EXPLICIT REPRESENTATION OF COMPACT CONFORMALLY FLAT HYPERSURFACES

We explicitly construct compact conformally Bat hypersurfaces in a sim ply connected, (n+1)-dimensional space form, where n is greater than 3 . We may assume that the ambient space is the standard (n+1)-sphere by a conformal diffeomorphism of a simply connected space form into the sphere. From this viewpoint we give a global parameterization of compa ct conformally hat hypersurfaces, and we establish relation between tw o types of hypersurfaces; one has umbilic points and the other has not . It is known that each compact conformally Rat hypersurface in a spac e form is conformally equivalent to a classical Schottky manifold. In order to determine the conformal types of our hypersurfaces. we explic itly represent conformal diffeomorphism of these hypersurfaces to corr esponding Schottky manifolds. In particular, we clarify the relation b etween our results and Pinkall's results.