# Questions tagged [conformal-geometry]

The conformal-geometry tag has no usage guidance.

190
questions

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### Conformal maps between two given domains

Consider two domains
$$
\begin{aligned}
D_1&=\{x=(x_1,x_2,...,x_n)\in\mathbb{R}^n:x_n\leq 0\},\\
D_2&=\{x=(x_1,x_2,...,x_n)\in\mathbb{R}^n:x_n\leq \psi(x_1,x_2,...,x_{n-1})\},
\end{aligned}
$$
...

3
votes

1
answer

131
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### Do we have uniformization theorems for fractional dimensional spaces?

The Riemann mapping theorem in $\mathbb{R}^2$ is known not to generalize well in higher dimensions and is basically trivial in lower dimensions.
I’m interested in how it generalizes for fractional ...

1
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0
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### Metric balls in Teichmüller space are topological balls

Let $X$ be a topological surface of finite type and $\mathcal{T}_X$ be the corresponding Teichmüller space. Let $B$ be a ball with respect to the Teichmüller metric on $\mathcal{T}_X$ (i.e., the ...

2
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3
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### Let $M$ be a manifold with a conformal structure and a volume measure. How can one reconstruct a metric on $M$ from this data in a geometric way?

Let $(M,g)$ be a (semi-)Riemannian manifold, and let $(M,[g])$ be the conformal class thereof. The following two sets are naturally torsors for the collection of positive-valued functions on $M$:
The ...

2
votes

0
answers

94
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### Test function in Simon Brendle's paper about Yamabe flow

I'm recently reading Simon Brendle's Convergence of the Yamabe flow in dimension 6 and higher.In this paper,he constructs a family of test functions with Yamabe energy less than Yamabe constant of ...

7
votes

1
answer

462
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### Definition of the conformal metric

On page 16 of this lecture notes and in this lecture, at some point (somewhere at 1:37:45), Rod Gover defined the $\text{conformal metric}$ $\mathbb{g}$ on a conformal manifold $(M, [g])$. He ...

2
votes

1
answer

237
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### Modulus of a torus

Suppose I have a torus (you can pretend that it is the usual torus of revolution: the boundary of a tubular neighborhood of radius $r_2$ of a circle of radius $r_1.$ The question is: can one compute ...

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122
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### Conformal diffeomorphism of $\mathbb R^k$

Let $f$ be a conformal diffeomorphism between $\mathbb{R}^k$, where $k\geq 2$, and its Euclidean metric. It follows from complex analysis and Liouville's theorem that $f$ can only be affine. Now, ...

2
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0
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77
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### Conformal changes of metric and normal coordinates

Suppose that $(M,g)$ is a smooth Riemannian manifold of dimension $n\geq 2$. Let $p\in M^{\textrm{int}}$. Does there exist a small $\delta>0$ and a smooth function $c>0$ such that for the ...

4
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1
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182
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### Is every Riemannian metric conformally equivalent to one that is geodesically complete?

The question in the title seems a natural one to ask, and I suspect that it is already considered, even completely solved, somewhere in the literature. Although I prefer some explanation of the idea(s)...

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### Standard 2-instantons on the 4-sphere under conformal transformation

It is well-known that there is a standard SU(2) 1-instanton on the 4-sphere and any 1-instanton can be obtained from the standard one by conformal transformation. Is there any explicit (family of) &...

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### Does the conformal factor satisfy some equation independent of the vector field?

Let $(M, g)$ be an $(n+1)$-dimensional (pseudo-)Riemannian manifold and for any
$X \in \mathcal{X}(M)$ define its deformation tensor by
\begin{align*}
{} ^{(X)}\pi_{ab} := (\mathcal{L}_X g)_{ab} = \...

-3
votes

1
answer

229
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### Weyl tensor of a Riemannian metric $g$

Does Weyl tensor of a Riemannian metric $g$ give information about the conformally-flatness of $g$?

4
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223
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### CFT as an axiomatic field theory

I'm trying to understand CFT from a purely axiomatic-field-theoretical perspective. That is, there is a vector space $V$ associated to the circle, and an element of $V^{\otimes n}$ associated to every ...

4
votes

1
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173
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### Yamabe operator, conformal transformations and square of the Dirac operator

On an $(M,g)$ compact n-dimensional Riemannian manifold the Yamabe operator $Y = 4(n-1)/(n-2)L + s$, where $L$ is the Laplacian and $s$ is the scalar curvature is conformally covariant, which for me ...

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### Geometric characterizations of conformal maps

I have some fairly basic questions with regards to conformal structures/maps -- I apologize if they are on the basic side for MO, but I figured I might get some clarity here.
Suppose $X$ and $Y$ are ...

2
votes

1
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216
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### Explicit universal covering map for higher genus algebraic curves

Suppose I have a projective plane curve $C = V(F)$ defined over $\mathbb{C}$, where $F$ is some homogeneous polynomial in three variables. For the sake of simplicity, let's assume that $C$ is ...

3
votes

1
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170
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### Conformal groupoid

I asked this over on Math.SE but it remained completely silent for over a week so I've deleted it and am reposting it here (I'm not really sure which site it fits better). The question itself is ...

1
vote

1
answer

144
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### Uniqueness of "stretching" (subject to constraints) for a two-dimensional figure

The New York Times, reporting on Dennis Sullivan's Abel prize, recounts the incident that lured Sullivan from chemical engineering to mathematics:
One day during an advanced calculus lecture, the ...

2
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0
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70
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### Equicontinuity, Beltrami coefficients and Sequence of top/bottom semi-annuli

In the Beltrami equation literature, one approach to showing equicontinuity for pairs $(f_{n},\mu_{n})$ (where $\partial_{\bar{z}}f_{n}=\mu_{n}(z)\partial_{z}f_{n}$) is via the relations of moduli and ...

5
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116
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### Examples of compact Kähler manifolds whose Bochner curvature tensor has constant norm?

The Bochner curvature tensor is the Kähler analog of the Weyl curvature tensor in the curvature decomposition of a Kähler, discovered by Bochner in 1949. The article on Bochner-Kähler metrics by ...

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### Triangulations with discrete metrics and conformal equivalence

A discrete metric for a triangulation of a 2-dimensional manifold is a map associating $\mathbb{R}_+$-valued lengths to all edges, such that the triangle inequality holds on every triangle. In many ...

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### What are the generators and relations of the conformal cobordism category?

According to a definition by Segal, a $2$-dimensional CFT is a symmetric monoidal functor from the category of oriented conformal cobordisms to the cateogry of projective complex vectorspaces. Coming ...

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157
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### Divergence of conformal Killing vector fields on $S^2$ and the spherical harmonics

Can anyone think of a conformal Killing vector field $W$ on $S^2$ with the round metric that is not Killing such that its divergence is $L^2$-orthogonal to the spherical harmonics with $\ell = 1$?
One ...

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0
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92
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### Gap phenomenon vs Rigidity results for surfaces

I am trying to understand the differences between the rigidity results and gap results for a given surface immersed into some manifold. For instance, a Gap theorem proved here (Theorem 2.7) says ...

2
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### How to understand the constant rank theorem for semilinear elliptic equations

Let $u$ be a solution to the equation $$\Delta u=f(u,\nabla u)$$ where $f>0$ is a smooth function with $f f_{uu} \le 2f_u^2$. The seminal constant rank theorem states that if $D^2 u$ is positive ...

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### What does it mean for correlation functions to be dominated by the vacuum block for a 2D CFT?

In a 2D CFT, correlation functions dominated by the vacuum block have no conical defects. You can calculate the OPE and determine the correlation function using the D-S equations and cancel out UV ...

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### References Request: Bach tensor

Recently I want to study about Bach tensor in detail. From the fundemental definition and properties to conformal invariant. Is there any references to me about Bach tensor? If there are some origins ...

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2
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796
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### What is a simplified intuitive explanation of conformal invariance? [closed]

Can the concept of conformal map and conformal Invariance be explained in very general terms, preferably in high school/undergrad-level Mathematics? Abstracting away from the applications in physics (...

3
votes

1
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144
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### Finitely connected orientable surface

Let $(M,g)$ be a finitely connected orientable complete Riemannian surface, that is, $M$ is homeomorphic to a compact orientable surface $\Sigma$ minus $k \geq 1$ points. Do you have references or a ...

2
votes

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### Access to an old paper of Obata

I'm trying to access the following paper of Morio Obata:
Conformal changes of Riemannian metrics on a Euclidean sphere, in "Differential Geometry -- In honor of K. Yano", 1972, pp. 347--353: ...

2
votes

1
answer

212
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### Existence of divergence-free unit vector field in conformally rescaled euclidean metric

Question. Let $\Omega \subset \mathbf{R}^2$ be a convex polygonal domain, equipped with a Riemannian metric $g$. Under which conditions on $g$ is there a vector field $X$ in $\Omega$ with $\mathrm{div}...

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164
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### Does there exist an isometry between a regular polygon and a circle?

In order to define the question in a meaningful fashion, I am referring to a smooth manifold $\mathcal{M}$ within an $\epsilon$-neighborhood of a regular polygon $\mathcal{P}$ satisfying $$\max\{\|x-p\...

1
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0
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254
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### Conformal changes of metric and Ricci curvature

Let $(M,g)$ be a three dimensional smooth Lorentzian manifold and let $p$ be a fixed point in $M$ and let $S$ be a smooth symmetric tensor of rank two on $T_pM\times T_pM$. Does there exist a smooth ...

6
votes

4
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482
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### Generalizing contour integration to quaternions and bicomplex numbers

I am interested in the possibility of generalizing the notion of contour integration to the quaternions or bicomplex numbers. I am aware that the Frobenius theorem prevents the construction of a true ...

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### The space of conformal classes of the $n-$sphere

It is well-known that the space of conformal classes on the $2-$sphere is just a point. What is known about the space of conformal classes on the $n-$sphere? Is there any structure on it?

3
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### Range of divergence operator on the space of traceless symmetric $(0,2)$ tensors; conformal vector fields on an arbitrary metric on $S^2$

Let $\gamma$ be a metric on $S^2$.
I am trying to solve the following PDE on a $(0,2)$ symmetric traceless tensor $A$:
$$div_{\gamma} A = \omega$$
where $\omega$ is a 1-form.
It is known that there ...

1
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0
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78
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### conformal changes of Lorentzian metrics

Let $M= \mathbb R \times M_0$ with $M_0$ a smooth compact manifold with smooth boundary and let $g=-dt^2+g_0(t,x)$ be a Lorentzian metric on $M$, with $g_0$ a family of Riemannian metrics on $M_0$ ...

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### Self duality and anti-self duality of Weyl curvature in four dimension

I am trying to compute explicitly in terms of extrinsic curvature the self dual part $W^+$ and the anti-self dual part $W^-$ of the Weyl tensor $W$ associated with a codimension 1 submanifold into ...

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### If SO$(3,\mathbb C)$ is isomorphic to PGL$(2,\mathbb C)$, what objects do vectors in $\mathbb C^3$ represent in the context of Möbius geometry?

I hope this question isn't too basic or ambiguous for this site.
The following is an explicit isomorphism from $\mathrm{PGL}(2,\mathbb C)$ to $\mathrm{SO}(3,\mathbb C)$:
$$\left[\begin{matrix}p & ...

5
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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
votes

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})/\...

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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^...

0
votes

1
answer

135
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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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3
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### For globally conformally flat surfaces, does radial symmetry of conformal factor imply the surface is a sphere?

We know that for the unit sphere in $\mathbb{R}^3$, the standard metric on such a sphere can be written as $$g=\frac{4}{(1+x_1^2+x_2^2)^2}(dx_1^2+dx_2^2)$$by a stereographic projection map from the ...

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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-...

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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 ...

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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 ...

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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 (...