Eigenvalues of the Laplace operator with potential under the backward Ricci flow on locally homogeneous 3-manifolds
Let be the first eigenvalue of under the backward Ricci flow on locally homogeneous 3-manifolds, where is the scalar curvature. In the Bianchi case, we get the upper and lower bounds of . In particular, we show that when the the backward Ricci flow converges to a sub-Riemannian geometry after a proper re-scaling, approaches zero, where .