# Questions tagged [conformal-geometry]

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137
questions

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73 views

### 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 $f_1 \colon \mathbb D \to D$ ...

**1**

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**1**answer

65 views

### 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 ...

**3**

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154 views

### 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 ...

**2**

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**1**answer

65 views

### 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 ...

**13**

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**1**answer

282 views

### 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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175 views

### 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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28 views

### 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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26 views

### 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 ...

**5**

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**1**answer

251 views

### 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 ...

**3**

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**1**answer

196 views

### 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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238 views

### 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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37 views

### 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 ...

**4**

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135 views

### 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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74 views

### Link between Yamabe invariant and Yamabe equation

I am trying to understand the solution to the Yamabe problem as presented by Lee and Parker. It seems to me that the constant $\lambda$ which appears in the Yamabe equation $$\square\varphi = \lambda \...

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**1**answer

105 views

### Can an “annular” subset of an annulus be conformally equivalent to the whole annulus?

Assume we are given an annulus
$$A = \{ z \in \mathbb{C}: 1< |z| < R\}.$$
Let $\phi\colon A \to A$ be a univalent map such that the image of $\phi$ contains a curve around the unit disk. Does ...

**9**

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153 views

### A geometric characterization of quasicircles

I'm reading an article by complex analysists.
A Jordan curve $J$ in the extended complex plane $\hat{\mathbb{C}}=\mathbb{C} \cup \{\infty\}$ is called a quasicircle if there is a quasiconformal map ...

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46 views

### Modulus estimate with intersecting annuli (quasi-additivity)

In general for annulus $A\subset \mathbb{C}$ if $A_{1},A_{2}....\subset A$ are disjoint annuli inside it, then we have
$$mod(A)=\frac{1}{2\pi}\int_{A}\int_{A} \frac{1}{|z|^{2}}dz>\frac{1}{2\pi}\...

**6**

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**1**answer

117 views

### 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.
...

**4**

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**0**answers

101 views

### 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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**1**answer

78 views

### 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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**1**answer

76 views

### Euclidean length of hyperbolic geodesics for annuli with bounded geometry

I am wondering whether there are estimates for the Euclidean length of vertical hyperbolic geodesics for annuli with good geometry.
More precisely:
Take an annulus $A$, whose outer boundary $\gamma_{...

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**1**answer

251 views

### Hyperbolic 3 manifold with trivial deformation of flat conformal structures

Does there exist closed hyperbolic three manifold which is locally rigid in the space of its all conformally flat structures? If so could someone provide examples?

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**1**answer

149 views

### $L^{2}$ Betti number

Let $\tilde{X}$ be a non-compact oriented, Riemannian manifold adimits a smooth metric $\tilde{g}$ on which a discrete group $\Gamma$ of orientation-preserving isometrics acts freely so that the ...

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198 views

### Does $\nabla g=\omega(\cdot) g$ imply $\nabla$ is metric w.r.t a conformal rescaling of $g$?

This is a cross-post.
Let $E$ be a smooth vector bundle over a manifold $M$, where $\text{rank}(E) > 1,\dim M > 1$. Suppose that $E$ is equipped with a metric $g$ and an affine connection $\...

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100 views

### 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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171 views

### 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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**1**answer

198 views

### 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 ...

**18**

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**3**answers

1k views

### 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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574 views

### 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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96 views

### 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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**2**answers

626 views

### 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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votes

**1**answer

127 views

### 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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90 views

### Is every “higher-order” harmonic morphism conformal?

$\newcommand{\TM}{\operatorname{TM}}$
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
$\newcommand{\TM}{\operatorname{T\M}}$
$\newcommand{\TN}{\operatorname{T\N}}$
$\newcommand{\TstarM}{...

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601 views

### A conformal map whose Jacobian vanishes at a point is constant?

Let $f:M \to N$ be a smooth weakly conformal map between connected $d$-dimensional Riemannian manifolds, i.e. $f$ satisfies $df^Tdf =(\det df)^{\frac{2}{d}} \, \text{Id}_{TM}$.
Assume $d \ge 3$ ...

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473 views

### Is there a discrete lattice analogue of conformal transformations?

There is a simple discrete combinatorial analogue of manifolds and homeomorphisms: Replace manifolds by simplicial complexes and homeomorphisms by Pachner moves. Equivalence classes of manifolds under ...

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53 views

### Conformal factors and light rays

Suppose $(\mathcal{M},g)$ is a $3$-dimensional Riemannian manifold and let $\gamma \in \mathcal{M}$ denote an arbitrary curve in $\mathcal{M}$. Does there exist a conformal factor $c>0$ such that $...

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151 views

### 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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**1**answer

66 views

### 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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72 views

### Conformal $L^p$ rigidity of Riemannian manifolds

$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
$\newcommand{\R}{\mathbb{R}}$
$\newcommand{\CO}[1]{\text{CO}(#1)}$
$\newcommand{\dist}{\operatorname{dist}}$
$\newcommand{\g}{\mathfrak{g}}...

**7**

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**1**answer

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 ...

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votes

**2**answers

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 = \...

**2**

votes

**1**answer

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 ...

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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 ...

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**1**answer

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

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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 ...

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votes

**1**answer

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| \...

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**1**answer

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.
...

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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 ...

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81 views

### Regularity of weak solutions of an equation related to conformal maps

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 ...

**33**

votes

**7**answers

4k views

### Why is conformal invariance only possible for massless theories?

I'm conscious that this isn't necessarily a research level question, but I've asked this question on mathstackexchange, and received no answer. So I'm trying it here.
A usual mantra in field theories ...