Gap results for compact quasi-Einstein metrics

In this paper, we work on compact quasi-Einstein metrics and prove several gap results. In the first part, we get a gap estimate for the first nonzero eigenvalue of the weighted Laplacian, by establishing a comparison theorem for the weighted heat kernel. In the second part, we establish two gap results for the Ricci curvature and the scalar curvature, based on which some rigid properties can be derived.