English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We relate $L^p$ convergence of metric tensors or volume convergence to agiven smooth metric to Intrinsic Flat and Gromov-Hausdorff convergence forsequences of Riemannian manifolds. We present many examples of sequences ofconformal metrics which demonstrate that these notions of convergence do notagree in general even when the sequence is conformal, $g_j=f_j^2g_0$, to afixed manifold. We then prove a theorem demonstrating that when sequences ofmetric tensors on a fixed manifold $M$ are bounded, $(1-1/j)g_0 \le g_j \le Kg_0$, 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 measuredGromov-Hausdorff and volume preserving Intrinsic Flat sense to $(M,g_0)$.

Relating Notions of Convergence in Geometric Analysis