Convergence of Riemannian Manifolds

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新的认识。这种技术对于研究截面曲率或里奇曲率有界的流形特别有用，但是需要一个新的较弱的收敛概念来理解没有这种强条件的流形。 最近PI和Wenger博士应用了Ambrosio和Kirchheim博士的工作，引入了流形之间的新距离：内在平坦距离。 虽然收敛性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可求积的。应用Cheeger-Colding、Gromov和Perelman的工作，PI和Wenger博士已经证明了具有非负Ricci曲率的流形的Gromov-Hausdorff极限和内在平坦极限是一致的。 然而，在一般情况下，极限空间是不同的，不收敛于Gromov-Hausdorff意义下的序列仍然可能收敛于内在平坦意义下。 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提出，以改善她的弗里德曼模型的稳定性的结果。类空宇宙在弗里德曼宇宙学中被描述为一个各向同性的三维黎曼流形，从最初的大爆炸开始扩展的时间。 实际上，宇宙不是各向同性的，因为它被引力以非均匀的方式弯曲。弱引力透镜（由于尘埃）和强引力透镜（由于大质量物体）已经被哈勃观测到扭曲空间区域。因此，在某种意义上，宇宙最多接近弗里德曼模型。在先前的工作中，在强假设下，PI已经证明了几乎各向同性的黎曼流形（在某种程度上允许弱引力透镜和局部强引力透镜）在Gromov-Hausdorff意义上接近弗里德曼模型。 这是证明通过研究Gromov-Hausdorff限制越来越迷向流形。 现在，PI提出通过研究黎曼流形的内在平坦极限，证明在较弱的假设下，宇宙在内在平坦意义上接近弗里德曼模型。 使用内禀平坦距离不仅可以考虑到弱引力透镜和强引力透镜，而且还可以考虑到虫洞的可能存在。

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.---------------------------------------------------------------------------