Analysis and geometry on a bounded strictly pseudoconvex domain and its boundary

Let D be a bounded pseudoconvex domain in Cn . Let g = gij¯dzidzj¯ , be a Kahler metric on D. The purpose of this dissertation is to study some spectral problems and characterization problems on the domain D and its boundary ∂ D.;In the first part, we consider the case where D is strictly pseudoconvex and ∂D is a real ellipsoid. Given a contact form, the real ellipsoid is a strictly pseudoconvex pseudo-Hermitian CR manifold of hypersurface type. Under suitable conditions, we establish a CR version of Obata theorem on real ellipsoids. We also prove that a real ellipsoid must be biholomorphic to the unit sphere in Cn provided that the Webster pseudo scalar curvature satisfy a certain assumption.;In the second part, we explore the spectrum of the Laplace-Beltrami operator on the domain D with respect to a metric of hyperbolic type. We obtain sharp estimates for the upper bound of the infimum of the spectrum when D is endowed with either a Kahler-Einstein metric or the Bergman metric. We also provide examples of manifolds for which the precise value of the infimum of the spectrum can be computed.;In the last part, we establish an explicit approximation formula for the solution of Fefferman equation on D. In addition, we consider the case where the boundary of D is a real ellipsoid in C2 . Under the assumption that D possesses a C4 Kahler-Einstein potential, we prove that D must be biholomorphic to the unit ball in C2 .