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Suppose we have a surface $S$ of a finite genus, without boundary with a finite number of punctures. Suppose that this surface comes equipped with a hyperbolic metric of curvature $-1$.

Question 1: If I add a puncture somewhere, the new surface again has a unique hyperbolic metric of curvature $-1$. Is there a way to see how this metric changes from the previous one, e.g. in terms of Fenchel–Nielsen coordinates?


Since it is a very hard questions, I add a more concrete situation I am interested in.

Question 2: Suppose that $S$ instead comes equipped with a family of hyperbolic metrics, and a pants decomposition, so that one geodesic length tends to zero (so there is a long collar cylinder nearby). Suppose also that all the other Fenchel Nielsen coordinates are fixed. Let me add a puncture on the shrinking geodesic. I would like to understand why this does not cause other Fenchel-Nielsen coordinates to change much (and that the only change is that a new pair-of-pants appears instead of this vanishing geodesic, but all the rest will be intact in the limit).

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    $\begingroup$ If you are hoping for some explicit formula, then this is unknown even for punctured spheres (say, adding a puncture to a 4 times punctured sphere). One can prove some quantitative results but you would have to be more specific in your request. $\endgroup$
    – Misha
    Commented Sep 8, 2019 at 4:58
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    $\begingroup$ @Misha I see. I have a family of curves where one geodesic length tends to zero (so there is a long collar cylinder nearby), and let other Fenchel Nielsen coordinates are fixed. Let me add a puncture on this vanishing geodesic. I would like to understand why this does not cause other Fenchel-Nielsen coordinates to change much (and that the only change is that a new pair-of-pants appears instead of this vanishing geodesic, but all the rest will be intact). $\endgroup$ Commented Sep 9, 2019 at 11:30
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    $\begingroup$ I suggest you revise your question and state what precisely you would like to know. $\endgroup$
    – Misha
    Commented Sep 9, 2019 at 16:57

2 Answers 2

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For question one: one possible qualitative approach is to consider the "thick-thin" decomposition of the given metric on $S$.


For question two: the hyperbolic metric on a surface and the "moduli of annuli" speak to each other in useful ways. (See Minksy's paper, referred to below, for references.) In particular, suppose that $\gamma$ is a curve in $S$. Suppose that, in $\rho_n$, your family of metrics, the length of $\gamma$ tends to zero. Then the modulus of the maximal round collar $A$ of $\gamma$ tends to infinity as $n$ does. If we cut this collar by $\gamma$, then we obtain two annuli $A^\pm$. Since modulus is additive, a symmetry argument implies that the moduli of $A^\pm$ also tend to infinity as $n$ does.

Let $S^\circ$ be the punctured surface. (The puncture is chosen on $\gamma$, and we should avoid allowing it to "twist" with respect to the family of metrics.) The metric $\rho_n|S^\circ$ is not complete. However (by uniformisation) there is a complete finite volume hyperbolic metric $\sigma_n$ in the conformal class of $\rho_n|S^\circ$. Recall that the modulus of an annulus is a conformal invariant. So the moduli of $A^\pm$ are the same when measured with respect to any one of $\rho_n$, $\rho_n|S^\circ$, and $\sigma_n$. Thus the lengths of the core curves $\gamma^\pm$ of $A^\pm$ are shrinking to zero (with respect to $\sigma_n$) as $n$ tends to infinity.

The desired result (the metrics $\sigma_n$, away from the punctured annulus cobounded by $\gamma^\pm$, settle down) now follows from the "product-like" nature of the "thin part of Teichmuller space". See Minksy's paper Extremal length estimates and product regions in Teichmuller space, in particular Sections 2.4 and 2.5.

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I don't know whether I'm understanding your problem correctly. It seems that you want to pinch a curve. Fenchel-Nielsen coordinates are....a coordinate system for the Teichmuller space of a compact surface. What you're doing is pinching a curve, namely, sending the length of a fixed curve to $0$. You can't say that the other FN coordinates are affected or aren't affected by the length of this curve. For example, $(x, y, z)$ is a coordinate system for $\mathbb{R}^3$. When you send $z \to 0$, $x$ and $y$ are certainly unaffected by $z$. Each coordinate is free. How other FN coordinates are changing is completely determined by the way you are pinching that fixed curve, and they can change in any way you want.

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  • $\begingroup$ no, the curve is not compact. Still we can measule the lengths of geodesics (this is what I mean saying ``Fenchel Nelsen'' coordinates). And I add a puncture, not pinching.. Sorry for confusion $\endgroup$ Commented Aug 19 at 6:27

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