Eigenvalues under the backward Ricci flow on locally homogeneous closed 3-manifolds

TL;DR

This paper investigates how the first eigenvalue of the Laplace-Beltrami operator behaves under the normalized backward Ricci flow on closed 3-manifolds, establishing bounds and convergence properties.

Contribution

It introduces new monotonic quantities under the backward Ricci flow and analyzes eigenvalue evolution, especially in convergence to sub-Riemannian geometries.

Findings

Eigenvalues tend to zero when the flow converges to sub-Riemannian geometry.

Constructed monotonic quantities provide bounds for eigenvalues.

Analyzed eigenvalue behavior on locally homogeneous closed 3-manifolds.

Abstract

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.