On totally real spheres in complex space

Let M ,N be two totally real and real analytic submanifolds in Cn . We say that M and N are biholomorphically equivalent if there is a biholomorphic mapping F defined in a neighborhood of M such that F (M ) = N . As a standard fact of complexification, one knows that all totally real and real analytic embeddings of M in C n are biholomorphically equivalent if M is of maximal dimension n . However, the topology of the manifold plays a major role in the existence of totally real immersions or embeddings. For instance, R.O. Wells [23] proved that if an n-dimensional compact and orientable manifold M admits a totally real embedding in Cn , then its Euler number must vanish. It was also observed by Wells that if M is a manifold of dimension n and it admits a totally real immersion in Cn , then its complexified tangent bundle T cM = TM⊗C is trivial. Conversely, the triviality of T cM also implies the existence of totally real immersions of M in Cn . This was obtained by M.L. Gromov in [11] through the method of convex integration. A stronger result due to J.A. Lees [17] says that M also admits Lagrangian immersions in Cn . The sphere S k : x 2 1 + . . . + x 2 k+1 = 1 in R k+1 gives us a trivial totally real embedding of S k in Ck+1. On the other hand, the works of Gromov [11], AhernRudin [1] and Stout-Zame [21] tell us that S k admits a totally real and real analytic embedding in Ck if and only if k = 1, 3. Our main result is the following. Theorem 1.1. If k ≤ 4, all totally real and real analytic embeddings of S k in Cn are biholomorphically equivalent. If k ≥ 5 and nk = k + 2[ k−1 4 ], there exist totally