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Proceedings - Mathematical Sciences

, Volume 124, Issue 3, pp 383–409 | Cite as

On the stability of the L p -norm of the Riemannian curvature tensor

  • Soma MaityEmail author
Article

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 ∈ (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.

Keywords

Riemannian functional critical point stability local minima. 

Mathematics Subject Classification.

53C21, 58E11, 58C15 

Notes

Acknowledgements

The author would like to thank Harish Seshadri for suggesting this problem and for his guidance, Atreyee Bhattacharya and H. A. Gururaja for some useful discussions related to this article. This work was supported by CSIR and partially supported by UGC Center for Advanced Studies.

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Copyright information

© Indian Academy of Sciences 2014

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia

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