Songbo Hou

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 ${\lambda }^{+}(t)=\max \{\lambda (t),0\}$.

In this paper, we study the evolving behaviors of the first eigenvalue of Laplace-Beltrami operator under the normalized backward Ricci flow, construct various quantities which are monotonic under the backward Ricci flow and get upper and lower bounds. We prove that in cases where the backward Ricci flow converges to a sub-Riemannian geometry after a proper rescaling, the eigenvalue evolves toward zero.

In this paper, we study the evolving behaviors of the first eigenvalue of Laplace-Beltrami operator under the normalized Ricci flow of model geometries. In every Bianchi class, we estimate the derivative of the eigenvalue. Then we construct monotonic quantities under the Ricci flow and obtain upper and lower bounds for the eigenvalue.