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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 ...

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Finding a Particular Möbius Transformation from $D \to D$

I'm well aware that the Möbius transformations that take the unit disk to itself, $f:D \to D$, are given by $$ f(z) = \frac{e^{i \theta}(z-\alpha)}{1-\bar{\alpha}z}, $$ where $\theta\in [0,2\pi)$ ...
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Checking whether $w = f(z)$ is conformal and its mapping?

I was thinking about the conformal map $w =f(z)$ in which $f$ satisfies $w = f(z) = \frac{df}{dz} - e^z$. Is ths conformal, how can i find the conformal mapping of $z$ plane? Similarly what will be ...
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How to see the image of the conformal map tangent?

Prove that : $w=\text{tan}(z)$ map the region $\{-\frac{\pi}{2}<Rez<\frac{\pi}{2} \}$ to the region $w=u+iv$, where two rays $\{ u=0,|v|\geq 1\}$ are excluded. I know this conformal map can be ...
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Riemann mapping theorem. Why using rotations?

When proofing the Riemann mapping theorem that every simply connected domain $\mathbb{C} \neq D \subset \mathbb{C}$ is conformal equivalent to the unit disk $\mathbb{D}$, we can construct a Möbius ...
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Conformal mapping of periodic bladed boundary

I am new to conformal mapping and I am trying to find the mapping of the vertical boundary with tilted cuts to a horizontal boundary with perpendicular cuts (See the image). Any suggestion is ...
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Fitting a conformal / holomorphic function

Suppose we have some 2D points $x_i$ (which we may take to be complex numbers) and some corresponding 2D points $y_i$. We seek a function $f:\mathbb{C}\rightarrow \mathbb{C}$ such that $f(x_i)\approx ...
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Conformal mapping interpretation

It says on Alfors, Complex Analysis page 73, chapter 2.3 that: Suppose that an arc $\gamma$ with the equation $z = z(t), \alpha \leq t \leq \beta$, is contained ina region $\omega$, and let $f(z)$ be ...
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Notation regarding generalized Minkowski space

In section 12 of the book Surfaces in classical geometries: A treatment by moving frames by Gary R. Jensen, Emilio Musso and Lorenzo Nicolodi (see preview here), Möbius geometry is described. They ...
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Why does orientation preserving maps respect CR?

From this post it is written that "orientation preserving conformal maps respect CR. The matrix of the map must be a constant multiplied by some matrix of rotation that has a positive determinant. ...
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Conformal compactification of spacetimes

I’m looking answers and/or references concerning the following questions about conformal compactification. Given a $d$-dimensional spacetime $(M,g)$ (or simply just for $d=4$): Does the conformal ...
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Transformation law for the Ricci curvature

In 4 dimensions, for a conformal change of metric $g=e^{2u}g_0$ the Ricci curvature tensor $\operatorname{Ric}$ satisfies the transformation law \begin{equation}\tag{1} \operatorname{Ric}_g = \...
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Conformal energy and first fundamental form

In my reference in mesh processing I came across the following equation Where $\bf{I}$ is the first fundametal form of some surface patch $\bf{x}$, I don't know the exact meaning of the notation $\bf{...
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Circles Riemann sphere

I was able to prove that the circle passing through the north pole of the Riemann sphere is mapped into a straight line. How can I prove that the circle on the Riemann sphere corresponds to a circle ...
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Are conformal maps between Riemannian manifolds real-analytic?

Let $M,N$ be oriented smooth ($C^{\infty}$) $n$-dimensional Riemannian manifolds, and let $f:M \to N$ be a smooth orientation-preserving weakly* conformal map. Do there exist real-analytic ...
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Inversion in a sphere preserves circles (proof)

Inversion in the unit sphere, for a vector $x$, is defined by $$\frac1x = \frac x{x^2} = \Big(\frac1{x\cdot x}\Big)x$$ How can we prove that a circle's inversion in the sphere is also a circle? (I ...
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Is $\frac{w-1}{w+1}$ a conformal mapping from $\mathbb{C}\setminus(-\infty,0]$ onto $\mathbb{C}\setminus\big((-\infty,-1]\cup[1,+\infty)\big)$?

Is the meromorphic function $g$ defined by $g(w) := \frac{w-1}{w+1}$ a conformal mapping from the singly slit plane $\mathbb{C}\setminus(-\infty,0]$ onto the doubly slit plane $\mathbb{C}\setminus\big(...
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In what $precise$ sense is Minkowski space asymptotically flat?

I've brought this question over from the physics stack exchange, where it didn't generate interest. We say a manifold $(M,g)$ is conformally compact if it is the interior of some $(\overline M, \...
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If $u:U\to C$ is a harmonic function, then $u\circ F$ is harmonic on $V$.

$Lemma:$ Let $V$ and $U$ be open sets in $C$ and $F:V\to U$ a holomorphic function. If $u:U\to C$ is a harmonic function, then $u\circ F$ is harmonic on $V$. $proof.$ The thrust of lemma is purely ...
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The change of area after conformal mapping in a single connected region?

Consider a simply connected rectangular region $D$ with a uniform rectangular mesh. Then, after the conformal mapping $f$ ,we can obtain a single connected region $G$. At this time, Where are the ...
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Prove $|f^{\prime}(a)|< \frac{Im f(a)}{Im a} $ for analytic self mapping on upper half plane

$f:H \to H$ is a analytic mapping, where $H$ is the upper half plane, then $a \in H$, prove the inequality : $|f^{\prime}(a)|< \frac{Im f(a)}{Im a} $. I tried to use Schwarz Lemma to solve it, but ...
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Why the conditions $w(0)=0$ and $w(2)=\infty$ map the region $|z-1|<1$ onto the region $\Re w>0$?

I want to find a linear fractional transformation which maps the region $D$ of the $z$-plane onto the region $G$ of the $w$-plane, where $D=\{z;|z-1|<1\},~G=\{w;\Re w>0\}$ This is an exercise ...
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Algebraic trick to map $|z|<2$

Suppose that we want to find the image of the region $|z|<1$ under the mapping $w=\frac z{z+1}$. Since $z=\frac{-w}{w-1}$ we should have $|\frac w{w-1}|=|\frac{u(u-1)+v^2-iv}{(u-1)^2+v^2}|<1$ or ...
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Conformal factor between Euclidean metric and metric on Poincaré Ball of arbitrary radius

Usually, a Poincaré Ball is given as the set $ \mathbb{D}^n = \{ x \in \mathbb{R}^n : \|x\|^2 < 1\} $ Let $g_{E,x}$ be the Euclidean Riemannian metric induced at $x \in \mathbb{R}^n$ -- in that ...
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Image from upper half plane to unit disc [closed]

Let $z,w$ be points in the upper half plane $\mathbb{H}$. Let $\pi$ denote the Cayley map from $\mathbb{H}$ to the unit disc $\mathbb{D}$ that sends $w$ to the center of $\mathbb{D}$. Then $$\pi(z)=\...
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Definitions of analytic, regular, holomorphic, differentiable, conformal: what implies what and do any imply that a function is a bijection?

I'm looking back at some complex analysis and have gotten myself a little muddled in all of the definitions analytic/ regular/ holomorphic/ differentiable/ conformal... In particular, at the moment I'...
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Characterizing all analytic map from some simply connected region $\Omega$ to $B(0,1)$ with prescribed values

suppose $\Omega = \{re^{i\theta}:0<r<\infty, |\theta|<\pi/4\}$, (i) show that there exists an analytic mapping from $\Omega$ to $B(0,1)$ such that $g(1)=0, g(2)=1/2$ (ii)show that there ...
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Finding a suitable transformation function for the picture

I haven't taken complex analysis yet so there may be many words to sift through and not much concrete mathematical notation. I'll try my best though. Given a local Euclidean unit square grid how ...
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Producing isothermal coordinates from a solution of the Beltrami Equation

Suppose that $f \colon U \to V$ is a diffeomorphism of planar domains. Its differential $Df$ can be pointwise expressed the sum of a complex-linear and complex-anti-linear mapping: Given a tangent ...
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How do you figure out what hyperbolic tilings are compatible with a given surface?

By the uniformization theorem, any Riemann surface $H$ is conformally equivalent to one of constant curvature. Let's say that $H$ is conformally equivalent to a surface with curvature $-1$. Therefore,...
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Square conformal models for 2 dimensional Euclidean geometry? [closed]

The Poincare disk model is one example of a conformal disk model of 2 dimensional Hyperbolic geometry. I'm wondering if there are conformal square models for 2 dimensional Euclidean geometry. What are ...
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Geodesics of sphere mapped to square question

I'm trying to accurately draw the mappings depicted. Finitely many geodesics (the curved lines) of the 2-sphere are being mapped to the unit square. I'm not sure if that is captured in the notation ...
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Existence of an injective holomorphic function with larger derivative

Let $\Omega$ be a simply connected region with $0\in \Omega$, $\Omega \neq \mathbb{C}$. Suppose that $f$ is an one-to-one holomorphic function from $\Omega$ into the open unit disc $D$ such that $f(\...
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Proof: Mobius Transformation Is Conformal (Poles Included)

I'm studying for an exam in complex functions analysis and I've come across a proof which states that the bi-linear Mobius function: $\omega(z)=\frac{az+b}{cz+d}$ such that $ad-bc \neq 0$ and $c \neq ...
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Find a conformal map 2

$\textbf{Exercise}$ Find a conformal mapping which maps the domain $D$ onto the open unit disc, where $D$ is the intersection of $\vert z \vert <1 $ and $\vert z-1 \vert <1 $. I knew that ...
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For a map $F: \mathbb{H} \rightarrow \mathbb{D}$, prove that $|F(z)| \leq |z-i|/|z+i|$ for all $z \in \mathbb{H}$.

For notation let $\mathbb{H}$ denote the upper-half plane and $\mathbb{D}$ the open unit disk. There was exercise on which I was stuck on: Let $F: \mathbb{H} \rightarrow \mathbb{C}$ be a ...
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What is $O(\rho^m)$ in this paper by Fefferman-Graham?

The following is an excerpt from Page 10 of this paper by Fefferman-Graham. We prepare to define ambient spaces in the even-dimensional case. Let $S_{IJ}$ be a symmetric 2-tensor field on an open ...
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Are $\{ z: 1<|z|<2\}$ and $\{z: |z|>1\}$ conformally equivalent?

I don't know whether a ring, i.e. the domain $\{z: 1<|z|<2\}$ and the exterior of the closed unit disk $\{z: |z|>1\}$ are conformally equivalent? I have tried to look on some topological ...
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Calculation to show $|\mathrm{d}r|^2_{\bar g} = 1$ implies sectional curvatures tend to $-1$.

$\textbf{tl;dr:}$ Given that $r$ is a definining function for the boundary of a conformally compact manifold, how does one show that the sectional curvatures tend to $-1$ if $|\mathrm{d}r|^2_{\bar g} =...
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1answer
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Probability of hitting curve in the plane

Consider the open unit disk $\mathbb{D} \subset \mathbb{R}^2$, and consider a Brownian motion in the plane starting at the origin. Let $\gamma : \mathbb{R} \to \mathbb{R}^2$ be a smooth planar curve ...
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Fixed point of conformap mapping of the unit disc

Let $S$ be the open unit square in the first quadrant of the complex plane. ($S = \{x+iy : 0 < x,y < 1\}$) Let $f : S \rightarrow S$ be a conformap mapping. Prove or disprove : (1) $f$ can ...
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How to calculate an angle between two lines - Conformal mapping

I have a rounded-triangle in $z$-plane that conformally mapped into unit disk in $\zeta$-plane using the Schwarz-Cristoffell method. By definition, the conformal mapping will preserve the local angle ...
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Geometric interpretation of Joukowski transformation

The Joukowski map $J : HD \rightarrow H$ is defined by $$f(z) = -\frac{1}{2}(z + z^{-1})$$ where $HD = \{x+iy : |x+iy| < 1, y > 0\}, H = \{x+iy : y > 0\}.$ It can be shown that $f$ is a ...
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Conformal mapping of the unit circle minus a smaller circle

Map the region inside the circle $|z| = 1$ and outside the circle $|z-1/2| = 1/2$ conformally onto the unit disk. I was thinking of using some scaling and shifting to get from the unit circle to ...
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Conformal mapping between disk and the complement of a spiral

The Riemann mapping theorem guarantees the existence of a biholomorphic mapping between the unit disk and the complement in the complex plane of an (archimedean or logarithmic) spiral ... is it known ...
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Geodesic curvature change under conformal metrics

Suppose that $\sigma_0$ is a fixed metric on a compact riemannian 2-manifold $M$ with boundary $\partial M$. Let $\sigma=\rho \sigma_{0}$, where $\rho=e^{2\varphi}$ with $\varphi \in C^{\infty}(M)$, ...
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A conformal map whose Jacobian vanishes at a point is constant?

Let $f:M \to N$ be a smooth weakly conformal map between connected $d$-dimensional Riemannian manifolds, i.e. $f$ satisfies $df^Tdf =(\det df)^{\frac{2}{d}} \, \text{Id}$. Assume $d \ge 3$ and ...
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A counter example for Liouville's theorem when the Jacobian is changing signs

The famous Liouville's theorem states the following: Let $\Omega$ be a domain in $\mathbb{R}^n$, and let $f \in W_{loc}^{1,n}(\Omega,\mathbb{R}^n)$ satisfy $Jf=\det df \ge 0$ a.e. on $\Omega$ or $\...
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Is every conformal diffeomorphism isotop (or homotop) to an isometry?

Assume that $M$ is a compact simply connected Riemannian manifold and $f$ is a conformal diffeomeorphism of $M$. Is it true to say that $f$ is homotopic (or isotopic) to an isometry of $...
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Schwarz- Christoffel Formula Question: Finding a transformation which maps the upper half of the w-plane for the following -

I need to find a transformation which maps the upper half of the w-plane inside the triangle with the vertices at -1, 0, and i using the Schwarz-Christoffel formula... Thus far I have drawn the ...
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Bilinear transformation at infinity

I have been asked to find the bilinear tranformation which maps the points $z_1=i\sqrt3$, $z_2=-i\sqrt3$, $z_3=1$ into $w_1=\infty$, $w_2=0$, $w_3=1$ Using the formula for finding the cross ratio of ...