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 maps onto open right half plane

On the Big Rudin there is the conformal map $$\varphi(z) = \frac {1+z}{1-z}$$ which sends $\{-1, 0, 1\}$ to $\{0, 1, \infty\}$. The book says: The segment $(-1, 1)$ maps onto the positive real ...
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38 views

Easy application of the Riemann Mapping Theorem

Riemann Mapping theorem Every simply connected region $\Omega \subset \mathbb C$ is conformally equivalent to the open unit disk (except $\Omega = \mathbb C$) What are application of this ...
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26 views

Conformal map from disk with smaller disk removed to upper half plane

I'm working on a problem that was a previous complex qualifying exam at my university. I believe I have a solution, but I'm not entirely confident in it. The problem is as follows: Find a ...
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27 views

Inversion map is a Conformal map

I'm studying PDE by Evans book and I need to show that the inversion map $f:\mathbb{R}^n-\{0\}\to \mathbb{R}^n$, defined by $$f(x)=\frac{x}{\|x\|^2}$$ is conformal. So I have a hint, show that ...
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15 views

Constructing a Mobius transformation that acts on any two points of the upper half complex plane:

I would like to construct a Mobius transformation that sends any two points $z_1$ and $z_2$ from the upper half of the complex plane to i and to $iR^+$, i.e., given any two points $z_1$ and $z_2$, ...
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37 views

conformal map disc with two removed points

I need to find all the bijective conformal maps from $D = \{ |z| < 1, z\neq \pm 1/2 \}$ onto itself. Since this set is not simply connected, I think that the $180°$ degree rotation is the only ...
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29 views

Conformal mapping of two annuli to the punctured unit disc

What is the general procedure for finding a holomorphic bijection from the region $ \Omega = \{z \in \mathbb{C}: |z - a| > 1, |z + a| > 1 \}$ to the punctured unit disc?
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40 views

Find the group of conformal automorphisms of $U=\lbrace z\in \mathbb{C}: \vert z-1\vert>1\rbrace$

Well $\phi$ is an automorphism of $U$ $\iff$ $1/ \phi$ is an automorphism of $U^C=\lbrace z\in \mathbb{C}:\vert z-1\vert<1\rbrace$ $\iff$ $1/\phi -1$ is an automorphism of the unit disc $\iff$ ...
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83 views

Non-conformal Schwarz-Christoffel integral

Using "conformal" to mean a holomorphic bijection, the Riemann Mapping theorem guarantees the existence of a conformal map from the upper half-plane $\mathbb{H}=\{z=x+iy\in\mathbb{C}:y>0\}$ to the ...
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29 views

Question about a Möbius transformation/Conformal map

I have a question about a conformal mapping. The map $f(z)=\frac{1+z}{1-z}$ takes the unit disk to the right half plane. Composing this map with $z^2$ gives $f(z)=(\frac{1+z}{1-z})^2$, which I think ...
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Conformal Mapping from Equilateral triangle to Isosceles Right Triangle

This is an exercise problem. Does there exist a conformal mapping from an equilateral triangle onto an isosceles right triangle such that, under correspondence of boundary, vertices are mapped to ...
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How does the Schrodinger's potential transformer if the metric conformally transformers?

Given Schrodinger's equation $$ (-\nabla^2+V)\psi=E\psi $$ and the conformal transformation $\tilde{g}_{mn}=e^{2\phi}g_{mn}$, how does the Schrodinger's potential $V$ transformer if the metric ...
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29 views

What is the formal name for the conformal laplacian?

\begin{align} L=R-4\dfrac{n-1}{n-2}\nabla^k\nabla_k \end{align} What is the formal name for $L$? I have seen it referred to as the conformal laplacian, however I thought I once read $L$ with a formal ...
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19 views

Proving maps conformal via a scaling factor

I'm in a differential geometry class and I just attended a review session where the TA gave an example problem about conformal maps on the board: Find a constant $k$ such that $x(u,v) = ...
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8 views

find conformal mapping

I need to find conformal mapping from area outside the two circles $|z-1|=1$,$|z+1|=1$ onto a half plane. We want to find trans’ that take $Z=0→W=∞$. such trans’ is $t(z)=1/z$ Now we find images of ...
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32 views

Green's function for Laplace operator in a conformally flat metric?

Given the Laplace–Beltrami operator $\nabla^2$, does there exists a closed form for the greens function $G$ such that $\nabla_x^2G(x,y)=-\delta(x,y)$, and $$ \nabla_x^2\iiint_{y^3}G(x,y)f(y)dy^3=-f(x) ...
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A name for a particular covering map?

The quotient space of $\mathbb C$ obtained by identifying points differing by a Gaussian integer is topologically a torus. The map that takes each point in $\mathbb C$ to its corresponding point in ...
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Four complex numbers $z_1,z_2,z_3,z_4$ lie on a generalized circle if and only if they have a real cross ratio $[z_1,z_2,z_3,z_4]\in\mathbb{R}$

Let $[z_1,z_2,z_3,z_4]$ denote the cross ratio of the complex numbers $z_1,z_2,z_3,z_4\in \mathbb{C}$. Show that the distinct points $z_1,z_2,z_3,z_4\in\widehat{\mathbb{C}}$ lie on a generalized ...
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Simplify $Im \left(\frac{az+b}{cz+d}\right)$

Let $z \in \mathbb{H}$, where $\mathbb{H}$ denotes the half plane $\mathbb{H}=\{z \in \mathbb{C}:Im(z)>0\}$. Let \begin{equation*} f(z)=\frac{az+b}{cz+d} \end{equation*} which is called a Mobius ...
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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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70 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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35 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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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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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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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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51 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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68 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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144 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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80 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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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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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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97 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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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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101 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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76 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. ...