Holomorphicity of parabolic stable minimal surfaces of high codimension

Abstract

A classical theorem of Micallef says that if $F:(\Sigma ,g)\to {\mathbb{R}}^4$ is a stable minimal immersion of an oriented $2$-dimensional complete Riemannian manifold (that is parabolic) into ${\mathbb{R}}^4$, it is necessarily holomorphic with respect to some parallel orthogonal complex structure on ${\mathbb{R}}^4$. We generalize this theorem by replacing ${\mathbb{R}}^4$ with ${\mathbb{R}}^{2+2k}$ for any codimension $2k$, under the additional hypothesis that the normal bundle $N\Sigma$ is equipped with a complex structure that is compatible with the induced metric and parallel with respect to the induced connection. This is a necessary assumption for such a theorem to hold, and it is automatically satisfied in the classical case $k=1$. We also briefly discuss possible further generalizations of such a result to other calibrations and to Smith maps.

16 pages, comments welcome