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?

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?

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Mirzakani 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?

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?