# Questions tagged [conformal-geometry]

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### Conformal welding of rectifiable curves

In classical conformal welding theory, we start with a homeomorphism $h$ of the unit circle and try to find a Jordan domain $D$ together with two conformal isomorphisms $f_1 \colon \mathbb D \to D$ ...
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### Assuming the conformal factor is radially decreasing, prove or disprove the uniqueness of geodesic joining origin and points on the boundary of ball

Let $u$ be a radially decreasing function defined on $\mathbb{R}^n$. We consider the metric $g=e^{2u}\delta$ where $\delta$ is the standard Euclidean metric on $\mathbb{R}^n$. Let $B_r$ be the ball ...
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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 ...
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### Conformal maps between simply connected domains with piecewise real algebraic boundary

Between polygons in $\mathbb C\cup\{\infty\}$ (including the "single side polygons", hemispheres, disks) the Schwartz-Christoffel mappings give arguably explicit conformal maps. For polygons with few ...
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### Regularity of conformal maps

In order to define what it means for a map $f \colon \Omega \subseteq \mathbb R^n \to \mathbb R^n$ to be conformal, it is sufficient to require that $f$ is everywhere differentiable. Does conformality ...
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### Conformal map from a 7-sided polyhedron to a square pyramid

I have a right-angled square pyramid, $A$, whose height and base-length are $l$. By 'right-angled', I mean that the apex of $A$ lies vertically above one of the vertices in its base. Now supposed I ...
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### Counterexample to a generalization of planar convex hulls

I am looking for a counter example to the following conjecture: for every finite compact region $\mathcal{R}\subset\mathbb{R}^2$ that is defined by a rectifiable Jordan curve and every fixed and ...
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### Conformal map from quadrilateral to a sector of a circle [duplicate]

Can anyone advise me on how to derive a conformal map for this mapping? I am familiar with how to apply Schwarz-Christoffel from the upper half plane to the quadrilateral, but how do I then map from ...
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### Reconciling Sullivan's theorem with the hyperbolic structure of the Figure–8 knot complement

I am interested in the 3-manifolds with hyperbolic structures from the physics (gravity) perspective. I encounter this paper https://arxiv.org/pdf/hep-th/9812206.pdf whose Eq. (9) mentions a theorem ...
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### Conformal map onto a circle, from an identification space composed of five squares

I am looking to derive a conformal map for the problem illustrated in this image. I've read a bit about how to map a square onto a circle, but I'm struggling to extend the concepts for the domain at ...
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### Is every conformal manifold equivalent to a flat one with cone singularities?

Consider a polytope with a $2$-dimensional surface and the corresponding metric on this surface (coming from the embedding in $3$-dimensional Euclidean space). Intrinsically the metric is flat ...
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### Triangulations of conformal manifolds

I'm seeking for a discrete representation of $2$-manifolds with a conformal structure (i.e. metric modulo scalar prefactor). The topology of a $2$-manifold is determined by the combinatorics of a ...
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### What is the value of the partition function of CFT on a compact conformal manifold?

Is the value of the partition function of a 2d CFT on a compact conformal manifold well defined? Or is there some kind of "anomaly" that makes it dependent on a bulk or some other kind of further ...
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### Is the conformal compactification of $M \setminus \{ p \}$ unique?

Let $(M,c)$ be a compact conformal manifold and $p \in M$. $M$ is a conformal compactification of $M \setminus \{ p \}$, because the embedding $M \setminus \{p\} \hookrightarrow M$ is an isometry. ...
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### Left passage probability of $SLE_8$?

Schramm's formula on left passage probabilities of $SLE_k$ is stated for $k \in [0,8)$ in theorem 2 here. However, after the statement he remarks that the formula simplifies to $1/2$ for $k = 8$. It ...
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### Rigid motions between two spheres [closed]

It has been well known that every Mobius transformation can be constructed by stereographi projection of the complex plane onto a sphere, followed by a rigid motion of the sphere and projection back ...
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### Conformal group and cobordism

In this post, I am exploring my thoughts on the implementation of conformal symmetry group structure and cobordism relations. Namely, I like to know what has been done and explored in the past? on ...
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### Completeness is a conformal invariant

In this article about completeness of semi-riemannian manifolds the author writes (in section 2.2) that it is unkown if the following statement holds: A compact indefinite manifold which is conformal ...
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### Hodge theory, conformal manifolds and Fredholm modules-understanding the proof of one Lemma

I would like to understand the proof of Lemma 1, page 339 in this book. Very briefly, the context is as follows: we have even dimensional oriented conformal manifold with the Hodge star operator ...
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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 ...
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### Are conformal maps between Riemannian manifolds real-analytic?

This is a cross-post. Let $M,N$ be oriented smooth ($C^{\infty}$) $n$-dimensional Riemannian manifolds, and let $f:M \to N$ be a smooth orientation-preserving weakly* conformal map. Do there ...
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### numerically approximating the conformal map between two curvilinear triangles to high precision

Here is a triangular region $T$ whose two curved edges are complicated analytic curves that I know only numerically, but can compute to any desired precision: And here is a simpler region $H$ whose ...
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### Conformal mappings that preserve angles and areas but not perimeters?

Conformal mappings from $U$ to $V$, both subsets of $\mathbb{C}$, locally preserve angles. But, in general, such mappings neither preserve areas nor preserve perimeters. Q. Are there examples of ...
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### Is the conformal compactification of a convex-cocompact hyperbolic manifold $M$ conformally diffeomorphic to the convex core of $M$?

I am not a hyperbolic geometer, so I apologize if I get anything wrong here, and please correct me. The conformal compactification $\overline {\mathbb{H}^n}$ of hyperbolic $n$-space $\mathbb{H}^n$ ...
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### Reference request: normal form of k-differentials and flat surfaces at a puncture

Let $f(z)$ be a holomorphic function defined on a punctured neighborhood of $z=0$ with non-essential singularity of degree $d$ at $0$ (namely, $f(z)=z^dh(z)$, where $h(z)$ is a holomorphic function ...
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### Equality on $\partial \mathbb{H}$ of lifts for isotopy to a conformal map

Let $\mathbb{H} \subset \mathbb{C}$ be the upper half plane. First recall the following statement: if $f^* \colon \mathbb{H} \rightarrow \mathbb{H}$ is quasi-conformal (qc), then there exists an ...
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$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\CO}{\text{CO}(#1)}$ $\newcommand{\dist}{\operatorname{dist}}$ $\newcommand{\g}{\mathfrak{g}}... 1answer 490 views ### Proofs that the conformal group in dimension$\ge 3$is a Lie group Let$M$be a smooth manifold of dimension$\ge 3$, equipped with a conformal structure (or a Riemannian metric). Then, the group of conformal diffeomorphisms is a finite dimensional Lie group. A ... 2answers 530 views ### Is there a conformal diffeomorphism between R3 minus a line and R2 x S1? There exists a conformal diffeomorphism between$\mathbb{R}^3$and$S_3$(less a point): $$g = dr^2+r^2\left(d\theta^2 + \sin^2\theta\ d\phi^2\right)$$ $$r = R \tan \frac{\alpha}{2}$$ $$g = \... 1answer 104 views ### conformal mapping and rational function Let E be an infinite compact subset of the complex plane \mathbb{C} such that \overline{\mathbb{C}}\setminus E is simply connected. By Riemann mapping theorem, there exists a unique exterior ... 0answers 93 views ### How do conformal maps affect curvature? Let (\overline{M}^{n+1}, \langle \cdot, \cdot \rangle) be a riemannian manifold with riemannian connection \overline{\nabla} and consider M^n \subset \overline{M} an orientable hypersurface with ... 1answer 156 views ### Vanishing of determinant of Cotton York tensor suppose \Omega \subset \mathbb{R}^3 is a Riemannian manifold with g= dx_1^2 + dx_2^2 + c(x_1,x_2,x_3)dx_3^2. Is it true that det(CY)=0? Thanks 0answers 57 views ### Will a slightly differently shaped torus make this guess about plane sections of a torus true? Jacobi's elliptic functions and plane sections of a torus After Greg Egan posted an excellent answer to a question of mine, which I accepted, I posted my own answer, linked above. The question ... 1answer 155 views ### \zeta(2n) and Levy processes I am missing some steps in the final derivation of a probabilistic computation of the even values of \zeta. They show the Cauchy distribution is relate to a certian Levy process:$$ |\mathbb{C}_1| \... 1answer 218 views ### Uniformisation for non simple closed curves Given a simple closed curve in the plane, there is a homeomorphism from the unit open disk to the interior of the curve. The homeomorphism can be taken conformal, this is the uniformisation theorem. ... 0answers 92 views ### conformal mapping and its residue Let$T$be a closed rectifiable Jordan curve in$\mathbb{C},G$be the interior of$T,$and$\Phi$be a conformal map of$G$onto the unit disk$\mathbb{D}.$My question is the following: for$n\in ...
Let $\Omega\subset R^d$ be a nice bounded domain (say, the unit ball). Consider the following equation for $f:\Omega\to R^d$, $$\operatorname{div}((\det \nabla f)^{1-2/d} \nabla f) = 0.$$ Suppose ...