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

Let $λ(t)$ be the first eigenvalue of $-Δ+aR\, (a>0)$ 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 $λ(t)$. In particular, we show that when the the backward Ricci flow converges to a sub-Riemannian geometry after a proper re-scaling, $λ^{+}(t)$ approaches zero, where $λ^{+}(t)=\max\{λ(t),0\}$.

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