Timeline for Conformal invariants of planar pairs of pants
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 13, 2014 at 23:43 | vote | accept | Pablo Lessa | ||
| May 12, 2014 at 20:43 | answer | added | Igor Rivin | timeline score: 1 | |
| May 12, 2014 at 17:55 | history | edited | Pablo Lessa | CC BY-SA 3.0 |
typo
|
| May 12, 2014 at 17:21 | comment | added | Dylan Thurston | Without much work, you could do at least a little better than that by considering non-concentric circles: take the inner circle of the annulus to be tangent to your two small circles. The extremal length for such an annular domain can also be calculated. | |
| May 12, 2014 at 17:10 | comment | added | Pablo Lessa | The circular annulus centered at $0$ with radii $m = \max(x+r_1, x+r_2)$ and $1$ has modulus $M = \log(1/m)/2\pi$. Let $\ell$ be the length of the outer boundary geodesic. The characterization of $M$ by extremal length yields $\ell^2/2\pi \le 1/M$. Similarly one can bound the distance $d$ between the outer boundary and some other boundary from above by using the modulus of this annulus. Then using the Collar lemma on the surface obtained by pasting two copies of the pants one get something like $\sinh(d)\sinh(\ell/2) \ge 1$ so this gives a lower bound for $\ell$. | |
| May 12, 2014 at 17:00 | comment | added | Dylan Thurston | What explicit bounds do you get? What's known about the extremal length for these domains? | |
| May 12, 2014 at 17:00 | answer | added | Dylan Thurston | timeline score: 3 | |
| May 12, 2014 at 14:59 | history | asked | Pablo Lessa | CC BY-SA 3.0 |