Timeline for Given a local metric which is $C^1$-close to another, can we extend it globally while preserving the approximation?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 27, 2018 at 11:05 | comment | added | Asaf Shachar | On further thought, since in my case $|g-g_0|_{C^0} \approx \epsilon^2$, your construction in fact gives the right error bound. Thanks again. | |
| Aug 27, 2018 at 11:03 | vote | accept | Asaf Shachar | ||
| Aug 27, 2018 at 9:32 | comment | added | Asaf Shachar | The reason why $U_{\epsilon}=B_{\epsilon}(p)$ is not random or mysterious by the way: I am trying to approximate a given metric with Euclidean metric in a $C^1$-sense. By looking at the metric expansion in normal coordinates, we see that every metric is $C^1$-close to being Euclidean- and the "closeness" (or error) depends linearly on the distance from the point of origin. | |
| Aug 27, 2018 at 9:11 | comment | added | Asaf Shachar | Not really; To be more precise, In my "application", $U$ depends on $\epsilon$ (or if you want, then $\epsilon$ depends on $U$; they are "coupled"). Specifically, $U_{\epsilon}=B_{\epsilon}(p)$ where $B_{\epsilon}(p)$ is the $\epsilon$-ball around $p$, w.r.t. the given metric $g$. Thus, your construction gives an error which is on the order of $1$, instead of $\epsilon$. I am elaborating on the exact details in the question now. | |
| Aug 26, 2018 at 22:05 | comment | added | Deane Yang | Just to make sure, you want a way to do this such that the new $\epsilon$ does not depend on the size of $U$? | |
| Aug 26, 2018 at 15:47 | comment | added | Asaf Shachar | Since I did not explicitly required something which is independent in the size of $U$, I think I will accept your answer. (It certainly clarified something for me). However, I will leave the question open for a while, to see if anyone has an idea regarding this refined version. | |
| Aug 26, 2018 at 15:43 | comment | added | Asaf Shachar | Thanks, that is nice. However, in your construction the error depends also on the size of $U$, not only on $\epsilon$: $\tilde{g} - g_0= \chi (g-g_0)$, so $|d\tilde{g} - dg_0|\le |d\chi| |g-g_0| + |dg-dg_0| \le (|d\chi|+1)\epsilon$. My problem is that $|d\chi|$ should be roughly like $\frac{1}{\text{diam}(U)}$. So, if $U$ is small (say the size of $\epsilon$) we have a problem. I wonder whether there is a way to do something which does not depend on the size of $U$. | |
| Aug 26, 2018 at 15:15 | history | answered | Deane Yang | CC BY-SA 4.0 |