On homogeneous hypersurfaces in C3

Abstract

We consider a family (Formula presented.), with (Formula presented.), (Formula presented.), of real hypersurfaces in a complex affine n-dimensional quadric arising in connection with the classification of homogeneous compact simply connected real-analytic hypersurfaces in (Formula presented.) due to Morimoto and Nagano. To finalize their classification, one needs to resolve the problem of the embeddability of (Formula presented.) in (Formula presented.) for (Formula presented.). In our earlier article we showed that (Formula presented.) is not embeddable in (Formula presented.) for every t and that (Formula presented.) is embeddable in (Formula presented.) for all (Formula presented.). In the present paper, we improve on the latter result by showing that the embeddability of (Formula presented.) in fact takes place for (Formula presented.). This is achieved by analyzing the explicit totally real embedding of the sphere (Formula presented.) in (Formula presented.) constructed by Ahern and Rudin. For (Formula presented.), the problem of the embeddability of (Formula presented.) remains open