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

Author

Listed:
  • 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)

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.

Suggested Citation

  • Shahroud Azami & Rawan Bossly & Abdul Haseeb & Abimbola Abolarinwa, 2024. "First Eigenvalues of Some Operators Under the Backward Ricci Flow on Bianchi Classes," Mathematics, MDPI, vol. 12(23), pages 1-16, December.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:23:p:3846-:d:1537762
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