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Spectrum of the laplacian on quaternionic k¨ahler manifolds

Shengli Kong University of California, Irvine Peter Li University of California, Irvine Detang Zhou Universidade Federal Fluminense-UFF

Differential Geometry mathscidoc:1609.10053

Journal of Differential Geometry, 78, (2), 295-332, 2008
Let M 4n be a complete quaternionic K¨ahler manifold with scalar curvature bounded below by .16n(n + 2). We get a sharp estimate for the first eigenvalue λ1(M) of the Laplacian, which is λ1(M) ≤ (2n + 1)2. If the equality holds, then either M has only one end, or M is diffeomorphic to R × N with N given by a compact manifold. Moreover, if M is of bounded curvature, M is covered by the quaterionic hyperbolic space QHn and N is a compact quotient of the generalized Heisenberg group. When λ1(M) ≥ 8(n+2) 3 , we also prove that M must have only one end with infinite volume.
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@inproceedings{shengli2008spectrum,
  title={SPECTRUM OF THE LAPLACIAN ON QUATERNIONIC K¨AHLER MANIFOLDS},
  author={Shengli Kong, Peter Li, and Detang Zhou},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160909104736082263710},
  booktitle={Journal of Differential Geometry},
  volume={78},
  number={2},
  pages={295-332},
  year={2008},
}
Shengli Kong, Peter Li, and Detang Zhou. SPECTRUM OF THE LAPLACIAN ON QUATERNIONIC K¨AHLER MANIFOLDS. 2008. Vol. 78. In Journal of Differential Geometry. pp.295-332. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160909104736082263710.
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