# 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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### Identifying a curve on a closed surface of genus 4

The notation is the one used in the attached picture. Take a closed, orientable surface $\Sigma_4$ of genus $4$, obtained as the identification space of a polygon with $16$ sides in the usual way. The ...
156 views

### Euler class of vertical tangent bundle of the surface bundle over circle

Suppose $\Sigma$ is an oriented genus $g>1$ surface and $h:\Sigma\to \Sigma$ is a diffeomorphism preserving a point $p$. Let $M$ be the surface bundle over $S^1$ obtained by gluing $\Sigma\times I$ ...
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### Maximum for sum of distances on a constant curvature surface

(This is a modification of a previous question where my hopes were too general.) Let $X$ be a closed constant curvature surface and consider the function $f_n : X \times \cdots \times X \to \mathbb{R}$...
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### Leaves of bounded genus

Let $\mathcal{F}$ be a codimension one foliation in a closed $3$-manifold $M$. Does there exist an upper bound for the genus of the compact orientable leaves? That is, does there exist $G >0$ such ...
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### Cubic lattice representation of a solid torus knot using the surface as a boundary

For physics simulation reasons, I would like to respresent a solid torus knot as a collection of integer points sat on a cubic lattice. If I were to do this using a sphere, I would do this by saying ...
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### Intersections of geodesics in an "almost flat" plane

Let $g$ be a complete metric on $\mathbb{R}^2$, such that: Outside of a compact connected set $K\subset \mathbb{R}^2$, the curvature of $g$ vanishes. The integral of the Gaussian curvature in $K$ is ...
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### Upper bound for the first eigenvalue of the Laplacian on surfaces with boundary

Let $\Sigma$ be a compact smooth surface with boundary. Define $$\Lambda(\Sigma) := \sup \{ \lambda_1(\Sigma,g) \operatorname{Area}(\Sigma,g) : g \text{ is a smooth Riemannian metric on \Sigma} \}$$ ...
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Let $\Sigma \subset \mathbb{R}^3$ be a compact embedded surface with boundary $\partial \Sigma$ and $i:\Sigma\setminus \partial\Sigma \to \mathbb{R}^3 \setminus \partial\Sigma$ the inclusion. Is the ...
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### Closed-form examples of CMC surfaces

Besides the trivial cases of cylinders and spheres, are there any other known examples of non-zero constant mean curvature surfaces which can be represented explicitly in a closed form? I am ...
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### Vanishing of Goldman bracket requires simple-closed representative?

Let $\Sigma$ be a connected oriented surface, and $[-,-]\colon \Bbb Z\big[\widehat\pi(\Sigma)\big]\times Z\big[\widehat\pi(\Sigma)\big]\to Z\big[\widehat\pi(\Sigma)\big]$ be the Goldman Bracket. Note ...
1 vote
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### Uniqueness of graph embeddings on surfaces for highly connected graphs

A theorem of Whitney says that if $i_1$ and $i_2$ are two embeddings of a 3-connected planar graph into $S^2$, then there exists a homeomorphism $h : S^2 \to S^2$ such that $h \circ i_1 = i_2$. Is ...
1 vote
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### Diffeomorphism induced by small perturbation

Consider the surface $S_{\epsilon}$ defined as: \begin{align} %S&=\{\vec x \in \mathbf{R}^3: x=0\}, \\ S_{\epsilon}&=\{\vec x \in \mathbf{R}^3:\epsilon (x^2 + y^2 + z^2 - 1) + x=0\}. \end{...
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### Topology change induced by small perturbation

Consider the surface $S_{\epsilon}$ defined as: \begin{align} %S&=\{\vec x \in \mathbf{R}^3: x=0\}, \\ S_{\epsilon}&=\{\vec x \in \mathbf{R}^3:f_{\epsilon}(\vec x)\equiv\epsilon ((x^2 + y^2 - ...
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### A closed curve can be homotopic to remove all intersections with a filling $\Gamma$ if it has zero geometric intersection numbers with $\Gamma$

Let $\Sigma$ be a compact oriented connected bordered surface other than the pair of pants. Let $\Gamma:=\{\gamma_i\}$ be a finite collection of simple closed curves on $\Sigma$ such that each ...
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### Every surface of sufficiently large genus separates

Let $M^3$ be a smooth closed orientable manifold. Does there exist a non negative integer $g_0$ such that every closed orientable embedded surface $\Sigma \subset M$ of genus $g \geq g_0$ represents ...
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### Kneser theorem about the Klein bottle

I know that in $1923$ H. Kneser showed that a continuous flow in a Klein bottle without singular points has a periodic trajectory. The original article is this, but does anyone know another old or new ...
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### Restriction function as a Morse function

Let $\Sigma$ be a closed surface smoothly embedded in $\mathbb R^3$. For any Morse function $h:\mathbb R^3 \to \mathbb R$, can we isotope $\Sigma$ so that the restriction of $h$ on $\Sigma$ is also a ...
192 views

### Factoring a projection of surface groups through a free group

Consider the surface group $S_g=\langle a_1,b_1,a_2,b_2,\dots,a_g,b_g \mid [a_1,b_1][a_2,b_2]\cdots[a_{g},b_{g}]=1\rangle$, which is the fundamental group of the closed orientable genus-$g$ surface. ...
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### Does a random walk on a surface visit uniformly?

Let $S$ be a smooth compact closed surface embedded in $\mathbb{R}^3$ of genus $g$. Starting from a point $p$, define a random walk as taking discrete steps in a uniformly random direction, each step ...
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### Rigidity for convex surfaces in elliptic/hyperbolic space

From Alexandrov's work we know that any metric on the sphere with lower curvature bound $\kappa$ (in the sense of Alexandrov) can be realized as a closed convex surface (i.e. boundary of a compact ...
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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 ...
1 vote
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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 ...
299 views

### 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 ...
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 ...