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Mathematics > Metric Geometry

Title: Contrasting Various Notions of Convergence in Geometric Analysis

Abstract: We explore the distinctions between $L^p$ convergence of metric tensors on a fixed Riemannian manifold versus Gromov-Hausdorff, uniform, and intrinsic flat convergence of the corresponding sequence of metric spaces. We provide a number of examples which demonstrate these notions of convergence do not agree even for two dimensional warped product manifolds with warping functions converging in the $L^p$ sense. We then prove a theorem which requires $L^p$ bounds from above and $C^0$ bounds from below on the warping functions to obtain enough control for all these limits to agree.
Comments: 7 figures by Penelope Chang of Hunter College High School
Subjects: Metric Geometry (math.MG); Differential Geometry (math.DG)
MSC classes: 53C23
Cite as: arXiv:1803.06582 [math.MG]
  (or arXiv:1803.06582v1 [math.MG] for this version)

Submission history

From: Christina Sormani [view email]
[v1] Sat, 17 Mar 2018 22:11:43 GMT (742kb,D)