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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Is the complex tangent Injective?
So if we consider the Analytic function $f(z)=\tan(z)$ when $z\in \{z\in\mathbb{C}\text{ : } -\pi/2<Re(z)<\pi/2 \}$ is it injective ? Its derivative is non zero however this only guarantees ...
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A problem in the proof of mapping formual (H →P) in stein Complex Analysis chapter8
We know that $F$ is a conformal map of $\mathbb{H}$ to $P$ ($P$ is a polygon)
Let$$
h_{k}(z)=(F(z)-a_{k})^{ \frac{1}{\alpha_{k}}}
$$
Then
$$
\frac{F^{''}(z)}{F^{'}(z)}=-\beta_{k} \frac{h^{'}_{k}(z)}{...
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Green's function of the conformal Laplacian
I am reading T. Parker, S. Rosenberg, "Invariants of conformal Laplacians", J. Differential Geom. 25(2): 199-222 (1987). I would like to understand how Green function changes if the metric ...
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Conformal coordinates - Isothermal coordinates
Given a smooth and closed surface $S$ parameterized by coordinates $u,v$ such that its metric can be written in the form $ ds^2 = \phi(u,v)^2(du^2 + dv^2)$ implies that the coordinates $u,v$ are ...
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Isometries as a subgroup of conformal diffeomorphisms
I have a question that, if true, does not seem trivial to prove, but if false, I am unable to find a counterexample. The question is as follows:
Given a 3-dimensional Riemannian manifold $M$, is the ...
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Conformal map of two intersecting circles
While studying for my Complex Analysis course, I came across the following exercise:
You are given the following region
$$\Omega = \left\{ z \in \mathbb{C} \mid |z - i| < \sqrt{2} \text{ or } |z + ...
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Doubt in Mobius geometry
I am currently studying Mobius geometry. I found a group in Mobius geometry called Mobius group which contains Mobius transformations. I have the following doubt.
Dose this group contain ...
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Area of image of holomorphic function on the disk [duplicate]
I'm working on an old qualifying exam problem, which asks me to show that if $f:\mathbb{D}\rightarrow \mathbb{C}$ is holomorphic and injective and $f'(0)=1$, then the area of $f(\mathbb{D})$ is at ...
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Comparing areas between conformal metrics in $\mathbb{R}^2$
I would like to ask for a reference on the following subject:
Let $f:\mathbb{R}^2\longrightarrow\mathbb{R}$ be a radial positive function, i.e., $$f(x,y)=\lambda(x^2+y^2)$$ for some $C^\infty$ ...
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Torus equation in conformal geometric algebra
How would you define a torus using conformal geometric algebra?
Since CGA has circles as a primitive, It seems to me that we should be able to able to define a torus as a circle C rotated around a ...
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Integral representation of conformal map on the upper half plane
It is a fact that there is a one to one correspondence between the space $M(k)$ of finite, signed Borel measures on $\mathbb{S}^1$ with total mass equal to $2$ and total variation equal to some $2 \...
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Stein and Shakarchi Complex Analysis Chapter 8, Proof of Lemma 1.3
I am confused about the proof to Lemma 1.3 in Stein and Shakarchi's Complex Analysis Chapter 8 (Conformal Mappings).
Here is a screenshot of the Lemma and its proof:
My question is is the following: ...
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Image of upper half-plane under sin(z) is not the half strip (Stein and Shakarchi Complex Analysis, Chapter 8)
In section 1.2 of Chapter 8 Conformal Mappings of Stein and Shakarchi's Complex Analysis, it is said that the map $f(z) = \sin(z)$ takes the upper half-plane conformally onto the half-strip $\{w = x + ...
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Schwarz-Pick Theorem Exercise
I am trying to solve the following problem, which I believe should be solved using the Schwarz-Pick Theorem.
Let $\mathbb{D}$ be the open unit disk and let $f\colon \mathbb{D}\xrightarrow{} \{z\in \...
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Dirichlet problem with conformal maps
The problem I am struggling to solve is from Brown and Churchill's Complex Variables and Applications and is intended to be solved using conformal maps:
Derive an expression for steady state ...
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Conformal mapping concerning a branch cut
Map the region $\Omega = \{z=re^{i \theta} \in \mathbb{C}: r >0, \frac{\pi}{4} < \theta < \frac{7\pi}{4} \} \setminus [-1,0]$ onto $\mathbb{H}$.
Thus, the $\Omega$ is the complex plane with ...
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conformal maps of S2 to compact topologically equivalent region in R3?
How do I conformally, compactly map the unit sphere in $R^3$?
I know that if I have a compact, simply connected region of the plane, $R^2$, I can create a conformal map onto a new region in the same ...
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Definition of an inner product.
I was solving some problems on isometries on the upper half plane and I came up with the following post: Excercise in isometries of the half upper plane here the answer states the following inner ...
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Conformal map between a domain minus an arc and the unit circle
I was trying to understand the paper computability of Brolin-Lyubich measure and, in the last counter-example of the paper (page 25), they say about a conformal map between the set $(\mathbb{C}\...
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Prove that there is no conformal mapping between $\mathbb{D}\setminus\{0,1/2\}$ and $\mathbb{D}\setminus\{0,1/4\}$
I have to prove that there is no conformal mapping between $\mathbb{D}\setminus\{0,1/2\}$ and $\mathbb{D}\setminus\{0,1/4\}$. I honestly have no idea. Intuitively, I guess that the problem has to do ...
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Conformal map a quarter disc into a horizontal strip
I'm interested in mapping the quarter disk $\{z = x+iy \in \mathbb{C}: 0 < x < 1, 0 <y<1 \}$, i.e. the intersection of $\mathbb{D}$ with the first quadrant to the horizontal strip $\{z=x+...
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What is C minus infinitely many disjoint "Jordan Balls" conformally equivalent to?
So I recently came across a Theorem due to Koebe proved in 1918 that states that an M connected domain $\mathbb{D}$ in $\mathbb{C}$, that is a a domain such that $\partial \mathbb{D}=\bigcup_{i=1}^{k}\...
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Show that the conformally complete Schwarzschild spacetime is asymptotically flat at null infinity
I am trying to show that the conformal factor used to conformally complete the
Schwarzschild spacetime renders it asymptotically flat at null infinity (according to the mathematical definition given ...
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Surjectiity of a conformal map
I'm stuck on this exercise:
Let $\left\{ z \in \mathbb{C} : z = x + iy, \; y > 0 \right\}$
Check whether the application $\phi : {\mathbb H}→ {\mathbb C}$ where ${\mathbb C}$
given
$\phi(z) = z^4 - ...
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Construct an explicit biholomorphism between two domains
This problem is from my homework: Construct an explicit biholomorphism between $D_1=\mathbb{C}-\{-x\pm\sqrt{-1}\pi\mid x\ge1\}$ and $D_2=\{x+\sqrt{-1}y\mid -\infty<x<\infty, -\pi<y<\pi\}$.
...
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How to Find a Conformal Mapping from a Crescent-Shaped Region to an Annulus?
I am trying to find a conformal mapping from
$\{𝑧:∣𝑧∣<1\}\cap\{𝑧:|z−\frac{1}{2}∣>\frac{1}{2}\}$ onto an annulus.
This question is the very last problem from an old complex analysis exam.
I ...
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Determining a conformal mapping to the unit disk
Let
$$\Omega = \left\{z \in \mathbb{C} : |z - 1| > 1, |z - 2| < 2, \Im(z) > 0\right\}.$$
We want to determine a conformal map which maps $\Omega$ to the unit disk. We solved it by composing $...
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Using conformal mapping to solve Laplace equation
Given the circles $$C_1 : x^2 + y^2 = 1,\quad C_2 : 5x^2 - 4x + 5y^2 = 0$$ let $D$ be the finite region between $C_1$ and $C_2$. Using the conformal mapping $$w = \frac{z-2}{2z-1}$$ solve the problem $...
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Let $D=\{z\in \Bbb C; 0<\Im z<\pi, z\not\in [0,\pi i/4]\cup [3\pi i/4,\pi i]\}$. How to find a conformal map from $D$ to the upper half plane?
Let $D=\{z\in \Bbb C; 0<\Im z<\pi, z\not\in [0,\pi i/4]\cup [3\pi i/4,\pi i]\}$. Here $\Im z$ is the imaginary part of $z$. How to find a conformal map from $D$ to the upper half plane?
My ...
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Combining the two conformal mapping equations
The mapping function ${Z_{\rm{1}}}$, which transforms the outside region of a unit circle (in the $\zeta$-plane) onto the outside region of the unit circle with two unequal radial aligned cracks (in ...
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Conformal map between $D-[\alpha,1)$ and $D-[0,1)$
Let $\alpha$ be real, $0 \leq \alpha < 1$. Let $U_{\alpha}$ be the open set obtained from the unit disc by deleting the segment $[\alpha, 1)$
Find an isomorphism of $U_{\alpha}$ with the unit disc ...
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Freitag and Busam Complex analysis conformal map condition theorem
Theorem I.5.15 of the book (Freitag&Busam complex analysis) says: A map $f:D\to D'$, where $D,D'$ open in $\mathbb{C}$, is locally conformal if and only if it is analytic and its derivative is ...
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Can two conformal mappings be combined?
A conformal mapping function ${z_1} = {\omega _{\rm{1}}}(\zeta )$ within a complex domain maps the unit circle in the $\zeta$-plane to the $z_1$-plane. Similarly, there is another function ${z_2} = {\...
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Understanding the Harmonic Energy for Maps Between Riemann Surfaces？
I am trying to understanding this functional, the thing I got confused about is the meaning of term $u_z\overline{u}_{\overline{z}}+ \overline{u}_{z}u_{\overline{z}}$ ?
Because harmonic map is almost ...
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Conformal equivalence from $D = \{z : 0 < \arg z < 3\pi / 2\}$ to $S = \{z: 0 < \Im z < 1\}$
I would like to verify if the conformal equivalence from $D = \{z : 0 < \arg z < 3\pi / 2\}$ to $S = \{z: 0 < \Im z < 1\}$ I have found is correct.
First, consider the map $f: D\to \mathbb{...
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Can a Lambert conformal conic projection be constructed geometrically?
A gnomonic projection can be constructed geometrically by putting a light source in the middle of a semi-transparent globe to project an image of half of tyat globe onto a plane. A stereographic ...
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Clarification on one-to-one mapping of a disk in complex analysis
I have a some difficulties understanding the proof in section 7.2 of "Fundamentals of Complex Analysis with Applications to Engineering, Science, and Mathematics" by Edward Saff and Arnold ...
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Existence of extremal map in teichmuller class
Let $R$ be a Riemann surface, and define two quasiconformal maps $f_i : R \rightarrow R_i, i=1,2$ to be equivalent if there exists a conformal map $c : R_0 \rightarrow R_1$ and a homotopy $g_t : R \...
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Analyzing conformality in a mapping function for unequal cracks
The mapping function for transforming the exterior of the unit circle with unequal cracks on both sides (in the $\zeta$-plane) to the exterior of the unit circle (in the $z$-plane) is as follows:
$${\...
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Why the name linear fractional map?
Fractional linear transformation is a map from extended complex plane to itself, defined by:
\begin{equation}
z\to \frac{az+b}{cz+d}
\end{equation}
with $ad-bc\ne0$.
Wikipedia says that "a linear ...
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Conformal Maps and Laplace Equation [closed]
So, I'm revisiting Complex Analysis mainly focused on the application of conformal maps in PDE's. Using Churchill's book (Complex Variables and Applications), the way to solve was using the fact that ...
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Why are there only six linearly independent global conformal Killing vector fields on the two-sphere with the round metric?
The title pretty much sums it up. For a 2-manifold with metric $\gamma_{AB}$, a vector field $Y^A$ is said to be a conformal Killing vector field if $\nabla_A Y_B + \nabla_B Y_A = \nabla_C Y^C \gamma_{...
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Is this conformal map correct?
The question is: does there exist a linear fractional (Möbius) transformation that maps the set $U = \{z \in \Bbb{C} \mid |z-1|<1, |z-i| < 1\}$ onto the quarter plane $\operatorname{Im}z>0, \...
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Conformal Killing fields satisfy a third order differential equation
Let $(M,g)$ be a $n$-dimensional Riemannian manifold with smooth metric $g$.
A smooth vector field $X$ is conformal Killing if
$$\nabla_iX_j + \nabla_jX_i - \frac{2}{n} \text{div} X \, \,g_{ij} = 0$$
...
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Determining Conformal Map from Conformal Factor
I am working on a project and need to apply a certain (Lorentzian) conformal transformation, $\psi:\mathbb{R}^2\mapsto \mathbb{R}^2$, to a figure which I have generated numerically in Mathematica. I ...
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Area between four parabolas
Say I have a region with four "corners" that are connected by parabolas (like in the picture below). Is there a nice way to compute the enclosed area? To make things simple, say the ...
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What does $dzd\bar{z}$ mean?
I've seen people write $g(z) = dzd\bar{z}$ to refer to the standard conformal metric on the complex plane $\mathbb{C}$ (e.g., on Wikipedia). I thought this was referring to the tensor product $dz \...
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Find a conformal mapping from Q (in the picture) onto the upper half plane set D = {z: Im(z) > 0} [closed]
So I got stuck trying to find a conformal mapping from Q onto the upper half plane set D = {z: Im(z) > 0}.
I've ultimately arrived at the conclusion that maybe $w = z^3$ could work as the first ...
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Is this family of functions normal?
I wanna tackle the following exercise:
Let $\mathcal{F}$ be the class of all functions $f \in \mathcal{H}(\mathbb{D})$ satisfying $f(0)=1$ and $\Re(f) > 0$. Show that $\mathcal{F}$ is a normal ...
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Conformal map from upper-half plane to the plane with several rays removed
I'm having trouble solving the following exercise in complex analysis. Let $L_k:z=r\mathrm{e}^{\mathrm{i}\frac{2k\pi}{n}}(1\leqslant r<\infty)$ be a ray for $k=1,\cdots,n$, and $\Omega=\mathbb C\...
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