UC Riverside

In the field of several complex variables, the Greene-Krantz Conjecture has been of interest for decades.

Conjecture 0.0.1 (Greene-Krantz Conjecture) Let Ω be a smoothly bounded domain in C^n. Suppose there exists {φ_j} ⊂ Aut(Ω) such that {φ_j(p)} accumulates at a boundary point q∈∂Ω for some p∈Ω. Then ∂Ω is of finite type at q.

Proof of the conjecture would allow us to start classifying bounded domains in C^n. We already have classification for bounded domains in one dimension.

The Riemann Mapping Theorem states that there are only two simply connected domains. Specifically, every proper, simply connected open subset in C that is not all of C is biholomorphic to the disc. While it would be nice to generalize this to higher dimensions, in C^2, the ball and bidisc are not biholomorphic to each other. To be one step closer in classifying all bounded domains in C^n, we will add some restrictions, like studying bounded domains with non-compact automorphism group.

In this paper, we will do that and prove a special case of the Greene-Krantz Conjecture in C^2, which can be extended to the case where we have a domain with smooth boundary in C^n.

Theorem 0.0.2 Let Ω be a bounded convex domain in C^2 with C2 boundary. Suppose there is a sequence {φ_j} ⊂ Aut(Ω) such that {φ_j(p)} accumulates at a boundary point for some p ∈ Ω. If q ∈ ∂Ω is an orbit accumulation point, then q is of finite type.