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Consider a planar pair of pants $$P = \left\lbrace z \in \mathbb{C}: |z| \le 1, |z-x| \ge r_1, |z+x| \ge r_2 \right\rbrace$$ where $-1 < -x-r_2 < -x+r_2 < 0 < x-r_1 < x+r_1 < 1$.

There is a unique conformal hyperbolic metric $e^{2u} (d x^2 + d y^2)$ on $P$ such that all boundary components are geodesics. This corresponds to the unique solution $u:P \to \mathbb{R}$ to $\Delta u = e^{2u}$ with constant normal derivative equal to the inverse of the radius (with a minus sign on the outer boundary and a plus sign on the two others) on each boundary component.

The lengths of the boundary components for this unique metric is a complete conformal invariant for $P$. By this I mean that two pants are conformally equivalent if and only if the three lengths are equal.

My questions:

  1. Given a pair of pants can one estimate what the three lengths of the boundary geodesics will be?
  2. Are there known relationships to other conformal invariants such as modules of rings, etc?
  3. Do you know of any reference where this particular conformal invariant is considered/studied?

So far I've made some progress on question 1. By considering different anulii and using the definition of extremal length I can bound certain hyperbolic distances from above (be it the length of a boundary curve, or the distance between two of them). Using the Collar Lemma this also yields lower bounds.

However I'm guessing something better is known (?).

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  • $\begingroup$ What explicit bounds do you get? What's known about the extremal length for these domains? $\endgroup$
    – Dylan Thurston
    Commented May 12, 2014 at 17:00
  • $\begingroup$ 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 $\ell$. $\endgroup$
    – Pablo Lessa
    Commented May 12, 2014 at 17:10
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    $\begingroup$ 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. $\endgroup$
    – Dylan Thurston
    Commented May 12, 2014 at 17:21

2 Answers 2

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There are several references that consider the relation between extremal length and hyperbolic length. Usually they consider closed surfaces, but you can put yourself in that situation by doubling along the boundary. Here are some relevant papers:

Matsuzaki, Katsuhiko. Bounded and integrable quadratic differentials: hyperbolic and extremal lengths on Riemann surfaces. Geometric complex analysis (Hayama, 1995), 443–450, World Sci. Publ., River Edge, NJ, 1996. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.163.7078

Maskit, Bernard. Comparison of hyperbolic and extremal lengths. Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 381–386.

Rafi, Kasra. Thick-thin decomposition for quadratic differentials. Math. Res. Lett. 14 (2007), no. 2, 333–341. http://www.math.toronto.edu/~rafi/Papers/Thick-Thin.pdf

I think the Maskit paper might be most relevant.

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  • $\begingroup$ Thanks! Maskit's work is relevant. He gives upper and lower bounds on the quotient of the extremal length the free homotopy class of a closed geodesic and hyperbolic length of the geodesic on a closed hyperbolic surface. In particular they go to zero and infinity together (though at infinity the quotient might be unbounded). $\endgroup$
    – Pablo Lessa
    Commented May 12, 2014 at 21:29
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Actually, pairs of pants (a.k.a. triply connected planar domains) are special, and it is unwise to derive results for them from "the general case".

See this nice paper by T. Sugawa (1996) (for posterity: Various Domain Constants Related to the Uniform Perfectness).

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  • $\begingroup$ Thanks for the reference! Corollary 5.5 looks nice but I'm pretty sure he's working on the maximal conformal hyperbolic metric on the open domain. Hence it seems to me that this would give only upper bounds for the situation I'm interested in. Am I right? $\endgroup$
    – Pablo Lessa
    Commented May 12, 2014 at 21:30
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    $\begingroup$ @PabloLessa actually, I don't think so, I think this is more general. $\endgroup$
    – Igor Rivin
    Commented May 12, 2014 at 23:55

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