I will introduce the harmonic level set method developed by Stern in 2019.
This technique has been used to prove the positive mass theorems in various
settings, for example, the Riemannian case, the spacetime case, the
hyperbolic case, and the positive mass theorem with charge. I will focus on
the positive mass theorem for asymptotically hyperbolic manifolds. We give a
lower bound for the mass in the asymptotically hyperbolic setting. In this
setting, we solve the spacetime harmonic equation and give an explicit
expansion for the solution. We also prove some rigidity results as
corollaries. This is joint work with Bray, Hirsch, Kazaras, and Khuri.