Questions tagged [conformal-geometry]
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137
questions
2
votes
0answers
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
vote
1answer
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
votes
0answers
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
votes
1answer
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
votes
1answer
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 ...
-3
votes
1answer
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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
votes
1answer
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
votes
1answer
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 ...
4
votes
2answers
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 ...
2
votes
0answers
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
votes
0answers
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 ...
2
votes
0answers
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 $$\square\varphi = \lambda \...
2
votes
1answer
105 views
Can an “annular” subset of an annulus be conformally equivalent to the whole annulus?
Assume we are given an annulus
$$A = \{ z \in \mathbb{C}: 1< |z| < R\}.$$
Let $\phi\colon A \to A$ be a univalent map such that the image of $\phi$ contains a curve around the unit disk. Does ...
9
votes
0answers
153 views
A geometric characterization of quasicircles
I'm reading an article by complex analysists.
A Jordan curve $J$ in the extended complex plane $\hat{\mathbb{C}}=\mathbb{C} \cup \{\infty\}$ is called a quasicircle if there is a quasiconformal map ...
1
vote
0answers
46 views
Modulus estimate with intersecting annuli (quasi-additivity)
In general for annulus $A\subset \mathbb{C}$ if $A_{1},A_{2}....\subset A$ are disjoint annuli inside it, then we have
$$mod(A)=\frac{1}{2\pi}\int_{A}\int_{A} \frac{1}{|z|^{2}}dz>\frac{1}{2\pi}\...
6
votes
1answer
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
votes
0answers
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 ...
0
votes
1answer
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 ...
2
votes
1answer
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_{...
3
votes
1answer
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?
2
votes
1answer
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 ...
5
votes
1answer
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 $\...
4
votes
0answers
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 ...
7
votes
0answers
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 ...
4
votes
1answer
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
votes
3answers
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 ...
11
votes
2answers
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 ...
4
votes
0answers
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 ...
2
votes
2answers
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 ...
2
votes
1answer
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 $\mathbb{H}^n$ ...
1
vote
0answers
90 views
Is every “higher-order” harmonic morphism conformal?
$\newcommand{\TM}{\operatorname{TM}}$
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$\newcommand{\TN}{\operatorname{T\N}}$
$\newcommand{\TstarM}{...
13
votes
3answers
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 $d \ge 3$ ...
10
votes
3answers
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 ...
1
vote
0answers
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 $...
6
votes
0answers
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 ...
1
vote
1answer
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 ...
4
votes
0answers
72 views
Conformal $L^p$ rigidity of Riemannian manifolds
$\newcommand{\M}{\mathcal{M}}$
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$\newcommand{\CO}[1]{\text{CO}(#1)}$
$\newcommand{\dist}{\operatorname{dist}}$
$\newcommand{\g}{\mathfrak{g}}...
7
votes
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 ...
12
votes
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):
$$
g = dr^2+r^2\left(d\theta^2 + \sin^2\theta\ d\phi^2\right)
$$
$$
r = R \tan \frac{\alpha}{2}
$$
$$
g = \...
2
votes
1answer
104 views
conformal mapping and rational function
Let $E$ be an infinite compact subset of the complex plane $\mathbb{C}$ such that $\overline{\mathbb{C}}\setminus E$ is simply connected. By Riemann mapping theorem, there exists a unique exterior ...
2
votes
0answers
93 views
How do conformal maps affect curvature?
Let $(\overline{M}^{n+1}, \langle \cdot, \cdot \rangle)$ be a riemannian manifold with riemannian connection $\overline{\nabla}$ and consider $M^n \subset \overline{M}$ an orientable hypersurface with ...
2
votes
1answer
156 views
Vanishing of determinant of Cotton York tensor
suppose $\Omega \subset \mathbb{R}^3$ is a Riemannian manifold with $g= dx_1^2 + dx_2^2 + c(x_1,x_2,x_3)dx_3^2$. Is it true that $det(CY)=0$?
Thanks
2
votes
0answers
57 views
Will a slightly differently shaped torus make this guess about plane sections of a torus true?
Jacobi's elliptic functions and plane sections of a torus
After Greg Egan posted an excellent answer to a question of mine, which I accepted, I posted my own answer, linked above. The question ...
2
votes
1answer
155 views
$\zeta(2n)$ and Levy processes
I am missing some steps in the final derivation of a probabilistic computation of the even values of $\zeta$. They show the Cauchy distribution is relate to a certian Levy process:
$$ |\mathbb{C}_1| \...
7
votes
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.
...
3
votes
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 ...
4
votes
0answers
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
7answers
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 ...
