4维Lorentz空间中类空极值图的曲率估计  

Curvature estimates for spacelike stationary graphs in the 4-dimensional Lorentz space

作　　者：倪睿洁 王长平 王鹏[1] Ruijie Ni;Changping Wang;Peng Wang

机构地区：[1]福建师范大学数学与统计学院,福州350117

出　　处：《中国科学:数学》2025年第1期93-106,共14页Scientia Sinica:Mathematica

基　　金：国家自然科学基金(批准号:11831005和12371052)资助项目。

摘　　要：本文研究4维Lorentz空间R_(1)^(4)中的非平坦类空极值图的曲率估计.首先证明如下的Bernstein型定理:若R_(1)^(4)中的类空极值图满足K≤0,则K=K^(⊥)≡0.进一步,设非平坦完备类空极值图的Gauss映射(ϕ,ψ)=(Ae^(-β(z)),Be^(-β(z)),其中A,B为非零常数,本文证明:当β(z)是非常值多项式时,K整体有界当且仅当K⊥整体有界当且仅当β(z)是非常值线性函数;当β(z)是形如g(z)e^(P(z))+Q(z)的超越整函数时,其中g(z),P(z),Q(z)均为多项式,则K与K⊥均无界;对于其他的一般情形,Gauss曲率K与法曲率K⊥至少有一个是无界的.In this paper,we investigate the curvature estimates for non-at spacelike stationary graphs in the 4-dimensional Lorentz space R_(1)^(4).We rst prove the following Bernstein-type theorem:If a spacelike stationary graph in R_(1)^(4)satis es K≤0,then K=K^(⊥)≡0.Furthermore,let(ϕ,ψ)=(Ae^(-β(z)),Be^(-β(z))be the Gauss map of a non-at complete spacelike stationary graph,where A and B are non-zero constants.We show that whenβ(z)is a non-constant polynomial,K is globally bounded if and only if K^(⊥)is globally bounded if and only ifβ(z)is a non-constant linear function.Whenβ(z)is a transcendental entire function of the form g(z)e^(P(z))+Q(z),where g(z),P(z)and Q(z)are all polynomials,both K and K?are unbounded.For other general cases,at least one of the Gauss curvature K and the normal curvature K^(⊥)is unbounded.

关 键 词：4维Lorentz空间 类空极值图 整函数 曲率估计 

分 类 号：O177[理学—数学] O186.1[理学—基础数学]