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Results tagged with riemann-surfaces
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user 11142
Riemann surfaces(Riemannian surfaces) is one dimensional complex manifold. For questions about classical examples in complex analysis, complex geometry, surface topology.
4
votes
Closed simple curves in $\mathbb{R}\mathbb{P}^2$
Well, if you take the double cover, under your assumptions the lift is two simple closed curves in $S^2,$ the complement of which will be two disks and an annulus, so the original curve bounds a disk …
- 96.4k
6
votes
Simple Closed Hyperbolic Geodesics on Punctured Spheres
Unless I misunderstand the question, the answer is no. The homotopy class of the circle is determined by the partition it determines on the set of marked points, so there are only finitely many homoto …
- 96.4k
3
votes
Automorphisms of genus 6 surfaces
The question is purely combinatorial - the Hurwitz bound comes from the observation that the quotient of a surface by its automorphism group is a hyperbolic orbifold, and the hyperbolic orbifold of sm …
- 96.4k
3
votes
On determinants of Laplacians on Riemann surfaces
A fairly complete treatment is given by Christian Grosche:
Grosche, C., Path integrals, hyperbolic spaces and Selberg trace formulae, Singapore: World Scientific. xi, 280 p. (1996). ZBL0883.58003.
( …
- 96.4k
1
vote
Mapping-Class Groups of Subsurfaces of a Hyperbolic Surface
I don't really understand the question, perhaps, but if the homeomorphism fixes the boundary, you can extend it by identity to the rest of the surface. This seems to be a homomorphism. Having it be in …
- 96.4k
8
votes
How can I calculate the period matrix of this Riemann surface?
You can gain infinite enlightenment by reading the very cool paper:
Gianni, Patrizia; Seppälä, Mika; Silhol, Robert; Trager, Barry, Riemann surfaces, plane algebraic curves and their period matrices, …
- 96.4k
2
votes
Riemann Theta Function On Hyperbolic Riemann Surfaces
To amplify on ElucisusFTW's answer, the relationship between the hyperbolic metric and the complex structure is the so-called "accessory parameters" problem, and very little is known about it, except …
- 96.4k
3
votes
Can the limit set of an infinitely generated Schottky group have positive area?
Yes. This is due Z.-X. He, and in greater generality to Rich Schwartz, see
Schwartz, Richard, The limit sets of some infinitely generated Schottky groups, Trans. Am. Math. Soc. 335, No.2, 865-875 (19 …
- 96.4k
2
votes
Can the limit set of an infinitely generated Schottky group have positive area?
Almost surely such examples are constructed by Stratmann and Urbanski:
Stratmann, Bernd O.; Urba\'nski, Mariusz, Pseudo-Markov systems and infinitely generated Schottky groups, Am. J. Math. 129, No. …
- 96.4k
4
votes
Elementary Proof of Riemann-Roch for Compact Riemann Surfaces
There is a U of Chicago REU by a Valeriya Talovikova which does everything from the beginning in 10 pages.
- 96.4k
2
votes
Examples of discrete subgroups of $PSL_2(\mathbf{R})$ with finite covolume and which are not...
For explicit examples of such groups see Louis Funar's notes, page 5 (these are Hecke groups, mentioned by Anton in his comment).
- 96.4k
1
vote
Does there exist a non-hyperelliptic Riemann surface with automorphism group $C_2\times A_4$?
There is a non-hyperelliptic (fixed-point free) involution of the surface of genus $5,$ with the quotient a surface of genus $3.$ Further, $A_4$ does come up as the automorphism group thereof, see
S. …
CommunityBot
- 1
2
votes
Accepted
Is every closed Sasakian 3-manifold a circle bundle on a Riemann surface?
A complete topological classification is due to Geiges, and can be found in this 2001 paper by Guilfoyle. (the first Theorem in the paper).
- 96.4k
2
votes
Classification of open subset of $\mathbb{R}^{3}$
The result in two dimensions the OP is thinking of is Grotsch's uniformization by slit domains (I think Grotsch only did finitely many boundary components). Koebe has conjectured that any planar domai …
- 96.4k
10
votes
Accepted
Frucht's type theorem for Riemann surface
You seem to be asking about the group of isometries, not the fundamental group. If so, for every $n$ and every finite group $G$ there is a compact hyperbolic manifold of dimension $n$ whose isometry g …
- 96.4k