On the Stability of the $L^p$-Norm of the Riemannian Curvature Tensor

Proceedings Mathematical Sciences

Soma Maity¹ 

[1] Department of Mathematics, Indian Institute of Science, Bangalore 0 0, India$$

关键词: Riemannian functional;    critical point;    stability;    local minima.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

We consider the Riemannian functional $mathcal{R}_p(g)=int_M|R(g)|^p dv_g$ defined on the space of Riemannian metrics with unit volume on a closed smooth manifold 𝑀 where $R(g)$ and $dv_g$ denote the corresponding Riemannian curvature tensor and volume form and $pin (0,∞)$. First we prove that the Riemannian metrics with non-zero constant sectional curvature are strictly stable for $mathcal{R}_p$ for certain values of 𝑝. Then we conclude that they are strict local minimizers for $mathcal{R}_p$ for those values of 𝑝. Finally generalizing this result we prove that product of space forms of same type and dimension are strict local minimizer for $mathcal{R}_p$ for certain values of 𝑝.

【 授权许可】

Unknown