On the Classification by Morimoto and Nagano

Abstract

We consider a family M 3 t , with t ą 1, of real hypersurfaces in a complex affine 3-dimensional quadric arising in connection with the classification of homogeneous compact simply-connected real-analytic hypersurfaces in C n due to Morimoto and Nagano. To finalize their classification, one needs to resolve the problem of the CR-embeddability of M 3 t in C 3 . In our earlier article we showed that M 3 t is CR-embeddable in C 3 for all 1 ă t ă b p2`?2q{3. In the present paper we prove that M 3 t can be immersed in C 3 for every t ą 1 by means of a polynomial map. In addition, one of the immersions that we construct helps simplify the proof of the above CR-embeddability theorem and extend it to the larger parameter range 1 ă t ă ?5{2.t cannot be real-analytically CR-embedded in C 7 . On the other hand, since S 3 is a totally real submanifold of Q 3 , any real-analytic totally real embedding of S 3 in C 3 Mathematics Subject Classification: 32C09, 32V30. Keywords: immersions and embeddings of CR-manifolds in complex space.