Eigenvalues under the Ricci flow of model geometries

TL;DR

This paper investigates how the first eigenvalue of the Laplace-Beltrami operator evolves under the normalized Ricci flow across different model geometries, providing bounds and monotonic quantities.

Contribution

It introduces new estimates and monotonic quantities for the eigenvalue evolution under Ricci flow in various geometric models.

Findings

Derived bounds for eigenvalues during Ricci flow

Constructed monotonic quantities for eigenvalue analysis

Analyzed eigenvalue behavior across Bianchi classes

Abstract

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.