A New Gap Theorem Result for Proper Holomorphic Mappings Between Complex Balls

In 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.Mathematics, Department o