规范ε—Ricci流下一类几何算子特征值的研究

   考虑度量餍足标准ε-Ricci流的闭的n维黎曼流形,给出一类几多算子-Δ+cR的特征值的发展方程,其中常数c≥1/4,R是流形上的数量曲率。作为使用,在闭曲面上证明了这种几多算子的特征值沿着标准ε-Ricci流保持单调性,从而推行

推戴了前人的相关研讨了局。 

  关键词标准ε-Ricci流;特征值;单调性;几多算子

中图分类号O186.1 文献标识码A 文章编号 2095-2457(2017)32-0017-002

AbstractAn n dimensional closed Riemannian manifold with the metric which satisfied the normalizedε-Ricci flow will be considered in the paper. The evolution of eigenvalues for geometric operator will be obtained. As an application, along the normalizedε-Ricci flow the monotonicity of eigenvalues can be proved on closed surfaces. These results generalizes our predecessors’ results on Ricci flow.

Key wordsThe normalizedε-Ricci flow; Eigenvalue; Monotonicity; Geometric operator

1 预备学问

3 结语

本文利用几多分析的方法,对标准ε-Ricci流下一类常见的几多算子的特征值进行研讨,得到了闭曲面上该算子特征值的单调性。文中的了局推行

推戴了文献4中的相关了局,也对ε-Ricci流及流形上几多算子特征值相关问题的进一步研讨有很好的启示意义。

参考文献

1PERELMAN G.The entropy formula for the Ricci flow and its geometric applicationsDB/OL.(2002-11-11)2012-11-25.http//arxiv.org/abs/math/0211159.

2MA Li.Eigenvalue monotonicity for the Ricci-Hamilton flow J.Ann.Global Anal.Geom.,2006,29(3)287-292.

3CAO Xiaodong. Eigenvalues of -R on manifolds with nonnegative curvature operatorJ.Math.Ann.,2007,337(2)435-441.

4CAO Xiaodong.First eigenvalues geometric operators under the Ricci flowJ.Proc.Amer.Math.Soc.,2008,136(11)4075-4078.

5儲亚伟,朱茱.ε-Ricci流上一类几多算子特征值的单调性J.阜阳师范学院学报自然科学版,2009,26(1)22-24.

6方守文,朱鹏.Ricci流下具有位能的共轭热方程Harnack量的熵J.扬州大学学报自然科学版,2012,15(2)14-16.

7方守文.延拓的Ricci流下具有位能的热方程Harnack估量J.扬州大学学报自然科学版,2013,16(2)13-15.

8FANG Shouwen,XU Haifeng,ZHU Peng.Evolution and monotonicity of eigenvalues under the Ricci flowJ.Sci.China Math.,2015,58(8)1737-1744.

9FANG Shouwen,ZHAO Liang,ZHU Peng.Estimates and monotonicity of the first eigenvalues under the Ricci flow on closed surfacesJ.Commun.Math.Stat.,2016,4(2)217-228.

10FANG Shouwen,YANG Fei.First eigenvalues of geometric operators under the Yamabe flowJ.Bull.Korean Math.Soc.,2016,53(4)1113-1122.

11FANG Shouwen,YANG Fei,ZHU Peng.Eigenvalues of geometric operators related to the Witten Laplacian under the Ricci flowJ.Glasg.Math.J.,2017,59(3)743-751.