Journal of Informatics and Mathematical Sciences Vol. 8, No. 5, pp. 335–346, 2016 ISSN 0975-5748 (online); 0974-875X (print) Published by RGN Publications http://www.rgnpublications.com Research Article Eigenvalues variation of the p -Laplacian under the Yamabe Flow on SM Shahroud Azami Department of Mathematics, Faculty of Sciences, Imam Khomeini International University, Qazvin, Iran azami@sci.ikiu.ac.ir Abstract. Let ( M, F ) be a closed Finsler manifold. Studying geometric flows and the eigenvalues of geometric operators are powerful tools when dealing with geometric problems. In this article we will consider the eigenvalue problem for the p-laplace operator for Sasakian metric acting on the space of functions on SM. We find the first variation formula for the eigenvalues of p-Laplacian on SM evolving by the Yamabe flow on M and give some examples. Keywords. Yamabe flow; Finsler manifold; p-Laplace operator MSC. 53C44; 58C40 Received: January 6, 2016 Accepted: January 24, 2016 Copyright © 2016 Mohd Shareduwan Mohd Kasihmuddin, Mohd Asyraf Mansor and Saratha Sathasivam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction For a closed Finsler manifold ( M, F ), the eigenvalues of geometric operators is important in geometric analysis. In studying of the p-Laplace equation several parts of mathematics for instance: Calculus of Variation, Partial Differential Equation, Potential Theory and Analytic Function have a momentous impress. Recently, there are many research about properties of the eigenvalues of p-Laplacian on Finsler manifolds and Riemannian manifolds to estimate the spectrum in terms of the other geometric structures of the manifold ( [6, 8, 14, 16]). Also, geometric flows have been a topic of active research interest in mathematics and other sciences ( [10–13]). Yamabe flow ( [7,9,17]) which is extension of Hamilton’s Ricci flow ( [4,10]) is the best known example of a geometric evolution equation. The Yamabe flow is related to