On the Classification of Homogeneous Hypersurfaces in Complex Space

Abstract

We discuss a family M n t , with n ≥ 2, t > 1, 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 C n due to Morimoto and Nagano. To finalize their classification, one needs to resolve the problem of the embeddability of M n t in C n for n = 3, 7. We show that M 7 t is not embeddable in C 7 for every t and that M 3 t is embeddable in C 3 for all 1 < t < 1 + 10 −6 . As a consequence of our analysis of a map constructed by Ahern and Rudin, we also conjecture that the embeddability ofThe group SO(n + 1, R) acts on Q n , and the orbits of this action are the sphere S n = Q n ∩ R n+1 as well as the compact strongly pseudoconvex hypersurfacesThese hypersurfaces play an important role in the classical paper [13], where the authors set out to determine all compact simply connected real-analytic hypersurfaces in C n homogeneous under an action of a Lie group by CR-transformations. They showed that every such hypersurface is CR-equivalent to either the sphere S 2n−1 or, for n = 3, 7, to the manifold M n t for some t. However, the question of the existence of a real-analytic CR-embedding of M n t in C n for n = 3, 7 was not clarified, thus the classification in these two dimensions was not fully completed. 1350064-1 Int. J. Math. 2013.24. Downloaded from www.worldscientific.com by YALE UNIVERSITY on 02/04/15. For personal use only. 1350064-7 Int. J. Math. 2013.24. Downloaded from www.worldscientific.com by YALE UNIVERSITY on 02/04/15. For personal use only.

On homogeneous hypersurfaces in ${\mathbb C}^3$

“…This was proved by analyzing the holomorphic continuation F : C 4 Ñ C 3 of the explicit polynomial totally real embedding of S 3 in C 3 constructed in [AR]. Also, in [I,Conjecture 3.1] we stated that the map F should in fact yield an embedding for all 1 ă t ă b p2 `?2q{3. In the present paper we confirm the conjecture and therefore obtain the following result:…”

“…More precisely, in [I,Theorem 3.1] we showed that M 3 t embeds in C 3 for all 1 ă t ă 1 `10 ´6. This was proved by analyzing the holomorphic continuation F : C 4 Ñ C 3 of the explicit polynomial totally real embedding of S 3 in C 3 constructed in [AR].…”

“…The family M n t was studied in our earlier paper [I]. In particular, in [I,Corollary 2.1] we observed that a necessary condition for the existence of a real-analytic CRembedding of M n t in C n is the embeddability of the sphere S n in C n as a totally real submanifold. The problem of the existence of a totally real embedding of S n in C n was considered by Gromov (see [G1] and [G2, p. 193]), Stout-Zame (see [SZ]), Ahern-Rudin (see [AR]), Forstnerič (see [F1], [F2], [F3]).…”

“…The problem of the existence of a totally real embedding of S n in C n was considered by Gromov (see [G1] and [G2, p. 193]), Stout-Zame (see [SZ]), Ahern-Rudin (see [AR]), Forstnerič (see [F1], [F2], [F3]). It has turned out that S n admits a smooth totally real embedding in C n only for n " 3, hence [I,Corollary 2.1] implies, in particular, that M 7 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 realanalytic totally real embedding of S 3 in C 3 (which is known to exist, for instance, by [AR]) extends to a biholomorphic map defined in a neighborhood of S 3 in Q 3 .…”

On the Classification by Morimoto and Nagano

“…, z n`1 q P C n`1 : |z 1 | 2 `¨¨¨`|z n`1 | 2 " tu X Q n , t ą 1, which are simply-connected for n ě 3. These hypersurfaces are all nonspherical (see [I1,Remark 2.2]) and pairwise CR-nonequivalent (see [KZ,Example 13.9], [BH,Theorem 2]). They are the boundaries of Grauert tubes around S n (note that Q n can be naturally identified with the tangent bundle T pS n q).…”