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

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• 作者：Akito Futaki (1)
  Haizhong Li (2)
  Xiang-Dong Li (3)
  
• 关键词：Witten–Laplacian ; Eigenvalue ; Shrinking Ricci solitons ; Self ; similar shrinker ; Diameter
• 刊名：Annals of Global Analysis and Geometry
• 出版年：2013
• 出版时间：August 2013
• 年：2013
• 卷：44
• 期：2
• 页码：105-114
• 全文大小：154KB
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• 作者单位：Akito Futaki (1)
  Haizhong Li (2)
  Xiang-Dong Li (3)
  
  1. Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan
  2. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China
  3. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 55, Zhongguancun East Road, Beijing, 100190, People’s Republic of China
  

文摘

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.