Extrinsic Monge–Ampère Rigidity for the Affine Quadric

Ler M be a connected closed complex submanifold of dimension n of C^m, m≥n+1, and put N (z) = ∥z∥^2. Assume that N |M has minimum 1, that its minimum set S is a smooth real n-manifold, and that U = arcosh N satisfies the homogeneous complex Monge–Amp`ere equation on W \S for some neighbourhood W of S in M . We prove that M is a unitary image of the affine quadric C^{n+1} ⊂C^m.