Ji, Shanyu2019-09-182019-09-18May 20142014-05May 2014https://hdl.handle.net/10657/4865In this dissertation, we study how rigidity properties of proper holomorphic mappings from complex balls of dimension $n$ to complex balls of dimension $N$ allow us to classify all such maps for particular values of $n$ and $N$, up to equivalence by automorphism on the boundary. For maps $F:\mathbb B^n\rightarrow \mathbb B^N$, when $N$ takes values in certain intervals, all maps are equivalent to $(G,0)$, where $G$ is a proper holomorphic map from $\mathbb B^n$ to $\mathbb B^M$, with $M < N$. These intervals are established by the First, Second, and Third Gap Theorems. The cases where $N$ lies on the upper boundary of one of these gaps are more difficult than cases where $N$ is smaller. We review the cases where $N < 3n-3$ and where $3n < N < 4n-6$, and then we prove a classification theorem for the case where $N= 3n-3$. In Chapter 10, we preemptively simplify our calculation by examining the coefficients of the $(1,1)$ terms of the Taylor expansion of the codimension components $\phi_{k\ell}$ of a normalized map. Given any $p$, our map $F$ is linear fractional along some affine subspace through $p$, which we use to show that all but one of the coefficients we investigate in this chapter are 0. We assume the geometric rank $\kappa_0 = 2$, because the smaller cases are covered in previous papers, and larger $\kappa_0$ has been shown to be impossible (cf. [HJX06, Corollary 1.3]). In Chapter 11, we perform a long calculation to show that $\deg(F)\leq 2$ along a certain Segre variety. This takes two steps. First, we show that $\deg(F)\leq 3$, and then we use an explicit statement of $F$ as a rational map to show $\deg(F)\leq 2$. Our calculation allows us to apply two theorems, [HJX06,Theorem 5.4] and [Leb11, Theorem 1.5], to prove that $F$ must be equivalent to the Generalized Whitney Map.application/pdfengThe author of this work is the copyright owner. UH Libraries and the Texas Digital Library have their permission to store and provide access to this work. Further transmission, reproduction, or presentation of this work is prohibited except with permission of the author(s).CR geometryComplex analysis2019-09-18Thesisborn digital