Eigenvalues of Weighted-Laplacian under the extended Ricci flow

2016/04/20 by Abolarinwa, Abimbola
#35P30 #53C21 #53C44 #58J35 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1604.05884

Abstract

Let Δ_φ= Δ-∇ φ∇ be a symmetric diffusion operator with an invariant weighted volume measure dμ= e^-φ 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 Δ_φ 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≥ 3. Our results are natural extension of some known results for Laplace-Beltrami operator under various geometric flows.

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