## An Obata singular theorem for stratified spaces

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- by Ilaria Mondello PDF
- Trans. Amer. Math. Soc.
**370**(2018), 4147-4175 Request permission

## Abstract:

Consider a stratified space with a positive Ricci lower bound on the regular set and no cone angle larger than $2\pi$. For such stratified space we know that the first non-zero eigenvalue of the Laplacian is larger than or equal to the dimension. We prove here an Obata rigidity result when the equality is attained: the lower bound of the spectrum is attained if and only if the stratified space is isometric to a spherical suspension. Moreover, we show that the diameter is at most equal to $\pi$, and it is equivalent for the diameter to be equal to $\pi$ and for the first non-zero eigenvalue of the Laplacian to be equal to the dimension. We finally give a consequence of these results related to the Yamabe problem. Consider an Einstein stratified space without cone angles larger than $2\pi$: if there is a metric conformal to the Einstein metric and with constant scalar curvature, then it is an Einstein metric as well. Furthermore, if its conformal factor is not a constant, then the space is isometric to a spherical suspension.## References

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## Additional Information

**Ilaria Mondello**- Affiliation: UPMC Université Paris 6, Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586
- Address at time of publication: Université Paris Est Créteil, UFR Sciences et Technologies, Laboratoire d’Analyse et Mathématiques Appliquées, 61, avenue du Gévéral de Gaulle, 94010 Créteil Cedex, France
- Email: ilaria.mondello@u-pec.fr
- Received by editor(s): February 1, 2016
- Received by editor(s) in revised form: October 19, 2016
- Published electronically: December 29, 2017
- Additional Notes: This work was supported by a public grant overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” program (reference: ANR-10-LABX-0098)
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**370**(2018), 4147-4175 - MSC (2010): Primary 53A30, 58C40
- DOI: https://doi.org/10.1090/tran/7105
- MathSciNet review: 3811523