We show that positivity of energy for stationary, or strongly uniformly Schwarzschildian, asymptotically flat, non-singular domains of outer communications can be proved using Galloway's null rigidity theorem.
We show that positivity of energy for stationary, asymptotically flat, non-singular domains of outer communications is a simple corollary of the Lorentzian splitting theorem.
We prove regularity for a class of boundary value problems for first order elliptic systems, with boundary conditions determined by spectral decompositions, under coefficient differentiability conditions weaker than previously known. We establish Fredholm properties for Dirac-type equations with these boundary conditions. Our results include sharp solvability criteria, over both compact and non-compact manifolds; weighted Poincaré and Schroedinger-Lichnerowicz inequalities provide asymptotic control in the non-compact case. One application yields existence of solutions for the Witten equation with a spectral boundary condition used by Herzlich in his proof of a geometric lower bound for the ADM mass of asymptotically flat three dimensional manifolds.
The paper, "Boundary value problems for Dirac--type equations, with applications", described just above, was judged too long by the referees, so it was split into two parts: the theory, and the applications. Here we have kept the analysis part of that paper, the applications will be published elsewhere, if at all.