A characterization of Cayley Hypersurface and Eastwood and Ezhov   conjecture
  Eastwood and Ezhov generalized the Cayley surface to the Cayley hypersurface in each dimension, proved some characteristic properties of the Cayley hypersurface and conjectured that a homogeneous hypersurface in affine space satisfying these properties must be the Cayley hypersurface. We will prove this conjecture when the domain bounded by a graph of a function defined on $\R^n$ is also homogeneous giving a characterization of Cayley hypersurface. The idea of the proof is to look at the problem of affine homogeneous hypersurfaces as that of left symmetric algebras with a Hessian type inner product. This method gives a new insight and powerful algebraic tools for the study of homogeneous affine hypersurfaces. 
Yuncherl Choi, Hyuk Kim