The aim is to define what it means to be a meromorphic section of a quaternionic holomorphic vector bundle over a compact Riemann surface and then prove a version of the Riemann-Rock theorem for divisors that generalizes the classical theorem. A meromorphic section of a quaternionic spin bundle provides Weierstrass data (modulo period conditions) for a conformal map into Euclidean three space with prescribed mean curvature half density.
