Abstract
In the paper we first derive the evolution equation for eigenvalues of geometric operator \(-\Delta _{\phi }+cR\) under the Ricci flow and the normalized Ricci flow on a closed Riemannian manifold M, where \(\Delta _{\phi }\) is the Witten–Laplacian operator, \(\phi \in C^{\infty }(M)\), and R is the scalar curvature. We then prove that the first eigenvalue of the geometric operator is nondecreasing along the Ricci flow on closed surfaces with certain curvature conditions when \(0<c\le \frac{1}{2}\). As an application, we obtain some monotonicity formulae and estimates for the first eigenvalue on closed surfaces.
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Acknowledgments
The authors would like to thank Professor Kefeng Liu and Professor Hongyu Wang for constant support and encouragement. The work is supported by PRC Grant NSFC (11371310, 11401514, 11471145), the University Science Research Project of Jiangsu Province (13KJB110029), the Natural Science Foundation of Jiangsu Province (BK20140804), the Fundamental Research Funds for the Central Universities (NS2014076), and Qing Lan Project.
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Fang, S., Zhao, L. & Zhu, P. Estimates and Monotonicity of the First Eigenvalues Under the Ricci Flow on Closed Surfaces. Commun. Math. Stat. 4, 217–228 (2016). https://doi.org/10.1007/s40304-015-0083-9
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DOI: https://doi.org/10.1007/s40304-015-0083-9