On the minimal eigenvalue of the Laplacian operator for $p$-forms in conformally flat Riemannian manifolds

Abstract:

Let $(M,g)$ be a compact orientable conformally flat Riemannian manifold and $^p{\lambda _1}$ the minimal eigenvalue of the Laplacian operator for $p$-forms. We prove that if there exists a positive constant $K$ such that $\rho \geqslant Kg$, where $\rho$ is the Ricci tensor of $M$, then $^p{\lambda _1} \geqslant Kp(n - p + 1)/(n - 1)$ for each $p$, $1 \leqslant p \leqslant n/2$, $(n = \dim M)$; moreover if the equality holds for some $p$ then $M$ is of constant curvature $\sigma = K/(n - 1)$.