Timeline for riemannian length of an element of the fundamental group of a manifold
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 15, 2014 at 0:45 | comment | added | Otis Chodosh | Frank Morgan download.springer.com/static/pdf/25/… and Brian White ams.org/journals/proc/1984-090-02/S0002-9939-1984-0727239-0/… . Recently, Robert Young proved that there is a lower bound for $\frac{A_{2\Gamma}}{A_\Gamma}$ arxiv.org/abs/1312.0966! | |
| Aug 15, 2014 at 0:44 | comment | added | Otis Chodosh | This is off topic, but there's an interesting higher dimensional version of this question: e.g. consider a curve $\Gamma$ in say $\mathbb{R}^4$ and find the least area $A_{\Gamma}$ of surfaces $\Sigma$ with boundary $\Gamma$. Now, find the least area $A_{2\Gamma}$ of surfaces with boundary $2\Gamma$. For essentially the same reason you give, $A_{2\Gamma}\leq 2 A_\Gamma$. However, strict inequality can hold in general! The first example was given by L.C. Young. This was later generalized by ... | |
| Aug 14, 2014 at 22:14 | answer | added | Tim Carson | timeline score: 0 | |
| Jun 27, 2011 at 16:16 | comment | added | Pietro Majer | (sorry, I see I have unwillingly down-voted this question) | |
| Jun 27, 2011 at 16:15 | history | edited | Pietro Majer | CC BY-SA 3.0 |
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| Jun 24, 2011 at 17:40 | comment | added | Sergei Ivanov | Generically, a geodesic loop realizing the minimum length does not close up smoothly at $p$, and then the double loop is not a geodesic and hence not a minimizer. So the statement is almost never true unless you minimize in a free homotopy class. | |
| Jun 24, 2011 at 17:02 | comment | added | Noam D. Elkies | When $\pi_1$ has an element $\alpha$ of order 2, $\alpha^2$ can have length 0 (that's what R.Kent was suggesting). There are intermediate possibilities too, such as $\alpha$ of order 3 implies $\alpha$ and $\alpha^2$ have the same length; this doesn't happen in two dimensions but does in three (e.g. lens space). | |
| Jun 24, 2011 at 16:58 | answer | added | Sam Nead | timeline score: 9 | |
| Jun 24, 2011 at 16:34 | comment | added | unkown | under what conditions the statement is true ? and can the length of \alpha^2 be less the length of \alpha ? | |
| Jun 24, 2011 at 16:01 | comment | added | Autumn Kent | The statement is false. Think projective plane. | |
| Jun 24, 2011 at 15:54 | history | asked | unkown | CC BY-SA 3.0 |