Since there is no Uniformization Theorem in several complex variables, there is a desire to classify all of the simply connected domains. We use a result of Zimmer and a localization technique of Lin and Wong to extend a result of Cheung et al. In particular, we show that if a domain with $C^{1,1}$ boundary on a Kobayashi hyperbolic complex manifold contains a totally real boundary point and covers a compact manifold, then its universal cover must be the Euclidean ball.

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.

We present two results. The first is a converse to a theorem first proved by Wongwhich says the ratio of intrinsic measures approaches 1 near the boundary of a strongly pseudoconvex domain; we show that for a particular type of domain the boundary is strongly pseudoconvex if the ratio of intrinsic measures approaches 1 near the boundary. The argument is primarily one from Zimmer using the scaling method. What we did is show that the ratio of intrinsic measures is a function which respects this scaling process. Our second contribution was done in an attempt to use one particular step of Huang and Xiao’s proof of the S.-Y. Cheng conjecture to settle the Ramadanov conjecture. While unsuccessful in this regard, we were able to make this step more direct and we ultimately show that if the Bergman metric is asymptotically Kähler-Einstein enough near the boundary of a C ∞ strongly pseudoconvex domain Ω then the boundary ∂Ω must be spherical. This result is of interest on its own but it also provides a more direct proof of the S.-Y. Cheng conjecture and may be used in further work on the Ramadanov conjecture.

Because the Riemann Mapping Theorem does not hold in several complex variables, it is of interest to fully classify the simply connected domains. By considering convex, bounded domains with noncompact automorphism groups, we can define a rescaling sequence based

on the boundary-accumulating automorphism orbit. If this orbit converges nontangentially we prove the accumulation point is of finite type in the sense of D’Angelo. This both provides a partial proof to the Greene-Krantz conjecture and also classifies such domains as polynomial ellipsoids.

The problem of characterizing bounded domains in Cn is can be related to the automorphism group and the geometry of the boundary. It is a conjecture of Greene and Krantz that if a smoothly bounded domain has a noncompact automorphism group, then the boundary is of finite type at any automorphism accumulation point. While there have been numerous supporting results, the conjecture is as yet unsolved. The purpose of this dissertation is to provide another result in support of the Greene-Krantz conjecture. Specifically, if the boundary of a smoothly bounded convex domain admits an iterated automorphism orbit nontangentially, then it is of finite type.

The structure of an open complete Riemannian manifold (Mn,g) with nonnegative

sectional curvature has been studied extensively and well understood. There are

two classical results due to Gromoll-Meyer [9] and Cheeger-Gromol [4]. Gromoll and Meyer proved that a complete open manifold (Mn,g) with positive sectional curvature is diffeormorphic to Rn. On the other hand, Cheeger and Gromoll proved that a complete open manifold (Mn,g) with nonnegative sectional curvature admits a totally geodesic compact submanifold S such that Mn is diffeomorphic to the normal bundle of S in Mn.

It is natural to imagine that these results and many others can easily be attained in

Ricci curvature case. However, in this case, there are relatively few structural results

except in a lower dimensional case n = 2 where all notions of curvature coincide. In

[17], Shen proved that a complete open Riemannian manifold with nonnegative Ricci

curvature and maximum volume growth is proper (admits an exhaustion function).

Regarding Shen’s result, it was observed by Wong and Zhang [21] that a complete open

K¨ ahler manifold with positive bisectional curvature and maximum volume growth can be embedded as a complex submanifold in a complex Euclidean space of higher dimension. Their observation is a partial result of a weaker version of Yau’s conjecture

which states that a complete open K¨ ahler manifold with positive bisectional curvature

can be embedded as a complex submanifold in a complex Euclidean space of higher

dimension. The original Yau’s conjecture [20] states that: a complete open K¨ ahler

manifold with positive bisectional curvature is biholomorphic to complex Euclidean

space.

Here, we exhibit that a complete open K¨ ahler manifold with positive bisectional

curvature can be embedded as a complex submanifold in a complex Euclidean space of

higher dimension if the volume of a cone of rays from a fixed base point is asymptotic

to the volume of a geodesic ball centered at the same point. The volume growth

condition we consider here is weaker than the maximum volume growth condition.

One of the most important problems in the field of several complex variables is the Greene-Krantz conjecture:

Conjecture Let D be a smoothly bounded domain in C^{n with non-compact automorphism group. Then the boundary of D is of finite type at any boundary orbit accumulation point.}

The purpose of this dissertation is to prove a result that supports the truthfulness of this conjecture:

Theorem Let D be a smoothly bounded convex domain in C^{n. Suppose there exists a point p in D and a sequence of automorphisms of D, f _{j, such that f j(p) &rarr q in the boundary of D non-tangentially. Furthermore, suppose Condition LTW holds. Then, the boundary of D is variety-free at q.}}

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.