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

Details Levi-flat Minimal Hypersurfaces in Two-dimensional Complex Space Forms Levi-flat Minimal Hypersurfaces in Two-dimensional Complex Space Forms

1999-09-27

arxiv.org/abs/math/9909159v1

Adv. Stud. Pure Math., 37, Math. Soc. Japan, Tokyo, 2002, 1--44

Mathematics

Differential Geometry

AMS-TeX 2.1, 35 pages, uses amsppt.sty

Scientific paper

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 complex 2-space with its standard flat 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.

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