四川大学学报(自然科学版)
四川大學學報(自然科學版)
사천대학학보(자연과학판)
JOURNAL OF SICHUAN UNIVERSITY(NATURAL SCIENCE EDITION)
2010年
3期
420-424
,共5页
K(a)hler仿射流形%Bernstein 性质
K(a)hler倣射流形%Bernstein 性質
K(a)hler방사류형%Bernstein 성질
K(a)hler affine manifold%Bernstein property
设x∶M→An+1是由定义在凸域Ω(∪)An上的某局部严格凸函数 xn+1=f(x1,...,xn)给出的超曲面. 记ρx=det((e)2f)/((e)xi(e)xj)(x)-1/n+2.假设M,ɡ是一完备的Hessian 流形且具有非负的李奇曲率, 作者证明了如果ρ满足△ɡρ=β(‖▽ρ‖2ɡ)/ρ(β≠1)则M一定是椭圆抛物面.
設x∶M→An+1是由定義在凸域Ω(∪)An上的某跼部嚴格凸函數 xn+1=f(x1,...,xn)給齣的超麯麵. 記ρx=det((e)2f)/((e)xi(e)xj)(x)-1/n+2.假設M,ɡ是一完備的Hessian 流形且具有非負的李奇麯率, 作者證明瞭如果ρ滿足△ɡρ=β(‖▽ρ‖2ɡ)/ρ(β≠1)則M一定是橢圓拋物麵.
설x∶M→An+1시유정의재철역Ω(∪)An상적모국부엄격철함수 xn+1=f(x1,...,xn)급출적초곡면. 기ρx=det((e)2f)/((e)xi(e)xj)(x)-1/n+2.가설M,ɡ시일완비적Hessian 류형차구유비부적리기곡솔, 작자증명료여과ρ만족△ɡρ=β(‖▽ρ‖2ɡ)/ρ(β≠1)칙M일정시타원포물면.
Let x:M→An+1 be a locally strongly convex hypersurface, given by the graph of a convex function xn+1=f(x1,...,xn) defined in a convex domain Ω(<) An. The Hessian metric ɡ on M is considered, defined by ɡ=∑((e)2f)/((e)xi(e)xj)dxidxj. Denote ρ(x)=(det((e)2f)/((e)xi(e)xj)(x))-1/(n+2).Suppose (M,ɡ) is a complete Hessian manifold with nonnegative Ricci curvature. It is proved that if ρ satisfies △ɡρ=β‖▽ρ‖2ɡ/ρ(β≠1), then x(M) must be an elliptic paraboloid.