Geometric Function Theory in the Unit Ball of $\mathbb{C}^n$

This monograph is a detailed development of geometric function theory in several complex variables from basic principles, focusing on the Euclidean unit ball as the domain. A thorough treatment covering decades of progress, the text presents subjects central to the field including the families of convex, starlike, and spirallike mappings, Loewner theory, and extension operators, as well as myriad additional topics such as linear invariance, Denjoy-Wolff theory, invariant metrics, holomorphic semigroups, the Andersen-Lempert theorem, Gromov hyperbolicity, and a Herglotz-type representation. Many results from the one-variable theory are found, often as consequences of higher-dimensional results. All background needed beyond what is found in standard introductory graduate courses is provided, including introductory material on holomorphic mappings of several variables, ensuring the book is valuable to interested graduate students and experienced researchers alike.