Questions tagged [conformal-geometry]
The conformal-geometry tag has no usage guidance.
194
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Growth/Decay of conformal Killing fields in cone metrics
Let $\gamma$ be a smooth metric on $S^2$ of positive curvature. Consider the metric
$$g= dr^2 + r^2 \gamma$$
on $[1,\infty) \times S^2$.
Does there exist a nontrivial conformal Killing field vanishing ...
2
votes
1
answer
84
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Parameterizing Teichmüller spaces of punctured surfaces
Let $S_{g,n}$ denote a genus $g$ surface with $n$ punctures. There is a map $F$ from the Teichmüller space of the punctured surface $T(S_{g,n})$ to the Teichmüller space of the compact surface $T(S_{g}...
7
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1
answer
659
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Analogy of Liouville conformal mapping theorem with Mostow rigidity?
I often hear mention of two theorems, Mostow's rigidity theorem and Liouville's theorem on conformal mappings, which superficially sound similar: they say that a set of geometric structures is, if ...
2
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1
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134
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A formula for the cross-ratio in terms of hyperbolic data
Let $(\zeta_i) \subset \hat{\mathbb{C}}$, for $i = 1, \ldots, 4$, be $4$ distinct points on the Riemann sphere $\hat{\mathbb{C}}$.
We will use the following convention for the cross-ratio $CR$ of ...
2
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0
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87
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Convergence of diffeomorphisms
Let $(\Sigma, g)$ be a compact $n$-dimensional Riemannian manifold without boundary. Let $F_i$ be a sequence of diffeomorphisms of $\Sigma$ and $u_i$ be a sequence of positive scalar functions.
...
0
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1
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113
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Expansion of metric near boundary of 3 dimensional Poincaré-Einstein/hyperbolic manifolds
In Mazzeo-Alexakis, there is a brief discussion that if $(M^3,g) \sim \mathbb{H}^3/\Gamma$ (for $\Gamma$ convex cocompact), then the metric can be expanded near the topological boundary as
$$g = \frac{...
2
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3
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262
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Let $M$ be a manifold with a conformal structure and a volume measure. How can one reconstruct a metric on $M$ from this data in a geometric way?
Let $(M,g)$ be a (semi-)Riemannian manifold, and let $(M,[g])$ be the conformal class thereof. The following two sets are naturally torsors for the collection of positive-valued functions on $M$:
The ...
4
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1
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244
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Conformal maps between two given domains
Consider two domains
$$
\begin{aligned}
D_1&=\{x=(x_1,x_2,...,x_n)\in\mathbb{R}^n:x_n\leq 0\},\\
D_2&=\{x=(x_1,x_2,...,x_n)\in\mathbb{R}^n:x_n\leq \psi(x_1,x_2,...,x_{n-1})\},
\end{aligned}
$$
...
3
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1
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140
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Do we have uniformization theorems for fractional dimensional spaces?
The Riemann mapping theorem in $\mathbb{R}^2$ is known not to generalize well in higher dimensions and is basically trivial in lower dimensions.
I’m interested in how it generalizes for fractional ...
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0
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76
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Metric balls in Teichmüller space are topological balls
Let $X$ be a topological surface of finite type and $\mathcal{T}_X$ be the corresponding Teichmüller space. Let $B$ be a ball with respect to the Teichmüller metric on $\mathcal{T}_X$ (i.e., the ...
7
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1
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705
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Definition of the conformal metric
On page 16 of this lecture notes and in this lecture, at some point (somewhere at 1:37:45), Rod Gover defined the $\text{conformal metric}$ $\mathbb{g}$ on a conformal manifold $(M, [g])$. He ...
2
votes
1
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331
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Modulus of a torus
Suppose I have a torus (you can pretend that it is the usual torus of revolution: the boundary of a tubular neighborhood of radius $r_2$ of a circle of radius $r_1.$ The question is: can one compute ...
0
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146
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Conformal diffeomorphism of $\mathbb R^k$
Let $f$ be a conformal diffeomorphism between $\mathbb{R}^k$, where $k\geq 2$, and its Euclidean metric. It follows from complex analysis and Liouville's theorem that $f$ can only be affine. Now, ...
2
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0
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105
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Conformal changes of metric and normal coordinates
Suppose that $(M,g)$ is a smooth Riemannian manifold of dimension $n\geq 2$. Let $p\in M^{\textrm{int}}$. Does there exist a small $\delta>0$ and a smooth function $c>0$ such that for the ...
4
votes
1
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215
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Is every Riemannian metric conformally equivalent to one that is geodesically complete?
The question in the title seems a natural one to ask, and I suspect that it is already considered, even completely solved, somewhere in the literature. Although I prefer some explanation of the idea(s)...
4
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1
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260
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Can every diffeomorphism be rescaled into a volume preserving one?
This is a cross-post. Let $D \subseteq \mathbb{R}^2$ be the closed unit disk, and let $f:D \to D$ be a diffeomorphism.
Does there exist a smooth $h \in C^{\infty}(D)$ such that $h\cdot f$ is an ...
5
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0
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100
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Standard 2-instantons on the 4-sphere under conformal transformation
It is well-known that there is a standard SU(2) 1-instanton on the 4-sphere and any 1-instanton can be obtained from the standard one by conformal transformation. Is there any explicit (family of) &...
4
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0
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108
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Does the conformal factor satisfy some equation independent of the vector field?
Let $(M, g)$ be an $(n+1)$-dimensional (pseudo-)Riemannian manifold and for any
$X \in \mathcal{X}(M)$ define its deformation tensor by
\begin{align*}
{} ^{(X)}\pi_{ab} := (\mathcal{L}_X g)_{ab} = \...
-3
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1
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312
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Weyl tensor of a Riemannian metric $g$
Does Weyl tensor of a Riemannian metric $g$ give information about the conformally-flatness of $g$?
4
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0
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273
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CFT as an axiomatic field theory
I'm trying to understand CFT from a purely axiomatic-field-theoretical perspective. That is, there is a vector space $V$ associated to the circle, and an element of $V^{\otimes n}$ associated to every ...
4
votes
1
answer
238
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Yamabe operator, conformal transformations and square of the Dirac operator
On an $(M,g)$ compact n-dimensional Riemannian manifold the Yamabe operator $Y = 4(n-1)/(n-2)L + s$, where $L$ is the Laplacian and $s$ is the scalar curvature is conformally covariant, which for me ...
2
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173
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Geometric characterizations of conformal maps
I have some fairly basic questions with regards to conformal structures/maps -- I apologize if they are on the basic side for MO, but I figured I might get some clarity here.
Suppose $X$ and $Y$ are ...
6
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4
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530
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Generalizing contour integration to quaternions and bicomplex numbers
I am interested in the possibility of generalizing the notion of contour integration to the quaternions or bicomplex numbers. I am aware that the Frobenius theorem prevents the construction of a true ...
2
votes
1
answer
258
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Explicit universal covering map for higher genus algebraic curves
Suppose I have a projective plane curve $C = V(F)$ defined over $\mathbb{C}$, where $F$ is some homogeneous polynomial in three variables. For the sake of simplicity, let's assume that $C$ is ...
4
votes
1
answer
184
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Conformal groupoid
I asked this over on Math.SE but it remained completely silent for over a week so I've deleted it and am reposting it here (I'm not really sure which site it fits better). The question itself is ...
1
vote
1
answer
147
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Uniqueness of "stretching" (subject to constraints) for a two-dimensional figure
The New York Times, reporting on Dennis Sullivan's Abel prize, recounts the incident that lured Sullivan from chemical engineering to mathematics:
One day during an advanced calculus lecture, the ...
2
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0
answers
71
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Equicontinuity, Beltrami coefficients and Sequence of top/bottom semi-annuli
In the Beltrami equation literature, one approach to showing equicontinuity for pairs $(f_{n},\mu_{n})$ (where $\partial_{\bar{z}}f_{n}=\mu_{n}(z)\partial_{z}f_{n}$) is via the relations of moduli and ...
5
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124
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Examples of compact Kähler manifolds whose Bochner curvature tensor has constant norm?
The Bochner curvature tensor is the Kähler analog of the Weyl curvature tensor in the curvature decomposition of a Kähler, discovered by Bochner in 1949. The article on Bochner-Kähler metrics by ...
2
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67
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Triangulations with discrete metrics and conformal equivalence
A discrete metric for a triangulation of a 2-dimensional manifold is a map associating $\mathbb{R}_+$-valued lengths to all edges, such that the triangle inequality holds on every triangle. In many ...
7
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0
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160
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What are the generators and relations of the conformal cobordism category?
According to a definition by Segal, a $2$-dimensional CFT is a symmetric monoidal functor from the category of oriented conformal cobordisms to the cateogry of projective complex vectorspaces. Coming ...
2
votes
2
answers
1k
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What is a simplified intuitive explanation of conformal invariance? [closed]
Can the concept of conformal map and conformal Invariance be explained in very general terms, preferably in high school/undergrad-level Mathematics? Abstracting away from the applications in physics (...
1
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0
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204
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Divergence of conformal Killing vector fields on $S^2$ and the spherical harmonics
Can anyone think of a conformal Killing vector field $W$ on $S^2$ with the round metric that is not Killing such that its divergence is $L^2$-orthogonal to the spherical harmonics with $\ell = 1$?
One ...
2
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2
answers
281
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General conformal transformation extension to hyperbolic manifold
I have a hyperbolic manifold with boundary a conformal sphere. Can I extend any conformal transformation of the boundary to the interior of the ball? I know how to do with Moebius but I wonder if ...
11
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2
answers
1k
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Jacobi's elliptic functions and plane sections of a torus
In $\mathbb R^3$ with Cartesian coordinates $(x,y,z),$ revolve the circle $(x-\sqrt 2)^2+z^2 =1,\ y=0$ about the $z$-axis.
This yields a torus embedded in $3$-space that is conformally equivalent to ...
1
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0
answers
94
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Gap phenomenon vs Rigidity results for surfaces
I am trying to understand the differences between the rigidity results and gap results for a given surface immersed into some manifold. For instance, a Gap theorem proved here (Theorem 2.7) says ...
2
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0
answers
104
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How to understand the constant rank theorem for semilinear elliptic equations
Let $u$ be a solution to the equation $$\Delta u=f(u,\nabla u)$$ where $f>0$ is a smooth function with $f f_{uu} \le 2f_u^2$. The seminal constant rank theorem states that if $D^2 u$ is positive ...
2
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3
answers
288
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For globally conformally flat surfaces, does radial symmetry of conformal factor imply the surface is a sphere?
We know that for the unit sphere in $\mathbb{R}^3$, the standard metric on such a sphere can be written as $$g=\frac{4}{(1+x_1^2+x_2^2)^2}(dx_1^2+dx_2^2)$$by a stereographic projection map from the ...
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0
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What does it mean for correlation functions to be dominated by the vacuum block for a 2D CFT?
In a 2D CFT, correlation functions dominated by the vacuum block have no conical defects. You can calculate the OPE and determine the correlation function using the D-S equations and cancel out UV ...
1
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0
answers
96
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References Request: Bach tensor
Recently I want to study about Bach tensor in detail. From the fundemental definition and properties to conformal invariant. Is there any references to me about Bach tensor? If there are some origins ...
3
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1
answer
155
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Finitely connected orientable surface
Let $(M,g)$ be a finitely connected orientable complete Riemannian surface, that is, $M$ is homeomorphic to a compact orientable surface $\Sigma$ minus $k \geq 1$ points. Do you have references or a ...
2
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0
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114
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Access to an old paper of Obata
I'm trying to access the following paper of Morio Obata:
Conformal changes of Riemannian metrics on a Euclidean sphere, in "Differential Geometry -- In honor of K. Yano", 1972, pp. 347--353: ...
2
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1
answer
237
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Existence of divergence-free unit vector field in conformally rescaled euclidean metric
Question. Let $\Omega \subset \mathbf{R}^2$ be a convex polygonal domain, equipped with a Riemannian metric $g$. Under which conditions on $g$ is there a vector field $X$ in $\Omega$ with $\mathrm{div}...
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0
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171
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Does there exist an isometry between a regular polygon and a circle?
In order to define the question in a meaningful fashion, I am referring to a smooth manifold $\mathcal{M}$ within an $\epsilon$-neighborhood of a regular polygon $\mathcal{P}$ satisfying $$\max\{\|x-p\...
1
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0
answers
297
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Conformal changes of metric and Ricci curvature
Let $(M,g)$ be a three dimensional smooth Lorentzian manifold and let $p$ be a fixed point in $M$ and let $S$ be a smooth symmetric tensor of rank two on $T_pM\times T_pM$. Does there exist a smooth ...
2
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0
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124
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The space of conformal classes of the $n-$sphere
It is well-known that the space of conformal classes on the $2-$sphere is just a point. What is known about the space of conformal classes on the $n-$sphere? Is there any structure on it?
3
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0
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124
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Range of divergence operator on the space of traceless symmetric $(0,2)$ tensors; conformal vector fields on an arbitrary metric on $S^2$
Let $\gamma$ be a metric on $S^2$.
I am trying to solve the following PDE on a $(0,2)$ symmetric traceless tensor $A$:
$$div_{\gamma} A = \omega$$
where $\omega$ is a 1-form.
It is known that there ...
1
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0
answers
82
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conformal changes of Lorentzian metrics
Let $M= \mathbb R \times M_0$ with $M_0$ a smooth compact manifold with smooth boundary and let $g=-dt^2+g_0(t,x)$ be a Lorentzian metric on $M$, with $g_0$ a family of Riemannian metrics on $M_0$ ...
0
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1
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538
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Self duality and anti-self duality of Weyl curvature in four dimension
I am trying to compute explicitly in terms of extrinsic curvature the self dual part $W^+$ and the anti-self dual part $W^-$ of the Weyl tensor $W$ associated with a codimension 1 submanifold into ...
7
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If SO$(3,\mathbb C)$ is isomorphic to PGL$(2,\mathbb C)$, what objects do vectors in $\mathbb C^3$ represent in the context of Möbius geometry?
I hope this question isn't too basic or ambiguous for this site.
The following is an explicit isomorphism from $\mathrm{PGL}(2,\mathbb C)$ to $\mathrm{SO}(3,\mathbb C)$:
$$\left[\begin{matrix}p & ...
24
votes
3
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3k
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How to find a conformal map of the unit disk on a given simply-connected domain
By the classical Riemann Theorem, each bounded simply-connected domain in the complex plane is the image of the unit disk under a conformal transformation, which can be illustrated drawing images of ...