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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Show that $\mathbb{C} \setminus [-1,1]$ and $\mathbb{D}\setminus\{0\}$ are conformally equivalent

So I have found this thread: $\mathbb{C}\setminus[-1,1]$ is conformally equivalent to $\mathbb{E}\setminus\{0\}$ it says that the function $f(z)=\frac{1}{2}(z+\frac{1}{z})$ from $\mathbb{C} \setminus [...
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Image of Upper Unit Semi Circle under Joukowsky Transformation

I'm trying to understand how the Joukowski Transformation would map the following region: $$\{z|0<arg(z)<\pi , |z|<1\}$$ with the Joukowski Transformation being : $w = \frac{1}{2}(z+\frac{1}{...
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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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Derive formula for conformal curvature

So I'm working through Needham's Visual Differential Geometry and Forms, and what is suggested as a simple exercise is to reduce the general Gaussian curvature formula $$K = -\frac{1}{AB}\left(\...
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The relationship between interior and exterior Riemann maps

Let $U\subset\widehat{\mathbb{C}}$ be a compact Jordan domain with $\partial U$ connected. Is there a relationship between the exterior Riemann map associated to $U$, $$\varphi:\mathring{(\mathbb{D}^c)...
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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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Solving Laplace Equation using conformal mapping

I know solution for Laplace Equation in region $A = \{x+iy:0\le y \le1\}$ with boundary conditions $u(x,0) = 0, u(x,1) = 1$ is $u(x,y) = y$. I have been given a new question about a region $B = \{z \...
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Inverse of Conformal Map of Semiinfinite strip to the unit circle, why the inverse doesn't work?

I am dealing with a composition of two conformal maps $\gamma$ and $\omega$ to transform a vertical strip, S = {(x,y) : −π/2 ≤ x ≤ π/2, y ≥ 0}, to the unit circle. The resulting map is, my $\gamma$ ...
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Finding a Möbius transformation, why do I need to expand area?

I'm currently studying for an re-exam in complex analysis, and got a question regarding Möbius transformation. The exam-question is following: Find a conform and bijective mapping from $A := ${$ z: 0 &...
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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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Improving the symmetry of the solution of minimal field equation

Suppose $u$ is a smooth function on $S^2$ and $$ X_1=x_2\frac{\partial}{\partial x_3}-x_3\frac{\partial}{\partial x_2}, X_2=x_3\frac{\partial}{\partial x_1}-x_1\frac{\partial}{\partial x_3},X_3=x_1\...
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How to find conformal maps - Complex analysis [closed]

I would like to know how one can find a conformal map that maps a given set onto another (not $\mathbb D\rightarrow\mathbb D$ or $\mathbb H\rightarrow \mathbb H$ since those are clear). For example, ...
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$ -\Delta_{S^2}\phi=3\frac{1-a^2}{(1-ax_3)^2}\phi$ only admits zero solution.

Let $\phi$ be a smooth solution of the following PDE on $S^2$: $$ -\Delta_{S^2}\phi=3\frac{1-a^2}{(1-ax_3)^2}\phi. $$ Prove that $\phi\equiv 0$. Note that PDE is close to the eigenvalue problem of $-\...
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Axially symmetric solution of a PDE on $S^2$

Suppose $u$ is a smooth solution of the following PDE on $S^2$: $$ -\frac{1}{3}\Delta_{S^2} u+1=e^{2u}. $$ Prove that the PDE only admits a constant solution, i.e., $u\equiv 0$ on $S^2$. With loss ...
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Conformal mappings / multiple transformations

Let find a conformal mapping from the first quadrant $S_1 = \{ x+iy | x>0, y>0 \}$ to the quarter circke disk $M = \{ x+iy | x>0, y>0, x^2+y^2 <1 \}$ $T_1(z) = z^2 $ $T_2(z) = \frac{1+...
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On conformality of a mapping in a region.

My question is that to show the linear fractional transformation $ f(z)=\frac{2z-1}{2-z} $ maps the cicrle $ C:|z|=1$ into itself. Also to prove that $f(z)$ is conformal in $D=\{z:|z|\leq1\}$. First ...
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Change of Variables in Liouville Measure

I am trying to understand the conformal covariance of Liouville measure and have been following this lecture notes. In page 30, under "informal proof", the author wrote: When we use the map ...
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Do these conformal Killing vectors have a name? If not, what should we call them?

I have been investigating conformal Killing vectors on pseudo-Riemannian manifolds, that is, vectors which obey $$ \mathcal{L}_X g = \lambda g$$ where $g$ is the metric and $\lambda$ is some function. ...
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Deriving a formula for Lie algebra of Conformal Field theory

I'm learning some conformal field theory. I'm trying to use the formula $$ \partial_{\mu} \epsilon_\nu+\partial_\nu \epsilon_\mu=\frac{2}{d}(\partial \epsilon) \eta_{\mu \nu} $$ to derive the equation ...
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Isometry in hyperbolic space determined by induced conformal homeomorphsm on the boundary?

The proposition in the book of Introduction to Geometric Topology[Martelli] states that technically, it didn't state there is a one-to-one correspondence between the isometry inside and conformal map ...
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If $X$ is a conformal Killing vector field, what is $\nabla_X \text{Rc}$?

Suppose $$ \nabla_a X_b+ \nabla_b X_a = \Omega g_{ab} $$ for some smooth, positive function $\Omega$ on a (pseudo-)Riemannian manifold $(M, g)$. Can one find a nice expression, independent of $X$, for ...
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Scalar fields with definite weight under conformal maps as sections of some bundle

Let $(M,g)$ be a smooth Riemannian manifold. A scalar field in $(M,g)$ is merely a map $\phi:M\to \mathbb{R}$. Under a diffeomorphism $f:(M,g)\to (M,g)$ it transforms to $\phi' = \phi\circ f$. We then ...
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conformal mapping $\mathbb{C}^{*} \rightarrow \mathbb{C} \backslash \{p\}$

The task was whether $\mathbb{C}^{*}$ can be conformally mapped to $\mathbb{C} \backslash \pm 1$ Then I read that: https://matmor.unam.mx/~robert/cur/2010-1%20CA/CA6.pdf $\mathbb{C}^{*}$ is ...
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Introductory Holography Resources

I'm looking for mathematically rigorous introductory resources (as rigorous as an introduction can be) on the subject of Holographic Duality in physics. I require something that adequately covers Anti-...
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Can any two annuli be sub-conformally mapped to each other?

It's a well known result that two annuli can only be conformally mapped to each other under the condition of the ratio of their inner to outer radii are the same. An explanation of this can be found ...
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Boundary preserving conformal map between surfaces

Consider two compact, simply connected surfaces $D$ and $D'$ in $\mathbb{R}^3$ with non-empty boundaries, and endowed with Riemannian metrics $g$ and $g'$, respectively. Does there exist a conformal ...
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Prove the image of the Joukowski map is a hyperbola, and at which points is it conformal.

Consider the map $$f(z):=\frac{1}{2}(z+\frac{1}{z})$$ and show that $f(B)$ is a hyperbola where $B=\{re^{i \theta}: \text{$r>1$ is variable and $\theta$ fixed}\}$ So I know I have to substitute in ...
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The negative gradient flow of Einstein-Hilbert functional in a fixed conformal class is Yamabe flow

In the book "Hamilton's Ricci flow" written by Bonnett Chow, Peng Lu and Lei Ni, there is an exercise pullzed me: Show that $n \ge 3$, the negative gradient flow of $$E(g)=\int_{\mathcal{M}} ...
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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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Understanding Blaschke factors

It is known that the automorphisms of the unit disk in the complex plane $\mathbb{D}$ = $\{z \in \mathbb{C}:|z| < 1\}$ all take the form: $$\psi_\alpha(z)= e^{i\theta}\frac{z-\alpha}{1-\overline\...
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Applying the Ricci identity on the Cotton tensor

The Schouten tensor is given by $$ P_{i j}=\frac{1}{n-2}\left(R_{i j}-\frac{1}{2(n-2)} R g_{i j}\right).$$ I want to compute the divergence of the Bach tensor in dimension $4$. From this post, we know ...
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Conformal mapping conserving K

In full isometry we have differential lengths and angles between them both conserved for 2D surfaces in $\ \mathbb R^3$. In partial isometry like in case of Chebychev Net "rhombus" lengths ...
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Show that the tranformation $ w = z + \frac{1}{z}$ transform r = constant in the z plane into a family of ellipses in the w plane.

Here we have, $$w = z + \frac{1}{z}$$ $$\therefore u + iv = x + iy + \frac{1}{x + iy}$$ On solving and comparing the real and imaginary parts we get, $$ u = x(1 + \frac{1}{x^2 + y^2})$$ and $$ v = y(1 ...
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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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Three dimensional (pseudo)-Riemannian space with submaximal conformal symmetry

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 ...

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