Classical boundary-value problem in Riemannian quantum gravity and Taub–Bolt–anti-de Sitter geometries

Abstract

For an SU (2) × U (1)-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 (L) 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.

Unique continuation results for Ricci curvature and applications

“…We point out that the the same result, and proof, also hold in the case of Einstein metrics on bounded domains, via Theorem 3.1; the condition (5.1) is of course replaced by L X A = 0. For some examples and discussion in the bounded domain case, see [1], [2].…”

“…If t s is the geodesic defining function for g s , (with boundary metric γ), then the Fefferman-Graham expansion givesḡ s = dt 2 s + (γ + t 2 s g (2),s + · · · + t n s g (n),s ) + O(t n+1 ). The estimate (5.40) implies that t s = t + sO(t n+2+α ) + O(s 2 ), so that modulo lower order terms, we may view t s ∼ t. Taking the derivative of the FG expansion with respect to s at s = 0, and using the fact that X is Killing on (∂M, γ), together with the fact that the lower order terms g (k) , k < n, are determined by γ, it follows that, forκ as in (5.37), (5.42)κ = 1 2 L X g (n) , at ∂M .…”

Formation of Higher-dimensional Topological Black Holes

“…We take the Riemannian Taub-NUT metric with a cosmological constant (k rather than Λ with our conventions) [6,1]…”

P 2 and P 1 sigma models in supergravity: Bianchi type IX instantons and cosmologies

“…In this section we briefly describe how one obtains two one-parameter family of metrics from (4.4) by making one of the roots regular (for more details, see [1,2,3,4]). The condition of regularity of (4.4) at any point ρ bolt where ∆ = 0 works out to be [15]:…”