The conformal-geometry tag has no usage guidance.

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### 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 ...

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68 views

### Extremal length of graphs in surfaces

Given a surface $\Sigma$ with conformal structure $\omega$, the extremal length of a homotopy class $\gamma$ of curves in $\Sigma$ is defined to be
$$
\sup_{g \in \omega} ...

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63 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 ...

**4**

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156 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. ...

**4**

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116 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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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 ...

**2**

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34 views

### A conformal mapping onto a region bounded by convex contours (Ahlfors)

I want to solve the following exercise (from Ahlfors' text, page 261). I've tried posting it on MSE, and placed a generous bounty, but I couldn't get any answers there.
*3. Using Ex. 2, show that ...

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65 views

### What are “minimal lines” in complex geometry?

Schwarz reflection across a real analytic arc $C$ in $\mathbb{R}^2$ is usually defined analytically. Thinking of the arc as the image of an interval in $\mathbb{R}$ under an invertible holomorphic map ...

**2**

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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 ...

**1**

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140 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 ...

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72 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 $\tilde{F}=\sigma F.$ ...

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99 views

### Poisson Kernel and Triangles

The Poisson Kernel is an approximation to the identity, meaning $P_r(\theta) \approx \delta(\theta)$; here is the formula on $\mathbb{D}$:
$$ P_r(\theta) = \sum_{n \in \mathbb{Z}} r^{|n|} ...

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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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33 views

### conformal deformation with fixed boundaries

For a flat plane with certain boundary, e.g., a rectangular patch, is it possible to conformally displace or deform such patch to a curved bump with exact same boundary? In this thesis, Dr. Keenan ...