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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How to find conformal mapping?

I know how to find conformal mapping but here I am a little confused that how can I estimate conformal mapping from the given range and domain information? A proper guide will be appreciated what ...
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mapping complement of half disk to unit disk

Let $K=\{ Im z \geq 0, |z| \leq 1 \}$ be the closed set of the intersection of the closed disk with upper half plane. I need to find a conformal map that takes complement of $K$, i.e. $\mathbb{C} - K$ ...
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conformal map from a disk minus a radial segment to unit disk

I need to find a conformal map from the disk minus a radial segment $\Omega$ to the unit disk $\mathbb{D}$, where $\Omega = \{ z \in \mathbb{C}: |z|<1, z \notin [\frac{1}{2},1) \}$. I have tried ...
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conformal map $\{ Re(z) > 0 : |z-1|>1 \}$ onto unit disk

How many conformal mapping we apply to map the exterior of the disk $\{z : |z-1|>1 \}$ in the right half plane to the unit disk?
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On the proof about the dimension of the conformal group of a manifold

I have been reading the book "Transformation Groups in Differential Geometry" by S. Kobayashi. More concretely, I am trying to understand the proof of the Theorem 6.1 of Chapter IV. Theorem ...
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Finding the explicit expression of a conformal map

I want to find a conformal map $f$ between $D = \{z\in\Bbb C : |Arg(z)|<\frac \pi4\}\setminus[1,+\infty)$ and $D(0,R)$, for $R$ a real positive number that I have to compute as well. It also has to ...
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Schwarz Lemma application problem

Schwarz lemma says that "Let ${\displaystyle \mathbf {D} =\{z:|z|<1\}}$ be the open unit disk in the complex plane ${\displaystyle \mathbb {C} }$ centered at the origin, and let ${\...
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Conformal Killing Fields on Curved Manifolds

Introduction A conformal Killing vector field $\xi^\mu$ is one that satisfies $$\mathcal{L}_\xi g_{\mu\nu}\equiv\nabla_\mu\xi_\nu+\nabla_\nu\xi_\mu=\kappa g_{\mu\nu}$$ for some scalar field $\kappa$. ...
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Conformal map from unit disk to strip $\{z \in \mathbb{C} \mid |\operatorname{Im}z| < \pi/2 \}$ satisfying $f(0) = 0$ and $f'(0) = 2$

I have the following problem: Let $D \subset \mathbb{C}$ denote the unit disk and let $S$ denote the strip $\{z \in \mathbb{C} : |\operatorname{Im}z| < \pi/2\}$. Find a conformal map $f$ from $D$ ...
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Holomorphic function not conformal?

Is it true that holomorphic functions are not conformal? By my understanding a holomorphic function is angle-preserving. But if I plot the lines $t+i$ and $1+it$ under the map $x \mapsto x^2$ then the ...
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Reference for a Liouville type theorem

Does anyone know a reference for the following theorem? Let $U\subseteq \mathbb{R}^n$ be a domain and $u: U\rightarrow \mathbb{R}^n$ a differentiable mapping such that $Du(x) \in SO(n)$ for all $x\in ...
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Schwarz-Christoffel formula for a half-plane

I can't understand the example that was given in the book Schwarz-Christoffel Mapping by Tobin Driscoll and Lloyd Trefethen. It's formula 2.5 at page 12. By using Schwarz–Christoffel formula for a ...
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Conformal maps of the three-torus

I'm interested in conformal maps of the three-torus $\mathbb T^3$, or (I think relatedly) of $\mathbb R^2 \times S^1$. (Of course if you allow the diameters of $\mathbb T^3$ to be different, then, $\...
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Renormalizing continuous curve via conformal transformation

Let $\gamma:[0,\infty) \to \bar{\mathbb{H}}$ continuous curve with $\gamma (0)=0$ and $\gamma (0, \infty) \subset \mathbb{H}$ where $\mathbb{H} = \{z \in \mathbb{C}: \Im (z) > 0 \}$. The curve $\...
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Show that a torus is conformally equivalent to a plane

With the metric of the torus given by $ds^2=a^2d\theta^2+(b+asin\theta)^2d\phi^2$, I'm asked to find the conformal transformation which proves that a torus is conformally equivalent to a plane. I must ...
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Is there an analytic function whose imaginary (or real) part 1) equals zero on the real axis 2) equals 1 on the imaginary axis?

The boundary conditions must only be satisfied in the first quadrant. I don't know how to even start to solve this problem, since non of my originally thought simple analytic functions satisfied these ...
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Derivation for finding the coefficients of a Möbius transformation defined by three points

Somewhere on the internet (I don't quite remember where) I came across the following formulation for finding the four parameters $a$, $b$, $c$, and $d$ of a Möbius transformation given three points $...
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Why does $w(z)=z^2$ represent a conformal map from the $z$ plane to the $w$ plane?

Doesn't every straight line $y=kx$ in the $z$-plane become a straight line $u=v\cdot2m/(1-m^2)$ in the $w$-plane such that each point having a phase angle $\theta$ in the $z$-plane has a phase angle $...
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CGA Representation of an Ellipse

In conformal geometric algebra (CGA), I know the representation of a circle $s$ with center $c$ is: $$ S = H(c) - \frac{1}{2}r^2n_{\infty} $$ That $H(c)$ is the representation of $c$ (i.e., $n_0 + c +‌...
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Mobius transformation sending spheres to concentric spheres

In the complex plane it is known that given two disjoint circles we can find a Mobius transformation that maps them to concentric circles. My question: Given two disjoint $(n-1)$-spheres $Y$ and $Z$ ...
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The Cross Ratio of a Mobius Transformation - Definition Clarification

so the Mobius transformation $f(z)$ of an extended complex plane preserves the following cross ratio $$\frac{(w_1-w_4)(w_3-w_2)}{(w_1-w_2)(w_3-w_4)}=\frac{(z_1-z_4)(z_3-z_2)}{(z_1-z_2)(z_3-z_4)}$$ ...
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Intuition/references for why Möbius transformations are uniquely defined by three points?

Assume that I have a Möbius transformation $$M(z)=\frac{az+b}{cz+d}$$ with real coefficients $a$, $b$, $c$, and $d$, and complex argument $z$. I have been interested in gaining intuition on why this ...
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Solutions of Cauchy-Riemann equations have limit at infinity

Let $f:\mathbb{R}^n\to \mathbb{R}^n$, $n\geq 3$ be a $\mathcal{C}^3$, sense preserving function which is a solution to the Cauchy-Riemann system $$Df^T(x)Df(x)=J_f^{2/n}(x)I,$$ where $Df$ is the ...
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Showing that Mobius Transformation is a Conformal Mapping.

I just want to check that my understanding and derivations are correct (some details are omitted). So, let's define a map $f:\hat{\mathbb{C}}\to\hat{\mathbb{C}}$ given by $f(z)=\frac{a+bz}{c+dz}$ ...
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I need help in a mapping problem

I have a problem that I'm having trouble understanding it. In the book it says to find the inverse Laplace transform of the function $$\mathcal L^{-1}\left(\frac{1}{s(1+e^{as})}\right)$$. In order to ...
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How to relate these two solutions to the special conformal scalar?

Given the transformation $T_\lambda (z)$ defined by $$ z\mapsto z'=T_\lambda (z)=\frac{z}{1+i\lambda z} $$ We want a function $u(z,\bar z)$ such that $u(z,\bar z)=u(z',\bar {z}')$. On Taylor Series, $$...
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Visualizing conformal map from unit disk to upper half plane

I understand that $$w=\frac{z-i}{z+i}$$ maps the open upper half plane $\mathbb{H}$ to the open unit disk $\mathbb{D}$, so $$z=i\frac{1+w}{1-w}$$ maps from $\mathbb{D}$ to $\mathbb{H}$. My question is ...
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Generalizing Contour Integration to Quaternions

I have recently entertained the possibility of defining complex contour integration for the quaternions. I am somewhat aware that the Frobenius theorem dictates that no division algebra can exist in $\...
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Show that the map $w(z)=\frac{1}{1-z^2}$ is a Conformal Mapping..

Show that the mapping $$w(z)=\frac{1}{1-z^2}$$ is a Conformal Mapping. My try- $$w(z)=\frac{1}{1-z^2}$$ $$ w'(z)=\frac{2z}{(1-z^2)^2}$$ $$ w'(z)\neq 0 \ \forall\ z\neq 0$$ So $w(z)$ is a Conformal ...
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Trouble finding conformal maps between these 2 areas

I have been trying my complex analysis assignment and couldn't solve these type of questions. So, I discussed with friends and they are also struggling with it. Assignmnet discussion by prof. is ...
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Find the points at which the mapping defined by $f(z)= nz+ z^n (n\in \mathbb{N})$ is not conformal

The following question is from Complex variables with applications by ponnusamy and silvermann and I had earlier tried some problems and I am asking some of them which I couldn't solve. Find the ...
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Question 11.8 Section Conformal mappings ponnusamy and silvermann

I am not able to solve the following question and so I am asking for help here. Theorem 11.2 : If f(z) is analytic at $z_0$ with $f(z_0) \neq 0$ , then f(z) is 1-1 in some nbd of $z_0$. Question : ...
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Stereographic Projection: Does Conformality Imply Circle Preservation?

This may be a silly question. I have learned to prove that circles are preserved at the infinitesimal scale, however, does this ALONE imply that circles are mapped as circles for the stereographic ...
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Projecting Sphere to Rectangle/Square using Mercerator Projection

I am learning some projection technique where we can project a 3d object like globe to a 2d. I have the 3d coordinates of points on the surface of sphere same as globe. Here is a reference where a ...
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Is there an analogue of the moduli space of the torus in semi-Riemannian signature?

I'm starting to study Riemann surfaces and already met the fact that Riemann surfaces have both a complex structure and a conformal structure that are, in fact, very closely related. If we consider a ...
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Is every topological three-sphere conformal to the unit three-sphere?

Let's suppose we have some space M that is topologically a three-sphere (but arbitrary in specifics) with metric $g_{\mu\nu}$. The affirmative solution to the Yamabe problem leads me to believe that ...
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How to prove Sphere $S^3$ minus point is conformal to submanifold with boundary of $\mathbb R\times (M,g)$?

How to prove Sphere $S^3$ minus point is conformal to submanifold with boundary of $\mathbb R\times (M,g)$ with $M$ is 2 dimensional compact simple manifold ? Definition: A manifold $(M,g)$ with ...
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Does there exist a conformal mapping that transforms a Euclidean Surface into a hyperbolic tessellation?

I'm quite new to the idea of conformal mappings, and having a slightly fundamental understanding of hyperbolic grids and tesselations I thought that it may be possible to transform a Euclidean grid or ...
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Does every conformal structure on a torus come from one of these specific embeddings in $\mathbb R^3$?

In the answers to this question: https://mathoverflow.net/questions/53999/ they say that every Riemann surface can be realized as a smooth surface embedded in $\mathbb R^3$. Question: Does every ...
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What is the image of y>0 under the map $w= e^{z}$

What is the image of y>0 under the map $w= e^{z}$ It is and entire circunference of radius 1? or only half circunference of radius 1? I see this video https://www.youtube.com/watch?v=d18M390rMaI ...
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How do we find the range of $z^z, \: z \in \mathbb{C}$?

I was recently exploring domain colouring, when the transformation $z \mapsto z^z$ changed the complex plane thus: This made me wonder how we could mathematically describe the coloured 'boomerang' ...
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Scale Invariance of the Einstein-Hilbert Functional

Define the normalised Einstein Hilbert functional for an n-dimensional Psuedo-Riemannian manifold ($M, g$) as $\mathcal{E}(g) = \frac{{\int_{\mathcal{M}}S_g \Omega_g}}{\left({\int_{\mathcal{M}} \...
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Area of Domain as.Sum of Power Series [duplicate]

Suppose that f is a one-to-one analytic function mapping the disc $|z|<1$ onto a bounded domain D. Show that the area of D is given by $$A(D)=\pi \sum_{n=1}^{\infty} n|a_n|^2$$, where $\sum_{n=0}^{\...
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Area of domain of a complex variable function

Suppose that f is a one-to-one analytic function mapping the disc $|z|<1$ onto a bounded domain D. Show that the area of D is given by $$A(D)=\int \int_{|z|<1} {|f'(z)|}^2 dxdy$$ This is a ...
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What are the conformal killing vector fields on $S^2$ equipped with the round metric?

What are the conformal killing vector fields on $S^2$ equipped with the round metric? (expressed in the standard coordinates $(\theta, \phi)$) The space of conformal killing vector fields make a 6 ...
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Conformal maps and möbius transformations

Is it possible to find a möbius transformation mapping $\mathbb{C}$ to the unit disc $D(0,1)$ my attempt: I think that since $\mathbb{C}$ isn't bounded by any circlines but the unit disc is bounded by ...
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Is the structure of “solid angle” conformal maps more interesting in higher dimensions than just conformal maps?

In the complex plane there exists a very large set of smooth conformal maps, (The famous Riemann Mapping Theorem states that between any two simply connected open sets in $\mathbb{C}$ there exists a ...
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Generators in the sense of the theory of canonical transformations

I was studying this paper 1 and I’m stuck at eq.13. I have really no idea why they wrote that equation. I’m especially confused about $K=pq$ and $X=q^2$. Maybe I have to study something about that, ...
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Conformal Mapping with Neuman Conditions

I'm trying to solve the added question with Neumann conditions instead of Dirichlet. Also, I could not figure it out how to obtain $x^2$ at (4). I would be pleased if anyone could help. Thank you.
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Special conformal transformation scaling factor and determinate

From Conformal Field Theory, Francesco, Mathieu, Senechal. The special conformal transformation (Eq.4.15) $$x'^\mu =\frac{x^\mu-b^\mu \vec{x}^2}{1-2\vec{b}\cdot \vec{x}+\vec{b}^2 \vec{x}^2}$$ Show ...

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