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Isoperimetric properties of higher eigenvalues of elliptic operators

Alexander Grigor'yan Shing-Tung Yau

Differential Geometry mathscidoc:1912.43580

American Journal of Mathematics, 125, (4), 893-940, 2003
We prove, in the setting of a measure energy space (M, ,(, )), that if the smallest eigenvalue 1 () of the generator of the Dirichlet form in any precompact open set M admits the estimate 1 () ()- where is a measure absolutely continuous with respect to and > 0 then a similar estimate holds for the kth smallest eigenvalue: k () const (k/ ()) . As an application, we obtain an upper estimate of the stability index of a minimal surface in [inline-graphic xmlns: xlink=" http://www. w3. org/1999/xlink" xlink: href=" 01i"/] via the total curvature.
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@inproceedings{alexander2003isoperimetric,
  title={Isoperimetric properties of higher eigenvalues of elliptic operators},
  author={Alexander Grigor'yan, and Shing-Tung Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224204210716305144},
  booktitle={American Journal of Mathematics},
  volume={125},
  number={4},
  pages={893-940},
  year={2003},
}
Alexander Grigor'yan, and Shing-Tung Yau. Isoperimetric properties of higher eigenvalues of elliptic operators. 2003. Vol. 125. In American Journal of Mathematics. pp.893-940. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224204210716305144.
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