Teichmuller space for surface with cone pointsCoordinates on Teichmuller spaceIs there a concept of Combined Teichmuller space for surfaces with both geodesic boundary and punctures/cuspswhen is the Teichmuller space a group?Coordinates for Teichmuller space for compact conformal surfacesReference for the result that the systol map from Teichmuller space to curve complex is coarsely LipschitzLimits at infinity of fellow-travelling sequences in Teichmuller space,Geometric quantization of Teichmuller spaceTeichmuller uniqueness theorem with marked pointsReference request for quantum Teichmuller space
Teichmuller space for surface with cone points
Coordinates on Teichmuller spaceIs there a concept of Combined Teichmuller space for surfaces with both geodesic boundary and punctures/cuspswhen is the Teichmuller space a group?Coordinates for Teichmuller space for compact conformal surfacesReference for the result that the systol map from Teichmuller space to curve complex is coarsely LipschitzLimits at infinity of fellow-travelling sequences in Teichmuller space,Geometric quantization of Teichmuller spaceTeichmuller uniqueness theorem with marked pointsReference request for quantum Teichmuller space
$begingroup$
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 $theta_1,dots, theta_b in (0,2pi)$ such that $$chi(M) - sum_j (2pi - theta_j) < 0$$ (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).
reference-request hyperbolic-geometry teichmuller-theory
asked Aug 12 at 6:08
user470881user470881
1176 bronze badges
$endgroup$
add a comment |
$begingroup$
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 $theta_1,dots, theta_b in (0,2pi)$ such that $$chi(M) - sum_j (2pi - theta_j) < 0$$ (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).
reference-request hyperbolic-geometry teichmuller-theory
asked Aug 12 at 6:08
user470881user470881
1176 bronze badges
$endgroup$
add a comment |
$begingroup$
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 $theta_1,dots, theta_b in (0,2pi)$ such that $$chi(M) - sum_j (2pi - theta_j) < 0$$ (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).
reference-request hyperbolic-geometry teichmuller-theory
asked Aug 12 at 6:08
user470881user470881
1176 bronze badges
$endgroup$
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 $theta_1,dots, theta_b in (0,2pi)$ such that $$chi(M) - sum_j (2pi - theta_j) < 0$$ (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).
reference-request hyperbolic-geometry teichmuller-theory
reference-request hyperbolic-geometry teichmuller-theory
asked Aug 12 at 6:08
user470881user470881
1176 bronze badges
asked Aug 12 at 6:08
user470881user470881
1176 bronze badges
asked Aug 12 at 6:08
user470881user470881
1176 bronze badges
asked Aug 12 at 6:08
user470881user470881
1176 bronze badges
asked Aug 12 at 6:08
user470881user470881
1176 bronze badges
1176 bronze badges
add a comment |
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
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.
answered Aug 12 at 20:06
Alexandre EremenkoAlexandre Eremenko
53.8k6 gold badges154 silver badges273 bronze badges
$endgroup$
$begingroup$
This seems to be the kind of thing I'm looking for. Thanks so much!
$endgroup$
– user470881
Aug 12 at 20:34
add a comment |
$begingroup$
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.
answered Aug 12 at 16:35
Igor RivinIgor Rivin
80.5k9 gold badges116 silver badges311 bronze badges
$endgroup$
$begingroup$
Thank you very much!
$endgroup$
– user470881
Aug 12 at 20:35
add a comment |
$begingroup$
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.
answered Aug 12 at 9:26
Sam NeadSam Nead
11.8k2 gold badges42 silver badges75 bronze badges
$endgroup$
$begingroup$
Thank you very much!
$endgroup$
– user470881
Aug 12 at 20:35
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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.
answered Aug 12 at 20:06
Alexandre EremenkoAlexandre Eremenko
53.8k6 gold badges154 silver badges273 bronze badges
$endgroup$
$begingroup$
This seems to be the kind of thing I'm looking for. Thanks so much!
$endgroup$
– user470881
Aug 12 at 20:34
add a comment |
$begingroup$
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.
answered Aug 12 at 20:06
Alexandre EremenkoAlexandre Eremenko
53.8k6 gold badges154 silver badges273 bronze badges
$endgroup$
$begingroup$
This seems to be the kind of thing I'm looking for. Thanks so much!
$endgroup$
– user470881
Aug 12 at 20:34
add a comment |
$begingroup$
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.
answered Aug 12 at 20:06
Alexandre EremenkoAlexandre Eremenko
53.8k6 gold badges154 silver badges273 bronze badges
$endgroup$
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.
answered Aug 12 at 20:06
Alexandre EremenkoAlexandre Eremenko
53.8k6 gold badges154 silver badges273 bronze badges
answered Aug 12 at 20:06
Alexandre EremenkoAlexandre Eremenko
53.8k6 gold badges154 silver badges273 bronze badges
answered Aug 12 at 20:06
Alexandre EremenkoAlexandre Eremenko
53.8k6 gold badges154 silver badges273 bronze badges
answered Aug 12 at 20:06
Alexandre EremenkoAlexandre Eremenko
53.8k6 gold badges154 silver badges273 bronze badges
53.8k6 gold badges154 silver badges273 bronze badges
$begingroup$
This seems to be the kind of thing I'm looking for. Thanks so much!
$endgroup$
– user470881
Aug 12 at 20:34
add a comment |
$begingroup$
This seems to be the kind of thing I'm looking for. Thanks so much!
$endgroup$
– user470881
Aug 12 at 20:34
$begingroup$
This seems to be the kind of thing I'm looking for. Thanks so much!
$endgroup$
– user470881
Aug 12 at 20:34
$begingroup$
This seems to be the kind of thing I'm looking for. Thanks so much!
$endgroup$
– user470881
Aug 12 at 20:34
add a comment |
$begingroup$
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.
answered Aug 12 at 16:35
Igor RivinIgor Rivin
80.5k9 gold badges116 silver badges311 bronze badges
$endgroup$
$begingroup$
Thank you very much!
$endgroup$
– user470881
Aug 12 at 20:35
add a comment |
$begingroup$
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.
answered Aug 12 at 16:35
Igor RivinIgor Rivin
80.5k9 gold badges116 silver badges311 bronze badges
$endgroup$
$begingroup$
Thank you very much!
$endgroup$
– user470881
Aug 12 at 20:35
add a comment |
$begingroup$
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.
answered Aug 12 at 16:35
Igor RivinIgor Rivin
80.5k9 gold badges116 silver badges311 bronze badges
$endgroup$
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.
answered Aug 12 at 16:35
Igor RivinIgor Rivin
80.5k9 gold badges116 silver badges311 bronze badges
answered Aug 12 at 16:35
Igor RivinIgor Rivin
80.5k9 gold badges116 silver badges311 bronze badges
answered Aug 12 at 16:35
Igor RivinIgor Rivin
80.5k9 gold badges116 silver badges311 bronze badges
answered Aug 12 at 16:35
Igor RivinIgor Rivin
80.5k9 gold badges116 silver badges311 bronze badges
80.5k9 gold badges116 silver badges311 bronze badges
$begingroup$
Thank you very much!
$endgroup$
– user470881
Aug 12 at 20:35
add a comment |
$begingroup$
Thank you very much!
$endgroup$
– user470881
Aug 12 at 20:35
$begingroup$
Thank you very much!
$endgroup$
– user470881
Aug 12 at 20:35
$begingroup$
Thank you very much!
$endgroup$
– user470881
Aug 12 at 20:35
add a comment |
$begingroup$
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.
answered Aug 12 at 9:26
Sam NeadSam Nead
11.8k2 gold badges42 silver badges75 bronze badges
$endgroup$
$begingroup$
Thank you very much!
$endgroup$
– user470881
Aug 12 at 20:35
add a comment |
$begingroup$
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.
answered Aug 12 at 9:26
Sam NeadSam Nead
11.8k2 gold badges42 silver badges75 bronze badges
$endgroup$
$begingroup$
Thank you very much!
$endgroup$
– user470881
Aug 12 at 20:35
add a comment |
$begingroup$
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.
answered Aug 12 at 9:26
Sam NeadSam Nead
11.8k2 gold badges42 silver badges75 bronze badges
$endgroup$
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.
answered Aug 12 at 9:26
Sam NeadSam Nead
11.8k2 gold badges42 silver badges75 bronze badges
answered Aug 12 at 9:26
Sam NeadSam Nead
11.8k2 gold badges42 silver badges75 bronze badges
answered Aug 12 at 9:26
Sam NeadSam Nead
11.8k2 gold badges42 silver badges75 bronze badges
answered Aug 12 at 9:26
Sam NeadSam Nead
11.8k2 gold badges42 silver badges75 bronze badges
11.8k2 gold badges42 silver badges75 bronze badges
$begingroup$
Thank you very much!
$endgroup$
– user470881
Aug 12 at 20:35
add a comment |
$begingroup$
Thank you very much!
$endgroup$
– user470881
Aug 12 at 20:35
$begingroup$
Thank you very much!
$endgroup$
– user470881
Aug 12 at 20:35
$begingroup$
Thank you very much!
$endgroup$
– user470881
Aug 12 at 20:35
add a comment |
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