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Abstract:
Corvino, Corvino and Schoen, Chrusciel and Delay have shown the existence of a
large class of asymptotically flat vacuum initial data for Einstein’s field equations which
are static or stationary in a neighborhood of space-like infinity, yet quite general in the
interior. The proof relies on some abstract, non-constructive arguments which makes it
difficult to calculate such data numerically by using similar arguments.
A quasilinear elliptic system of equations is presented of which we expect that it can
be used to construct vacuum initial data which are asymptotically flat, time-reflection
symmetric, and asymptotic to static data up to a prescribed order at space-like infinity.
A perturbation argument is used to show the existence of solutions. It is valid when the
order at which the solutions approach staticity is restricted to a certain range.
Difficulties appear when trying to improve this result to show the existence of solutions
that are asymptotically static at higher order. The problems arise from the lack of
surjectivity of a certain operator.
Some tensor decompositions in asymptotically flat manifolds exhibit some of the difficulties
encountered above. The Helmholtz decomposition, which plays a role in the preparation
of initial data for the Maxwell equations, is discussed as a model problem. A method
to circumvent the difficulties that arise when fast decay rates are required is discussed. This
is done in a way that opens the possibility to perform numerical computations.
The insights from the analysis of the Helmholtz decomposition are applied to the York
decomposition, which is related to that part of the quasilinear system which gives rise
to the difficulties. For this decomposition analogous results are obtained. It turns out,
however, that in this case the presence of symmetries of the underlying metric leads to
certain complications. The question, whether the results obtained so far can be used again
to show by a perturbation argument the existence of vacuum initial data which approach
static solutions at infinity at any given order, thus remains open. The answer requires
further analysis and perhaps new methods.
