Minimal immersion of surfaces in quaternionic projective spaces

Abstract

For a minimal immersion of a surface in a quaternionic Kähler manifold a concept of non-degeneracy is defined. Then using a theorem on elliptic differential systems we show a non-degenerate surface is in a sense generic, and around each point with possible exception of an isolated set of degenerate points we can define a smooth Darboux frame. The frame is continuous at a degenerate point. Next, by reducing the structure group we define a symmetric sextic form of type $(6,0)$ and we show in the case that ambient space is $HP^{n}$ this form is a holomorphic (abelian) differential. The last section is a brief note on the relation of our work to Glazebrook's twistor spaces for $HP^{n}$.