I am reading a paper on Riemann surfaces and the author used the fact that

$\{$Riemann surfaces with genus $g$ and $n$ punctures$\}$ is in one-to-one correspondence with $\{$ hyperbolic surfaces with genus $g$ and $n$ cusps $\}$. The paper says to equip a Riemann surface with a complete hyperbolic metric, we need to move punctures to infinity. How do we do that?

Can someone help me understanding why this is true, please?

  • $\begingroup$ Hey, do you remember which paper it was? $\endgroup$ – draks ... Feb 28 '16 at 15:04
  • $\begingroup$ I don't remember but if I find it, I'll let you know. $\endgroup$ – yaa09d Feb 29 '16 at 9:45
  • $\begingroup$ I am not able to comment since reputation is low. How about infinitely many punctures? Also, given a compact Riemann surface with boundary, we take punctures both at the boundary and in the interior, can we do the uniformization so that the geodesic curvature of the boundary of the resulting surface is zero? $\endgroup$ – Xiaoxiang Chai Oct 30 '17 at 21:19
  • $\begingroup$ P.S. The name of the paper is Moduli spaces of hyperbolic surfaces and their Weil–Petersson volumes by Norman Do $\endgroup$ – yaa09d May 1 '19 at 22:04

First suppose that $n=0$ on a genus $g \geq 2$ surface. Then the surface can be endowed with a hyperbolic metric (this is essentially the uniformization theorem). Note that $g \geq 2$ is necessary, e.g. because of Gauss-Bonnet theorem.

Now, suppose this surface is compactly embedded in some Euclidean space. If you make a puncture in this surface, the metric will be no longer complete at that point (the geodesics just stop there without any reason). In order to save the metric's completeness, we need to move these punctures off to infinity.

To give a very crude illustration, take a circle and make a puncture in it. You get an open interval and the geodesics will hit the interval's boundary in finite time. In order to save the situation you must expand the interval so that it actually becomes the real line (thought of as Riemannian manifold with flat metric) with boundary off at infinity. Now all is good since you only hit the boundary in infinite time and the metric is complete.

For surfaces, this procedure creates spikes -- you can imagine that the manifold is a gum that deforms as you pull out a point (representing the puncture) off towards infinity.

enter image description here

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  • 1
    $\begingroup$ Thanks for the answer. I still didn't get the concept of moving a puncture to infinity, so could you recommend me a reference, please? $\endgroup$ – yaa09d Sep 7 '12 at 4:20
  • $\begingroup$ How would the procedure that you described for the punctured disk be performed on a disk with n punctures? Perhaps it would be better to make it a sphere with n+1 punctures, I can imagine stretching out a neighborhood of each puncture to make a Gabrial's horn sort of thing, but I'm not sure what the explicit map would be. Should I try to figure it out or is it highly non-trivial? $\endgroup$ – Eben Kadile Oct 2 '18 at 4:46

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