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

Shahroud Azami, Rawan Bossly (), Abdul Haseeb () and Abimbola Abolarinwa
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Shahroud Azami: Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin 34148-96818, Iran
Rawan Bossly: Department of Mathematics, College of Science, Jazan University, P.O. Box 114, Jazan 45142, Saudi Arabia
Abdul Haseeb: Department of Mathematics, College of Science, Jazan University, P.O. Box 114, Jazan 45142, Saudi Arabia
Abimbola Abolarinwa: Department of Mathematics, University of Lagos, Akoka, Lagos 101017, Nigeria

Mathematics, 2024, vol. 12, issue 23, 1-16

Abstract: 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.

Keywords: Ricci flow; p-Laplacian operator; eigenvalue (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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