Evolution of the first eigenvalue of buckling problem on Riemannian manifold under Ricci flow

Among the eigenvalue problems of the Laplacian, the biharmonic operator eigenvalue problems are interesting projects because these problems root in physics and geometric analysis. The buckling problem is one of the most important problems in physics, and many studies have been done by the researchers about the solution and the estimate of its eigenvalue. In this paper, first, we obtain the evolution equation of the first nonzero eigenvalue of the buckling problem on closed Riemannian manifold (compact and without boundary Riemannian manifold) along the unnormalized Ricci flow and normalized Ricci flow and by using them, we prove that the first nonzero eigenvalue and some quantities dependent to this eigenvalue are monotonic along the Ricci flow, under the some geometric conditions. Then, on special manifold such as homogeneous, 3- dimensional, 2-dimensional manifolds, we study the evolutionary behavior of this eigenvalue. Especially in the 2-dimensional state, depending on the value of the scalar curvature along the normalized Ricci flow, we find the quantities dependent on the first eigenvalue that are monotonic under the normalized Ricci flow. Finally, we give examples of soliton states and Einstein manifolds, and we obtain the evolution of the first eigenvalue of the buckling problem under the Ricci flow on these examples.