Abstract

The classification of bounded domains in ${ℂ}^n$, with $n>1$, is related to the geometric properties of the boundary. A conjecture of Greene and Krantz relates the geometry of the boundary with the group of biholomorphic self mappings of the domain. The Greene–Krantz conjecture, if true, can tell us much about the classification of smoothly bounded domains in ${ℂ}^n$. Much work has been done to attempt to solve this conjecture, though it has yet to be proved or disproved. However, there are numerous partial results which support the conjecture. In this paper, we prove a special case of the conjecture:

Theorem: Suppose $\Omega \subset {ℂ}^n$ is a bounded convex domain with $C^{\infty }$ boundary. Suppose there exists $\phi \in Aut\left(\Omega \right)$ and $p\in \Omega$ such that for the sequence of iterates $\left\{{\phi }^j\right\}\subset Aut\left(\Omega \right)$ we have ${\phi }^j\left(p\right)\to x\in \partial \Omega$ nontangentially. Then $x$ is of finite type.

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Mathematical Subject Classification 2010

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