Characterization of equality in Zhong-Yang type (sharp) spectral gap estimates for metric measure spaces

Abstract: We prove that a compact $RCD^*(0,N)$ (or equivalently $RCD(0,N)$) metric measure space, $\left(X, d, m \right)$, with $\diam X \le d$ and its first (nonzero) eigenvalue of the Laplacian (in the sense of Ambrosio-Gigli-Savar\'{e}) , $\lambda_1 = \frac{\pi^2}{d^2}$, has to be a circle or a line segment with diameter, $\pi$. 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 $\Ric \ge 0$, $(0,N)-$ Bakry-\'{E}mery 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.