Conformal Rigidity and Non-rigidity of the Scalar Curvature on Riemannian Manifolds

被引:0
作者
Jaeyoung Byeon
Sangdon Jin
机构
[1] KAIST,Department of Mathematical Sciences
[2] KAIST,Stochastic Analysis and Application Research Center
来源
The Journal of Geometric Analysis | 2021年 / 31卷
关键词
Conformal; Rigidity; Non-rigidity scalar curvature; Linearized operator; Riemannian manifold; 53C24; 53C21;
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摘要
For a compact smooth manifold (M,g0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(M,g_0)$$\end{document} with a boundary, we study the conformal rigidity and non-rigidity of the scalar curvature in the conformal class. It is known that the sign of the first eigenvalue for a linearized operator of the scalar curvature by a conformal change determines the rigidity/non-rigidity of the scalar curvature by conformal changes when the scalar curvature Rg0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{g_0}$$\end{document} is positive. In this paper, we show the sign condition of Rg0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{g_0}$$\end{document} is not necessary, and a reversed rigidity of the scalar curvature in the conformal class does not hold if there exists a point x0∈M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0 \in M$$\end{document} with Rg0(x0)>0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{g_0}(x_0) > 0.$$\end{document}
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页码:9745 / 9767
页数:22
相关论文
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