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

TL;DR

This paper studies how the first eigenvalue of a modified Laplace operator evolves under the backward Ricci flow on locally homogeneous 3-manifolds, providing bounds and asymptotic behavior insights.

Contribution

It derives bounds for the eigenvalues under the backward Ricci flow and analyzes their asymptotic behavior in the Bianchi case.

Findings

Established upper and lower bounds for eigenvalues.

Showed eigenvalues approach zero under certain geometric limits.

Connected eigenvalue behavior to convergence to sub-Riemannian geometry.

Abstract

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