# Questions tagged [surfaces]

A surface is a two-dimensional topological manifold. The term can also be used to describe a smooth surface, depending on the context.

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### Question in the proof of Hilbert's theorem

I'm researching about Hilbert's theorem which says that there isn't isometric immersion of a complete surface with constant negative Gaussian curvature in $\mathbb{R}^3$. I'm taking as a reference the ...

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### Degree one self-map of $\Bbb R^2\big\backslash \big\{(n,0):n\in \Bbb Z\big\}$ not homotopic to any self-homotopy equivalence

Consider the surface $\Sigma=\Bbb R^2\big\backslash \big\{(n,0):n\in \Bbb Z\big\}$. Does there exist a proper
map $f\colon \Sigma\to \Sigma$ of degree $1$ and not homotopic to
any self-homotopy ...

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30 views

### Dirichlet domains of cusped hyperbolic surfaces

I'm used to think of cusped hyperbolic surfaces as equipped with ideal (all vertices are ideal) fundamental domain, but this is not in general a Dirichlet domain.
Is is true that any such surface has ...

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### Are the “generalized Catalan numbers” of Dumitrescu–Mulase the “moments” of some “multivariate Wigner semicircle distribution”?

The classical Catalan numbers
$$ C_n = \frac{1}{n+1} \binom{2n}{n}, $$
well-known for their numerous combinatorial interpretations (the second volume of Stanley's Enumerative Combinatorics famously ...

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44 views

### Are there canonical exhaustions for bordered surfaces?

If $S$ is a boundaryless connected surface then $S$ has a canonical exhaustion: a sequence of compact connected surfaces $(F_n)$ such that $F_n\subset intF_{n+1}$, $S=\bigcup F_n$ and if
$W$ is a ...

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111 views

### Algebraic context for Mednykh's formula?

Let $S$ be a closed orientable surface and let $G$ be a finite group, then Mednykh's formula says that
$$
\sum_{V}d(V)^{\chi(S)} = |G|^{\chi(S) - 1} |\text{Hom}(\pi_1 S, G)|
$$
where the sum is over ...

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106 views

### Can the infinite jungle gym surface be expressed by an exhaustion of compact surfaces with one boundary component?

Is it possible to write the infinite jungle gym surface as the increasing union of compact surfaces whose boundaries consist of only one simple closed curve? I believe that this is true. This surface ...

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159 views

### The Schoenflies Theorem on two dimensional surfaces

Let $S$ be a surface and $U$ an open connected subset of $S$. If the frontier of $U$ in $S$ is a two sided circle $C$, then the closure of $U$ in $S$ is a surface whose boundary is $C$. I would like ...

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163 views

### Is it possible to prove that surfaces with compact boundary are homeomorphic by glueing disks to the boundary components?

Let $S_1$ and $S_2$ be two surfaces with compact boundary and the same number of boundary components. Let $M_1$ and $M_2$ be the surfaces obtained by glueing closed disks to the boundary of $S_1$ and $...

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130 views

### Abelian variety corresponding to a vector space

I would like to know what the following statement means:
"Let $B_t$ be the Abelian subvariety in $J_t$ corresponding to the $\mathbb{Q}$-vector subspace $H^1(C_t,\mathbb{Q})_{van}$ in the space $...

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### Subgroups of Mod(S) generated by Dehn twists depend only on intersection numbers?

$\DeclareMathOperator\Mod{Mod}$Let $S$ be a closed surface and $\Mod(S)$ be its mapping class group.
It is a well known fact, proved in the Primer on Mapping class groups for example, that the ...

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139 views

### Lifting of a proper map in the cover is a proper map

Let $M$ be an orientable surface without boundary$($I am not assuming $M$ is compact, it can be non-compact$)$. Let $\Phi: M\to M$ be a proper homotopy-equivalnce$($A proper homotopy-equivalence can ...

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108 views

### Pair of pants decomposition for non-orientable surfaces

Pair of pants decomposition on orientable surfaces is a classic way to decompose orientable surface to simples surfaces.
Is there a way to decompose non-orientable surfaces into pair of pants?
...

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246 views

### Constant Gaussian curvature disks

This question has also been posted on MSE, but maybe here is the right place to post it.
Is it true that if $D$ is a Riemannian $2$-disk having constant Gaussian curvature equal to $1$ and whose ...

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248 views

### Homotopy equivalence preserving all geometric intersection numbers

This question again might be silly, like the last post(deleted). Let me know I will delete it.
Problem: Let $\Sigma$ be a surface without boundary and $f:\Sigma\to \Sigma$ be
a proper homotopy ...

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### Hyperbolic length of curve that does not enter a collar

Let $\Sigma$ be a compact surface of genus at least $1$ with one boundary component, equipped with a hyperbolic metric so that the boundary is geodesic. There is a version of the collar lemma that ...

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517 views

### All non-compact simply connected $2$-manifolds with boundary

There are two corresponding posts MSE and MSE by me without any answers.
Problem: Let $\Sigma$ be a non-compact simply-connected $2$-dimensional manifold,
with boundary. Then, up to homeomorphism $\...

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55 views

### Let $S$ be a surface, $K$ compact in $S$ with finitely many components. Does the frontier of a component of $S-K$ have finitely many components?

Let $S$ be a connected surface and $K$ a compact subset of $S$ with finitely many connected components. Let $U$ be a connected component of $S-K$. Does the frontier of $U$ in $S$ have finitely many ...

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107 views

### Space of embedded minimal surfaces of fixed genus in a generic $3$-manifold

Let $M^3$ be a closed, connected and oriented smooth $3$-manifold, and fix an integer $g \geq 1$. Is it true that for a generic set of Riemannian metrics on $M$ the set of closed, connected and ...

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148 views

### Infinitely many simple closed geodesics in any compact orientable surface but the sphere

My question is the following: if $(\Sigma, g)$ is any compact orientable Riemannian surface of genus $g \geq 1$, is it true that there are infinitely many closed, simple and geometrically distinct ...

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### flips on labelled fatgraphs and mapping classes

A fatgraph $G$ is a graph with a cyclic ordering of the edges at each vertex. A labelled fatgraph $(G,L)$ is a fatgraph together with a labelling $L$ of each edge. A labelled fatgraph spine $(G,L,e)$ ...

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### Are there any special relations among the 27 lines on a cubic surface?

If we consider 3 skew lines and the hyperboloid formed by them, any other line on the hyperboloid intersects these three lines. Moreover, the lines orthogonal to the plane containing a point on the ...

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### When do the lengths of simple closed curves determine a hyperbolic surface?

Consider hyperbolic metrics on $\Sigma_g$ a closed orientable surface of genus $g$. Let $[\gamma_1] , \cdots, [\gamma_n]$ be a finite collection of isotopy classes of simple closed curves on $\Sigma_g$...

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218 views

### Group of surface homeomorphisms is locally path-connected

I think the following is true and I need a reference for the proof. (Given a closed surface $S$, i.e. a compact 2-dimensional topological manifold (without boundary), we endow $S$ with a distance ...

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182 views

### Untangling two simple closed curves on a surface

Let $S$ be a smooth surface and $\gamma_1, \gamma_2$ be two transversal simple closed curves on it. Suppose moreover that there exists a simple closed curve $\gamma_1'$ on $S$ isotopic to $\gamma_1$ ...

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101 views

### Homogeneous metric surfaces

I am looking for a reference for this result.
Let $S$ be a metric space such that
it is homeomorphic to a two-dimensional manifold,
it is 2-homogeneous: given two pairs of points
$(x,y)$ and $(x',y')...

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124 views

### Cusps of hyperbolic surfaces under finite covers

The following statement seems true, but I don't know a proof or a reference for it (and I would like one).
Let $\Gamma< \operatorname{PSL}(2,\mathbb R)$ be a nonuniform lattice with one cusp. We ...

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### Representing relative homology classes orientable surfaces with boundary

Let $S$ be compact oriented surface without boundary. Then it is a classical result that a primitive class $\gamma \in H_1(S; \mathbb{Z})$ is always represented by a simple closed curve. It implies ...

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### Whether or not two distinct points in Teichmuller space induce absolutely continuous volume forms on the unit tangent bundle of a surface?

Let $S$ be a closed orientable surface of genus greater than two. Let $g$ and $g'$ be metrics two of constant curvature. I guess we an think of these as two points in the Teichmüller space $\mathcal{T}...

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### Borel conjecture and arbitrary surface

Before starting my question I want to write something that I already know.
Borel Conjecture: Any homotopy equivalence between two closed
aspherical manifolds is homotopic to a homeomorphism.
Now, my ...

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102 views

### Problem of Thickening an Arc in a Topological $ 2 $-Manifold

Let $ M $ be a topological $ 2 $-manifold (possibly with boundary), $ C $ an arc in the interior of $ M $ (i.e., an injective continuous function from $ [- 1,1] $ into $ \operatorname{Int}(M) $), and $...

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230 views

### Is Gauss map of a free boundary convex disk a diffeomorphism?

I asked this question on MSE, but obtained no answer. Maybe this is the right place to post it.
Let $D$ be a properly embedded free boundary disk in the closed unit ball $\mathbb{B}^3$ of $\mathbb{R}^...

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### Spines of Teichmuller space of a non-orientable surface

Let $S_{g,n}$ be the orientable surface of genus $g$ and $n$ punctures. Denote $\Gamma_{g,n}$ be the mapping class group of $S_{g,n}$ and $\mathcal T_{g,n}$ the Teichmüller space of $S_{g,n}$.
In http:...

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358 views

### The largest group acting on a non-orientable surface of genus 5

Let $N_5$ denote the non-orientable surface of genus 5.
In Conder's database https://www.math.auckland.ac.nz/~conder/BigSurfaceActions-Genus2to101-ByGenus.txt we can see that the biggest finite group $...

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81 views

### What can we say about the complement of two homotopic simple closed curves on a compact orientable surface?

Do two homotopic simple closed curves separate a compact orientable surface?
If they are disjoint they do and bound a cylinder. But if they intersect? What can we say about the components of their ...

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122 views

### A computation in a commutative Frobenius algebra

This is already posted here https://math.stackexchange.com/questions/3695584/a-computation-in-a-commutative-frobenius-algebra but I didn't get any answers.
Given a commutative Frobenius algebra (in ...

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### Convergence of free boundary minimal surfaces

I suspect the following statement is true:
Let $(M^3,g)$ be a compact and orientable Riemannian $3$-manifold with nonempty boundary. Let $\{\Sigma_n\}_{n \geq 1}$ be a sequence of compact and ...

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319 views

### An orientable surface that cannot be embedded into $\Bbb R^3$? [duplicate]

I previously asked this question on MSE, without success.
By Whitney's embedding theorem, every 2-dimensional manifold (aka. a surface) can be embedded into $\Bbb R^4$.
Now, Wikipedia states in this ...

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204 views

### Space of representations of surface group into Lie groups

In the context of Goldman's paper The symplectic nature of fundamental groups of surfaces:
Consider a closed oriented surface $S$ with fundamental group $\pi$, and let $G$ be a connected Lie group. ...

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157 views

### Continuity of $r\mapsto\int_{\Sigma\cap B_r(x)}f^2d\mu$

Let $\Sigma$ be an embedded smooth surface in $\mathbb{R}^3$, and let $f:\Sigma\to\mathbb{R}$ be a smooth function. Suppose $f$ is square-integrable on $\Sigma$, with
\begin{align}
0<\int_{\Sigma}f^...

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### How do I find the portion of a cell/voxel lying within a defined surface?

We have a 3-dimensional grid of voxels (or cells), with individual voxels being of volume $dx\,dy\,dz$ where $dx=dy=dz=1$.
A cone-like surface is defined by some function, $z = f(x, y)$, which in ...

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### When is a regular surface in $\mathbb{R^4}$ contained in a $3$-dimensional vector subspace of $\mathbb{R^4}$?

Introduction: Let $S$ be a regular connected surface embedded in $\mathbb{R^3}$. We know that $S$ is contained in a plane (a $2$-dimensional vector subspace of $\mathbb{R^3}$) iff every point of $S$ ...

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275 views

### Permuting $n$ points in a $2$-manifold

Given $n$ points on a connected $2$-manifold $M$, I'd like to consider the homotopy classes of paths that "permute" these points.
Edit (Clarifying what I mean by this):
Given a set of $n$ ...

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179 views

### Closed simple curves in $\mathbb{R}\mathbb{P}^2$

EDIT: The well known Jordan curve theorem says: let $C\subset S^2$ be a closed simple curve on the 2-sphere. Then its complement $S^2\backslash C$ consists of two connected components, both ...

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### A generalization of Jordan-Schoenflies theorem on simple plane curves

The well known Jordan-Schoenflies theorem says: let $C\subset \mathbb{R}^2$ be a closed simple curve. Then there exists a homeomorphism $f\colon\mathbb{R}^2\to \mathbb{R}^2$ such that $f(C)$ is the ...

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360 views

### Flat metric on compact surface minus a point

Let $T^2$ be a compact smooth surface and let $p\in T^2$. Suppose that $T^2$ admits a symmetric $\left(0,2\right)$-tensor which is a flat Riemannian metric restricted to $T^2-\{p\}$. Is it true that $\...

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164 views

### Approximation of a Sobolev surface by a smooth surface

I was quite sure that the answer to the following question is known, and was surprised not to find any reference:
Let $M$ be a compact, oriented $2$-dimensional manifold with boundary. Let $f:M\to R^...

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204 views

### Immersion in $\mathbb R^3$ of a Klein bottle with Morse-Bott height function without centers

Can the Klein bottle be immersed in $\mathbb R^3$ so that the associated height function be of Morse-Bott type and have no centers?
That is, the height function would have only Bott-type extrema and ...

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237 views

### Immersion of non-orientable surface in $\mathbb R^3$ with conditions on the height function

EDIT: The answer is trivially positive; the question arose from my misunderstanding of the figure below.
Can a non-orientable closed surface of odd genus be immersed in $\mathbb R^3$ so that the ...

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122 views

### Are these two arguments incompatible?

I want to understand why (if so) these two arguments are not incompatible. And if that's the case, which one is wrong.
First we have this paper (by Honda, Kazez and Matic). We look at the last Lemma ...