Timeline for Harmonic coordinates on asymptotically flat manifold
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 31, 2018 at 18:25 | comment | added | Paul | @WillieWong so we agree that the result is false? We have always existence with the decay $max(1-\tau, 2-n+\epsilon)$ and the dimension don't matter. | |
| Jan 31, 2018 at 17:03 | comment | added | Willie Wong | For existence: using the trivial embeddings $W^{k,p}_\tau \subset W^{k,p}_{\sigma}$ if $\tau < \sigma$, if $1 - \tau < 2-n$ you at least have the existence of harmonic coordinates with "worse decay". In terms of the elliptic theory you can't really do better: just think about solving the Poisson equation $\triangle \phi = f$ on $\mathbb{R}^n$ for $n \geq 3$. Even if $f$ has compact support, the best you can generally expect is that $\phi$ decays like the Newton potential at rate $r^{2-n}$. So the obvious interpretation is really the best you can do. | |
| Jan 31, 2018 at 16:50 | comment | added | Willie Wong | The requirement of injectivity is presumably useful to obtain quantitative bounds, as well as for proving uniqueness results (for example, I think the uniqueness of the harmonic coordinates for sufficiently large $\tau$ (or sufficiently small $1-\tau$) is used in the proof of Theorem 9.5 in Lee and Parker). | |
| Jan 31, 2018 at 16:40 | history | edited | Paul | CC BY-SA 3.0 |
added 438 characters in body
|
| Jan 31, 2018 at 6:53 | history | edited | Paul | CC BY-SA 3.0 |
added 8 characters in body
|
| Jan 30, 2018 at 21:17 | history | edited | Paul | CC BY-SA 3.0 |
added 882 characters in body
|
| Jan 30, 2018 at 21:00 | comment | added | Paul | @AntonPetrunin I am going to edit my post. | |
| Jan 30, 2018 at 20:59 | comment | added | Paul | @WillieWong I agree that it should work for the PMT. However, I would like a general staement. Moreover I don't understand why they need $1-\tau$ should be negative since we care about surjectivity and not injectivity. | |
| Jan 30, 2018 at 19:34 | comment | added | Willie Wong | You are looking at the wrong end point, I think for the Lee and Parker statement. The replacement by $1-\tau + \epsilon$ when $n = 3$ is because $\tau > (n-2)/2$, the condition they did suppose, allows $\tau < 1$ when $n = 3$, in which case $1-\tau > 0$. // I think for practical purposes you can suppose that $1 - \tau > 2 - n$ throughout, since for the PMT the mass term appears exactly at the level $\tau = n- 2 < n-1$ and similarly for the Yamabe problem (see Theorem 6.5 in Lee and Parker). | |
| S Jan 30, 2018 at 18:03 | history | suggested | Yury Ustinovskiy | CC BY-SA 3.0 |
fixed repeated word in the title
|
| Jan 30, 2018 at 16:32 | review | Suggested edits | |||
| S Jan 30, 2018 at 18:03 | |||||
| Jan 30, 2018 at 16:01 | history | asked | Paul | CC BY-SA 3.0 |