Classical Boundary-value Problem in Riemannian Quantum Gravity and Self-dual Taub-NUT-(anti)de Sitter Geometries

The classical boundary-value problem for positive-definite solutions of the Einstein field equations is studied with an arbitrary cosmological constant, in the case of a compact (S) boundary given a biaxial Bianchi-IX positive-definite three-metric, specified by two radii (a, b). For the simplest, four-ball, topology of the manifold with this boundary, the regular classical solutions are found within the family of Taub-NUT-(anti)de Sitter metrics with self-dual Weyl curvature. For arbitrary choice of positive radii (a, b), we find that there are three solutions for the infilling geometry of this type on the four-ball. We obtain exact solutions for them and for their Euclidean actions. The conditions determining whether a solution can be real or must be complex are determined in terms of the boundary data (a, b). The case of negative cosmological constant is further investigated numerically – the (real) actions are found to have a “catastrophic” structure with a “catastrophe manifold” that intersects with itself much like a Klein bottle. The same boundary-value problem with more complicated interior topology containing a “bolt” is investigated in a forthcoming paper.