Questions tagged [conformal-geometry]
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194
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Factorization of conformal maps between annuli
Consider two doubly-connected open subsets $A$ and $A'$ of the Riemann sphere. We assume these two domains to be of same modulus (the moduli space being one real parameter), i.e. we assume that there ...
8
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Is there a manifold structure on a space of conformal maps?
I would be very grateful for any information or pointers for the following:
1) Fix an open subset $U$ of $\mathbb{CP}^1$. a) Does the set of all holomorphic maps from $U$ to $\mathbb{C}$ (with the ...
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2
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Non-bijective conformal maps between annuli
I need to answer the following question, hopefully in the negative.
Question: Does there exist a conformal map $f$ of degree $1$ from the annulus $\{1<|z|<R\}$ to the punctured disk $\{0<|...
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4
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Looking for a reference on conformal mapping on $\Bbb R^n$
A mapping $T: \Bbb R^n\to \Bbb R^n$ is said to be conformal if it is bijective and preserves angles, i.e.,
if $x, y: [0,1]\to \Bbb R^n$ are curves with $x(t_0)=y(t_0)$ then
$$\cos (Tx(t_0),Ty(t_0))= \...
3
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1
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What is the orbit of the standard conformal structure on $S^2$ under $\operatorname{SL}(3,\mathbb{R})$?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}$Consider the group $\GL_+(3,\mathbb{R})$ acting on $\mathbb{R}^3$. It induces an action of $\GL_+(3,\mathbb{R})/\...
5
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An application of Leray-Schauder degree theory for Nirenberg problem on the 2-sphere
I'm studying the article "The scalar curvature equation on 2- and 3-spheres" by Chang, Gursky and Yang and I'm particulary interested in the 2-sphere case.
They prove that if $K:S^2\...
3
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2
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Classification of conformal diffeomorphisms of Minkowski space, part 2
This is a continuation of Classification of conformal diffeomorphisms of Minkowski space
Consider $\mathbb{R}^{n+1}$ equipped with the Minkowski (sign indefinite) metric:
$$g=(x^0)^2-(x^1)^2-\dots -(x^...
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1
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Classification of similarity transformations of Minkowski space
Consider $\mathbb{R}^{n+1}$ equipped with the Minkowski (sign indefinite) metric:
$$g=(x^0)^2-(x^1)^2-\dots -(x^n)^2.$$
Is there a classification of diffeomorphisms $F\colon \mathbb{R}^{n+1}\tilde\to ...
2
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0
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The conformal map from interior of ellipse to interior of the unit disk (property check)
Based on example 5 (page 546) and example 7 (page 550) of the book "Applied and computational complex analysis. Volume 3, Wiley, 1986" written by Peter Henrici, if $a,b>0$ satisfies $a^2-...
2
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Uniformization theorem with boundary in the non-compact case
Let $\Sigma$ be a simply connected (and therefore orientable) smooth $2$-manifold with non-empty and connected boundary. Suppose that the interior $\operatorname{int}(\Sigma)$ is endowed with a ...
17
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1
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Geometric interpretation of the Weyl tensor?
The Riemann curvature tensor ${R^a}_{bcd}$ has a direct geometric interpretation in terms of parallel transport around infinitesimal loops.
Question: Is there a similarly direct geometric ...
3
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2
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252
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Regularity of a conformal map
Let $D$ be a domain in $\mathbb{C}$ with $n$ boundary components. From the work of Koebe, we know that $D$ can be conformally mapped to a parallel slit domain of a specified angle of inclination (...
12
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How large can you draw an island on a map?
A cartographer friend asked me this question: could you classify (shapes of) islands by how much space they occupy on a map (comparatively to how much space is occupied by water) if you draw them as ...
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0
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Continuity of conformal grafting, wrt the (infinite type) surface
Say I have two closed hyperbolic surfaces $X,Y$ and a smooth, $(1+\epsilon)$-bilipschitz map $f : X \to Y$ for some small $\epsilon$. Pick a simple closed curve $c \subset X$, and let $X',Y'$ be the ...
7
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Curvature of complete conformal metrics on the open unit disk
Let $D$ be the unit disk in the complex plane, and assume that $g$ is a Riemannian metric on $D$ which is complete and conformal to the standard Euclidean metric. Can it be the case that the Gaussian ...
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2
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compatification of $\mathbb{R}^2$ under a conformal metric
I want to understand the following question:
Let $\delta$ be the Euclidean metric in $\mathbb{R}^2$. Is there any criteria for smooth function $u$ such that $(\mathbb{R}^2, e^{2u}\delta)$ can be ...
3
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1
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278
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Conformal mapping between two right-angled triangles
I want to derive a conformal mapping $f\!:\!A\!\to\! B$ where $A=\{ (x,y)\ |\ x\!\in\![0,1]\ \text{ and }\ 0 \leq y \leq x \}$ and $B=\{ (x,y)\ |\ x\!\in\![0,1]\ \text{ and }\ 0 \leq y \leq \frac{...
1
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1
answer
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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 ...
2
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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 $f_1 \colon \mathbb D \to D$ ...
2
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1
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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 ...
14
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1
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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 ...
0
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0
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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 ...
3
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1
answer
345
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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 ...
0
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0
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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 ...
5
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1
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405
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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 ...
4
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2
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353
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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 ...
2
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0
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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 ...
4
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223
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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 ...
2
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0
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125
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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 $$\square\varphi = \lambda \...
2
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1
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192
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Can an "annular" subset of an annulus be conformally equivalent to the whole annulus?
Assume we are given an annulus
$$A = \{ z \in \mathbb{C}: 1< |z| < R\}.$$
Let $\phi\colon A \to A$ be a univalent map such that the image of $\phi$ contains a curve around the unit disk. Does ...
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A geometric characterization of quasicircles
I'm reading an article by complex analysists.
A Jordan curve $J$ in the extended complex plane $\hat{\mathbb{C}}=\mathbb{C} \cup \{\infty\}$ is called a quasicircle if there is a quasiconformal map ...
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0
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Modulus estimate with intersecting annuli (quasi-additivity)
In general for annulus $A\subset \mathbb{C}$ if $A_{1},A_{2}....\subset A$ are disjoint annuli inside it, then we have
$$mod(A)=\frac{1}{2\pi}\int_{A}\int_{A} \frac{1}{|z|^{2}}dz>\frac{1}{2\pi}\...
8
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1
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Uniqueness theorem for conformal mapping
Let $f$ and $g$ be analytic functions in the unit disk $D$, continuous in the closed disk and locally univalent, $f'(z)\neq 0,\; g'(z)\neq 0,\; z\in D$.
Assume that each has only finitely many ...
6
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1
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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.
...
4
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0
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143
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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 ...
0
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1
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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 ...
2
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1
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203
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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_{...
5
votes
1
answer
502
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Adding segments to an annulus - a question regarding the conformal modulus
Let $A \subset \mathbb{C}$ be an topological annulus, i.e. a region of $\mathbb{C}$ bounded by two disjoint Jordan curves.
Let $B \subset \mathbb{C}$ be a quadrilateral, i.e. a topological disc with ...
3
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1
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509
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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
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1
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$L^{2}$ 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
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1
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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 $\...
5
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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 ...
7
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0
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229
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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 ...
4
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1
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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 ...
12
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3
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608
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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 ...
11
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2
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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 exist ...
13
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3
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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 $d \ge 3$ ...
4
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0
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135
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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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0
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Is every "higher-order" harmonic morphism conformal?
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