Rigidity for the spectral gap on $RCD(K,\infty)$-spaces

Abstract

We consider a rigidity problem for the spectral gap of the Laplacian on an $RCD(K,\infty)$-space (a metric measure space satisfying the Riemannian curvature-dimension condition) for positive $K$. For a weighted Riemannian manifold, Cheng--Zhou showed that the sharp spectral gap is achieved only when a $1$-dimensional Gaussian space is split off. This can be regarded as an infinite-dimensional counterpart to Obata's rigidity theorem. Generalizing to $RCD(K,\infty)$-spaces is not straightforward due to the lack of smooth structure and doubling condition. We employ the lift of an eigenfunction to the Wasserstein space and the theory of regular Lagrangian flows recently developed by Ambrosio--Trevisan to overcome this difficulty.