Levi-Flat Minimal Hypersurfaces in Two-dimensional Complex Space Forms

Abstract

The purpose of this article is to classify the real hypersurfaces in complex space forms of dimension 2 that are both Levi-flat and minimal. The main results are as follows:

When the curvature of the complex space form is nonzero, there is a 1-parameter family of such hypersurfaces. Specifically, for each one-parameter subgroup of the isometry group of the complex space form, there is an essentially unique example that is invariant under this one-parameter subgroup.

On the other hand, when the curvature of the space form is zero, i.e., when the space form is $\mathbb{C}^2$ with its standard metric, there is an additional 'exceptional' example that has no continuous symmetries but is invariant under a lattice of translations. Up to isometry and homothety, this is the unique example with no continuous symmetries.