The boundary effects on the Bose-Einstein condensation with a nonvanishing chemical potential on an ultra-static space-time are studied. High temperature regime, which is the relevant regime for the relativistic gas, is studied through the heat kernel expansion for both Dirichlet and Neumann boundary conditions. The high temperature expansion in the presence of a chemical potential is generated via the Mellin transform method as applied to the harmonic sums representing the free energy and the depletion coefficient. The effects of boundary conditions on the relation between the depletion coefficient and the temperature are analyzed. Both charged and neutral bosons are considered.

The Bose-Einstein condensation for an ideal Bose gas and for a dilute weakly interacting Bose gas in a manifold with nonnegative Ricci curvature is investigated using the heat kernel and eigenvalue estimates of the Laplace operator. The main focus is on the nonrelativistic gas. However, special relativistic ideal gas is also discussed. The thermodynamic limit of the heat kernel and eigenvalue estimates is taken and the results are used to derive bounds for the depletion coefficient. In the case of a weakly interacting gas Bogoliubov approximation is employed. The ground state is analyzed using heat kernel methods and finite size effects on the ground state energy are proposed. The justification of the c-number substitution on a manifold is given.

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The momentum and Hamiltonian constraints of vacuum Einstein equations, within the Bowen‐York formalism, for two interacting black holes in close separation, with anti‐parallel spins and anti‐parallel linear momenta is studied. An analytical solution using perturbation theory is given. The location and the shape of the apparent horizon which generically depend on all the parameters, angles, and the separation between the black holes are also computed. The solution only works for distances far away from the black holes. To gain more insight close into the black holes, one has to go to the higher orders in perturbation theory, which is a rather cumbersome process. But the solution presented here can be of some use for numerical computations as the latter should match our result for the described problem.