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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Where does $\;f(z)=\displaystyle \frac{1}{z} +e^{2i \gamma}z\;$ maps the unit disk $|z|<1$ in $w$- plane? given that $0 \leq \gamma \leq 2\pi$

Where does mapping $\;f(z)=\displaystyle \frac{1}{z} +e^{2i \gamma}z\;$ maps the unit disk $|z|<1$ in $w$- plane? given that $0 \leq \gamma < 2\pi$ Solution i tried - Given mapping is $$\;f(z)=\...
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Linear conformal mapping transfers the upper complex plane to new upper or lower complex plane.

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Schwarz-Christoffel Mapping

The Schwarz-Christoffel Mapping is a conformal map between interior of sphere and interior of square. But will it be a conformal map between exterior of sphere and exterior of square? Disc to square ...
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Continuous extension of conformal map to accessible point

Let $G \subset \mathbb{C}$ be a simply connected bounded domain. By the Riemann mapping theorem there is a holomorphic bijection $f : G \to \mathbb{D}$. An accessible point of $\partial G$ is an ...
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Existence of conformal equivalence between doubly connected domain and annulus

The following math overflow post https://mathoverflow.net/questions/261535/mapping-the-doubly-connected-domain-to-an-annulus provides a sketch proof of the fact that any (non-degenerate) doubly ...
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What does it mean that an equation is conformally invariant?

What does it mean that an equation is conformally invariant? I'am reading a PDE paper which says that the following critical Yamabe equation is conformally invariant $$-\Delta u = \frac{n(n-2)}{4}|u|^{...
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Mapping of a Disk onto a Half-Plane

look at this text: In example 3 The following map is discussed: $$w=-{{z+1}\over{z-1}}$$ The author claims that this linear fractional transformation maps the unit disk onto the right halfplane but ...
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Why is this called the conformal tensor?

Suppose you have a Riemannian manifold $(M, g)$ and a diffeomorphism $f: M \longrightarrow M$. Define $E^f:= f^*g - g$. I can understand that $E^f$ defines a $(0, 2)$-tensor which in a sense keeps ...
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Why $f$ preserve angles means that $f$ preserve radii ? (Proof of Liouville Theorem in Riemannian Geometry)

Picture below is from 5.2 Theorem of chapter 8 of do Carmo's Riemannian Geometry. I can't understand the red line. Why $f$ preserve angles means that $f$ map radii into radii ? $U\subset \mathbb R^n$ ...
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Finding unit normal vector to conformally expanding space embedded in codimension 2 manifold in terms of conformal parameter.

I have a manifold $M^n$ of dimension $n$ that I'm treating as embedded within an $n+1$ dimensional manifold $N^{n+1}$, $M^{n}\subset N^{n+1}$. Now I also have an $n-1$ dimensional manifold $\Sigma^(n-...
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How to construct a specific conformal map

For my current research I need a conformal map from the infinite strip with a specified branch cut to the upper half plane. I already have a conformal map from the full plane with a branch cut from $-...
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Estimate about a Conformal Map

currently I am trying to solve the following problem: Let $f: \Omega \to \Omega^{\prime}$ be a conformal map (holomorphic with non-vanishing derivative). In the pdf I am reading the author states the ...
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Heegaard Floer homology of a genus two diagram of $S^3$

I am reading the introductory paper "Heegaard diagrams and holomorphic disks" by Ozsváth and Szabó (https://arxiv.org/abs/math/0403029v1, Section 2.2), and I do not understand one of the ...
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Why is the Schwarz-Christoffel mapping biholomorphic?

Let $a$ and $b$ be two real numbers such that $0 < a+b< 1$. For $\mathbb{H}$ the upper half plane and $T$ a triangle, let $f : \mathbb{H} \rightarrow T$ be the map defined by $$ f(z) = \int_0^z \...
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conformal mapping (from unit disc onto any simply-connected domain)

From Riemann mapping theorem, we know that if the domain $\Omega$ is simply-connected, there is a conformal mapping from the exterior of unit disk $\Delta$ onto the exterior of $\Omega$ of the form $$ ...
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Is the conformal Killing factor always an eigenvalue of the Laplacian?

Suppose $(M, g)$ is a pseudo-Riemannian manifold and $\xi_{\mu}$ is a conformal Killing field, i.e. $$ \nabla_{\nu} \xi_{\mu} + \nabla_{\mu} \xi_{\nu} = \kappa g_{\mu \nu} $$ for some smooth ...
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Resources/explanations for conformal geometry with “null cones at infinity”

In the Wikipedia article on conformal geometry https://en.m.wikipedia.org/wiki/Conformal_geometry there’s a section in Mobius geometry that says it’s the study of pseudo Euclidean spaces with either a ...
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local model of conformal structure

I'm reading the book The Geometry of Four-Manifolds by Donaldson and Kronheimer. In the page 147, the book states a fact that: consider the conformal structure $\mathcal{C} $ of $4$-dimensional ...
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`Conformal class' in Riemannian geometry vs Complex Analysis

Recently I found myself a bit confused about the definition of conformality. In Riemannian geometry, we say that two metrics on a manifold $M$, $g$ and $h$ are in the same conformal class if there ...
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Area Theorem (conformal mapping)

I am trying to understand the above theorem. I don't understand how we get the first line of the proof. 1- Why is the area of $E_r$ defined in this way? 2- How is $E_r$ two-dimentional? 3- How does a ...
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conformal mapping from a cut plane to unit disk

I came across the following question and I'm having a hard time figuring out how to construct such conformal mapping. Question: Find a conformal mapping $f$ which maps the cut plane $D_{1,\pi}$ onto ...
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Determining a function that conformally maps a half strip to a half disk

I am trying to determine a function that maps (preferably conformal map) that maps an infinite strip is the 2nd quadrant, between $y=\pi/6, 7\pi/6$, to a half disk (upper plane, quad 1/2) of radius ...
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Analytic maps from lower half plane onto itself.

I know that the bilinear map $f(z)=\frac{az+b}{cz+d},a,b,c,d\in\mathbb{R},ad−bc>0$, maps the upper half plane onto itself. But I wonder what is the map which maps lower half plane onto itself such ...
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Coefficients of inverse conformal metric

I'm trying to understand some proof written in local coordinates. The situation is as follows: We have a $2$-dim Riemannian manifold $(M,g)$ and a conformal diffeomorphism $f:M\rightarrow M$ such that ...
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Is every conformal transformation given by a conformal Killing field?

I am reading A Mathematical Introduction to Conformal Field Theory, Second Edition by Schottenloher. In the first chapter, he classifies the conformal transformations on $\Bbb R^{p,q}$ by classifying ...
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Intuitive way of describing a conformal distribution

I was wondering if anyone knows a way to describe a conformal distribution in an intuitive way, preferably to someone who doesn't have much previous knowledge of differential geometry. I know that a ...
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Do conformal maps preserve structure/ordering?

I am curious if conformal maps preserve structure. What I mean is suppose I have the unit disk $\mathbb{D}$ and that I am mapping it to the upper half plane. As I walk around the boundary of the ...
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Finding the Image of a Complex Transformation

I want to find the image of the complex transformation $f(z)=w=2z+\frac{1}{z}$ on the exterior of the unit circle in the complex plane. This is clearly a conformal mapping except at the points $z=0$ ...
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Image of real line under Moebius transformation: center and radius?

I am having a problem due to what no doubt must be an embarrassing error on my part. I have a Moebius transformation $\phi(z)=(a z+b)/(c z+d)$ with $a,b,c,d$ complex. As $t$ goes over the reals, $\phi(...
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Complex Conjugate of a Function for a given mapping

I have a complex function which actually is a Green Function of a Laplacian Operator in the Upper plane (y>0). I want to use a conformal mapping to map a circle into an upper half-plane (y>0) ...
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Show that any mapping from the extended complex plane to the Riemann Sphere back to the extended complex plane is a Möbius Transformation [duplicate]

Definitions: Extended Complex Plane $\mathbb{C}^\infty = \mathbb{C} \cup \{\infty\}$. Stereographic projection: A mapping from a sphere in $\mathbb{R}^3$ to the extended complex plane. Möbius ...
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In the conformal mapping, can I choose the exactly same points corresponding to the domain?

D = {$z \in \mathbb{C}: |z| < 1 $} and $\Omega$ = {$z \in \mathbb{C}: |z-\sqrt3i| < 2$} $\cap $ {$z \in \mathbb{C}: |z+\sqrt3i| < 2$}. I want to get the explicit solution of this conformal ...
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Proving $u$ is harmonic with the Dirichlet Problem

I'm studying the Dirichlet problem in the Complex Analysis book from Stein and Shakarchi. For the exercises I was a bit confused so I looked up examples and came across this this useful page. I ...
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a question on mapping properties of complex functions

I want to show that the mapping given by $$w = f(z) = - \frac{1}{2} \left( z + \frac{1}{z} \right)$$ is a bijective mapping from the upper half disc to the upper half plane. One-to-one case is ...
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Horizontally weakly conformal maps between Riemannian manifolds applied to complex functions on $\mathbb{R}^3$

I know that a map $\phi:(M,g)\to (N,h)$ between Riemannian manifolds is horizontally weakly conformal at $p\in M$ if and only if the differential $d\phi_p=0$ or $d\phi_p$ maps the horizontal space $\{\...
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Conformal map from $\Omega = \{z \in \mathbb{C} | \operatorname{Re}(z)>0\} \setminus [0,1]$ to unit disk

Find a conformal map $f$ from $\Omega = ${$z \in \mathbb{C} | \operatorname{Re}(z)>0$}$ \setminus [0,1]$ to unit disk $\mathbb{D}$. Also make sure that $f(2) = 0$. I drew $\Omega$ (see image below,...
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3 votes
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Is every transitive action conformal?

Suppose that a Lie group $ G $ acts transitively on a manifold $ M $. If $ G $ is compact then we can equip $ M $ with a Riemannian metric with respect to which $ G $ acts by isometries. What if $ G $ ...
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Are meromorphic functions generally (anti)conformal

All analytic functions are conformal whenever their derivatives aren’t $0$. $x^{-1}$ is anticonformal (preserves angles but changes orientations) whenever its derivative isn’t $0$ or $\infty$ Are ...
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Proving that a map with the holomorphic branch of the square root is a conformal map

Let $\Omega = \mathbb{C} \setminus \{z=iy \in \mathbb{C} \mid y \in \mathbb{R}, |y| \geq 1 \} $. Also let $R : \Omega \rightarrow \mathbb{C}:z\mapsto (z^2+1)^{1/2}$ be the holomorphic branch of the ...
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Maximal compact in conformal group is isometry group

Let $ M $ be a compact Kahler manifold. Then is it true that the maximal compact subgroup of the conformal group is the isometry group? This seems true for the Riemann surface case. If $ Conf(M) $ ...
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Volume of a Riemannian manifold with a scaled metric

I'm confused with the volume of an n-dimensional Riemannian manifold with a scaled metric. Specifically, let $M$ be a Riemannian manifold with Riemannian metric $g$ and its volume denoted by $vol_g(M)$...
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The problem of conformal mapping of a set onto the upper half-plane.

I have a some trouble with solving the following problem: Map area conformally to upper half-plane $S$ = {$z \in \mathbb{C}: Im(z) > 0$} $\cup $ {$z\in \mathbb{C}: |z+i| > \sqrt2 $}. I tried to ...
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Different definitions for the Teichmüller space of puctured spheres

My question is about understanding of equivalence of two different definition for the Teichmüller space of hyperbolic surfaces $\mathbb{S}^2 \setminus P$, where $|P| \geq 3$. The first definition for ...
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Schwarz-Christoffel like mapping

I'm trying to prove the result stated in the accepted answer of this SE post. That is, let $f(z) = \frac{1}{z} \prod_{j=1}^n (z - a_j)^{\lambda_j}$, with the $a_j \in S^1$ distinct and oriented ...
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3 votes
1 answer
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conformal map from disc to funny domain

I am working on numerical experiment about the Riemann mapping theorem. In order to test my program, I would like to have some test function on highly non-symmetric and non convex shape, funnier than ...
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Show that any biholomorphic map from the upper half plane to itself is a Mobius transform

Show that any biholomorphic map from $U = \{z \in \mathbb{C} : Im(z)>0\}$ to $U$ is a Mobius transformation. I know this statement to be true, just having a hard time directly proving this, I ...
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2 votes
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Harmonic function extension and Conformal maps

Let $\bigcup_{j=1}^n [0,a_i] = S$ be a 'star' where $a_i \in \mathbb{C}$ and the $[0,a_i]$ denote the line segments from $0$ to $a_i$ in the plane, all of the $a_i$ here are distinct and nonzero. ...
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is there a mapping of a sphere to a pseudosphere

I'm starting to study differential geometry. is there a mapping brings a sphere to a pseudosphere? the Riemann sphere mapping from the sphere to the complex plane makes me think there is one, but I ...
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6 votes
1 answer
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Are conformal maps on a hyperbolic surface isometries?

Let $S$ be a closed surface of genus $g\geq 2$ and $h$ a hyperbolic Riemannian metric (constant curvature $-1$) on $S$. A conformal diffeomorphism $f \colon (S,h) \to (S,h)$ is a diffeomorphism that ...
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2 votes
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extension problem with motivation

I am motivated by the Lorenz curve used in economics and statistics - a proper multivariate generalization will help researchers (Arnold, Taguchi, etc.) assess multidimensional risk and inequality (...
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