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

Let λ (t) be the first eigenvalue of the operator − ∆ + a R b 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. [ABSTRACT FROM AUTHOR]