# Questions tagged [conformal-geometry]

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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} ...
131 views

### 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 vote
64 views

### 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 ...
243 views

### 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 ...
94 views

### 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 ...
462 views

### 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 ...
237 views

### 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 ...
122 views

### 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, ...
77 views

### 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 ...
182 views

### 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)...
95 views

### 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) &...
100 views

### 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} = \...
229 views

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

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

### 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 ...
173 views

### 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 ...
104 views

### 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 ...
216 views

### 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 ...
170 views

### 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
144 views

### 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 ...
70 views

### 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 ...
116 views

### 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 ...
54 views

### 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 ...
142 views

### 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 ...
1 vote
157 views

### 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 ...
1 vote
92 views

### 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 ...
95 views

### 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 ...
1 vote
53 views

### 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 ...
1 vote
86 views

### 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 ...
1 vote
796 views

### 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 (...
144 views

### 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 ...
114 views

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: ...
212 views

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 $\mathrm{div}... 1 vote 0 answers 164 views ### Does there exist an isometry between a regular polygon and a circle? In order to define the question in a meaningful fashion, I am referring to a smooth manifold$\mathcal{M}$within an$\epsilon$-neighborhood of a regular polygon$\mathcal{P}$satisfying $$\max\{\|x-p\... 1 vote 0 answers 254 views ### Conformal changes of metric and Ricci curvature Let (M,g) be a three dimensional smooth Lorentzian manifold and let p be a fixed point in M and let S be a smooth symmetric tensor of rank two on T_pM\times T_pM. Does there exist a smooth ... 6 votes 4 answers 482 views ### Generalizing contour integration to quaternions and bicomplex numbers I am interested in the possibility of generalizing the notion of contour integration to the quaternions or bicomplex numbers. I am aware that the Frobenius theorem prevents the construction of a true ... 2 votes 0 answers 97 views ### The space of conformal classes of the n-sphere It is well-known that the space of conformal classes on the 2-sphere is just a point. What is known about the space of conformal classes on the n-sphere? Is there any structure on it? 3 votes 0 answers 97 views ### Range of divergence operator on the space of traceless symmetric (0,2) tensors; conformal vector fields on an arbitrary metric on S^2 Let \gamma be a metric on S^2. I am trying to solve the following PDE on a (0,2) symmetric traceless tensor A:$$div_{\gamma} A = \omega$$where \omega is a 1-form. It is known that there ... 1 vote 0 answers 78 views ### conformal changes of Lorentzian metrics Let M= \mathbb R \times M_0 with M_0 a smooth compact manifold with smooth boundary and let g=-dt^2+g_0(t,x) be a Lorentzian metric on M, with g_0 a family of Riemannian metrics on M_0 ... 0 votes 1 answer 415 views ### Self duality and anti-self duality of Weyl curvature in four dimension I am trying to compute explicitly in terms of extrinsic curvature the self dual part W^+ and the anti-self dual part W^- of the Weyl tensor W associated with a codimension 1 submanifold into ... 7 votes 0 answers 298 views ### If SO(3,\mathbb C) is isomorphic to PGL(2,\mathbb C), what objects do vectors in \mathbb C^3 represent in the context of Möbius geometry? I hope this question isn't too basic or ambiguous for this site. The following is an explicit isomorphism from \mathrm{PGL}(2,\mathbb C) to \mathrm{SO}(3,\mathbb C):$$\left[\begin{matrix}p & ... 5 votes 4 answers 434 views ### 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 $$\cos (Tx(t_0),Ty(t_0))= \... 3 votes 1 answer 130 views ### What is the orbit of the standard conformal structure on S^2 under \operatorname{SL}(3,\mathbb{R})? \DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}Consider the group \GL_+(3,\mathbb{R}) acting on \mathbb{R}^3. It induces an action of \GL_+(3,\mathbb{R})/\... 5 votes 0 answers 103 views ### An application of Leray-Schauder degree theory for Nirenberg problem on the 2-sphere I'm studying the article "The scalar curvature equation on 2- and 3-spheres" by Chang, Gursky and Yang and I'm particulary interested in the 2-sphere case. They prove that if K:S^2\... 3 votes 2 answers 241 views ### Classification of conformal diffeomorphisms of Minkowski space, part 2 This is a continuation of Classification of conformal diffeomorphisms of Minkowski space Consider \mathbb{R}^{n+1} equipped with the Minkowski (sign indefinite) metric:$$g=(x^0)^2-(x^1)^2-\dots -(x^... 0 votes 1 answer 135 views ### 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 ...
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 ...