3维渐近平坦流形中的不稳定常平均曲率球面和非中心常平均曲率球面
Unstable CMC spheres and outlying CMC spheres in AF 3-manifolds
数学广义相对论里面的一个重要问题是常平均曲率曲面的存在唯一性问题. 在证明唯一性的时候,一般要要求球面是稳定的．稳定性条件能否去掉一直是未知的．在这篇文章中,我们引入一个常微分方程方法,在具有对称性的黎曼流形中构造常平均曲率曲面．作为应用,我们在度量形如 $g_{ij}=(1+ rac{1}{l})^{2}delta_{ij}+O(l^{-2})$ 渐近Schwarzschild流形中构造不稳定的常平均曲率球面和非中心的常平均曲率球面．不稳定的常平均曲率球面的存在性说明了庆杰和田刚文章“渐近平坦3维流形中常平均曲率球面分叶结构的唯一性”中的稳定性条件是不能去掉的.
One of the central problems in mathematical relativity is the existence and uniqueness of constant mean curvature surfaces. In proving the uniqueness, the stability condition is usually required. In this paper, we introduce a non linear ODE method to construct CMCsurfaces in Riemannian manifolds with symmetry. As an applicationwe construct unstable CMC spheres and outlying CMC spheres in asymptoticallySchwarzschild manifolds with metrics like $g_{ij}=(1+ rac{1}{l})^{2}delta_{ij}+O(l^{-2})$.The existence of unstable CMC spheres tells us that the stabilitycondition in Qing-Tian's work “On the uniqueness of the foliation of spheres of constant mean curvature in asymptotically flat 3-manifolds” can not be removedgenerally.
马世光
数学
渐近平坦流形常平均曲率曲面稳定性常微分方程黎曼流形
asymptotically flat manifold constant mean curvature surface stability ordinary differential equation Riemannian manifold
马世光.3维渐近平坦流形中的不稳定常平均曲率球面和非中心常平均曲率球面[EB/OL].(2016-03-02)[2025-08-03].http://www.paper.edu.cn/releasepaper/content/201603-36.点此复制
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