An Obata singular theorem for stratified spaces

TL;DR

This paper establishes a rigidity theorem for stratified spaces with positive Ricci bounds, showing that equality cases in spectral and geometric bounds characterize spherical suspensions and relate to the Yamabe problem.

Contribution

It proves an Obata-type rigidity theorem for stratified spaces, characterizing when the first eigenvalue attains its lower bound and linking diameter, eigenvalues, and conformal geometry.

Findings

First non-zero eigenvalue ≥ dimension for such spaces.

Equality in eigenvalue implies the space is a spherical suspension.

Diameter is at most π, with equality characterizing spherical suspensions.

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.