Annals of Global Analysis and Geometry

, Volume 44, Issue 2, pp 105-114

First online:

On the first eigenvalue of the Witten–Laplacian and the diameter of compact shrinking solitons

  • Akito FutakiAffiliated withGraduate School of Mathematical Sciences, University of Tokyo
  • , Haizhong LiAffiliated withDepartment of Mathematical Sciences, Tsinghua University Email author 
  • , Xiang-Dong LiAffiliated withAcademy of Mathematics and Systems Science, Chinese Academy of Sciences

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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.


Witten–Laplacian Eigenvalue Shrinking Ricci solitons Self-similar shrinker Diameter