## Arts & Sciences Electronic Theses and Dissertations

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Summer 5-15-2015

#### Author's School

Graduate School of Arts and Sciences

Mathematics

#### Degree Name

Doctor of Philosophy (PhD)

Dissertation

#### Abstract

\begin{itemize}

\item \textit{Chapter I.} In this chapter, finite type domains with hyperbolic orbit accumulation points are studied. We prove, in case of $\mathbb{C}^2$, they have to be (global) pseudoconvex domains, after an assumption of boundary regularity.

\item \textit{Chapter II.} In $\mathbb{C}^2$, we classify the domains for which $\rm Aut(\Omega)$ is noncompact and describe these domains by their defining functions. This chapter is based on the scaling method introduced by Frankel and Kim. One feature is that we are able to analyze the defining functions of infinite type boundary.

\item \textit{Chapter III.} The Schwarz lemmas are well-known characterizations of holomorphic maps and we exhibit two applications. For a sequence family of biholomorphisms $f_j$, it is useful to determine the location of $f_j(q)$ for a fixed point $q$ in source manifolds. With it, we extend the Fornaess-Stout's theorem on monotone unions of balls to ellipsoids. We also discuss the curvature bounds of complete K\"{a}hler metric on $\rtimes$ domains defined in Chapter II.

\item \textit{Chapter IV.} The Wong-Rosay theorem characterizes the strongly pseudoconvex domains of $\mathbb{C}^n$ by their automorphism groups. In this chapter, we generalize the Wong-Rosay theorem to the simply-connected complete K\"{a}hler manifold with a negative sectional curvature. One aim of this chapter is to exhibit a Wong-Rosay type theorem of manifolds with holomorphic non-invariant metrics.

\item \textit{Chapter V.} Let $\phi_j$ be a family of automorphisms of a bounded domain $\Omega$ in $\mathbb{C}^n$. For $q\in\Omega$, the locations of cluster points of $\lbrace\phi_j^{-1}(q)\rbrace$ have been unknown for a long time. We answer this question with a newly defined energy functional for automorphisms motivated by those in context of geometric flows. In the second part, we partially extend the theorem of Chapter I and give a counterexample which reveals the theorem of Chapter I cannot be fully extended.

\item \textit{Chapter VI.} In this Chapter, we introduce some partial results on Diederich-Fornaess index. This is a part of ongoing paper with Krantz \cite{KLiu005}.

\end{itemize}

English (en)

Steven Krantz

#### Committee Members

Quo-Shin Chi, Guido Weiss, John McCarthy, Ramanath Cowsik,