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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.
8
votes
A question in Möbius geometry
You seem to imply that the book from which you study does not DEFINE what a Möbius transformation is. Then discard this book and choose another one. Unfortunately the terminology is not settled. The s …
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
2
votes
Branched covering maps between Riemann surfaces
For infinite degree, the definition of "branched covering" can be somewhat ambiguous. But
$$z\mapsto \cos z: \mathbb{C}\to \mathbb{C}$$
$$\wp: \mathbb{C}\to S$$ are a simple examples of branched cove …
- 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
6
votes
Accepted
Holomorphic Gauss normal map
Gauss map is holomorphic (as a map to the Riemann sphere)
if the surface is minimal. This is Lemma 8.3 in the book of Osserman, A Survey of Minimal Surfaces. In fact, if you replace "embedded" by "imm …
- 91.8k
2
votes
Accepted
Existence of covering isomorphism
I suppose that "non-compact complex algebraic curve" means complex affine curve.
The following counterexample was proposed by my friend Fedor Pakovich.
Let $D=\mathbf{C}\backslash\{-1,1\}$.
Consider t …
- 91.8k
6
votes
An inf-sup estimate for holomorphic functions
This is not true: take $n=1$, $r=1$, $\eta(z)=e^{az},\; a>0,$
then
$$\max_{z\in B(0,r)}|\eta(z)|=e^a,$$
while
$$\min_{z\in B(0,1)}|\eta(z)|=e^{-a}.$$
Since $a>0$ is arbitrary, no $\kappa$ with require …
- 91.8k
4
votes
Related to the Schwarz Christoffel map
Take a curvilinear quadrilateral $A',B',C',D'$ satisfying all your conditions, that is $[A',B'],[C',D']$ are disjoint arcs
of the unit circle, while the other sides do not belong to the unit circle. Y …
- 91.8k
1
vote
Proof verification for a theorem about a harmonic function on the unit disc
This is a consequence of the Phragmen-Lindelof Principle, whose general formulation is the following: Let $D$ be a bounded region
in the plane, and $\zeta_0\in\partial D$. Let $u$ be a subharmonic fun …
- 91.8k
7
votes
Geometry of critical points of holomorphic maps in projective space
For n=1, every point in the critical divisor has degree $≤d-1$, where $d$ is the degree of the map, and the total degree of the critical divisor is $2d−2$, and any such divisor can occur.
To state it …
- 91.8k
4
votes
Accepted
Practically calculating the domain of a power series for function of several complex variables
The usual Cauchy-Hadamard formula has a generalization to several variables.
The numbers $r_1,\ldots,r_n$ are called conjugate radii of convergence if the
series converges in the open polydisk $B(r_1, …
- 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, …
CommunityBot
- 1
5
votes
What are parabolic bundles good for?
They arise in analytic theory of differential equations with regular singularities, Riemann Hilbert problem and Painleve equations.
MR1924757
Biswas, Indranil
A criterion for the existence of a flat c …
- 4,713
8
votes
Accepted
Ahlfors' proof of Bloch's theorem
He explains his choice in lines 8-9 on p. 364 of the paper:
"This metric has the curvature $-4$ for it is obtained from the hyperbolic metric by the transformation $w'=w^{1/2}$ ."
Remarks. By more sop …
- 91.8k
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 …
- 105k