Abstract.

In this talk I will present a maximal diameter theorem for metric measure spaces that satisfy the reduced Riemannian curvature-dimension condition $RCD^*(N-1,N)$. That is, if there is equality in the Bonnet-Myers diameter estimate, the space is a spherical suspension. The proof is an application of the Gigli-Cheeger-Gromoll splitting theorem, and the stability of Riemannian curvature-dimension bounds under coning. More precisely, the latter states that the $(K,N)$-cone over some metric measure space satisfies the reduced Riemannian curvature-dimension condition $RCD^*(KN,N+1)$ if and only if the underlying space satisfies $RCD^*(N-1,N)$. I will describe the main ideas of the proof.