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Near space-like infinity an initial value problem for the conformal Einstein equations is formulated such that: (i) the data and equations are regular, (ii) space-like and null infinity have a finite representation, with their structure and location known a priori, and (iii) the setting relies entirely on genera1 properties of conformal structures.A first analysis of this problem shows that the solutions develop in general a certain type of logarithmic singularity at the set where null infinity touches space-like infinity. These singularities form an intrinsic part of the solutions' conformal structure. Conditions on the free initial data near space-like infinity are derived which ensure that for solutions developing from these data singularities of this type cannot occur.

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The regularity of the solutions to the Yamabe Problem is considered in the case of conformally compact manifolds and negative scalar curvature. The existence of smooth hyperboloidal initial data for Einstein's field equations is demonstrated.

We discuss Einstein's equations in the context of normal conformal Caftan connections, derive a new conformal representation of the equations, and express the equations in a conformally invariant gauge. The resulting formulation of the equations is used to show the existence of asymptotically simple solutions to Einstein's equations with a positive cosmological constant. The solutions are characterized by Cauchy data on a space-like slice and by the intrinsic conformal structure on the conformal boundary at space-like and null infinity.