UC Irvine

This thesis includes the analysis on initial data sets with singularities which helps identify sufficient conditions on the singularity guaranteeing the positivity of mass, characterising the dominant energy condition on polyhedra, and showing the relation between boundary energy and interior energy.

The main contribution of this thesis is to provide both global and local perspectives of the relation between geometry and physics. First, we show a spacetime positive mass theorem with corners. Then, by putting Gromov's dihedral rigidity conjecture and fill-in conjecture into the context of general relativity, we can use the aforementioned theorem to provide partial solutions to these conjectures by constructing suitable extensions for compact initial data sets.