An Obata singular theorem for stratified spaces

Ilaria Mondello

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arXiv · 1511.08093

An Obata singular theorem for stratified spaces

Ilaria Mondello

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Abstract

Consider a stratified space with a positive Ricci lower bound on the regular set and no cone angle larger than 2$π$. 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 $π$, and it is equivalent for the diameter to be equal to $π$ 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$π$: 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.

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Ilaria Mondello. 2015-11-25. An Obata singular theorem for stratified spaces. https://arxiv.org/abs/1511.08093

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