Timeline for Lower regularity version of Moser's theorem on volume elements
Current License: CC BY-SA 3.0
11 events
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| Jul 16, 2018 at 5:22 | comment | added | Piotr Hajlasz | @FanZheng I am leaving tomorrow and I will be away for several weeks so I will not have a chance to look at your question. Sorry. | |
| Jul 16, 2018 at 3:32 | comment | added | Fan Zheng | @PiotrHajlasz The remark right before Section 2.4 of [2] says that their method doesn't give a $C^{1,\alpha}$ diffeomorphism that is identity on a neighborhood of $\partial\Omega$. Can this be fixed by a pasting argument (say first shrink $\Omega$ along its collar neighborhood)? If so, then the local result can be globalized by a ($C^{1,\alpha}$) partition of unity argument that changes the volume form in a coordinate chart, one at a time. | |
| Apr 26, 2018 at 13:54 | comment | added | Robert Bryant | @PiotrHajlasz: Thanks for the notice and the pointer to the references for this question. While I was sure that this kind of thing had to be known, it's not the kind of question I normally think about, so I didn't know the right references. | |
| Apr 25, 2018 at 22:16 | comment | added | Piotr Hajlasz | @RobertBryant Because of your answer you might find mine interesting. | |
| Apr 25, 2018 at 22:16 | comment | added | Piotr Hajlasz | @DeaneYang Because of your comments you might find my answer interesting. | |
| Nov 2, 2014 at 17:56 | comment | added | Deane Yang | If you can somehow make the flow also satisfy either a parabolic PDE or a transverse elliptic PDE, then you would achieve the regularity you want. | |
| Nov 2, 2014 at 17:55 | comment | added | Deane Yang | I suspect Robert is correct about the regularity. When you integrate a flow, you gain a derivative in the time direction only and not in any transverse ("spatial") directions. This can be seen from the proof of the existence and uniqueness of parameterized ODE's. | |
| Oct 30, 2014 at 20:13 | comment | added | Robert Bryant | @AnthonyQuas: Unfortunately, I don't know the answer in your more general case. Probably, though, if you have the $\omega_i$ being $C^\alpha$, you can get $\phi$ (and hence $X$) to be $C^\alpha$ in the above argument. However, whether the flow of $X$ will then be $C^{1+\alpha}$ is another question. I suspect that it is not, and that the proof with optimum regularity that you want may need to be done by some other means, so probably checking alvarezpaiva's suggested reference is better than what I was suggesting. | |
| Oct 30, 2014 at 17:33 | comment | added | Anthony Quas | Thanks very much for this. Sorry - this is far from what I know about - it will take me some time to process this. May I ask slightly more about this (apologies for not managing to express this fully the first time around): If the manifold is $C^{1+\alpha}$ and the volume forms are just $C^\alpha$, can you still get a $C^{1+\alpha}$ diffeomorphism? Does this break down at $\alpha=0$? | |
| Oct 30, 2014 at 16:21 | history | edited | Robert Bryant | CC BY-SA 3.0 |
added 3 characters in body
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| Oct 30, 2014 at 16:13 | history | answered | Robert Bryant | CC BY-SA 3.0 |