  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. 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 arbitrary dimension
are delineated. For the case of Schwarzschild, analytic formulae for the two
infilling black hole masses in arbitrary dimension have been obtained. This
should facilitate the 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.
