The celebrated positive mass theorem, first proven by Schoen and Yau and then by Witten in the 80’s, says that the ADM energy-momentum vector of an asymptotically flat initial data set with the dominant energy condition must be either future timelike or null. It led to the natural question to characterize initial data sets with null ADM energy-momentum, so-called the equality case. There have been some successful approaches that advance our knowledge on the equality case. It is known that in low dimensions the equality case can only be realized by slices of the Minkowski spacetime. In higher dimensions, an analogous result holds, provided a stringer decay rate for asymptotically flatness, and counterexamples are found without the stringer decay rate. In this mini course, I will describe a variational approach to the equality case, inspired by Bartnik’s quasi-local mass program and uniqueness of stationary spacetime. I will also discuss how that approach is extended to handle the equality case of the positive mass theorem for asymptotically (locally) hyperbolic manifolds. The mini course is based on joint work with Hyun Chul Jang, with Dan A. Lee, and with Daniel Martin.