Questions tagged [conformal-geometry]

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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 ...
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2 votes
1 answer
106 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 ...
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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 ...
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2 votes
0 answers
58 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 ...
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5 votes
0 answers
99 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 ...
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43 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 ...
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6 votes
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108 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 ...
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1 vote
0 answers
89 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 ...
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78 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 ...
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61 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 ...
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$L^\infty$ norm of Schwarzian derivative implies univalence?

On page 26 of these notes (Riemann surfaces, dynamics and geometry) there is a theorem 2.13 which says Let $f: \mathbb H \to \hat{\mathbb C}$ be a holomorphic map, satisfying $\|Sf\|_{\infty} <1/2$...
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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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0 answers
71 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 ...
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1 vote
2 answers
376 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 (...
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1 answer
131 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 ...
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2 votes
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110 views

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: ...
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2 votes
1 answer
164 views

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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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\...
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1 vote
0 answers
138 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 ...
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6 votes
4 answers
387 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 ...
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2 votes
0 answers
67 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?
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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 ...
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1 vote
0 answers
69 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 $M_0$ ...
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  • 2,858
0 votes
1 answer
232 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 ...
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7 votes
0 answers
244 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 & ...
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5 votes
4 answers
311 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))= \...
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3 votes
1 answer
121 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})/\...
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5 votes
0 answers
87 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\...
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3 votes
2 answers
162 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^...
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0 votes
1 answer
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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 ...
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2 votes
3 answers
217 views

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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2 votes
0 answers
144 views

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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1 vote
0 answers
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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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16 votes
1 answer
603 views

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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3 votes
2 answers
170 views

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 (...
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  • 813
1 vote
0 answers
77 views

Continuity of conformal grafting, wrt the (infinite type) surface

Say I have two closed hyperbolic surfaces $X,Y$ and a smooth, $(1+\epsilon)$-bilipschitz map $f : X \to Y$ for some small $\epsilon$. Pick a simple closed curve $c \subset X$, and let $X',Y'$ be the ...
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  • 457
7 votes
1 answer
206 views

Curvature of complete conformal metrics on the open unit disk

Let $D$ be the unit disk in the complex plane, and assume that $g$ is a Riemannian metric on $D$ which is complete and conformal to the standard Euclidean metric. Can it be the case that the Gaussian ...
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1 vote
2 answers
141 views

compatification of $\mathbb{R}^2$ under a conformal metric

I want to understand the following question: Let $\delta$ be the Euclidean metric in $\mathbb{R}^2$. Is there any criteria for smooth function $u$ such that $(\mathbb{R}^2, e^{2u}\delta)$ can be ...
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  • 1,060
3 votes
1 answer
166 views

Conformal mapping between two right-angled triangles

I want to derive a conformal mapping $f\!:\!A\!\to\! B$ where $A=\{ (x,y)\ |\ x\!\in\![0,1]\ \text{ and }\ 0 \leq y \leq x \}$ and $B=\{ (x,y)\ |\ x\!\in\![0,1]\ \text{ and }\ 0 \leq y \leq \frac{...
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2 votes
0 answers
87 views

Conformal welding of rectifiable curves

In classical conformal welding theory, we start with a homeomorphism $h$ of the unit circle and try to find a Jordan domain $D$ together with two conformal isomorphisms $f_1 \colon \mathbb D \to D$ ...
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1 vote
1 answer
75 views

Assuming the conformal factor is radially decreasing, prove or disprove the uniqueness of geodesic joining origin and points on the boundary of ball

Let $u$ be a radially decreasing function defined on $\mathbb{R}^n$. We consider the metric $g=e^{2u}\delta$ where $\delta$ is the standard Euclidean metric on $\mathbb{R}^n$. Let $B_r$ be the ball ...
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3 votes
0 answers
179 views

Can every diffeomorphism be rescaled into a volume preserving one?

This is a cross-post. Let $D \subseteq \mathbb{R}^2$ be the closed unit disk, and let $f:D \to D$ be a diffeomorphism. Does there exist a smooth $h \in C^{\infty}(D)$ such that $h\cdot f$ is an ...
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2 votes
1 answer
100 views

Conformal maps between simply connected domains with piecewise real algebraic boundary

Between polygons in $\mathbb C\cup\{\infty\}$ (including the "single side polygons", hemispheres, disks) the Schwartz-Christoffel mappings give arguably explicit conformal maps. For polygons with few ...
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  • 855
14 votes
1 answer
334 views

Regularity of conformal maps

In order to define what it means for a map $f \colon \Omega \subseteq \mathbb R^n \to \mathbb R^n$ to be conformal, it is sufficient to require that $f$ is everywhere differentiable. Does conformality ...
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  • 1,307
-3 votes
1 answer
188 views

Conformal map from a 7-sided polyhedron to a square pyramid

I have a right-angled square pyramid, $A$, whose height and base-length are $l$. By 'right-angled', I mean that the apex of $A$ lies vertically above one of the vertices in its base. Now supposed I ...
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0 votes
0 answers
30 views

Counterexample to a generalization of planar convex hulls

I am looking for a counter example to the following conjecture: for every finite compact region $\mathcal{R}\subset\mathbb{R}^2$ that is defined by a rectifiable Jordan curve and every fixed and ...
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0 votes
0 answers
27 views

Conformal map from quadrilateral to a sector of a circle [duplicate]

Can anyone advise me on how to derive a conformal map for this mapping? I am familiar with how to apply Schwarz-Christoffel from the upper half plane to the quadrilateral, but how do I then map from ...
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  • 167
5 votes
1 answer
335 views

Reconciling Sullivan's theorem with the hyperbolic structure of the Figure–8 knot complement

I am interested in the 3-manifolds with hyperbolic structures from the physics (gravity) perspective. I encounter this paper https://arxiv.org/pdf/hep-th/9812206.pdf whose Eq. (9) mentions a theorem ...
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3 votes
1 answer
299 views

Conformal map onto a circle, from an identification space composed of five squares

I am looking to derive a conformal map for the problem illustrated in this image. I've read a bit about how to map a square onto a circle, but I'm struggling to extend the concepts for the domain at ...
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  • 167
4 votes
2 answers
307 views

Is every conformal manifold equivalent to a flat one with cone singularities?

Consider a polytope with a $2$-dimensional surface and the corresponding metric on this surface (coming from the embedding in $3$-dimensional Euclidean space). Intrinsically the metric is flat ...
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