Eigenvalues of Weighted-Laplacian under the extended Ricci flow

TL;DR

This paper investigates how the first nonzero eigenvalue of a weighted Laplacian evolves under the extended Ricci flow on compact manifolds, establishing monotonicity, divergence in finite time, and extending known results in geometric analysis.

Contribution

It introduces new monotonic quantities and analyzes eigenvalue behavior under the extended Ricci flow, extending classical results for the Laplace-Beltrami operator.

Findings

Monotonicity of eigenvalues along the flow

Finite-time divergence of eigenvalues for dimensions n≥3

Identification of monotone quantities under the flow

Abstract

Let $\Delta_\varphi = \Delta -\nabla \varphi \nabla$ be a symmetric diffusion operator with an invariant weighted volume measure $d\mu = e^{-\varphi} dv$ on an $n$-dimensional compact Riemannian manifold $(M,g)$, where $g=g(t)$ solves the extended Ricci flow. In this article we study the evolution and monotonicty of the first nonzero eigenvalue of $\Delta_\varphi$ and we obatin several monotone quantities along the extended Ricci flow and its volume preserving version under some technical assumption. We also show that the eigenvalues diverge in a finite time for the case $n\geq 3$. Our results are natural extension of some known results for Laplace-Beltrami operator under various geometric flows.