Abstract
Let (M, g(t)) be a compact Riemannian manifold and the metric g(t) evolve by the Ricci flow. We derive the evolution equation for the eigenvalues of geometric operator −Δ ϕ + cR under the Ricci flow and the normalized Ricci flow, where Δ ϕ is the Witten-Laplacian operator, ϕ ∈ C ∞(M), and R is the scalar curvature with respect to the metric g(t). As an application, we prove that the eigenvalues of the geometric operator are nondecreasing along the Ricci flow coupled to a heat equation for manifold M with some Ricci curvature condition when \(c > \tfrac{1} {4}\).
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Fang, S., Xu, H. & Zhu, P. Evolution and monotonicity of eigenvalues under the Ricci flow. Sci. China Math. 58, 1737–1744 (2015). https://doi.org/10.1007/s11425-014-4943-7
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DOI: https://doi.org/10.1007/s11425-014-4943-7