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Boundaryvalue problems in quantum gravity and classical solutions

It is proved that TaubBolt infillings are doublevalued whereas TaubNut and EguchiHanson infillings are unique in arbitrary dimensions. In the case of trivial bundles, there are two or no Schwarzschild infillings. The condition of whether a particular type of infilling exists can be expressed as a limitation on squashing through a functional dependence on dimension in each case. The case of the EguchiHanson metric is solved in arbitrary dimension. The TaubNut and the TaubBolt are solved in four dimensions and methods for higher dimensions are discussed. For the case of Schwarzschild in arbitrary dimension, thermodynamic properties of the two infilling blackhole solutions are discussed and analytic formulae for their masses are obtained using higher order hypergeometric functions. Convexity of the infilling solutions and isoperimetric inequalities involving the volume of the boundary and the volume of the infilling solutions are investigated. In particular, analogues of Minkowski’s celebrated inequality in flat space are found and discussed. In Chapters 3, the Dirichlet problem is studied for an SU (2) x U(1)invariant S^{3} boundary within the class of selfdual TaubNut(anti) de Sitter metrics. Including complex ones there can be a total of three solutions for the infilling although there will be a unique real solution or no real solution depending on the boundary data  the two radii of the S^{3}. Exact solutions of the infilling geometries are obtained making its possible to find their Euclidean actions as analytic functions of the two radii of the S^{3}boundary. The case of L < 0 is investigated further. For reasonable squashing of the S^{3}, all three infilling solutions have realvalued actions which possess a “cusp catastrophe” structure with a “catastrophe manifold” that shows that the unique real positivedefinite solution dominates. The necessary and sufficient condition for the existence of the positivedefinite solution is found as a condition on the two radii of the S^{3}. In Chapter 4, the same boundaryvalue problem is studied for the TaubBoltantide Sitter metrics. Such metrics are obtained from the twoparameter TaubNUTanti de Sitter family. The condition of regularity results in two bifurcated oneparameter family. It is found that any axially symmetric S^{3}boundary can be filled in with at least one solution coming from each of these two branches. The infillings appear or disappear catastrophically in pairs as the values of the two radii of S^{3} are varied; this happens simultaneously for both branches. It is found that the total number in independent infillings is two, six or ten. When the two radii are of the same order and large this number is two. In the isotropic limit, i.e., for round S^{3} this holds for small radii as well. In Chapter 5, the Dirichlet problem is studied within Euclideanised Schwarzschildanti de Sitter and anti de Sitter metrics, i.e., for an S^{1} x S^{n} boundary. For such boundary data there exist two or no blackholes and always a unique anti de Sitter solution. The black holes have strictly positive and negative specific heats (and hence locally thermodynamically stable and unstable respectively). It is shown that for any radius of the cavity, the larger hole can be globally thermodynamically stable above a critical temperature by demonstrating that a phase transition occurs from hot AdS to SchwarzschildAdS within the cavity. This gives the HawkingPage phase transition in the infinite cavity limit. It is found that the case of five dimensions is special in that the masses of the two black holes, and hence other quantities of classical and semiclassical interest, can be obtained exactly as functions of cavity radius and temperature. It is also possible in this case to obtain the minimum temperature (below which no black holes exist) and the critical temperature for phase transition as analytic functions of cavityradius. In Chapter 6, cosmological and instanton solutions are found for CP^{1} and CP^{2} sigma models coupled to gravity with a possible cosmological constant.
