Ricci-flat Metrics with U(1) Action and the Dirichlet Boundary-value Problem in Riemannian Quantum Gravity and Isoperimetric Inequalities

TL;DR

This paper investigates Ricci-flat metrics with U(1) symmetry in higher dimensions, analyzing boundary-value problems, infilling solutions, and isoperimetric inequalities, with implications for black hole thermodynamics and geometric analysis.

Contribution

It provides new solutions and formulas for Ricci-flat metrics with U(1) action, including arbitrary-dimensional Eguchi-Hanson and Schwarzschild infillings, and explores isoperimetric inequalities in this context.

Findings

Taub-Bolt infillings are double-valued in non-trivial bundles.

Unique Taub-Nut and Eguchi-Hanson solutions in specific cases.

Analytic formulas for black hole masses in arbitrary dimensions.

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. 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.