Rudolf Zeidler (Münster)
We will present variants of the spacetime positive mass theorem in the spin setting: Firstly, its conclusion \(E \geq|P|\) holds for a single asymptotically flat (AF) end in a spin initial data set satisfying the dominant energy condition, even if the initial data set has other arbitrary complete (but not necessarily AF) ends.
Secondly, we can also treat incomplete situations provided that the incompleteness is controlled in terms of the Riemannian distance on the underlying manifold. Moreover, there are versions of these results in the presence of a compact boundary under suitable boundary conditions.
On a technical level they are based on a combination of Witten’s spinor proof of the positive mass theorem with recent ideas originating in approaches to Gromov’s band with estimates. However, to deal with the general initial data case, a curious new trick is required—we need to formally introduce a second timelike direction to the spinor bundle in addition to the one implicit in the initial data set.
Based on joint work with S. Cecchini and M. Lesourd.
