Invariant metrics on domains

The purpose of this thesis is to introduce a (topological) metric on the space of biholomorphic equivalence classes of pointed taut manifolds and to study certain properties and applications of this metric.; The metric is defined in terms of a certain extremal problem of holomorphic maps. It serves as a natural device to measure how different two domains are holomorphically. We will show that several holomorphic invariants are stable with respect to this metric.; Equipped with the metric, the space of equivalence classes of pointed taut manifolds becomes a metric space. It is shown that this space is unbounded. We will show that there exist isolated points in the space. We will also show that there exist infinitely many "geodesic segments" (that is, the shortest curves) connecting two arbitrarily close domains.; As one of the applications, we study the behavior of the automorphism groups of perturbations of taut manifolds and bounded domains. We prove upper semi-continuity of the isotropy subgroups of automorphism groups of respect to the topology induced by the metric. We also prove a semi-continuity result for the identity components of the automorphism groups.; Denoting a pointed domain with the notation (D,p), we prove that for a strongly pseudoconvex domain D, lim{dollar}sb{lcub}ptopartial D{rcub}{dollar}(D,p) = ({dollar}Bsp{lcub}n{rcub}{dollar},0), where {dollar}Bsp{lcub}n{rcub}{dollar} is the unit ball. We give an estimate for this convergence. On the basis of this estimate, we give estimates for the boundary behavior of the Kobayashi metric and the Caratheodory metric. These estimates are more precise than the known estimates, thus useful in problems which need subtle analysis of invariant metrics.; Finally we apply the aforementioned estimates of invariant metrics to the study of iterates of holomorphic maps. We prove that if a holomorphic self map of a bounded contractible strongly pseudoconvex domain in {dollar}doubcsp{lcub}n{rcub}{dollar} with {dollar}Csp3{dollar} boundary has no fixed point in the interior, then it has a fixed point on the boundary in certain sense. This implies an analogue of the classical Denjoy-Wolff Theorem for contractible strongly pseudoconvex domains in {dollar}doubcsp2{dollar} with smooth boundary. We also prove that if D {dollar}subsetsubset{dollar} {dollar}doubcsp{lcub}n{rcub}{dollar} is a contractible strongly pseudoconvex domain with smooth boundary, not biholomorphic to the ball, then every automorphism of D has a fixed point in D.