Timeline for A complete Riemannian metric for which the Ricci flow has no solution
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 9, 2014 at 1:09 | history | edited | Chih-Wei Chen | CC BY-SA 3.0 |
(9 June) Add a reference (Deane Yang's article). I apologize for missing this pioneer work at the first time.
|
| Jun 2, 2014 at 5:11 | comment | added | Chih-Wei Chen | Thank you Igor for your kind comment. In the beginning I just thought that Xu's precise result is not crucial for this post, and the focus is the convergence type at $t=0$. So I cited it quite unceremoniously. I agree with you that it would be better if I wrote the answer more carefully (both in phrase and in math), in particular I should indicate which Corollary I use. On the other hand, the reason that I didn't mention the assumptions of Xu ($Ric$ bounded from below and $Rm$ has averaged integral bound) is because both these conditions follow trivially from Shi's assumption ($Rm$ bounded). | |
| Jun 2, 2014 at 4:29 | comment | added | Igor Belegradek | @Chih-WeiChen: the way you phrase Xu's result is strange. What you call a pointwise bound on $Rm(0)$ is not an assumption because as you say it is alsways true. Why mention it? On the other hand, Xu's corollary 1.2 assumes a lower bound on Ricci plus some integral curvature bound. Not sure why you do not state it. Anyway, thank you for the answer! | |
| Jun 2, 2014 at 4:06 | comment | added | Chih-Wei Chen | For "pointwise bounded", I mean "bounded at every point by different values". For instance, $x^2$ is pointwise bounded on $\mathbb{R}$, but no uniform bound can be found. Therefore, a complete manifold with smooth (or at least $C^2$) metric has pointwise bounded $Rm$. (Is this not a usual terminology? sorry about unclearness.) For Xu's result, in the terminology explained above, he assumes pointwise bound on the curvature. The result I mentioned is Corollary 1.2 in page 3, there he said that the condition of Sobolev constant can be replaced by assuming Ricci curvature bounded from below. | |
| Jun 2, 2014 at 2:22 | comment | added | Deane Yang | I also don't understand the description "$Rm(0)$ is pointwise bounded but not uniformly bounded". Looking at Xu's paper, he proves a local existence under assumptions that don't require the sectional curvature to have a uniform pointwise bound. The theorem is not a generalization of Shi's, because Xu's proof (like mine) requires a uniform bound on the "local Sobolev constant". | |
| Jun 2, 2014 at 0:52 | comment | added | Igor Belegradek | When you say "pointwise bounded" do you mean that the bound is the same at all points? | |
| Jun 2, 2014 at 0:35 | history | edited | Chih-Wei Chen | CC BY-SA 3.0 |
More words about Xu's work are added. Some stupid typos are fixed. The last paragraph is revised.
|
| Jun 1, 2014 at 16:54 | comment | added | Chih-Wei Chen | Thank you for this nice comment! According to the answer of your post there, which says in certain sense that "solutions always exist", I guess that it is really not easy to find an example we want for the Ricci flow. | |
| Jun 1, 2014 at 15:09 | comment | added | Terry Tao | Hmm, the answer may indeed hinge on exactly what the notion of solution is - if one allows solutions that grow arbitrarily fast at infinity, there may well be no non-compact connected solutions, as was the case with the heat equation when I asked a similar question at mathoverflow.net/questions/72195/… | |
| Jun 1, 2014 at 14:59 | history | answered | Chih-Wei Chen | CC BY-SA 3.0 |