The conformal-geometry tag has no wiki summary.
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1answer
110 views
Are there compact Riemannian manifolds whith Q-curvature negative?
Are there known examples of compact Riemannian manifolds with Q-curvature negative?
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1answer
168 views
A question on the twistor space of a manifold
Let $M$ be either (a) self-dual conformal 4-manifold, or (b) hypercomplex $4n$-manifold.
In either case one can construct the twistor space $Z$ (in the case (b) $Z=\mathbb{C}\mathbb{P}^1\times M$ as a ...
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1answer
121 views
Complex function for mapping a circle to a superellipse
I was wondering if anyone knows an analytic complex function that would map a circle to a superellipse, or vice versa. Any ideas, comments, or functions are much appreciated!
Thanks,
Kayvan
8
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1answer
671 views
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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0answers
49 views
Bi-Lipschitz classification of germs of conformal metrics at a singularity
First let me introduce some definitions.
By a germ of conformal metrics at a singularity, or simply a germ, I mean a conformal Riemannian metric $g$ defined on a punctured neighborhood $U$ of $0$ in ...
3
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1answer
113 views
Ricci flow and conformal classes
Is it true that the conformal class of the metric is preserved under Ricci flow? I have seen it mentioned in an answer on this site. Is there an easy argument?
(This question was asked on MSE but it ...
3
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0answers
105 views
Continuous extension of Riemann maps and the Caratheodory-Torhorst Theorem
If $G\subsetneq\mathbb{C}$ is a simply-connected plane domain, then by the Riemann mapping theorem there is a conformal isomorphism $\newcommand{\D}{\mathbb{D}}\varphi:\D\to G$, where $\D$ is the unit ...
9
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1answer
189 views
On the conformal removability of Jordan curves
We say that a compact subset $E$ of the Riemann sphere $\mathbb{C}_\infty$ is (conformally) removable if every homeomorphism of $\mathbb{C}_\infty$ conformal outside $E$ is actually conformal ...
3
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1answer
187 views
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 ...
4
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2answers
168 views
The necessary and sufficient condition for $\textbf{global}$ conformal flatness of a n-dim (pseudo-)Riemannian manifold
There is a theorem :
1) 2-dim (pseudo-)Riemannian manifold must be local conformal flat;
2) 3-dim (pseudo-)Riemannian manifold is local conformal flat iff the Cotton tensor vanishes.
3) n-dim (n>3) ...
1
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1answer
89 views
How far do conjugated Mobius transforms move points?
I start with an automorphism $f$ of the complex unit disc $S^1 = \{ z \in \mathbb{C} : |z| \leq 1\}$. I assume that such a map is given by a Mobius transform, namely
$$
f(z) = \frac{z - ...
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1answer
142 views
Holomorphic maps on $\mathbb{R}^{n}$ (for n not necessarily even)
Edit according to the comment of user36931 I remove the "motivation" from the previous version and I add an statement to the first question
We consider the following two classes of smooth maps on ...
3
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2answers
213 views
In what condition is a conformal flat manifold flat?
$g^{\mu\nu}(x)=\Omega^{2}(x)g'^{\mu\nu}(x)$ is a conformal transformation.
If $g'^{\mu\nu}$ is flat, what kind of $\Omega(x)$ is choosed can make $g^{\mu\nu}$ flat.
We can think about any dimension ...
3
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1answer
156 views
Change of coordinates for Teichmüller space of the 4-holed sphere
The diagram below indicates 2 ways to use Fenchel-Nielsen coordinates to parameterize the Teichmüller space of conformal structures on the 4-holed sphere with totally-geodesic boundary, corresponding ...
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1answer
180 views
What's the height of the capped hyperbolic pants?
Consider a hyperbolic pair of pants with totally-geodesic boundaries of lengths $l_1,l_2,l_3$. Cap off boundary 1 with a conformal disk. The result is a conformal cylinder, which has a unique flat ...
2
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1answer
129 views
Coordinates for Teichmuller space for compact conformal surfaces
Fenchel-Nielsen coordinates give a coordinatization of Teichmuller space for compact conformal surfaces admitting a pants decomposition. But not all compact conformal surfaces (possibly with boundary, ...
5
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1answer
230 views
Green's function for *GJMS* operator
Consider a Riemannian manifold $(M^n, g)$ of dimension $n$ with a metric $g$. We assume $M$ to be closed (compact without boundary). Let's not assume any hypothesis on the Yamabe invariant of the ...
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0answers
154 views
Does the concept of connective constant make sense for any tiling of the plane?
First let me define what is the "connective constant" of a two dimensional lattice.
Let $c_{n}$ denote the number of $n$ step self-avoiding walks starting from a fixed origin point in the lattice. ...
3
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1answer
121 views
Global conformal equivalence of two regions of Minkowski spacetime
I am wondering whether the region $H:=\{(t,x):x^2−t^2<1\}$ of $(1+1)$-dimensional Minkowski spacetime, equipped with the restriction $g_H$ of the standard Minkowski metric $g=−\mathrm{d} t\otimes ...
2
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1answer
165 views
Why do holomorphic maps increase extremal length?
Let $\mathcal{L}$ denote extremal length. The following theorem appears in http://arxiv.org/abs/math/0505191.
Theorem
Let $U$ and $V$ be Riemann surfaces, and let $f:U\rightarrow V$ be a holomorphic ...
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0answers
156 views
What are Euler density and Weyl invariants?
I would like to know as to what is the definition and significance of what are called "Euler density" and "Weyl invariants" (of weight $-d$ on a $d-$manifold)
Do many (which?) of them vanish when ...
2
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1answer
220 views
hayman's result for $ A^2(D) $
Consider injective homolomorphic functions $f:\mathbb D\to \mathbb C$ on the unit disk $|z|\leq 1$, normalized by the conditions $f(0)=0$ and $f'(0)=1$.
Thus for $|z|\leq 1$ we have $ ...
3
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0answers
108 views
The Tangent Bundle of the Space of CR Structures on S^(2n+1)
Let $M$ be a smooth compact $n$-manifold without boundary, $g$ some choice of Riemannian metric on $M$, and $\omega_g$ the volume form gotten from $g$. Say you're interested in finding extrema for ...
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1answer
140 views
Inverse “Riemann mapping” [closed]
The Riemann mapping theorem states, that any simply connected domain $U \subset \mathbb C$ can be conformally mapped to the open unit disk $D$. I.e. there is a Diffeomorphism $\Psi: D \to U$ such that ...
5
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1answer
347 views
Proof of the general expression for anomaly in a CFT and its partition function
I think the statement is that for any dimensional CFT the following is true,
$$\langle T^{\mu}_\mu \rangle = \sum B_n I_n - 2(-1)^{d/2}AE_d,$$
where $E_d$ is the `"Euler density" and $I_n$ are ...
1
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1answer
263 views
Two different forms of Schwarz-Christoffel-Mapping of unit disk to rectangle. Are they identical?
I found two different equations for the Schwarz-Christoffel-mapping of a unit disk to a rectangle (which are the general form of the SC-mapping, I guess). The first, e.g. from Link, page 20, is
...
8
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1answer
320 views
Can one use Brownian motion to prove that two manifolds are not conformally equivalent?
Let me start by a very simple example; consider the following question:
"Let D1 be a square and D2 a rectangle (boundary included). View them
as subsets of the complex plane. Does there exist a ...
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0answers
114 views
Global conformally flat coordinates
I know that for a 2D manifold there can allways be found a local chart such that the metric is conformally flat. Is it possible globally? If the manifold is simply connected, like a disk, does it make ...
1
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0answers
66 views
Conformal Finsler metrics
Given a Finsler metric $F$ on a compact boudaryless manifold $M$ and $\sigma : M \longrightarrow \mathrm{R}$ a $C^2$ strictly positive function, define a new Finsler metric by $\tilde{F}=\sigma F.$ ...
7
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1answer
390 views
How unique is a conformal compactification?
I'm trying to understand the term "conformal compactification" which is often used in physics. I reckon that most places take this to mean a (sometimes specific) compact conformal completion. That is, ...
2
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1answer
292 views
Variant of the Riemann Mapping Theorem for $Conf(\mathbb H^2)$?
According to the Riemann mapping theorem it is possible to map a simply connected open subset $B \subset \mathbb C$ into any other $B' \subset \mathbb C$ by a (bi-)holomorphic mapping. Moreover, such ...
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1answer
183 views
Analytic curve on Riemann surface
Suppose there is a closed analytic curve $C$ on a Riemann surface $S$, that is the image of a map $\gamma$ from the equator $E$ of the Riemann sphere to the surface $S$ which is a restriction of a ...
9
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1answer
404 views
Surfaces in a 3-manifold with the same Gaussian curvature with respect to two ambient conformal metrics
Let $M$ be a 3-smooth manifold and $g_{1}$ and $g_{2}$ two conformal metrics on $M$. Consider an immersed surface S in $M$ and let $K_{1}$ and $K_{2}$ be the Gaussian curvatures of $S$ with respect to ...
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1answer
302 views
How to determine bounds on the extremal length around annuli?
I wish to determine bounds for the sum of moduli of a family of topological annuli in the complex plane. Towards that end I would like to ask a question about the closely related concept of extremal ...
1
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1answer
216 views
Harmonic/conformal map composition between manifolds in either order?
Suppose $\mathcal{M}$, $\mathcal{N}$, and $\mathcal{P}$ are Riemannian manifolds (compact and of dimension 2, if it matters). It seems well-known that if $\phi:\mathcal{M}\rightarrow\mathcal{N}$ is ...
5
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1answer
236 views
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 ...
4
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2answers
386 views
Conformal structure does not see conical singularities
the conformal structure does not see the conical singularities of a polyhedral surface.
This is a quote from the Preface of Quantum Triangulations (eds.: Carfora, Marzuoli).
The sentiment is ...
3
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1answer
340 views
eigenspinors of Dirac operator
$M$ compact manifold. Let $\lambda$ be an eigenvalue for the Dirac operator of multiplicity greater than 2. I'm interested in showing the existence of two linearly independant eigenspinors $u$ and $v$ ...
4
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0answers
355 views
Correlation functions of complex operators
One defines the "scaling dimension" (as opposed to "engineering dimension") of an operator $\cal{O}$ as $[\cal{O}]$ such that if $\cal{O}(t^{-1}x) = t^{[\cal{O}]}\cal{O}(x)$ then the Lagrangian in ...
1
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2answers
386 views
Conformal transformations and harmonic analysis on the sphere
Consider the $n$-dimensional sphere $S^n$. I'm especially interested in the $n=4$ case. The Hilbert space $L^2(S^n)$ can be decomposed into a direct sum of eigenspaces of the Laplacian, which are ...
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3answers
607 views
conformal diffeomorphism of sphere
I want to understand Prop 6 in the paper "Convergence of the Q-curvarture flow on $S^4$" by Simon Brendle. I understand that for every $p\in B^4$, where $B^4=\{x\in\mathbb{R}^5: |x|\leq 1\}$,
...
1
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1answer
223 views
Conformal Extension from a closed set to open
Let $Q = \{(x,y): x,y\geq 0\} $ be the 1st quadrant of $\mathbb R^2$, and $f$ is a function defined on it such that all the partial derivative(any order) of $f$ exists and continuous. By Whitney ...
1
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1answer
179 views
Conformal extension
Does there exist a conformal smooth extension of a smooth function? Smooth extension is guaranteed by Whitney extension theorem. does that theorem also says for conformality.
Precisely the ...
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2answers
209 views
general conformal transf. extension to hyperbolic manifold
I have an hyperbolic manifold with boundary a conformal sphere. Can I extend any conformal transformation of boundary to the interior of the ball? I know how to do with Moebius but I wonder if there ...
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1answer
765 views
Geometric Interpretation of $Q$-curvature
Let $(M,g)$ be a Riemannian manifold of dimension $n>2$. Thanks to the late T.Branson we have the following definition of the so-called $Q$-curvature:
$Q= \Delta R + ...
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2answers
321 views
Approximation of conformal mapping as a sum of elementary conformal mappings
Hi,
I would like to approximate any 2d conformal mapping, as a sum of elementary conformal mappings. So I would have some basis, a conformal mapping with some parameters, and by adding several ones ...
2
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0answers
286 views
Collection of charged line segments in 2D - where do electric field lines meet it?
Suppose we have a collection of charged line segments in 2D. I'd like to be able to do two things :
from an arbitrary point in the plane, follow the electric field and find where it meets the ...
11
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3answers
550 views
Conformal Welding Reference
I'm looking for a reference for the following fact: given two Riemann surfaces and an identification of their boundaries, once I topologically glue the surfaces together there exists a unique ...
3
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2answers
618 views
Conformal-symplectic geometry ?
I think in priciple it's possible to consider a theory of "conformal-symplectic manifolds", in an analogous fashion as the usual conformal geometry.
To spell out the spontaneous definitions: say ...
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1answer
1k views
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 ...
