Soft isoperimetric rigidity

In this talk, we consider the relation between isoperimetric profile and observable variance, where the observable variance is defined to be the supremum of the variance of 1-Lipschitz functions.  We have the detailed metric structure of a metric measure space such that its isoperimetric profile is not greater than that of a one-dimensional model and the observable variance coincides with that of the model.  As an application, we obtain a new type of splitting theorem for a complete Riemannian manifold with positive Bakry-Emery Ricci curvature.  This is a joint work with Hiroki Nakajima.