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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.
11
votes
Accepted
Does every smoothly embedded surface $\mathbb{R}^3$ inherit a natural complex structure, and...
Question 1: Looks good to me.
Question 2: is a duplicate of this question (the answer is: every conformal structure can be so realized).
Question 3/4. There are algorithms to compute the conformal s …
- 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
8
votes
The fundamental group of a closed surface without classification of surfaces?
Yes, the magic words are "The Poincare polygon theorem". For (considerably) more detail, see Fine and Rosenberger "Algebraic generalizations of discrete groups: A path to combinatorial group theory th …
- 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
7
votes
What are the possible automorphism groups of Riemann surfaces of low genus?
The canonical reference on the subject (though one I don't have in front of me) is:
Characters and Automorphism Groups of Compact Riemann Surfaces
Part of London Mathematical Society Lecture Note Ser …
- 96.4k
7
votes
Accepted
Questions on 3-manifolds with a given boundary
When you say "Riemann surface", do you mean "topological surface"? Does the Riemann surface structure have any significance?
I assume below that you mean "two-manifold"
Well, any three manifold c …
- 96.4k
6
votes
Compact surfaces with arbitrary gaps in spectrum
If you are allowed to choose a metric, together with the Riemann surface, this is related to the well-known result of Colin de Verdiere (you can choose the bottom of the spectrum as you like):
MR0932 …
- 96.4k
6
votes
Accepted
Is there a proof of the uniformization theorem using circle packing?
Firstly, the Rodin-Sullivan argument can not, in principle, be used to give a proof of uniformization, since it uses uniformization of domains of infinite type (due to Marden, if I recall) as an ingre …
- 96.4k
6
votes
Accepted
Is the cardinality of occuring torsion subgroups in cofinite lattices in SL(2,R) bounded?
There are triangle groups $(2, 3, n)$ for any $n>6,$ so I would say the answer is NO
- 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
5
votes
Equivalence of Harmonic Maps and Conformal Maps on Genus-0 closed surfaces
MR0164306 (29 #1603) Reviewed
Eells, James, Jr.; Sampson, J. H.
Harmonic mappings of Riemannian manifolds.
Amer. J. Math. 86 1964 109–160.
Example on p. 118
- 96.4k
5
votes
Accepted
Conformal Welding Reference
See MR1966191 (2005e:30012)
Hamilton, D. H.(1-MD)
Conformal welding. Handbook of complex analysis: geometric function theory, Vol. 1, 137–146, North-Holland, Amsterdam, 2002.
30C35
and other paper …
- 96.4k
5
votes
Teichmuller geodesics vs. geodesics in the hyperbolic plane
This subject is addressed in Kasra Rafi's very nice paper (particularly relevant to point 4).
- 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
4
votes
Complete metric on a Riemann surface with punctures
To see how you get complete metrics, you should know enough about hyperbolic geometry to know what an ideal triangle is. Once you do, note that any triangulation can be made of ideal triangles (usuall …
- 96.4k