Ricci-flat metrics with U(1) action and the Dirichlet boundary-value problem in Riemannian quantum gravity and isoperimetric inequalities

Abstract

The Dirichlet boundary-value problem and isoperimetric inequalities for positive definite regular solutions of the vacuum Einstein equations are studied in arbitrary dimensions for the class of metrics with boundaries admitting a U(1) action. In the case of trivial bundles, apart from the flat space solution with periodic identification, such solutions include the Euclideanized Schwarzschild metrics with an arbitrary compact Einstein-manifold as the base, whereas for non-trivial bundles the regular solutions include the Taub-Nut metric with a ℂPn base and the Taub-Bolt and the Euguchi–Hanson metrics with an arbitrary Einstein–Kähler base. We show that in the case of non-trivial bundles Taub-Bolt infillings are double-valued whereas Taub-Nut and Eguchi–Hanson infillings are unique. In the case of trivial bundles, there are two Schwarzschild infillings in arbitrary dimensions. The condition of whether a particular type of filling in is possible can be expressed as a limitation on squashing through a functional dependence on dimension in each case. The case of the Eguchi–Hanson metric is solved in arbitrary dimension. The Taub-Nut and the Taub-Bolt are solved in four dimensions and methods for higher dimensions are delineated. For the case of the Schwarzschild in arbitrary dimension, thermodynamic properties of the two infilling black-hole solutions are discussed and analytic formulae for their masses are obtained. These should facilitate further study of black-hole dynamics/thermodynamics in higher dimensions. We found that all infilling solutions are convex. Thus convexity of the boundary does not guarantee uniqueness of the infilling. Isoperimetric inequalities involving the volume of the boundary and the volume of the infilling solutions are then investigated. In particular, the analogues of Minkowski's celebrated inequality in flat space are found and discussed providing insight into the geometric nature of these Ricci-flat spaces.