Stability of the Positive Mass Theorem for Graphical Hypersurfaces of Euclidean Space
Stability of the Positive Mass Theorem for Graphical Hypersurfaces of Euclidean Space
复制标题
欧氏空间图解超曲面正质量定理的稳定性
DOI：
10.1007/s00220-014-2265-9
复制
发表时间：
2014
影响因子：
2.4
通讯作者：
Dan A. Lee
中科院分区：
文献类型：
--
作者：
Lan;Dan A. Lee
The rigidity of the positive mass theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and zero mass is Euclidean space. We prove a corresponding stability theorem for spaces that can be realized as graphical hypersurfaces in $${\mathbb{R}^{n+1}}$$Rn+1. Specifically, for an asymptotically flat graphical hypersurface $${M^n\subset \mathbb{R}^{n+1}}$$Mn⊂Rn+1 of nonnegative scalar curvature (satisfying certain technical conditions), there is a horizontal hyperplane $${\Pi\subset \mathbb{R}^{n+1}}$$Π⊂Rn+1 such that the flat distance between M and $${\Pi}$$Π in any ball of radius $${\rho}$$ρ can be bounded purely in terms of n, $${\rho}$$ρ, and the mass of M. In particular, this means that if the masses of a sequence of such graphs approach zero, then the sequence weakly converges (in the sense of currents, after a suitable vertical normalization) to a flat plane in $${\mathbb{R}^{n+1}}$$Rn+1. This result generalizes some of the earlier findings of Lee and Sormani (J Reine Angew Math 686:187–220, 2014) and provides some evidence for a conjecture stated there.
The rigidity of the positive mass theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and zero mass is Euclidean space. We prove a corresponding stability theorem for spaces that can be realized as graphical hypersurfaces in $${\mathbb{R}^{n+1}}$$Rn+1. Specifically, for an asymptotically flat graphical hypersurface $${M^n\subset \mathbb{R}^{n+1}}$$Mn⊂Rn+1 of nonnegative scalar curvature (satisfying certain technical conditions), there is a horizontal hyperplane $${\Pi\subset \mathbb{R}^{n+1}}$$Π⊂Rn+1 such that the flat distance between M and $${\Pi}$$Π in any ball of radius $${\rho}$$ρ can be bounded purely in terms of n, $${\rho}$$ρ, and the mass of M. In particular, this means that if the masses of a sequence of such graphs approach zero, then the sequence weakly converges (in the sense of currents, after a suitable vertical normalization) to a flat plane in $${\mathbb{R}^{n+1}}$$Rn+1. This result generalizes some of the earlier findings of Lee and Sormani (J Reine Angew Math 686:187–220, 2014) and provides some evidence for a conjecture stated there.