The author studies the Cauchy problem for Schrödinger maps from into a target manifold with a complex structure . The Schrödinger map equation is given by
where is the covariant derivative along the curve . A Schrödinger map is a function that solves the equation (1). To show that solutions of the Cauchy problem for the Schrödinger map equation (1) exist, the author first studies the following approximate equation for any
The equation (2) is a wave map, for which general existence theory is known. For appropriate initial data there is a sequence of local solutions of equation (2) that exist on the time intervals . The limit of these approximate solutions as solves the Schrödinger map problem. Energy estimates which bound the norms of solutions imply that is independent of . Then, for some fixed , the time interval of existence for each solution is , and their limit exist on this same interval. By a standard convergence argument it is proven that satisfies the Schrödinger map equation (1). The uniqueness of the Schrödinger map is also shown.