# Teichmuller space for surface with cone points

Working on my current research problem, Teichmuller spaces for surfaces with cone points have come into play. It's fairly easy to formulate some of the definitions, a few basic results, etc. To be precise, I am looking to understand the Teichmuller space of a closed surface $$M$$ with genus $$g$$ and $$b$$ cone points with fixed cone angles $$\theta_1,\dots, \theta_b \in (0,2\pi)$$ such that $$\chi(M) - \sum_j (2\pi - \theta_j) < 0$$ (this last condition ensures there is a metric of constant curvature $$-1$$ away from the cone points). It would be nice to include punctures and boundary components, but I'll take what I can get.

I'm wondering if people have fleshed out the details for these spaces, and if so what are the best/standard sources? Some things should quite naturally carry over (for instance, Fenchel Nielsen coordinates), but I'm sure there must be some non-obvious subtleties. I'm simply asking as it would be quite the tedious task to do all this myself (but I suppose I will if the answer to my question is negative).

Here are some recent papers:

Rafe Mazzeo, Hartmut Weiss arXiv:1509.07608 Teichmüller theory for conic surfaces

arXiv:1710.09781 Rafe Mazzeo and Xuwen Zhu, Conical metrics on Riemann surfaces, I: the compactified configuration space and regularity.

• This seems to be the kind of thing I'm looking for. Thanks so much! – user470881 Aug 12 at 20:34

The canonical reference, from which all follows, is Marc Troyanov's beautifully written paper:

Troyanov, Marc, Les surfaces euclidiennes à singularités coniques. (Euclidean surfaces with cone singularities), Enseign. Math., II. Sér. 32, 79-94 (1986). ZBL0611.53035.

If you are interested in hyperbolic surfaces with cone points, a very nice reference is Chris Judge's paper and references therein (though for the basics, there is not much difference between the Euclidean and hyperbolic cases).

Judge, Christopher M., Tracking eigenvalues to the frontier of moduli space. I: Convergence and spectral accumulation, J. Funct. Anal. 184, No. 2, 273-290 (2001). ZBL1005.58012.

• Thank you very much! – user470881 Aug 12 at 20:35

The first theorem you want (that the Teichmüller space exists) is given as a remark in "Three-dimensional Orbifolds and Cone-Manifolds" by Daryl Cooper, Craig Hodgson, and Steve Kerckhoff. See Chapter 4, Section 6, second paragraph. They give further references, that may have more precise statements.

It seems to me that this covers surfaces with geodesic boundary, as well (and even cone points at the boundary - doubling reduces everything to the closed case). Thus the second theorem you want (FN coordinates) follows.

• Thank you very much! – user470881 Aug 12 at 20:35