UC Riverside

This dissertation investigates rigidity in several complex variables through the study of intrinsic metrics on bounded domains in Cn. In particular, it focuses on geometric and analytic characterizations of the unit ball using the Carathéodory, Kobayashi, and Bergman metrics. These metrics are biholomorphically invariant and play a central role in understanding the complex geometry of domains independently of coordinate representations. The work is motivated by classical results of Lempert and Lu, as well as more recent developments concerning strongly pseudoconvex domains and curvature rigidity. The first main contribution concerns the Kähler structure of the Carathéodory metric on bounded strongly pseudoconvex domains. Lempert proved that on bounded convex domains the Carathéodory and Kobayashi metrics coincide, and an important consequence is that the Kobayashi metric is Kähler if and only if the domain is biholomorphic to the unit ball. More recently, Zimmer extended this rigidity phenomenon to strongly pseudoconvex domains by showing that if the Kobayashi metric is Kähler, then the universal cover of the domain is biholomorphic to the unit ball. In this dissertation, we establish an analogous characterization using the Carathéodory metric. Our approach builds upon the theory of complex geodesics developed by Lempert and later generalized by Bracci, Fornæss, and Wold. A key ingredient is the existence and behavior of complex geodesics near strongly pseudoconvex boundary points. By analyzing the relationship between extremal maps and intrinsic metrics, we prove that the Kähler condition for the Carathéodory metric imposes strong geometric rigidity on the domain. This result deepens the connection between intrinsic metrics and biholomorphic classification and provides a new perspective on metric rigidity phenomena in several complex variables. The second principal contribution generalizes Lu's classical uniformization theorem for the Bergman metric. Lu proved that if a bounded domain in Cn has a complete Bergman metric with constant negative holomorphic sectional curvature, then the domain must be biholomorphic to the unit ball. In this dissertation, we weaken the global curvature assumption and show that local constancy of the holomorphic sectional curvature is sufficient to obtain the same conclusion. Central to the proof is the theory of Bergman representative coordinates, which provides a local biholomorphic linearization of the domain and allows curvature information to be transferred effectively. By exploiting these coordinates and comparing the Bergman and Carathéodory metrics, we demonstrate that local geometric data can determine the global biholomorphic type of a domain. As an application, we extend observations of Cheung and Wong by relaxing convexity assumptions to strong pseudoconvexity in certain settings. In addition to these main results, the dissertation surveys foundational material on intrinsic metrics, curvature, and complex geodesics, and discusses broader connections to important open problems in complex geometry, including the Cheng-Yau conjecture and the Ramadanov conjecture. The results presented here suggest that comparisons among intrinsic metrics, especially through curvature and coincidence properties, may provide new tools for understanding rigidity and uniformization phenomena beyond the strongly pseudoconvex setting.