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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Geodesic parameterization under conformal mapping

Under a conformal deformation of the euclidean metric, say: $\hat{g}_{ij}=e^{\phi}\delta_{ij}$, where $\phi$ depends on the radial coordinate alone, I am struggling to see the following fact: "With ...
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63 views

Prove that $\Big|\frac{f(z)-f(w)}{f(z)-\overline{f(w)}}\Big|\le \Big|\frac{z-w}{z-\overline w}\Big|$

Let $\mathbb{H}$ denote the upper half plane of $\mathbb{C}$, i.e. \begin{equation*} \mathbb{H}=\{z \in \mathbb{C}: Im(z)> 0\} \end{equation*} Suppose $f:\mathbb{H}\to\mathbb{H}$ is analytic. ...
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Help me understand where the factor 2 comes in, in this conformal mapping.

I want to "Show that the mapping $f(z) = z + \frac{R^2}{z}$ takes the two concentric circles, $|z|=R$ and $|z| = R'> R$, onto a line segment and an ellipse." (These are depicted in a figure. The ...
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Understanding angle-preserving definition

My book (Real and complex analysis, by Rudin) gives the following definition: Let $A(z) = \frac z{|z|}$. Then we say $f$ preserves angles at $z_0$ if $$\lim_{r \to 0}e^{-i\theta} A[f(z_0 + ...
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set of arbitrary positive measure conformally equivalent to unit disk

Show that for any  $\epsilon$ > 0, there is a dense subset of $\mathbb{C}$ with measure less than $\epsilon$ and which is conformally equivalent to the unit disc. To make a dense set that has ...
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33 views

Three-Dimensional Metrics as Deformations of a Constant Curvature Metric?

I read the following paper Three-Dimensional Metrics as Deformations of a Constant Curvature Metric and discovered the following result: I have three questions: (1) Is $h$ also a conformally flat ...
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All rank two symmetric tensors are several conformally flat metrics summed together?

If given a rank two symmetric tensor $T_{mn}$ can I decompose it as \begin{align} T_{mn} = \sum_{i=1}^{M} \phi_ig_{mn}{}^{i}{} \end{align} where $\phi_i$ are the $i$th conformal factors and ...
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How to use the conformal property of bilinear transform to show that the left half plane is mapped to the unit circle? [duplicate]

Intuitively, conformal means angle preserving. How can I see that the left half plane is mapped to the unit disc with the angle of each element of the left half plane preserved?
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Conformal group, Minkowski metric

The conformal group is the subgroup of coordinate transformations that leave the metric invariant up to a scale factor. So under a transformation $x\rightarrow x'$ we have $g_{\mu\nu}(x)\rightarrow ...
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Characterization of locally conformally flat manifolds with Frobenius theorem

In Hamilton's Ricci Flow (by Chow, Lu, Ni, pp. 29-31, see here) they show that a Riemannian manifold $(M^n,g)$ is locally conformally flat iff the Weyl tensor vanishes (when $n\ge 4$) and iff the ...
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How to specify Schwarz Christoffel Mapping of an unbounded domain?

I am struggling to see how Extra Example 1 found here uses the Schwarz Christoffel theorem properly. The problem is to map the exterior of the blue domain onto the upper half plane. In using the ...
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41 views

Are all 2D tensors in a specified flat metric equal to that same metric conformally scaled?

I have a tensor $T_{mn}$ where its indices coorespond to a flat metric $g_{mn}$. I want $T_{mn}$ to be a new metric $\tilde{g}_{mn}$, such that $T_{mn}(g_{rs}) = \tilde{g}_{mn}$. A theorem says that ...
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36 views

How to extend a conformal map from a rectangle to the upper half plane to the entire plane meromorphically

I'm taking a look at Ahlfors's Complex Analysis, Third Edition. In Section 2.3 "Mapping on a Rectangle", the author talks about how to extend a conformal map from a rectangle to the upper half plane ...
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Represent the following curves in the z-plane in the form z(t)

Represent the following curve in the z-plane in the form z=z(t). $$y = {x^2}\quad 1 \le x \le 3 $$ I know the answer is $$z(t) = t + i{t^2}\quad 1 \le t \le 3 $$ however I don't have any idea how to ...
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21 views

Conformal Equivalence of Two Metrics

I am having difficulty understanding something in the book Introduction to Curvature by John M. Lee. Let $\sigma: S^n-N \rightarrow \mathbb{R}^n$ be the stereographic projection, $g_0$ be the metric ...
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43 views

Unique circle through two points perpendicular to a given line?

Suppose I have a line in the plane, and two points not on the line. How can I prove that there is a unique generalized circle (i.e. circle or line) passing through the two points which intersects the ...
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49 views

Conformal maps from the unit disc onto itself, given by two sets of three points on the boundary

I want to construct a conformal map from the closed unit disc onto itself that maps given three points on the unit circle to another given set of three points on the unit circle. I know that the word ...
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132 views

Conformal Mappings dealing with Slits

The goal is to find a conformal mapping of the domain $$U=\lbrace z: \vert z \vert <1, z\not\in [1/2,1)\rbrace$$ to the unit disc $D$. I would like to learn how to deal with the slit; I imagine in ...
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71 views

conformal map of unit disk slit

Map the unit disk slit along $(-1,-r ]$, $r \in (0, 1)$, onto the unit disk. Can anyone explain how to do the conformal map thoroughly since I have difficulty understanding it. Thanks
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Arc length change in a conformal map

During conformal mapping, corrosponding angles are conserved but lengths change. In the case of a Moebius Map, what scaling (magnification or reduction of differential arc lengths, i.e., dilation ) ...
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37 views

Find a Conformal Mapping from $U=\lbrace \vert z\vert >1 \rbrace - [1,\infty)$ to the Unit Disc

The conformal mapping $z\mapsto \frac{1}{z}$ doesn't map onto the unit disc, but it maps to the unit disc minus the interval $(0,1)\subset\mathbb{R}$. I tried using the fact that the circle is ...
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Biholomorphic map from a disk to its quarter

Could you tell me how to find a biholomorphic map from a unit disk $D$ to $\{ |z|<1 \ : \ \Re z >0, \ \Im z >0 \}$? I know that mapping of the form : $\frac{az+b}{cz+d}$ won't work. Also, ...
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How to construct Möbius map to tranform so 2D region to another?

I want to know what is the general step-by-step method to find a Möbius map $w(x) = \dfrac{ax+b}{cx+d}$ that transforms a region in a complex plane to another region (2D). I know it takes circle and ...
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Mapping from a point inside a disk to a point inside an annulus

How do I map any point inside a disk to a point inside an annulus ? Disk and annulus are concentric (at the origin). After some more deep thoughts: may be the origin should not be included in the ...
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33 views

Conformal transformation of complement of disk in upper half plane

Let $U$ be the complement in the half-plane $\operatorname{Im} z > 0$ of a disk of radius $a<1$ centered at $i$. I am looking for a conformal transformation that maps $U$ onto an annulus. Since ...
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88 views

Showing that stereographic projection is a homeomorphism

For any $n\geq 0$,the unit $n$-sphere is the space $S^{n}\subset \mathbb{R^{n+1}}$ defined by $$S^{n}=S^{n}(1) :=\left\{ (x_{1},...,x_{n+1}) \left\vert\,\sum_{i=1}^{n+1} x_{i}^{2}=1\right.\right\}$$ ...
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33 views

Does there exists known special cases of a zero Riemann tensor for 3D metrics?

In two dimensions, if one has a flat metric $g_{ab}$, then one can transform $g_{ab}$ to another flat metric $h_{ab}=e^{2\varphi}g_{ab}$, when $\nabla^2 \varphi =0$ and the Riemann tensor remains ...
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90 views

How to visualize bilinear transform of the form $S(z) = \frac {T}{2} \frac {z+1}{z-1}$

Note that this is the bilinear transform from a z-domain as appears in Z-transform to s-domain in Laplace transform Recall that bilinear transform has form $M(z) = \frac{az+b}{cz+d}$ with and has to ...
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71 views

Dirichlet Problem using conformal mapping

Using appropriate conformal maps, solve the Dirichlet problem (for Laplace's equation) for the following region and boundary condition: $U=\{\text{Im}(z)>0\cup \text{Im}(z)=0\}$, with boundary ...
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Harmonic maps and angle preservation

I have the following question: What are the angle preservation properties of harmonic maps? Conformal maps preserve angles exactly, but they distort lengths. In this sense a conformal map is the ...
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Conformal map from the inside of the unit disk to the inside of an ellipse [duplicate]

I lack intuition when it comes to some conformal mappings and I'm presently looking for a conformal map taking the inside of a disk, let's say the unit disk and sending it to the inside of an ellipse. ...
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1answer
44 views

Conformal/Biholomorphism equivalence classes in $\mathbb{C}^n$

Recently I have got interested in the topic of conformal equivalence classes of complex domains, mostly one-dimensional ones. Here by conformal map $f: U \rightarrow V$ I mean a complex holomorphic ...
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Clifford Algebras for Projective and Conformal Geometry

According to Clifford Algebra: A Visual Introduction, A Clifford Algebra over $\mathbb{R}^3$ may describe the rigid motions in space (namely, conjugation acts as a reflection by a plane). A ...
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Existence of such function

So we know that if $g(z)=\frac{z-c}{1-\overline{c}z}$ $(c\in\mathbb{C})$ $|g(z)|=1$ for $|z|=1$. Does there exist a function $f(z)$ satisfies the following properties: (1) $f$ is analytic in some ...
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Find a Harmonic Function which is $1$ on $|z-5i| = 4$ and $0$ on $Im(z) = 0$

I've been trying to solve the following problem: Find a function $\varphi$ which is harmonic in the upper half-plane exterior to the circle $|z-5i| = 4$, is $1$ on $|z-5i| = 4$, and $0$ on the real ...
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1answer
92 views

Does a conformal map take boundaries to boundaries?

I think it is a well-known result that conformal maps between sets in $\mathbb{C}$ take boundaries to boundaries. However, I looked around a little and I had trouble finding this result. Is it true? ...
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Enumerating Automorphisms of Upper Half Plane

I'm trying to find all conformal automorphisms of the upper half plane $\{\Im[z] \gt 0\}$, known to be $f(z) = \frac{az + b}{cz + d}$ where $a, b, c, d$ are real and $ad - bc \gt 0$. The main work ...
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Plotting the region $ -1 < Re(z) \le 1$ and $ -\pi/2 < Im(z) \le \pi/2$ before and after being transformed by $w=e^z$

Can someone please verify that my diagram plot based on the calculations below is correct -Thanks. $$w=e^z=e^x\cos(y)+ie^x\sin(y)$$ $$\implies u=e^x\cos(y)\space and \space v=e^x\sin(y) \tag{1}$$ ...
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1answer
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Transform $\Re(z)=1 \space , \Re(z)=\Im(z) \space and \space \Re(z)=-\Im(z)$ using the mapping $w=iz^2$

Can someone please verify whether i am doing this the right way -Thanks. $$w=iz^2=i(x^2+2xiy-y^2)=-2xy+i(x^2-y^2)$$ $$\color{green}{u=-2xy \tag{1}}$$ and $$\color{green}{v=x^2-y^2 \tag{2}}$$ ...
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Transformation of line $y=k=constant$ under the mapping $w=cos(z)$

I have been going over this post and found myself confused by the calculations given by the OP. $$\color{red}{ shouldn't \space this \space be \space done \space as \space follows:}$$ ...
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Conformal Mapping defined by Cubic Polynomial

Ahlfors studies the mapping defined by $\omega = a_0 z^3 + a_1 z^2 + a_2 z + a_3$ in Complex Analysis 3rd Edition at the bottom of page 95. First he says we can get rid of the quadratic term by the ...
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Determine the image of the strip $S$ consisting of all points $z$ with $\frac{-\pi}{2}\lt Re(z) \lt \frac{\pi}{2}$ and $Im(z)>0$ under $w=i\sin z$

$\color{green}{\text{transformation is}\space w=i\sin z}$ $$w=i\sin z = i\sin(x+iy)=\frac{1}{2}\left(e^{ix-y}-e^{-(ix-y)}\right)=-\cos(x)\sinh(y)+i\sin(x)\cosh(y)$$ $\therefore u = -\cos(x)\sinh(y) ...
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120 views

The action of the conformal mapping -(1/z)

I know that the mapping -1/z is conformal away from the origin, since the mapping would then be analytic and have a non-zero derivative everywhere in C. It apparently also maps the upper half plane ...
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Find the region in the w-plane to which the line y = 1 is transformed by $\frac{1}{z}$

I tried to do the following: $$w=\frac{1}{z}=\frac{x-iy}{x^2+y^2}$$ $\implies u = \frac{x}{x^2+y^2} and\space v = \frac{-y}{x^2+y^2}$ $\color{green}{need\space to\space transform\space the\space ...
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103 views

Conformal mapping for region between a square and a circle

I was wondering if anyone knew a mapping that would take the region between a square of length 4 and a unit circle, and map it into the region between two circles. It would be great if the mapping is ...
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47 views

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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1answer
66 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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51 views

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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1answer
46 views

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 ...