Convergence of Riemannian Manifolds

黎曼流形的收敛性

• 批准号: 1006059
• 负责人: Christina Sormani
• 金额: $ 16.3万
• 依托单位: Research Foundation Of The City University Of New York (Lehman)
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-01 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1006059&HistoricalAwards=false
• 关键词: Convergence; Riemannian; Manifolds

Convergence of Riemannian ManifoldsOver the past three decades mathematicians have gained deep new insight into Riemannian manifolds by applying the methods of Gromov-Hausdorff, Lipschitz and metric measure convergence. Such techniques have been particularly useful for studying manifolds with bounds on sectional or Ricci curvature, but a new weaker notion of convergence is needed to understand manifolds without such strong conditions. Recently the PI and Dr. Wenger have applied work of Drs. Ambrosio and Kirchheim to introduce a new distance between manifolds: the intrinsic flat distance. While the convergence is weaker than previous forms of convergence, the limit spaces, called Integral Current Spaces, are countably H^mrectifiable. Applying work of Cheeger-Colding, Gromov and Perelman, the PI and Dr. Wenger have shown that the Gromov-Hausdorff and intrinsic flat limits of manifolds with nonnegative Ricci curvature agree. However, in general the limit spaces are different and sequences which do not converge in the Gromov-Hausdorff sense may still converge in the intrinsic flat sense. The PI will study the properties of these limit spaces under a variety of conditions on the sequence of manifolds and prove stability theorems under these weaker conditions. In particular the PI proposes to improve her results on the stability of the Friedmann model.The spacelike universe is described in Friedmann cosmology as an isotropic three dimensional Riemannian manifold that expands in time starting from the initial Big Bang. In reality the universe is not isotropic because it is bent by gravity in a nonuniform way. Weak gravitational lensing (due to dust) and strong gravitational lensing (due to massive objects) has been observed by the Hubble to distort regions of space. The universe is thus, at best, close to the Friedmann model in some sense. In prior work, under strong assumptions, the PI has shown that a Riemannian manifold which is almost isotropic (in a way which allows for weak gravitational lensing and localized strong gravitational lensing) is close to the Friedmann model in the Gromov-Hausdorff sense. This is proven by studying the Gromov-Hausdorff limits of increasingly isotropic manifolds. Now the PI proposes to prove that under weaker assumptions, the universe is close to the Friedmann model in the intrinsic flat sense by studying the intrinsic flat limits of Riemannian manifolds. Using the intrinsic flat distance will not only allow for weak and strong gravitational lensing but also allow for the possible existence of wormholes.---------------------------------------------------------------------------

黎曼流形的收敛在过去的三十年里，数学家们通过应用Gromov-Hausdorff、Lipschitz和度量测度收敛的方法，对黎曼流形有了深刻的新的认识。这种技巧对于研究截面曲率或Ricci曲率有界的流形特别有用，但需要一个新的较弱的收敛概念来理解没有这样强条件的流形。最近，PI和Wenger博士应用Ambrosio博士和Kirchheim博士的工作，在流形之间引入了一种新的距离：固有平坦距离。虽然收敛比以前的收敛形式弱，但极限空间，称为积分流空间，是可数H^m可校正的。应用Cheeger-Colding，Gromov和Perelman的工作，PI和Wenger博士证明了具有非负Ricci曲率的流形的Gromov-Hausdorff极限和固有平坦极限是一致的。然而，一般来说，极限空间是不同的，不在Gromov-Hausdorff意义下收敛的序列仍然可能在固有平坦意义下收敛。PI将研究这些极限空间在流形序列上的各种条件下的性质，并在这些较弱的条件下证明稳定性定理。特别是，PI建议改进她关于Friedmann模型稳定性的结果。在Friedmann宇宙学中，类空宇宙被描述为一个各向同性的三维黎曼流形，它从最初的大爆炸开始在时间上扩展。事实上，宇宙并不是各向同性的，因为它被重力以一种不均匀的方式弯曲。哈勃望远镜观测到了弱引力透镜(由于尘埃)和强引力透镜(由于大质量物体)，从而扭曲了空间区域。因此，在某种意义上，宇宙充其量也就是接近弗里德曼模型。在以前的工作中，在强假设下，PI已经证明了在Gromov-Hausdorff意义下，几乎各向同性(允许弱引力透镜和局域强引力透镜)的黎曼流形接近于Friedmann模型。这一点通过研究日益各向同性流形的Gromov-Hausdorff极限得到了证明。现在PI建议通过研究黎曼流形的固有平坦极限来证明在较弱的假设下，宇宙在固有平坦意义上接近Friedmann模型。使用本征平坦距离不仅允许弱引力透镜和强引力透镜，而且还允许wormholes.---------------------------------------------------------------------------的可能存在