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Mirzakhani proved identities for the lengths of geodesic curves on Riemann surfaces of genus $g$ and with $n$ boundary components. She used these to provide an integration scheme over the corresponding moduli space.

Do some of these facts extend to the case of surfaces whose boundaries have marked points?

See here for more details.

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    $\begingroup$ Mirzakhani, not Mirzakani. $\endgroup$
    – abx
    Commented Aug 16, 2021 at 12:22
  • $\begingroup$ thanks, I've fixed it $\endgroup$
    – giulio bullsaver
    Commented Aug 16, 2021 at 12:23
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    $\begingroup$ Can you provide more specific details for people who don't know about this? What identities, links to the papers where this is done? $\endgroup$
    – Hollis Williams
    Commented Aug 16, 2021 at 16:58
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    $\begingroup$ I think you should still be more precise. What kind of effect of marked points on boundaries are you looking for? Are you interested in Weil-Petersson volumes of marked Riemann surfaces? Integration over the positions of marked points (with respect to the natural measure on a boundary induced by the hyperbolic metric) is straightforward. $\endgroup$
    – Timothy Budd
    Commented Aug 17, 2021 at 8:51
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    $\begingroup$ I am interested in anything is already known regarding Mirzakani-like identities for surfaces with marked points on the boundaries and integration over their moduli space. For instance, something that I would be interested in, is if there are identities involving the sum over non-closed geodesics over such surfaces and/or if this allows the reduction of the integral over moduli space of surfaces obtained pinching such geodesics. I am asking vague questions because I am interested in anything which has to do with bordered surfaces $\endgroup$
    – giulio bullsaver
    Commented Aug 17, 2021 at 9:42

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