极小模原理在一类四阶全对称张量不等式中的应用  

Application of the Minimal Norm Tensors to a Class of Totally Symmetric Fourth-order Covariant Tensors'Inequalities

作　　者：龚一帆 段德园 李虹[1] GONG Yifan;DUAN Deyuan;LI Hong(School of Mathematics,Yunnan Normal University,Kunming 650500,China)

机构地区：[1]云南师范大学数学学院,云南昆明650500

出　　处：《云南师范大学学报（自然科学版）》2024年第4期15-20,共6页Journal of Yunnan Normal University:Natural Sciences Edition

基　　金：云南师范大学博士科研启动基金资助项目(00800205020503105);云南省基础研究计划青年资助项目(00800206020616021);国家自然科学基金资助项目(12201554).

摘　　要：研究了四阶全对称张量的极小模张量,得到了其极小模张量和极小模的完整分类表达式,并证明了其极小模张量与trace-free分解的等价关系.进一步,利用极小模的非负性证明了在单位球∑^(n+1)中任意的一个极小超曲面上M^(n)上,|▽^(2)h|^(2)≥3/2[S×trh^(4)+n×S-(trh^(3))^(2)-2×S^(2)]+3S(S-n)^(2)/2(n+4),并发现其等号成立时M^(n)包含了∑^(n+1)中所有圆心是球心的大圆及其部分和Clifford环∑^(k)(√k/n)×∑^(n-k)(√n-k/n).In this paper,we study the minimal norm tensor for the totally symmetric fourth-order covariant tensors.Firstly,we obtain the general expression of the minimal norm tensor.Then,we prove the equivalence relation between the minimal norm tensor and the trace-free decomposition.Finally,by use of the non-negativity of minimal norm,we prove that on a minimal hypersurface of the unit sphere,the inequality |▽^(2)h|^(2)≥3/2[S×trh^(4)+n·S-(trh^(3))^(2)-2·S^(2)]+3S(S-n)^(2)/2(n+4)is always true,we also find that a great circle or a part of it or the Clifford torus satisfy the sharp condition in this inequality.

关 键 词：极小模原理 张量的trace-free分解 极小超曲面 Clifford环 

分 类 号：O186.12[理学—数学]