Relating Notions of Convergence in Geometric Analysis
Abstract
We relate $L^p$ convergence of metric tensors or volume convergence to a given smooth metric to Intrinsic Flat and GromovHausdorff convergence for sequences of Riemannian manifolds. We present many examples of sequences of conformal metrics which demonstrate that these notions of convergence do not agree in general even when the sequence is conformal, $g_j=f_j^2g_0$, to a fixed manifold. We then prove a theorem demonstrating that when sequences of metric tensors on a fixed manifold $M$ are bounded, $(11/j)g_0 \le g_j \le K g_0$, and either the volumes converge, $\operatorname{Vol}_j(M)\rightarrow \operatorname{Vol}_0(M)$, or the metric tensors converge in the $L^p$ sense, then the Riemannian manifolds $(M,g_j)$ converge in the measured GromovHausdorff and volume preserving Intrinsic Flat sense to $(M,g_0)$.
 Publication:

arXiv eprints
 Pub Date:
 November 2019
 arXiv:
 arXiv:1911.04522
 Bibcode:
 2019arXiv191104522A
 Keywords:

 Mathematics  Metric Geometry;
 Mathematics  Differential Geometry
 EPrint:
 40 pages