The conformal-geometry tag has no usage guidance.

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### 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 $v$ ...

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392 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 ...

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433 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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727 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 $B^4=\{x\in\mathbb{R}^5: |x|\leq 1\}$,
...

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**1**answer

230 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 ...

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**1**answer

207 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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**2**answers

213 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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**1**answer

872 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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344 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 ...

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**0**answers

293 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 ...

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636 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 ...

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**3**answers

785 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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**1**answer

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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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251 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 ...

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**1**answer

495 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 ...

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**2**answers

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 ...

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**1**answer

489 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 ...

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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 ...