Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

An Obata singular theorem for stratified spaces


Author: Ilaria Mondello
Journal: Trans. Amer. Math. Soc. 370 (2018), 4147-4175
MSC (2010): Primary 53A30, 58C40
DOI: https://doi.org/10.1090/tran/7105
Published electronically: December 29, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 53A30, 58C40

Retrieve articles in all journals with MSC (2010): 53A30, 58C40


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

DOI: https://doi.org/10.1090/tran/7105
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)
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society