Timeline for Gromov-Hausdorff limits of 2-dimensional Riemannian surfaces
Current License: CC BY-SA 3.0
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| S Apr 25, 2016 at 11:25 | history | suggested | CommunityBot |
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| Apr 25, 2016 at 11:06 | review | Suggested edits | |||
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| Apr 20, 2016 at 3:43 | answer | added | Igor Belegradek | timeline score: 8 | |
| Apr 19, 2016 at 15:04 | comment | added | Mikhail Katz | @Igor, in the end Yamaguchi is not needed for the solution. | |
| Apr 19, 2016 at 14:42 | history | edited | user21574 |
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| Apr 19, 2016 at 13:09 | answer | added | Mikhail Katz | timeline score: 10 | |
| Apr 14, 2016 at 9:49 | history | edited | asv | CC BY-SA 3.0 |
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| Apr 14, 2016 at 8:29 | history | edited | asv | CC BY-SA 3.0 |
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| Apr 13, 2016 at 18:52 | history | edited | asv | CC BY-SA 3.0 |
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| Apr 13, 2016 at 14:17 | vote | accept | asv | ||
| Apr 13, 2016 at 14:16 | vote | accept | asv | ||
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| Apr 13, 2016 at 13:27 | answer | added | Igor Belegradek | timeline score: 10 | |
| Apr 13, 2016 at 3:54 | comment | added | asv | @IgorBelegradek: I agree that some reading would be very helpful to me and I would greatly appreciate a reference. I just wanted to make sure that the above reference does contain the answer to my question since I am wondering about a specific fact rather than about more general question "how one can think about these matters". Anyway many thanks for your comments. | |
| Apr 13, 2016 at 1:30 | comment | added | Igor Belegradek | I suggest you look at the paper by Shioya-Yamaguchi I mentioned and get the conclusion yourself. I cannot help wondering how one can think about these matters and did not know about Yamaguchi's fibration theorem. I think some reading is in order. | |
| Apr 13, 2016 at 1:24 | comment | added | asv | @IgorBelegradek: I am not sure, what is the conclusion of the use of the collapsing theory? | |
| Apr 13, 2016 at 1:23 | comment | added | asv | @AlexandreEremenko: Thanks, corrected. | |
| Apr 13, 2016 at 1:22 | history | edited | asv | CC BY-SA 3.0 |
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| Apr 12, 2016 at 21:46 | comment | added | Igor Belegradek | I wish to add that if you wish to black box the collapsing theory (and you shouldn't!) you could just multiply your collapsing sequence by a circle, so that you now have a sequence of $3$-manifolds collapsing to a cylinder. Then Shioya-Yamaguchi's paper formally applies, see projecteuclid.org/euclid.jdg/1090347524. Of course this is an overkill. | |
| Apr 12, 2016 at 20:17 | comment | added | Alexandre Eremenko | Are your surfaces compact? You did not say this. If con-compact are permitted, take cylinders $I\times S$ where $I$ is an interval and $S$ is a small circle, and equip with the standard metric of curvature $0$. You obtain a segment in the limit. | |
| Apr 12, 2016 at 16:46 | comment | added | Igor Belegradek | If the limit is smooth (e.g. a circle), then by the Yamaguchi's fibration theorem $M_i$ is a fiber bundle over the limit (for large $i$). The only orientable closed surface that fibers over the circle is the torus (because the fiber is also a circle for dimension reasons). Reading Yamaguchi's more recent papers on low-dimensional collapse will help you understand the case when the limit is an interval. | |
| Apr 12, 2016 at 16:19 | history | edited | asv | CC BY-SA 3.0 |
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| Apr 12, 2016 at 16:08 | history | edited | asv | CC BY-SA 3.0 |
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| Apr 12, 2016 at 15:22 | history | edited | asv | CC BY-SA 3.0 |
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| Apr 12, 2016 at 15:16 | history | edited | asv | CC BY-SA 3.0 |
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| Apr 12, 2016 at 13:03 | history | asked | asv | CC BY-SA 3.0 |