SOME RESULTS OF EVOLUTION OF THE FIRST EIGENVALUE OF WEIGHTED p-LAPLACIAN ALONG THE EXTENDED RICCI FLOW

In this article we study the evolution and monotonicity of the first non-zero eigenvalue of weighted p-Laplacian operator which it acting on the space of functions on closed oriented Riemannian n-manifolds along the extended Ricci flow and normalized extended Ricci flow. We show that the first eigenvalue of weighted p-Laplacian operator diverges as t approaches to maximal existence time. Also, we obtain evolution formulas of the first eigenvalue of weighted p-Laplacian operator along the normalized extended Ricci flow and using it we find some monotone quantities along the normalized extended Ricci flow under the certain geometric conditions.