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The Dirichlet boundary-value problem and isoperimetric inequalities forpositive definite regular solutions of the vacuum Einstein equations arestudied in arbitrary dimensions for the class of metrics with boundariesadmitting a U(1) action. We show that in the case of non-trivial bundlesTaub-Bolt infillings are double-valued whereas Taub-Nut and Eguchi-Hansoninfillings are unique. In the case of trivial bundles, there are twoSchwarzschild infillings in arbitrary dimensions. The condition of whether aparticular type of filling in is possible can be expressed as a limitation onsquashing through a functional dependence on dimension in each case. The caseof the Eguchi-Hanson metric is solved in arbitrary dimension. The Taub-Nut andthe Taub-Bolt are solved in four dimensions and methods for arbitrary dimensionare delineated. For the case of Schwarzschild, analytic formulae for the twoinfilling black hole masses in arbitrary dimension have been obtained. Thisshould facilitate the study of black hole dynamics/thermodynamics in higherdimensions. We found that all infilling solutions are convex. Thus convexity ofthe boundary does not guarantee uniqueness of the infilling. Isoperimetricinequalities involving the volume of the boundary and the volume of theinfilling solutions are then investigated. In particular, the analogues ofMinkowski's celebrated inequality in flat space are found and discussedproviding insight into the geometric nature of these Ricci-flat spaces.Comment: 40 pages, 3 figure

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