All Questions
9 questions
2
votes
3
answers
478
views
Groups of conformal isomorphisms of simply connected surfaces
By the uniformization theorem every connected and simply connected surface $M$ is conformally equivalent to one of the following three surfaces:
open disk $D$, complex plane $\mathbb{C}$, or $2$-...
3
votes
1
answer
468
views
Relationship between two kinds of classifications of Riemann surfaces
There are two kinds of classifications of Riemann surfaces.
Classification 1: Let $M$ be a Riemann surface. We will call $M$:
elliptic iff $M$ is compact (= closed);
parabolic iff $M$ is not compact ...
- 438
2
votes
0
answers
358
views
Triangulating Riemann surfaces by using non-constant meromorphic functions
Let $X$ be a connected Riemann surface, i.e. $X$ is a one dimensional connected complex manifold (Hausdorff and second-countable as a topological space). The following is a classical result:
Theorem (...
- 395
9
votes
1
answer
321
views
Notational question about quadratic differentials in Strebel's book "Quadratic differentials"
In Kurt Strebel's book "Quadratic Differentials", in Chapter 2, $\S4$, he begins by saying:
"Every analytic function $\varphi$ is a domain $G$ of the $z$-plane defines, in a natural way, a field of ...
- 7,527
7
votes
2
answers
813
views
Criterion for deciding the conformal class of a metric on a complete surface
For orientable closed Riemannian surfaces $(S,g)$, there is a constant curvature metric $\overline{g}$ on $S$ that is conformal to $g$ in the sense that $\overline{g} = e^ug$ for some smooth function $...
- 173
1
vote
0
answers
215
views
Coordinate charts on converging Riemann surfaces
Let $S$ be a $2-$dim manifold and $q \in S$. Furthermore, let $j_{n}$ be a sequence of complex structures on $S$ converging in $C^{\infty}_{\text{loc}}$ to a complex structure $j$ on $S$ as $n\...
- 11
1
vote
1
answer
386
views
existence of positive curved line bundles on a compact Riemann surface
Can anyone suggest a proof of the existence of positive line bundles on a compact Riemann surface, avoiding Hodge decomposition. (I am aware of the method in Dror Varolin's book, but I consider that ...
- 2,106
4
votes
1
answer
505
views
Characterization of the moduli space of the pair of pants in terms of the modules of the extremal ring domains
Hi, I was thinking about the following question ; I will appreciate it if somebody can give me a full or partial answer or can at least cite any reference(s)/ papers etc :
By $ \bar{P} $ , we ...
- 1,471
3
votes
6
answers
1k
views
Dolbeault cohomology
Hello
I am trying to get a good book that explains the Dolbeault Cohomology, does anyone know of a good one?
- 143