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

Abstract

ABSTRACT. Let (M, g) be a compact orientable conformally flat Riemannian manifold and pXi the minimal eigenvalue of the Laplacian operator for p-forms. We prove that if there exists a positive constant K such that p > Kg, where p is the Ricci tensor of M, then pXi > Kp(n -p + l)/(n -1) f°r eacn P, 1 < p < n/2, (n = dim M); moreover if the equality holds for some p then M is of constant curvature a = K/(n -1).

The Higher Spin Dirac Operators on 3-Dimensional Manifolds

“…In section 5, we have the lower bound estimations for the first eigenvalue of the Laplace type operators. This estimation is a generalization of the one for the Dirac operator given in [1] or the Laplace-Beltrami operator in [7] and [10]. In the section 6, we consider the case of the 3-dimensional manifold of the constant curvature and show that some operators commute.…”