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Eigenvalues under the Backward Ricci Flow on Locally Homogeneous Closed 3-manifolds

Eigenvalues under the Backward Ricci Flow on Locally Homogeneous Closed 3-manifolds
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摘要 In this paper, we study the evolving behaviors of the first eigenvalue of the Laplace- Beltrami operator under the normalized backward Ricci flow, construct various quantities which are monotonic under the backward Ricci flow and get upper and lower bounds. We prove that in cases where the backward Ricci flow converges to a sub-Riemannian geometry after a proper rescaling, the eigenvalue evolves toward zero. In this paper, we study the evolving behaviors of the first eigenvalue of the Laplace- Beltrami operator under the normalized backward Ricci flow, construct various quantities which are monotonic under the backward Ricci flow and get upper and lower bounds. We prove that in cases where the backward Ricci flow converges to a sub-Riemannian geometry after a proper rescaling, the eigenvalue evolves toward zero.
作者 Song Bo HOU
机构地区 Department of Applied Mathematics
出处 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2018年第7期1179-1194,共16页 数学学报（英文版）
基金 Supported by NSFC(Grant No.11001268) Chinese Universities Scientific Fund(Grant No.2014QJ002)
关键词 Homogeneous 3-manifold backward Ricci flow EIGENVALUE estimate Homogeneous 3-manifold backward Ricci flow eigenvalue estimate
分类号 O186.12 [理学—基础数学]
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Acta Mathematica Sinica,English Series
2018年 第7期

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