. In the second half of my talk, I will focus on a related conjecture due to R. Schoen on the minimal surface proof of the positive mass theorem, its solution in three space dimensions in my joint work with O. Chodosh, as well as the proof of an important special case of the conjecture and counterexamples to the general conjecture in higher dimensions in joint work with T. Koerber.