In 1983, Schoen and Yau found that a positive lower bound on the spectrum of the linear operator involving the Laplacian and the Gaussian curvature implies a diameter bound on a 2-surface. In recent years, such a notion of spectral lower bound has been found useful in the aspherical conjecture, stable Bernstein conjecture, and positive mass conjecture in high dimensions. In this talk, I will present some basics of spectral scalar curvature and some rigidity results. In particular, I will show a positive mass theorem for an asymptotically hyperbolic 3-manifold with toroidal infinity. The talk is based on ongoing joint work with Yukai Sun, Yimin Chen, and Juncheol Pyo.
