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

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      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

       

      Soma Maity1

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

       
  • Proceedings – Mathematical Sciences | News

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