# Questions tagged [conformal-geometry]

A conformal structure is one that captures the idea of angles, but not lengths. Conformal geometry is the study of geometries with conformal structures, with special focus on transformations that preserves the angles (while possibly changing lengths). Examples of conformal transformations include the Mercator projection from cartography, and the Möbius transformations of the Riemann sphere.

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### Link between Univalence & Conformality

I am familiar with the definition that a holomorphic function is a conformal mapping of some domain if it has non-zero derivative on that domain (as this is an equivalence with angle-preserving). ...
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### Confusion about conformality of Möbius maps

I am aware that Möbius maps are conformal maps, that is, they preserve oriented angles. So I was thinking, say I have a circle in $C$. Then I can find a Möbius map that maps it to a a line (a circle ...
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### Proving that a product of Riemannian Manifolds is locally conformally flat

I like to proof an embedding Theorem for LCF-Manifolds. But im stuck on a little detail. The setting is the following. Given a locally conformally flat Manifold with Boundary. I want to attach a ...
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### Conformal annulus

Given $A(a,b) = \{ z \in \mathbb{C} : a < \left| z \right| < b \}$, with $0<a<b$ then $$A(a,b) \simeq A(a',b')$$ if and only if $a/b= a'/b'$. I don't get some parts of the following ...
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### Finding conformal map verification

I would like to know if this reasoning is correct: I want to find a conformal map from $\mathbb D$ to $\mathbb C$. My reasoning: First we seek a conformal map from $\mathbb H$ to $\mathbb D$ which is ...
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### conformal map of a sector to the right half plane

I have a sector $S_{\alpha} = \{ z \in \mathbb C \ | \ |arg(z)| \lt \alpha \}$ with $0 \lt \alpha \leq \pi$. My question is: Does the function $\phi: z \to z^2$ map the sector $S_\alpha$ to the right ...
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### conformal mapping from $0 \lt |arg(z)| \lt \pi$ to $B(0,1)$

Let $S_{\alpha} = \{ z \in \mathbb C \ | \ |arg(z)| \lt \alpha \}$ with $0 \lt \alpha \leq \pi$. I have to find a conformal mapping $\phi: S_{\alpha} \to B(0,1)$. What would be a good approach to ...
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### How to prove that Linear-Fractional Transformations map circles/straight lines onto circles/straight lines?

There this theorem that states the Linear Fractional Transformations map circles and straight lines to circles and straight lines. How can you prove this? This is what the theorem states in my lecture ...
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### Image of Half-Strip under mapping $w=\cosh (z)$

I'm trying to get the image of a half-strip under the conformal mapping $w=\cosh (z)$. Here is the half-strip: $\{z=x+iy | x>0, 0<y<\pi\}$ and here is what I have attempted: Start by setting ...
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### conformal mapping and bilinear transformation

I've got the next problem in complex analysis: find the bilinear transformation from A to B: $${A = [z: \text{Re } z \ge-1 ]},\quad B = [w: |w|\ge1]$$ first, I tried to determine what each of the ...
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### Invariance of the Poisson integral under inversions

Let us denote the Poisson kernel on $B_r$ by $$P(x,\zeta)=\frac{r^2-\vert x \vert^2}{r\omega_{n-1}\vert x-\zeta \vert^n}$$ where $x\in B_r$ and $\zeta\in\partial B_r$. Given a boundary function $f$ ...
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### How to prove that the Douady-Hubbard conformal map from the exterior of Mandelbrot Set to exterior of unit disc is actually holomorphic?

I was reading the Orsay Notes on Exploring the Mandelbrot Set. (https://pi.math.cornell.edu/~hubbard/OrsayEnglish.pdf) On Page 64, it is proven that the Mandelbrot Set is connected. I understood the ...
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### How exactly can we write an incomplete elliptic integral of the first kind as a sum of real and imaginary parts?

All I want to do is numerically map the upper-half plane $\mathbb H:=\{z\in\mathbb C:\Im(z)>0\}$ to the unit square $[0,1)^2$. How this can be done is described in the Wikipedia article of the ...
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### Constructing conformal mapping from union of three overlapping discs [closed]

How can I construct a conformal mapping from the union of three discs $$D = \{|z-i|<\sqrt 2\} \cup \{|z+i|<\sqrt 2\} \cup \{|z-\sqrt 2|<1\}$$ to the upper half plane? Can someone give me a ...
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### Constructing conformal mapping from $\mathbb{C} \setminus \{ \text{2 discs plus a line segment} \}$ [closed]

Find a conformal mapping of the region $D=\{z:|z−1|>\sqrt{2},|z+1|>\sqrt{2}\}$ minus strip [i,2i] onto upper half plane. I thought we should use 1/z, but when I do use it I got ellipse-like ...
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### How do we solve this rather simple ODE (Loewner equation with driving function $\sqrt t$)?

Remember the following result for the Loewner equation: If $\lambda:[0,\infty)\to\mathbb R$ is continuous, then for all $z\in\mathbb C\setminus\{\lambda(0)\}$ there is a uniqe $\zeta(z)\in(0,\infty]$ ...
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### Equivalent definitions for conformal map

I've encountered several definitions for conformal maps, and I was wondering whether they are equivalent or not. My goal is understanding how the concepts of conformality and holomorphy relate to each ...
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### How can we numerically simulate a Schramm-Loewner evolution?

I would like numerically simulate the curve generated by a Schramm–Loewner evolution, but there are two issues which I'm not able to resolve. Firs of all, let me recapulate the following result of the ...
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### When most generally are maps $\mathbb{R}^2 \rightarrow \mathbb{R}^2$ conformal?

I'm interested in continuous conformal maps $\mathbb{R}^2 \rightarrow \mathbb{R}^2$ that are ONLY angle preserving (not necessarily orientation preserving). I would like to write down a generic system ...
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### Equal area map from sphere to hemisphere

The Mercator projection is the unique conformal projection which maps parallels to horizontal lines and meridians to vertical lines. There is an interesting hemispheric analogue of this, which maps ...
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We know that spaces with maximal conformal symmetry are conformally flat, and the dimension of the Lie algebra of its conformal Killing vectors is $\frac{(n+1)(n+2)}{2}$, where $n$ is the dimension of ...