加权黎曼流形中超曲面的第一稳定特征值  

On the First Stability Eigenvalue of Hypersurfaces in the Weighted Riemannian Manifolds

作　　者：刘子健 刘建成 LIU Zi-jian;LIU Jian-cheng(School of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China)

机构地区：[1]西北师范大学数学与统计学院,兰州730070

出　　处：《西南师范大学学报（自然科学版）》2020年第4期36-40,共5页Journal of Southwest China Normal University(Natural Science Edition)

基　　金：国家自然科学基金项目(11761061).

摘　　要：加权黎曼流形(M^n+1,g,e^-fdv)在黎曼流形(M^n+1,g)上赋予一个加权体积dvf=e^-fdv,其中f是M^n+1上的光滑实值函数,dv为M^n+1的体积元,记Σn为加权黎曼流形(M^n+1,g,e^-fdv)中具有常加权平均曲率Hf的紧致无边超曲面,在截面曲率Sec≥c的条件下,研究了超曲面上加权稳定算子Jf的第一特征值问题,运用了不等式(a+b)^2≥a^21+k-b^2k等号成立当且仅当b=-k1+ka,其中任意的a,b∈R和k>-1,得到了超曲面上第一稳定特征值的一个上界.当f为常数时,加权黎曼流形也就回到了通常的黎曼流形,此时也得到了稳定算子J的第一非零特征值的上界,进而从这个上界来讨论超曲面的稳定性.In this paper,it's gived a weighted volume dvf=e^-fdv for weighted Riemannian manifold(M^n+1,g,e^-f dv)on Riemannian manifold(M^n+1,g),where f is the smooth and real function on M^n+1,dv is volume of M^n+1,Σn is a compact infinitesimal hypersurface on weighted Riemann manifold(M^n+1,g,e^-f dv)with constant weighted mean curvature Hf.Under the condition of section curvature Sec≥c,the first eigenvalue problem of the weighted stability operator Jf on the hypersurface is studied.The equality of inequality(a+b)^2≥a^21+k-b^2kis established if and only if b=-k1+ka,where any a,b∈R and k>-1,upper bound on the first stable eigenvalue on the hypersurface is obtained.When f is a constant,the weighted Riemannian manifold returns to the usual Riemannian manifold,and the upper bound of the first nonzero Eigenvalue of the stable operator J is obtainedan.Furthermore,the stability of the hypersurface can be discussed from the obtained upper bound.

关 键 词：加权黎曼流形 第一稳定特征值 加权稳定算子 Bakry-Emery-Ricci张量 

分 类 号：O186.12[理学—数学]