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Advances in Geometry

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Eigenvalues of the weighted Laplacian under the extended Ricci flow

Abimbola Abolarinwa
Published Online: 2018-10-25 | DOI: https://doi.org/10.1515/advgeom-2018-0022


Let φ = − ∇φ∇ be a symmetric diffusion operator with an invariant weighted volume measure = eφ 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.

Keywords: Witten–Laplacian; eigenvalues; Ricci flow; monotonicity; curvature

MSC 2010: Primary 53C21; 53C44; Secondary 35P30; 58J35


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About the article

Received: 2016-07-22

Revised: 2017-03-24

Published Online: 2018-10-25

Citation Information: Advances in Geometry, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2018-0022.

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