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

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.

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A poor man's positive energy 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.

Paper: PostScript (600 dpi, 273 K), Dvi (27 K), or Pdf (206 K)

Boundary value problems for Dirac-type equations, with applications

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.

Paper: PostScript (600 dpi, 1082 K), Dvi (400 K), or Pdf (617 K)

Boundary value problems for Dirac-type equations

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.

Paper: PostScript (600 dpi, 742 K), Dvi (265 K), or Pdf (503 K)

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.