Bull Braz Math Soc, New Series https://doi.org/10.1007/s00574-019-00167-8 Bounds for the First Eigenvalue of (- - R) Under the Ricci Flow on Bianchi Classes Fereshteh Korouki 1 · Asadollah Razavi 1 Received: 11 March 2019 / Accepted: 14 August 2019 © Sociedade Brasileira de Matemática 2019 Abstract In this paper, we analyse behavior of the first eigenvalue of the operator −Δ − R on locally homogeneous closed 3-manifolds along the normalized Ricci flow, moreover in each Bianchi class we find bounds for the corresponding eigenvalues. Keywords Eigenvalue · Ricci flow · Laplacian-type operator · Bianchi classes Mathematics Subject Classification 53C44 1 Introduction Ricci flow first was introduced in physics in 1980 by Friedan (1980, 1985) for renormalization of σ -models, then R.Hamilton independently defined it in 1982 in mathematics (Hamilton 1982) for solving Poincaré’s conjecture. Since then Ricci flow has had many applications in both physics and mathematics. In physics in par- ticular for gravity, relativity and black holes the reader can look through (Samuel and Chowdhury 2007; Olinyk 2009; Headrick and Wiseman 2006; Ye 1993; Miron and Anastasiei 1997). In mathematics the major application has been proving Poincaré’s conjecture through works of Perelman (2002, 2003). One of the main special fields of research has been investigating eigenvalues of some operators e.g (Headrick and Wiseman 2006; Samuel and Chowdhury 2007). The eigenvalues of Laplacian-type operator −Δ + cR, where c is a constant, under the Ricci flow are important tools to realize geometry and topology of manifolds. Particulary, finding the upper and lower bounds for these operator has become an interesting topic in recent years. See (Cao 2007, 2008; Cao et al. 2012; Li 2007). B Asadollah Razavi arazavi@uk.ac.ir Fereshteh Korouki f.korouki@math.uk.ac.ir 1 Shahid Bahonar University of Kerman, Kerman, Iran 123