Spectral constant rigidity of warped product metrics

A theorem of Llarull says that if a smooth metric $g$ on the $n$-sphere ${\mathbb{S}}^n$ is bounded below by the standard round metric and the scalar curvature $R_g$ of $g$ is bounded below by $n(n-1)$, then the metric $g$ must be the standard round metric. We prove a spectral Llarull theorem by replacing the bound $R_g\ge n(n-1)$ by a lower bound on the first eigenvalue of an elliptic operator involving the Laplacian and the scalar curvature $R_g$. We utilize two methods: spinor and spacetime harmonic function.