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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Conformal transformation of the divergence

Let $f$ be a smooth function on a $n$-dimensional Riemannian mainfold $(M, g)$, so that $\tilde{g} = e^{2f} g$ is a conformal transformation of $g$. I'm trying to show that the divergence operator ...
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40 views

Schwarz Lemma/Conformal mapping problem

Let $F:\mathbb{H}\rightarrow \mathbb{D}$ be holomorphic, where $\mathbb{H}$ is the upper half plane and $\mathbb{D}$ is the unit disc. Show that if $F(i)=0$, then $$|F(z)|\leq ...
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The image of a Joukowsky transform,

How can I show that the Joukowsky transform, $J(z)= z+\frac{1}{z}$ maps the set $R =\{(x,y)\in R^2: x^2+y^2>1, y>0\}$ conformally onto the upper half-plane? It's clear that the parts of the ...
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Compute $f''(0)$ for a holomorphic function on a square given $f'(0)$ and $f(0)$

Let $S$ be the square $\{x + iy: |x| < 1, |y| < 1\}$ and $f:S \rightarrow S$ a holomorphic function so that $f(0)= 0$ and $f'(0) = 1$. Find $f''(0)$. It seems like I need to use Cauchy's ...
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inversion of the circle $t \mapsto (3 + is) + e^{it} $ around the unit circle.

We know that inversion interchanges lines and circles, but it's very hard to The inversion map about the unit circle is just $\displaystyle z \mapsto \frac{1}{\overline{z}}$. As a Möbius ...
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37 views

Riemann Mapping Theorem, the concept of a Riemann mapping

If I construct a composition of mappings that map the upper half of the unit disk conformally to the entire unit disk, then this mapping is a Riemann mapping, by the Riemann Mapping Theorem, since ...
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24 views

Uniqueness of conformal mappings with different normalizations: three boundary points, or an interior point

Some stuff I've seen in lecture but am still a little shaky on: 1) To determine my mapping explicitly, it suffices to know where 3 distinct points on my pre-image object, say, the unit circle, gets ...
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Automorphisms of the upper half plane

STATEMENT: Suppose $(x_1,x_2,x_3)$ and $(y_1,y_2,y_3)$ are two pairs of three distinct points on the real axis with$$x_1<x_2<x_3 \;\;\;\;\text{and} \;\;\;\;\;y_1<y_2<y_3$$ Prove that ...
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28 views

Conformal Map for Circle to Circle

I am trying to find a conformal map that maps a circle in the $\zeta$ plane to a circle in the $z$ plane. As far as I know, a Mobius transformation is appropriate for this. These are the conditions ...
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19 views

Conformally mapping the unit disk to the upper-half plane

Within $\mathbb{C} \cup \{ \infty \}$, consider the unit-disk $\mathbb{D} = \{ z : |z|\leq 1 \}$ with three points labelled as $a$, $b$, $c$ on its boundary. I want to map $\mathbb{D}$ conformally ...
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conformal mapping, regions of the complex plane marked +/-, find the function f,

The picture shows what the function f: $\mathbb{C}\to\mathbb{C}\cup\infty$ does to the plane. The values 0 at 0, 1 at $\pm$1, and $\infty$ at $\pm i$ are specified. To elaborate on the picture: ...
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Mapping a region between two circles on a half plane,

I've tried this problem for awhile now but something strange is happening at the end of my solution. The question asks to map the region between $|z| =1$ and $|z-\frac{1}{2}| = \frac{1}{2}$ on a half ...
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41 views

Conformal mapping of part of an annulus

I have a question about conformal mapping. I am wanting to map annuli to some other simple domain (probably rectangular). I have an image of my problem below In image A. we see a standard annulus ...
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54 views

Conformal mapping circle onto square (and back)

I'm programming an implementation of the Peirce quincuncial map projection. The projection involves a stereographic projection of a hemisphere of the globe onto a circle (I've got that part), then ...
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25 views

Excercise (and consequence) for Explicit Riemann Mapping

Set $V:=D_1(e^{\frac{i\pi}{6}})\cap D_1(e^{-\frac{i\pi}{6}})$. Find the (unique) explicit conformal function $f:V\rightarrow \mathbb{C}$ such that $f(V)=\mathbb{D}$ with $f(1)=0$ and $f'(1)>0$. ...
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Excercise for Explicit Riemann Mapping

Set $U:=\{ z\in \mathbb{C}\mid \arg(z)\in(-\alpha,\alpha)\}$ with $\alpha\in(0,\pi)$.Give explicitly the conformal mapping $f:U\rightarrow \mathbb{D}$ such that $f(U)=\mathbb{D}$ with $f(1)=0$ and ...
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Conformality of a map

A conformal mapping is a map $f:U\to V$ with $U,V\subseteq\mathbb{C}$ such that the angles are locally preserved. This can be reformulated saying the jacobian matrix is everywhere a scalar multiple of ...
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31 views

Can the derivative of a conformal homeomorphism onto a bounded domain be unbounded?

Consider $f$ being a holomorphic homeomorphism of the closed unit disk into the complex plane (ie. $f$ must be a homeomorphism - which also imply bijectivity to its image - of the closed unit disk, it ...
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30 views

Analytic onto maps from D to D

We just characterized using the Schwarz Lemma the conformal self maps of the open unit disk. I am now trying to characterize the holomorphic onto maps from $\mathbb{D}$ onto $\mathbb{D}$. As a ...
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25 views

Möbius transformation $(-1,0,1)\to (i,-1,-i)$

Find Möbius transformation $S$ that maps points $(-1,0,1)$ to points $(i,-1,-i)$. And what's the image of real axis and upper half of imaginary axis $\{z\in\mathbb{C}| \operatorname{Im} z \geq 0 \} $ ...
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A conformal mapping from a sector to a strip

What is the simplest function that maps the sector $r < 1$, $0 < \theta < \pi$ conformally onto the strip $0 < u < \pi/2$, $v > 0$? Here, $r$, $\theta$, $u$, $v$ have their usual ...
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Conformal transformation of metric on $\mathbb{R}^n$

Let us define the following metric on $\mathbb{R}^n$: $$ g|_v(X, Y) := e^{-|v|^2} \langle X, Y\rangle,$$ where the brackets denote the standard scalar product. How does the resulting manifold look ...
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Flow of a Metric & Conformality

What is the flow of a metric mathematically? I want to be able to understand what it means to say that a metric preserves a conformal structure through the actual definition $\Theta_t^*g = ...
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Conformal mapping of nonsimply connected domains

The question asks: Map the complement of the arc $|z|=1$, $y\geq 0$ on the outside of the unit circle so that the points at $\infty$ correspond to each other. How would you construct such conformal ...
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Conformal classes and almost-complex structures

It is well-known that on closed oriented surfaces $S$, conformal classes of metrics on $S$ correspond bijectively to complex structures on $S$. My understanding is that this correspondance goes as ...
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Ricci SCALAR curvature

Are there any manifolds that have NULL SCALAR curvature but not null ricci curvature tensor? For dim>2, obviously. Edit: Are there, also, any manifolds of null scalar curvature but ricci curvature ...
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“Angle-preserving” equivalent to conformal?

I'd like to investigate the common turn of phrase that conflates "angle-preserving map" with "conformal map". Let $f:\Bbb R^2\to\Bbb R^2$ be a continuous function. I'll define $f$ to be ...
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Conformal cobblestones

Open-ended question: It strikes me that the cobblestone pattern known as Bogen, which can be seen in many European cities, is a fairly accurate representation of the conformal mapping defined by the ...
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46 views

Conformal mapping from a tetrahedron to a spherical sector

I have been trying to find a way to do a conformal mapping from tetrahedron, specifically a trirectangular tetrahedron, to a spherical sector, but being an engineer with no training in this subject it ...
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Check that the parametrization x(u,v)is conformal if and only if E=G and F=0.

Check that the parametrization x(u,v)is conformal if and only if E=G and F=0. I am slightly confused with what this question is asking me. Could someone please walk me through this question. I ...
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Stereographic projection is conformal — from the line element

I'm looking over some fairly basic stuff on complex methods and the book I'm using takes the formula for the stereographic projection: $$z = \cot(\beta/2)e^{i\phi} $$ as well as the line element on ...
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Ricci flow and conformal classes

Is it true that the conformal class of the metric is preserved under Ricci flow? Is there an easy argument?
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Weird conformal map problem

Construct a conformal map from the region $\omega$ = open disk of radius 1 centered at 0 minus the closed disk of radius 0.5 centered at 0.5 to $\mathbb{D}$ = disk radius 1 centered at 0. I really ...
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Let $f(z) = z + z^2$ and let $V = \displaystyle \{z \in \mathbb{C} : |z| < \frac{1}{2}, \frac{3\pi}{4} < arg\{z\} < \frac{5\pi}{4}\}$.

Let $f(z) = z + z^2$ and let $V = \displaystyle \{z \in \mathbb{C} : |z| < \frac{1}{2}, \frac{3\pi}{4} < arg\{z\} < \frac{5\pi}{4}\}$. $(a)$ Show that $f(V) \subset V.$ $(b)$ Let $f_n$ be ...
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Find sup$\{|f′(3)| : f$ maps $Ω$ analytically into the unit disk $\}.$

Let $Ω=\{z=x+iy∈C : |y|<x\}.$ Find sup$\{|f′(3)| : f$ maps $Ω$ analytically into the unit disk $\}.$ Okay. So I can find a conformal map from $Ω\rightarrow \mathbb{D}$. I used the map $f(z) = ...
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Conformal transformation

The problem is following. This is an Exerciese of Polchinski $2.6$ (explanation about conformal field transformation) Consider the flat Euclidean metric $\delta_{ab}$ in $d$ dimensions. An ...
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122 views

Finding the transfinite diameter of the level sets of complex logarithm

Given a simply-connected domain $|g(z)|\ge C$ how can I find the analytic conformal mapping guaranteed by the Riemann mapping theorem? In particular I'm interested in finding the transfinite diameter ...
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Conformal mapping between symmetric region and unit disc

Exercise 3 of VII.4 of Conway's Complex Analysis states Let $G$ be a simply connected region which is not the whole plane, and suppose that $\bar{z}\in G$ whenever $z\in G$. Let $a\in ...
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holomorphic functions with nonvanishing derivative on unit disk $D$

Let $f$ be a holomorphic function on the unit disk $D$. Suppose for any $z\in D$, $f'(z)\neq 0$. Then does $f$ have to be a conformal map from $D$ to $f(D)$?
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Find a conformal map from semi-disc onto unit disc

This comes straight from Conway's Complex Analysis, VII.4, exercise 4. Find an analytic function $f$ which maps $G:=$ {${z: |z| < 1, Re(z) > 0}$} onto $B(0; 1)$ in a one-one fashion. ...
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112 views

Conformal mapping of the domain bounded by a line segment and a circular arc

I am trying to construct a conformal map from the region $R$ which is the set of points in the complex plane bounded by the segment connecting $i$ and $1$ and the part of the unit circle in the first ...
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A hard Conformal Mapping problem

I am trying to construct a conformal map from $R = \{z \in \mathbb{C} : -1 < Re(z) < 1$ and $Im{(z)} > 0\} \cap \{z \in \mathbb{C} : |z| > 1\}$ to the unit disk $\mathbb{D}$. I am really ...
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Easy solution to Yamabe problem for surfaces

The Yamabe problem asks if, given a Riemannian manifold $(M,g_0)$, it is possible to find a conformal metric $g$ on $M$ with constant scalar curvature. I would like to know if there is some "easy" ...
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Conformal group in two dimensions

In Conformal field theory, physicist says, the conformal group in two dimensions is infinite dimensional, so the associated with the infinity of generators and infinity conserved charges provided. Is ...
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Analytical form of conformal mapping of simple closed curve

I want to create a mapping to map a simple closed curve to a unit disk, in analytic forms. The curve is simple enough, with several segments: each with an analytical form, but no general analytical ...
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an inequality derived from conformal automorphisms of unit disk

Let $f$ be a holomorphic function on $D(0,1)$ such that $|f(z)|<1$ for all $z\in D(0,1)$. I have obtained $$ \frac{|f(0)|-|z|}{1+|f(0)||z|}\leq |f(z)|\leq \frac{|f(0)|+|z|}{1-|f(0)||z|}. $$ Is it ...
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The proof of the Area Theorem for Conformal Maps

The Area Theorem: Suppose $f(z)$ is one-to-one and analytic on the punctured unit disk, and is given by $f(z) = 1/z + \sum_0^\infty a_nz^n$ Then $\sum_0^\infty n|a_n|^2 \le 1$ I'm reading the ...
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How do we check conformal equivalence of parametrized surfaces, e.g. parallel surfaces?

Suppose we have two parametrized surfaces in $\mathbb{R}^3$: $$ X,Y:\mathbb{R}^2 \rightarrow \mathbb{R}^3 $$ The induced metric on either surface is the pullback of the Euclidean metric $\bar g$ due ...
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Random points inside a convex polytope

Given a convex polytope, defined by set of vertices $P = \{\mathbf{x}^{(i)}\}_{i = 1}^n, x^{(i)} = (x^{(i)}_1, x^{(i)}_2, \dots, x^{(i)}_d): \operatorname{conv}(P) = P$. How to generate uniformely ...
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Conformal map two slits to circles

I am trying to find a conformal mapping that maps a double slitted plane onto a plane with two circles. The two slits are both located along the real axis with similar lengths. For a single slit ...