Spectral constant rigidity of warped product metrics

A theorem of Llarull says that if a smooth metric on the ‐sphere is bounded below by the standard round metric and the scalar curvature of is bounded below by , then the metric must be the standard round metric. We prove a spectral Llarull theorem by replacing the bound by a lower bound on the first eigenvalue of an elliptic operator involving the Laplacian and the scalar curvature . We utilize two methods: spinor and spacetime harmonic function.