Timeline for Gromov-Hausdorff limits of 2-dimensional Riemannian surfaces
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 15, 2016 at 6:18 | comment | added | Uri Bader | Thanks, Igor. I am not sorry for not making it a separate question... | |
| Apr 14, 2016 at 22:39 | comment | added | Igor Belegradek | @user89334: still the set is not closed: rescaling a flat torus exactly as above gives a point in the limit while the curvature is zero. More importantly one gets many singular limits spaces. | |
| Apr 14, 2016 at 17:55 | comment | added | Uri Bader | I asked this here because this is by-passing and I am not motivated enough to make a too big deal out of this question. As for the question, I meant that the curveture bound is LOWER, as iin the original 2-dim question. Sorry for the mistake and please don't bother with this too much. Just my curiosity. | |
| Apr 14, 2016 at 16:04 | comment | added | Igor Belegradek | @user89334: it always is better to ask a separate question (rather than ask in the comments). In this case the answer is no: start with your favorite closed negatively curved manifold and create a sequence scaling the metric by $1/i$. The limit is a point and the curvature tends to $-\infty$ so the negative upper curvature bound is there but the limit is of different dimension. | |
| Apr 14, 2016 at 14:51 | comment | added | Uri Bader | Igor, this is very nice, but the set of tools you use seems very specific. What about the following problem: is the set of all compact manifolds of a fixed dimension, upper bounded (negative) curvature and upper bounded diameter closed in the HG topology? | |
| Apr 13, 2016 at 18:53 | comment | added | asv | @IgorBelegradek: Done. | |
| Apr 13, 2016 at 17:44 | comment | added | Igor Belegradek | @semyonalesker: would you add the Gauss-Bonnet/Bishop argument (by editing your original question)? | |
| Apr 13, 2016 at 16:13 | comment | added | asv | @MikhailKatz: May I add also a simpler comment, though it might be obvious for experts. In order to exclude the case of the limiting space to be a point when $g\geq 2$, one has to use Gauss-Bonnet combined with the Bishop inequality. But the case of segment seems to be less elementary. | |
| Apr 13, 2016 at 14:48 | comment | added | Igor Belegradek | @MikhailKatz: On Gauss-Bonnet, what one has to rule out is small handles converging to points. One hopes that in the process curvature must tend to $-\infty$. If the handle is attached along a flat cylinder, then this seems doable: simply double along the boundary of the cylinder and argue that a closed hyperbolic surface cannot collapse to a point (which still requires more than Gauss-Bonnet, as far as I can see). Now in general the handle need not be attached along a flat cylinder and the curvature may go to $+\infty$ on the attaching region. Then I am not sure how to isolate that handle.. | |
| Apr 13, 2016 at 14:18 | comment | added | asv | Great!! Thanks a lot. Though I agree with the comment by Mikhail Katz... | |
| Apr 13, 2016 at 14:17 | vote | accept | asv | ||
| Apr 13, 2016 at 14:16 | vote | accept | asv | ||
| Apr 13, 2016 at 14:16 | |||||
| Apr 13, 2016 at 13:45 | history | edited | Igor Belegradek | CC BY-SA 3.0 |
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| Apr 13, 2016 at 13:41 | comment | added | Mikhail Katz | Very nice but I wonder if there might be a straightforward argument using Gauss-Bonnet in the 2-dimensional case. | |
| Apr 13, 2016 at 13:27 | history | answered | Igor Belegradek | CC BY-SA 3.0 |