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If we take a sequence of compact hyperbolic Riemann surface with k geodesic boundary components such that the lengths of the geodesic boundary components go to zero, then in the "limit", we should get a surface with k punctures/cusps. Is there a concept of distance ( in Teichmuller theory/ Riemannian geometry ) which would realize this convergence ? I know that there is a concept of Teichmuller space with genus g and k geodesic boundary components and also there is a concept of Teichmuller space with genus g and k cusps with finite area metrics. But is there a concept of Teichmuller space which will unify both of them, i.e. is distance will realize the above convergence and will contain both of the above Teichmuller spaces as subspace ?

Should we use Gromov-Housdorff convergence to realize the above convergence ?

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If you have a compact hyperbolic surface with geodesic boundary $\Sigma$, then you may double the surface along its boundary to get a closed hyperbolic surface $D\Sigma=\Sigma\cup_{\partial\Sigma}\Sigma$, which has an involution $\tau:D\Sigma\to D\Sigma$ which exchanges the two sides and fixes $\partial \Sigma$. One may then identify the Teichmuller space parameterizing hyperbolic structures on $\Sigma$ with geodesic boundary with the subspace of the Teichmuller space of $D\Sigma$ which is fixed under the involution $\tau$ (which as an element of the mapping class group acts by an isometric involution on Teichmuller space of $D\Sigma$).

The Weil-Petersson metric on Teichmuller space is an incomplete metric. Its completion is obtained by appending the Teichmuller spaces of Riemann surfaces with nodes, where some of the curves on the surface have been pinched (one may think of these as hyperbolic metrics with a double cusp, by letting the length of the curve approach zero). The Weil-Petersson metric extends to this completion, as proved by Masur.

Consider the Teichmuller space of $D\Sigma$, and the corresponding subspace associated to $\Sigma$ (the fixed point set of $\tau$). If one pinches some of the curves associated to the fixed point of $\tau$ on $D\Sigma$, then $\tau$ acts as an involution on the noded surface, and one may append the Teichmuller space of this noded surface invariant under $\tau$ to the Teichmuller space of $\Sigma$ with the Weil-Petersson completion. This gives you the moduli space you're describing of surfaces with geodesic boundary and punctures, and the Weil-Petersson metric gives a well-defined distance function on this space.

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  • $\begingroup$ Thanks Dr. Agol, I still have one naive question : are the "limiting" Riemann surfaces with double cusps topologically a 2- manifold ? Are they quotient of the upper half-plane $H$ ? $\endgroup$
    – Analysis Now
    Commented Mar 31, 2011 at 14:10
  • $\begingroup$ @Plus: you can think of these as removing the curve, and making two cusps (topologically). If you take a limit of surfaces with length approaching zero, then the limit is a surface with two cusps. Algebraic geometers think of these as noded surfaces, or an algebraic curve with a singularity, and these arise in the Deligne-Mumford compactification of Moduli space. But from the perspective of your question, one may think of it as a doubly cusped surface. Actually, one may probably think of the Weil-Petersson metric for manifolds with boundary, and dispense with the doubling/symmetry. $\endgroup$
    – Ian Agol
    Commented Mar 31, 2011 at 16:35

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