Abstract. Initial value problems involving hyperboloidal hypersurfaces are pointed out. Characteristic properties of hyperboloidal initial data and rigorous results concerning the construction of smooth hyperboloidal initial data are discussed.

INTRODUCTION

In this article I shall discuss some properties of “hyperboloidal hypersurfaces”. These occur naturally in a number of interesting initial value problems. I became first interested in them in the context of abstract existence proofs for solutions of Einstein's field equations which fall off in null directions in such a way that they admit the construction of a smooth conformal boundary at null infinity (Friedrich (1983)). But it appears to me that hyperboloidal hypersurfaces should also be of interest, in particular if questions concerning gravitational radiation are concerned, in various numerical studies.

Let us consider solutions to Einstein's field equations with vanishing cosmological constant and possibly massive sources of spatially compact support and long range fields like Maxwell fields. We call a space-like hypersurface in such a space-time “hyperboloidal” if it extends to infinity in such a way that it ends on null infinity. We assume that the hypersurface remains space-like in the limit when it “touches null infinity”. The standard examples of such hypersurfaces are the space-like unit hyperbolas in Minkowski space, which motivate the name hyperboloidal. In the standard picture of Minkowski space it is seen that these hypersurfaces are asymptotic to certain null cones.