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Pablo Lessa
Pablo Lessa
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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 minusa 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 (?).

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 minus 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 (?).

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 (?).

Source Link
Pablo Lessa
Pablo Lessa
  • 4.3k
  • 28
  • 37

Conformal invariants of planar pairs of pants

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 minus 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 (?).