##### Venue: Lecture Hall - I, Dept. of Mathematics

This thesis considers two themes, both of which emanate from and involve the Kobayashi and the Carath\\‘{e}odory metric. First we study the biholomorphic invariant introduced by B. Fridman on strongly pseudoconvex domains, on weakly pseudoconvex domains of finite type in $\\mathbf C^2$ and on convex finite type domains in $\\mathbf C^n$ using the scaling method. Applications include an alternate proof of the Wong-Rosay theorem, a characterization of analytic polyhedra with noncompact automorphism group when the orbit accumulates at a singular boundary point and a description of the Kobayashi balls on weakly pseudoconvex domains of finite type in $\\mathbf C^2$ and convex finite type domains in $\\mathbf C^n$ in terms of Euclidean parameters. Second a version of Vitushkin’s theorem about the uniform extendability of a compact subgroup of automorphisms of a real analytic strongly pseudoconvex domain is proved for $C^1$-isometries of the Kobayashi and Carath\\‘{e}odory metrics on a smoothly bounded strongly pseudoconvex domain.

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Last updated: 04 Aug 2020