Holomorphic curves with bounded Fubini-Study derivative

In this dissertation we present work on two problems in the theory of holomorphic curves. The first is to generalize to arbitrary dimension a result of Hayman and Clunie. This work concludes the efforts initiated by Eremenko by proving the result for curves whose Fubini-Study derivative has polynomial growth. The second problem is the computation of rates of growth for the derivative of a holomorphic self-map of n-dimensional complex projective space. This computation is related to the work of Gromov and others on "short" maps between compact manifolds.