Eigenvalues of the Laplace operator with potential under the backward Ricci flow on locally homogeneous 3-manifolds

Let $\lambda(t)$ be the first eigenvalue of $-\Delta+aR\, (a>0)$ under the backward Ricci flow on locally homogeneous 3-manifolds, where $R$ is the scalar curvature. In the Bianchi case, we get the upper and lower bounds of $\lambda(t)$. In particular, we show that when the the backward Ricci flow converges to a sub-Riemannian geometry after a proper re-scaling, $\lambda^{+}(t)$ approaches zero, where $\lambda^{+}(t)=\max\{\lambda(t),0\}$.