Positive Mass Theorem I'm currently a third year undergrad writing about Minimal Surfaces. In particular, trapped surfaces and black holes.
What does the Positive Mass Theorem have to do with this? And does the theorem directly predict the existence of black holes?
 A: The positive mass theorem is a theorem, so you make some assumptions and fix a deductive methodology and then get a conclusion.  Common assumptions include an energy condition and asymptotic flatness.  The conclusion is usually something like that the ADM mass is positive.
I think the idea is intuitively explained by an example.  If you look at the Schwarzschild solution you see that it has a parameter M.  Assume it is positive.  You also notice that for large $r$ the metric approximates that of Minkowski spacetime in spherical coordinates, that's an example of an asymptotically flat metric.  Then you notice that just looking at the metric at very large $r$ you can extract the parameter $M$ from the metric.  It's positive.  You identify that large scale behaviour as the total mass $M$ from the metric in the asymptotically flat region.  Then you wonder if you can use that same method to extract parameters from arbitrary spacetimes that are asymptotically flat.  You'd like to get positive numbers so you can interpret them as the total mass.  You need hypothesi to get your conclusion, so you make some up, then you deduce your conclusion, and thus you have a theorem.
Better theorems have weaker assumptions.  Trapped surfaces are historical ways to try to bound an event horizon, and an event horizon is what makes a black hole (not the singularity).  In general the positive mass theorem doesn't have to have anything to do with black holes, for instance a stationary static spherically symmetric uncharged star has an exterior solution that is just like the exterior of a stationary static spherically symmetric uncharged black hole, so the exterior asymptotically flat end has a positive ADM mass, but there is no trapped surface, no event horizon, and no black hole in the case of the stationary static spherically symmetric uncharged star.
