仿射 Kähler －Scalar 曲率为零的仿射 Kähler 流形

摘要

设x：M→An＋1是由定义在凸域Ω炒An上的某局部严格凸函数xn＋1＝f（x1，．．．，xn）给出的超曲面．考虑Hessian度量g ＝∑əə2fxiəxjdxidxj ．若（M，g）是具有非负李奇曲率的紧致Hessian流形且仿射Kähler－Scalar曲率为零，作者证明了如果Δρ≤nρ2，则函数f一定是二次多项式，其中ρ＝[det（fij）]－1n＋2．%Let x:M→An+1 be a locally convex hypersurface, given by the graph of a convex function xn+1 =f(x1,...,xn) defined in a convex domain ΩAn .The Hessian metric g on M is considered, defined by g =∑əə2 fxiəxj dxi dxj .Suppose ( M,g) is a compact Hessian manifold with nonnegative Ricci curvature , and with zero affine Kähler-Scalar curvature .It is proved that ifΔρ≤nρ2 , then f must be a quadratic polynomial , whereρ =[det(fij)] -1n+2 .