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Results tagged with complex-geometry
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user 25510
Complex geometry is the study of complex manifolds, complex algebraic varieties, complex analytic spaces, and, by extension, of almost complex structures. It is a part of differential geometry, algebraic geometry and analytic geometry.
32
votes
How would a topologist explain "every Riemann surface of genus $g$ is hyperelliptic if and o...
A 19th century topologist would explain this by dimension count. By Riemann-Hurwitz, a surface of genus $g$ covering the sphere
with $2$ sheets has $2g+2$ ramification points which gives $2g-1$ free c …
- 91.8k
22
votes
Accepted
Poincaré metric on the Riemann sphere minus more than two points
Yes. The density of the Poincare metric with respect to the spherical metric is
a positive continuous function which tends to infinity at the punctures. Thus it
is bounded from below by some positive …
- 91.8k
17
votes
Accepted
A "holomorphic" Peano curve?
Here it is:
MR0015154
Salem, R.; Zygmund, A.
Lacunary power series and Peano curves.
Duke Math. J. 12, (1945). 569–578.
- 91.8k
16
votes
Embed a bordered Riemann surface into punctured Riemann surfaces?
The answer is no. For $g=0$, the problem is equivalent to the following: is there a univalent function in the unit disk which takes given values at finitely many given points. It is well known that th …
- 91.8k
15
votes
Accepted
Interesting results for open Riemann surfaces
The results on open Riemann surfaces are not "rare". They are just well forgotten.
I only list a few books which deal with open Riemann surfaces:
MR0114911 (Zbl 0196.33801), MR0228671 (Zbl 0152.27401) …
- 91.8k
15
votes
Are real-analytic functions in $\mathbb{R}^2$ holomorphic after suitable change of coordinates?
The answer is negative. For the non-injective case, the reason is that non-constant complex analytic functions are open, discrete maps, while real analytic functions can be neither open nor discrete ( …
- 91.8k
13
votes
Accepted
A harmonic function
Yes, it can be found explicitly, though not in elementary functions but in terms of a combination of elementary and hypergeometric functions.
The problem is (almost) equivalent to finding a conformal …
- 91.8k
13
votes
Accepted
Spicing up Riemann surfaces course (revised)
Forster just touches the Riemann-Hilbert problem and fiber bundles. Expansion on this can be interesting
I recommend the books of Bolibrukh.
Applications of compact Riemann surfaces to solitons ("Ex …
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13
votes
Accepted
Gluing Riemann surfaces
The answer to the first question is "yes". This is called conformal gluing, and the proof is based on the following lemma due to Lavrentiev: Let $\phi$ be an increasing diffeomorphism of $[-1,1]$ onto …
- 91.8k
12
votes
Accepted
Universal covering map from $\mathcal{H}$ to $\mathbb{C}\setminus \mathbb{Z}\oplus i\mathbb{...
On the first question (the universal cover of the complement of a lattice). The missing points are in the image, so it is not the map that "behaves" but the inverse map.
The inverse map behaves in a v …
- 91.8k
12
votes
Accepted
The behaviour of holomorphic mapping of curves
You are asking too many questions, some of them are very difficult.
Here are some answers.
Image of a Jordan curve under a rational function. Take a circle for $\gamma$.
Every continuous function o …
- 91.8k
11
votes
Accepted
The holomorphic version of Galois theory
The thing you are asking was much studied in connection with Hilbert Problem 13.
The roots of a polynomial of degree exactly $d$ form an unordered $d$-tuple. The set of
unordered $d$-tuples is called …
- 91.8k
11
votes
Accepted
why do we need to study entire curves?
A good reference is S. Lang, Introduction to complex hyperbolic geometry.
Shortly, I can give the following reasons. Listed in historical order.
There are famous Picard theorems about entire curves …
- 91.8k
11
votes
Accepted
Three questions about three functions similar to $\sin,\cos$
There is no addition formula: functions satisfying an algebraic addition formula have been completely characterized,
Painlevé, P.
Sur les fonctions qui admettent un théorème d’addition, Acta Math. 27, …
- 91.8k
9
votes
Accepted
Reference request: uniformization theorem
On a basic level:
W. Abikoff, The uniformization theorem, Amer. Math. Monthly 88 (1981), no. 8, 574–592.
L. Ahlfors, Conformal invariants, last chapter.
S. Donaldson, Riemann surfaces, Oxford, 2011. V …
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