Timeline for Submanifolds of $\mathbb{R}^N$ whose local charts have uniformly bounded derivatives
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Nov 3, 2017 at 17:30 | comment | added | Rbega | If the manifold is graphical to order 1 on scale 1 (in the notation from my answer), then the mean curvature of the graph satisfies a uniformly elliptic quasi-linear PDE on a fixed size domain so one can use elliptic estimates to bootstrap up the regularity to that of H. In practice, to show that the manifold is graphical to order 1, one generally uses the second fundamental form condition described in my answer. | |
| Nov 3, 2017 at 15:47 | comment | added | macbeth | Oh, interesting. So (as I am sure @Rbega knows) I actually omitted an important hypothesis for the intrinsic case -- one needs bounded injectivity radius, or bounded-below volume of balls of a fixed radius, in order to get the theorem "bounded Ricci implies good charts". Is there a similar geometric hypothesis in the extrinsic case to get "bounded mean curvature implies graph of good function"? | |
| Nov 3, 2017 at 11:56 | comment | added | Rbega | Mean curvature isn't strong enough (e.g. one can have catenoidal necks) | |
| Nov 3, 2017 at 8:48 | history | edited | macbeth | CC BY-SA 3.0 |
added 487 characters in body
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| Nov 3, 2017 at 8:20 | history | answered | macbeth | CC BY-SA 3.0 |