2015/06/16 by Lakzian, Sajjad
#51Fxx #53C21 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1506.04936
We prove that a compact RCD^*(0,N) (or equivalently RCD(0,N)) metric measure space, (X, d, m ), with \diam X ≤ d and its first (nonzero) eigenvalue of the Laplacian (in the sense of Ambrosio-Gigli-Savaré) , λ1 = (π2)/(d2), has to be a circle or a line segment with diameter, π. This completely characterizes the equality in Zhong-Yang type sharp spectral gap estimates in the metric measure setting with Riemannian lower Ricci bounds. Among such spaces, are the familiar Riemannian manifolds with \Ric ≥ 0, (0,N)- Bakry-Émery manifolds, (0,n)- Ricci limit spaces and non-negatively curved Alexandrov spaces. Inspired by Gigli's proof of the non-smooth splitting theorem, the key idea in the proof of our result, is to show that the underlying metric measure space (perhaps minus a closed subset of co-dimension, 1) splits off an interval isometrically whenever there exists a weakly harmonic potential f whose gradient flow trajectories are geodesics (i.e. multiples of f are Kantorovich potentials at least for short time and on suitable domains). This is standard in Riemannian geometry due to the de Rham's decomposition theorem which is a key ingredient in the proof of the Cheeger-Gromoll's celebrated splitting theorem.