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On the Characterization of Convex Domains With Non-Compact Automorphism Group


In the field of several complex variables, one of the most important questions that remains to be answered is the voracity of the Greene-Krantz conjecture:

Conjecture 0.0.1. Let D be a smoothly bounded domain in Cn. Suppose there exists {gj} ⊂ Aut(D) such that {gj(x)} accumulates at a boundary point p ∈ ∂D for some x∈D. Then ∂D is of finite type at p.

The purpose of this dissertation is to prove a result that supports the truth- fulness of this conjecture:

Theorem 0.0.2. Let D be a bounded convex domain in Cn with C2 boundary. Suppose that there is a sequence {gj} ⊂ Aut(D) such that {gj(z)} accumulates at a boundary point some point z ∈ D. Then if p ∈ ∂D is such an orbit accumulation point, ∂D contains no analytic variety passing through p.

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