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

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/124/03/0383-0409

• # Keywords

Riemannian functional; critical point; stability; local minima.

• # Abstract

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 $p\in (0,\infty)$. 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.

• # Author Affiliations

1. Department of Mathematics, Indian Institute of Science, Bangalore 560 012, India

• # Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 3
June 2019