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Classical boundary-value problem in Riemannian quantum gravity and Taub-Bolt-anti-de Sitter geometries. (English) Zbl 1028.83508
Summary: For an $SU\left(2\right)×U\left(1\right)$-invariant ${S}^{3}$ boundary the classical Dirichlet problem of Riemannian quantum gravity is studied for positive-definite regular solutions of the Einstein equations with a negative cosmological constant within biaxial Bianchi-IX metrics containing bolts, i.e., within the family of Taub-Bolt-anti-de Sitter (Taub-Bolt-AdS) metrics. Such metrics are obtained from the two-parameter Taub-NUT-anti-de Sitter family. The condition of regularity requires them to have only one free parameter $\left(L\right)$ and constrains $L$ to take values within a narrow range; the other parameter is determined as a double-valued function of $L$ and hence there is a bifurcation within the family. We found that any axially symmetric ${S}^{3}$-boundary can be filled in with at least one solution coming from each of these two branches despite the severe limit on the permissible values of $L$. The number of infilling solutions can be one, three or five and they appear or disappear catastrophically in pairs as the values of the two radii of ${S}^{3}$ are varied. The solutions occur simultaneously in both branches and hence the total number of independent infillings is two, six or ten. We further showed that when the two radii are of the same order and large the number of solutions is two. In the isotropic limit this holds for small radii as well. These results are to be contrasted with the one-parameter self-dual Taub-NUT-AdS infilling solutions of the same boundary-value problem studied previously.
##### MSC:
 83C45 Quantization of the gravitational field 83C15 Closed form solutions of equations in general relativity