Eigenvalues of the weighted Laplacian under the extended Ricci flow

Abstract

Abstract Let ∆φ = ∆ − ∇φ∇ be a symmetric diffusion operator with an invariant weighted volume measure dμ = e −φ dν on an n-dimensional compact Riemannian manifold (M, g), where g = g(t) solves the extended Ricci flow. We study the evolution and monotonicity of the first nonzero eigenvalue of ∆φ and we obtain 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 n ≥ 3. Our results are natural extensions of some known results for Laplace–Beltrami operators under various geometric flows.