Timeline for On a family of $C^0$-convergent Riemann metrics
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 18, 2012 at 13:44 | comment | added | Liviu Nicolaescu | @ Deane This means that for any $p\in M$ and for any $2$-plane $\pi\subset T_pM$ the scalar $K^0(\pi)$ is the sectional curvature of $g$ along $\pi$. | |
| May 16, 2012 at 8:08 | comment | added | Deane Yang | Also, what exactly do you mean by the statement "$K^0$ is the sectional curvature of $g$"? | |
| May 16, 2012 at 8:05 | comment | added | Deane Yang | You're absolutely right, and Anton Petrunin has a much better answer to your question. | |
| May 16, 2012 at 1:58 | answer | added | Anton Petrunin | timeline score: 6 | |
| May 16, 2012 at 0:10 | comment | added | Liviu Nicolaescu | I have looked at the paper you mentioned. That paper uses only the assumption that $K^\varepsilon$ is uniformly bounded. In my case $K^\varepsilon$ is convergent to a smooth limit. The question is weather this limit is indeed the curvature of the limiting metric $g$ which we know is smooth. | |
| May 15, 2012 at 22:41 | comment | added | Deane Yang | Have you looked at the paper by Stefan Peters, "Convergence of riemannian manifolds", Compositio Math 62 (1987) 3-16? You can't conclude that the limiting sectional curvature is continuous but it is bounded almost everywhere. | |
| May 15, 2012 at 22:19 | history | edited | Liviu Nicolaescu | CC BY-SA 3.0 |
Added more details.; added 1 characters in body; added 1 characters in body; deleted 54 characters in body
|
| May 15, 2012 at 22:14 | comment | added | Liviu Nicolaescu | It's a bit better than the uniform convergence mentioned in my question. let me edit the question. | |
| May 15, 2012 at 20:46 | comment | added | Deane Yang | I just want to confirm that the sectional curvature converges only for co-ordinate 2-planes and not for other 2-planes? Others can answer your question better than me, but my experience has been that controlling the sectional curvature only along co-ordinate 2-planes (for a fixed set of co-ordinates) is not enough. | |
| May 15, 2012 at 20:04 | history | asked | Liviu Nicolaescu | CC BY-SA 3.0 |