Rigidity of the spectral gap for non-negatively curved RCD spaces

Rigidity of the spectral gap for non-negatively curved RCD spaces

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Christian Ketterer, Karlsruher Institut für Technologie

Online Talk 

Zhang and Yang proved the lower bound $(\pi/diam)^2$ for the spectral gap of a closed Riemannian manifold $M$ with non-negative Ricci curvature.  Hang and Wang then showed that this gap is saturated if and only if $M$ is isometric to the 1D circle with the same diameter.  By work of Jiang and Zhang (and also Cavalletti and Mondino) the estimate still holds in the more general setting of metric measure spaces with non-negative synthetic Ricci curvature and an upper dimension bound $N$, so-called RCD(0,N) spaces. 

In this talk I will present the corresponding rigidity result in the context of non-negatively curved, finite dimensional $RCD$ spaces. More precisely the spectral gap is saturated if and only if the space is isometric to the circle or to the interval of the same diameter. One application of this result is an almost rigidity statement for collapsing Riemannian manifolds. This is a joint work with Sajjad Lazian and Yu Kitabeppu.