#
Poor man, rich man: the positive energy theorem - a problem in Lorentzian geometry? and:
regularity of solutions of Witten's equation, with applications to positive energy theorems.

## A poor man's positive energy theorem: II. Null geodesics

**Author:** Piotr T. Chruściel

**Comments:** Latex2e, 24 A4 pages, http://arxiv.org/abs/gr-qc/0406077

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.

## A poor man's positive energy theorem

**Author:** Piotr T. Chruściel, Gregory Galloway

**Comments:** gr-qc/0402106, 8 pages

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.

## Boundary value problems for Dirac-type equations, with applications

**Author:** Robert Bartnik, Piotr T. Chruściel

**Comments:** math.DG/0307278, 88 pages

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.

## Boundary value problems for Dirac-type equations

**Author:** Robert Bartnik, Piotr T. Chruściel

**Comments:**To appear in Crelle Journal (Journal for Reine and Angewandte Mathematik), 58 pages

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.

You can find here my Erice paper on the definition of the ADM mass, and here
my work (in collaboration with M.Herzlich, J.Jezierski, G.Nagy and S.Leski) on the mass of asymptotically hyperbolic initial data sets, including the Trautman-Bondi mass.