# Variation of the first eigenvalue of $(p,q)$-Laplacian along the Ricci-harmonic flow

## TL;DR

This paper investigates how the first eigenvalue of the $(p,q)$-Laplacian changes monotonically along the Ricci-harmonic flow on closed manifolds, deriving variation formulas and constructing monotonic quantities.

## Contribution

It provides the first variation formula for the eigenvalue under Ricci-harmonic flow and introduces new monotonic quantities based on initial conditions.

## Key findings

- Derived the first variation formula for the eigenvalue.
- Constructed monotonic quantities along the flow.
- Identified conditions for monotonicity.

## Abstract

In this paper, we study monotonicity for the first eigenvalue of a class of $(p,q)$-Laplacian. We find the first variation formula for the first eigenvalue of $(p,q)$-Laplacian on a closed Riemannian manifold evolving by the Ricci-harmonic flow and construct various monotic quantities by imposing some conditions on initial manifold.