Abstract
We prove a lower bound estimate for the first non-zero eigenvalue of the Witten–Laplacian on compact Riemannian manifolds. As an application, we derive a lower bound estimate for the diameter of compact gradient shrinking Ricci solitons. Our results improve some previous estimates which were obtained by the first author and Sano (Asian J Math, to appear), and by Andrews and Ni (Comm Partial Differential Equ, to appear). Moreover, we extend the diameter estimate to compact self-similar shrinkers of mean curvature flow.
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Acknowledgments
A. Futaki’s research was supported by JSPS Grant-in-Aid for Scientific Research (A) No. 21244003 and Challenging Exploratory Research N0. 23654023. H. Li’s research was supported by NSFC No. 11271214 and Tsinghua University-K. U. Leuven Bilateral Scientific Cooperation Fund. X.-D. Li’s research was supported by NSFC No. 10971032, Key Laboratory RCSDS, CAS, No. 2008DP173182, AMSS Research Grant Y129161ZZ1, and a Hundred Talents Project of AMSS, CAS. Part of this work was done during the first author visited Mathematical Sciences Center of Tsinghua University in September–October 2011, by the invitation of Professor S.-T. Yau. The third author would like to thank Professors Dominique Bakry and Mu-Fa Chen for explaining their results obtained in [5, 11, 12] to him. We also thank Dr. Daguang Chen for helpful discussion.
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Futaki, A., Li, H. & Li, XD. On the first eigenvalue of the Witten–Laplacian and the diameter of compact shrinking solitons. Ann Glob Anal Geom 44, 105–114 (2013). https://doi.org/10.1007/s10455-012-9358-5
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DOI: https://doi.org/10.1007/s10455-012-9358-5