Communications of the
Korean Mathematical Society CKMS

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.

Keywords: Laplace, extended Ricci flow, eigenvalue

MSC numbers: 58C40, 53C44, 53C21