First Eigenvalues of Some Operators Under the Backward Ricci Flow on Bianchi Classes

Let λ(t) be the first eigenvalue of the operator −∆+aRb on locally three-dimensional homogeneous manifolds along the backward Ricci flow, where a,b are real constants and R is the scalar curvature. In this paper, we study the properties of λ(t) on Bianchi classes. We begin by deriving an evolution equation for the quantity λ(t) on three-dimensional homogeneous manifolds in the context of the backward Ricci flow. Utilizing this equation, we subsequently establish a monotonic quantity that is contingent upon λ(t). Additionally, we present both upper and lower bounds for λ(t) within the framework of Bianchi classes.