Timeline for Negatively curved metrics minimizing the length of a homotopy class of simple closed curves
Current License: CC BY-SA 3.0
9 events
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| Aug 13, 2015 at 13:29 | comment | added | BS. | @Ian : ok! I was under the wrong impression that you were sketching the proof of precompactness (it was kind of obvious to me that a precompact metric space is bounded, which is what you prove). Sorry. | |
| Aug 13, 2015 at 5:48 | comment | added | Ian Agol | By the precompactness theorem - see the third paragraph. | |
| Aug 10, 2015 at 21:19 | comment | added | BS. | Sorry Ian, maybe I'm being thick, but I don't understand why you can assume $g_i$'s and $h_i$'s are $\delta$-close among themselves. Could you elaborate ? | |
| Apr 27, 2015 at 20:47 | comment | added | Ian Agol | @SelimG: Yes, you're essentially correct. In fact, for a fixed $M$, there are boundedly many curves of bounded length. So a subsequence will preserve the homotopy class of $\gamma$. I should fix the answer to reflect this. | |
| Apr 27, 2015 at 16:32 | comment | added | Selim G | Dear Ian, I checked in detail this morning your argument. There is one thing which is not clear to me(in the last part). Two metric on $M$ are said to be $C$-close if there is a diffeomorphism $f$ of $M$ such that both $f$ and $f^{-1}$ are $e^C$-Lipschitz, but to get the bound on $L_g(\gamma)$ one would would have to be sure that $f$ preserves the class of $\gamma$. We might need here something like the finiteness of the mapping class group of $M$ to conclude right ? | |
| Apr 25, 2015 at 10:18 | comment | added | Selim G | Thank you Ian, Belgradek's paper answer the very first question I had in mind, namely how different can two negatively curved metric be on the same underlying manifold. | |
| Apr 25, 2015 at 10:16 | vote | accept | Selim G | ||
| Apr 23, 2015 at 16:12 | history | edited | Ian Agol | CC BY-SA 3.0 |
Added more exposition.
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| Apr 23, 2015 at 5:22 | history | answered | Ian Agol | CC BY-SA 3.0 |