Classification of Calabi hypersurfaces with parallel Fubini-Pick form

Classification of Calabi Hypersurfaces in $\bbr^{n+1}$ with parallel Fubini-Pick form

“…Definition 2.1. [23] Two graph hypersurfaces (x i , f (x i )) and (x i , f (x i )), defined respectively in domains Ω, Ω ⊂ R n , are called Calabi-affine equivalent if they differ only by an affine transformation φ ∈ SA(n + 1).…”

“…Accordingly, we have Definition 2.2. [23] Two smooth functions f and f respectively defined on domains Ω, Ω ⊂ R n are called affine equivalent (related with an affine transformation…”

“…In [22] Xu and Li introduced a large class of new canonical Calabi hypersurfaces, which are denoted by Q(c 1 , • • • , c r ; n), see also [23].…”

“…In section 4, properties of Calabi hypersurfaces with parallel cubic form in terms of a typical basis are presented, so that the classification problem of such hypersurfaces is divided into (n + 1) cases, namely: {C m } 0≤m≤n , depending on the decomposition of the tangent space into three orthogonal distributions, i.e., D 1 (of dimension one), D 2 and D 3 . Recall that Cases C 0 , C n and C n−1 were proved in [23] and [24], respectively. As a corollaries of theorems 3.6 and 3.3, the Cases C 1 and C n−1 will be settled in this section.…”

“…Xu and Li [22] presented a class of new canonical examples denoted by Q(c 1 , • • • , c r ; n) and classified the Calabi complete Tchebychev affine Kähler hypersurfaces with nonnegative Ricci curvature. Recently, in [23] and [24], the authors classified n(≤ 4)-dimensional Calabi hypersurfaces with parallel Fubini-Pick form.…”