Annals of Global Analysis and Geometry

, Volume 44, Issue 2, pp 105–114

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


DOI: 10.1007/s10455-012-9358-5

Cite this article as:
Futaki, A., Li, H. & Li, XD. Ann Glob Anal Geom (2013) 44: 105. doi:10.1007/s10455-012-9358-5


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 

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesUniversity of Tokyo Meguro-ku, TokyoJapan
  2. 2.Department of Mathematical SciencesTsinghua UniversityBeijing People’s Republic of China
  3. 3.Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingPeople’s Republic of China

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