Remarks on a construction of Duistermaat

Abstract

A construction of Duistermaat's, in which an auxiliary holomorphic curve in the line bundle $\mathcal O(4)$ on $\mathsf{P}_1$ is associated to a minimal surface in $R^3$, is described in a new way and generalised to minimal surfaces in $R^n$. If the minimal surface is complete and has finite total curvature then the auxiliary curve is algebraic.