Tagged Questions
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3
votes
1answer
105 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 ...
4
votes
1answer
141 views
+50
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 ...
1
vote
1answer
68 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, ...
1
vote
0answers
61 views
Green's function for *GJMS* operator
Consider a Riemannian manifold $(M^n, g)$ of dimension $n$ with a metric $g$. Let's not assume any hypothesis on the Yamabe invariant of the manifold.
A GJMS operator on $M^n$ is a differential ...
4
votes
0answers
134 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. ...
0
votes
0answers
57 views
Invertibility of Paneitz operator on a compact manifold without boundary
Let $(M,g)$ be a Riemannian manifold of any even dimension that we assume compact and without boundary and $P_g$ the Paneitz operator in the metric $g$. The question is the following:
how to define a ...
3
votes
1answer
68 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
votes
1answer
138 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 ...
1
vote
0answers
88 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
votes
1answer
211 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
votes
0answers
72 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 ...
0
votes
1answer
124 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
votes
1answer
289 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
vote
1answer
161 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
...
7
votes
1answer
266 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 ...
1
vote
0answers
80 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
vote
0answers
52 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 $\tilde{F}=\sigma F.$ ...
6
votes
1answer
303 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
votes
1answer
283 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 ...
1
vote
1answer
165 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
votes
1answer
381 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 ...
1
vote
1answer
269 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
vote
1answer
200 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
votes
1answer
207 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
votes
2answers
332 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
votes
1answer
316 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
votes
0answers
307 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
vote
2answers
342 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 ...
2
votes
3answers
495 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
vote
1answer
219 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
vote
1answer
160 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 ...
2
votes
2answers
195 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 ...
7
votes
1answer
665 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 + ...
1
vote
2answers
301 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
votes
0answers
269 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
votes
3answers
517 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
votes
2answers
545 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 ...
7
votes
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 ...
0
votes
0answers
229 views
Errata in “Conformal Mapping: Method and Applications” from Schinzinger
hi,
i'm studying the book "Conformal Mapping: Method and Applications" from Schinzinger and more precisely the chapter 6 concerning non-planar field.
In section 6.1.3 the author proposes to solve a ...
0
votes
1answer
434 views
Invariance of the cylindrical Laplace equation under conformal transform
hello,
it is often said that a conformal mapping preserves the Laplace equetion in 2D.
However, if this is true for the cartesian coordinates (x,y), where the laplacian is:
$$
\frac{\partial^2 ...
16
votes
2answers
2k views
A question about the proof of Mostow rigidity
I have recently been studying a proof of Mostow rigidity (along the lines of Mostow's original argument), and I'm left a little confused about something. We start with an isomorphism $\alpha: \Gamma ...
6
votes
1answer
455 views
Analogy of Liouville conformal mapping theorem with Mostow rigidity?
I often hear mention of two theorems, Mostow's rigidity theorem and Liouville's theorem on conformal mappings, which superficially sound similar: they say that a set of geometric structures is, if ...
29
votes
5answers
2k views
Intuition behind moduli space of curves
For a genus g compact smooth surface $M$, an algebraic structure is the same as a complex structure is the same as a conformal structure. So the moduli space of smooth curves should be the same as the ...
