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.
262
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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 ...
4
votes
1
answer
156
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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 $\Sigma\times I$ ...
- 545
4
votes
1
answer
115
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Topological type of complement of Heegaard curves in Heegaard surface $(\Sigma - \alpha - \beta)$
Suppose $(\Sigma, \alpha, \beta)$ is a genus-$g$ Heegaard diagram for a closed, oriented $3$-manifold $Y$, i.e. $\Sigma$ is an orientable genus-$g$ surface, and $(\alpha_1, \dots, \alpha_g)$ and $(\...
- 271
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Maximal orders and surface subgroups of even genus
Let $A$ be a quaternion algebra over a totally real number field $k$. Suppose that $A$ splits at exactly one real place of $A$. Let $\mathcal{O}$ be a maximal order in $A$. Then $\mathcal{O}$ contains ...
- 483
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Generically finite projection $\pi_L: X \to \mathbb{P}^2$ from plane $L$ and critical points
(In following we are working in "classical" complex setting: i.e. all involved schemes are considered to be varieties over $k=\mathbb{C}$)
Let $X \subset \mathbb{P}^n$ be irreducible surface ...
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votes
2
answers
246
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Surjections from genus $n$ surface group to free group of rank $n$
Let $\Sigma_n$ be a genus $n$ surface, let $\mathcal{H}_n$ be a genus $n$ handle body, and let $F_n$ be a free group of rank $n$. Fix an identification of $\pi_1(\mathcal{H}_n)$ with $F_n$. I know ...
- 73
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votes
2
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128
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Generate $\mathrm{Mod}(S_g)$ by two Dehn twists
Let $S_g$ be a closed orientable surface of genus $g>1$.
How can one prove that its mapping class group $\mathrm{Mod}(S_g)$
is not generated by two Dehn twists?
A pair of simple closed curves in $...
- 713
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votes
0
answers
27
views
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 $f_n : X \times \cdots \times X \to \mathbb{R}$...
- 933
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votes
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69
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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 ...
- 2,001
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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 ...
5
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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 ...
- 381
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129
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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
$$\Lambda(\Sigma) := \sup \{ \lambda_1(\Sigma,g) \operatorname{Area}(\Sigma,g) : g \text{ is a smooth Riemannian metric on $\Sigma$} \}$$
...
- 2,001
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votes
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Links and non-orientable surfaces
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 ...
- 413
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answer
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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 ...
- 151
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answers
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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 ...
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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 ...
- 933
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0
answers
63
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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{...
- 413
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votes
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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 - ...
- 413
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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 ...
- 793
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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 ...
- 2,001
4
votes
2
answers
81
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Characterization of a non-trivial non-peripheral element of the free homotopy classes of a compact bordered surface
Let $\Sigma$ be a compact orientable connected $2$-manifold with a non-empty boundary. Let $\widehat \pi(\Sigma)$ denote the set of free homotopy classes of
curves in $\Sigma$. We say $x\in \widehat \...
- 793
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answer
104
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Given $f$ from the cylinder $C$ to the interval constant on one boundary, is there a $r:C\to C$ constant on a boundary with $f\circ r = f$?
My question might be trivial, but my lack of knowledge of this particular subject has not enabled me to find the answer. What I want to know is the following. Let $I=[0,1]$ and $C=S^1\times I$ be the ...
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241
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Why do we define the pants complex and the pants decomposition? [closed]
Why do we define the pants complex?
I learned for the first time in A Presentation for the mapping class group of a closed orientable surface (by A. Hatcher and W. Thurston)
that we have definition of ...
- 119
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Earliest known proof of "Any degree one self-map of an orientable connected finite-type non-compact surface is homotopic to a homeomorphism"
I attended a talk where the speaker said the following is due to Nielsen. I searched here and there but couldn't find the corresponding paper, if any. So, what is the earliest known proof of the ...
- 793
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Groups of non-orientable genus 1 and 2
The non-orientable genus (aka crosscap-number) $\overline{\gamma}(G)$ of a finite group $G$ is the minimum non-orientable genus among all its connected Cayley graphs (and $0$ if $G$ has a planar ...
- 141
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votes
2
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773
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Fixed-point free diffeomorphisms of surfaces fixing no homology classes
One of my graduate students asked me the following question, and I can't seem to answer it. Let $\Sigma_g$ denote a compact oriented genus $g$ surface. For which $g$ does there exist an orientation-...
- 303
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1
answer
147
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Intersection pairing on $H_1$ of surfaces
Let $X$ be a surface (real, not complex). With coefficients in a ring $R$, there is an intersection pairing $\langle -,- \rangle:H_1(X;\mathbb R) \otimes H_1(X;\mathbb R) \to R$. Is this pairing ...
- 1,045
2
votes
1
answer
111
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Classification of degree one map between two closed orientable surfaces without using induction on the genus
A theorem of Edmonds (see Theorem 3.1. of "Deformation of Maps to Branched Coverings in Dimension Two") says that
Theorem 1: A degree-one map between closed orientable surfaces is homotopic ...
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answer
107
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Isotopic homeomorphisms of surface induces same map on the space of ends
Let $\Sigma$ be a non-compact orientable connected two-manifold without boundary. Let $f,g\colon \Sigma\to \Sigma$ be two homeomorphisms. Suppose there is a homotopy $H\colon \Sigma\times [0,1]\to \...
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votes
2
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373
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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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Area of a deformation of a closed surface
Let $(M^3,g)$ be a complete Riemannian manifold. Fix a two-sided immersion $\varphi : \Sigma^2 \to M$ from a closed surface into $M$, with unit normal $N$. Given $f \in C^\infty(\Sigma)$ and $\alpha : ...
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votes
3
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956
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How to generate all triangulations of an orientable surface?
$\newcommand{\comb}{\mathrm{comb}}$Consider an orientable surface $S$ with punctures and boundaries (each boundary having at least a marked point).
A triangulation, up to orientation preserving ...
- 1,305
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0
answers
136
views
Doubly ruled surfaces in hyperbolic 3-space
A well-known theorem of classical surface theory states that the only doubly ruled surfaces in Euclidean 3-space are planes, 1-sheeted hyperboloids and hyperbolic paraboloids. There are a number of ...
- 2,287
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128
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End space of non-compact 2-manifolds described with proper rays
I am wondering if the classification of noncompact surfaces given by Ian Richards can be stated with proper rays instead of nested sequences of connected open subsets with compact boundary. I asked ...
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261
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"Classification" of (orientable) 3-manifolds with genus-g-surface as their boundary
This is in some sense a generalization of the question I asked some time ago. I am very sorry if this question is too basic for MathOverflow, but I just started learning about some more detailed ...
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votes
1
answer
182
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Inequivalent free $\Bbb Z/n\Bbb Z$-actions on orientable compact bordered surface
Let $S_{g,b}$ denote the orientable connected compact surface of genus $g$ with $b$ boundary components. A group homomorphism $\varphi\colon G\to \text{Homeo}^+(S_{g,b})$ is said to be free $G$-...
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139
views
Minimal surface enclosing balls
(This question is tangentially related to an earlier
question I posed: Minimal surface enclosing two congruent balls.)
Let $B_1,\ldots,B_k$ be unit-radius balls in $\mathbb{R}^3$, with pairwise
...
- 147k
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answer
190
views
Minimal surface enclosing two congruent balls
Let $B_1$ and $B_2$ be two unit-radius balls in $\mathbb{R}^3$ whose centers are separated
by a distance $d \ge 2$.
Q. For sufficiently small $d$,
is the minimal area surface enclosing $B_1$ and $B_2$...
- 147k
2
votes
1
answer
102
views
Can we always find coordinates on a surface such that $K=K(u-v)$?
Let $(M^2,g)$ be a 2-dimensional Riemannian manifold. For any point $p\in M^2$ can we always find coordinates $(u,v)$ in a neighborhood $U$ of $p$ such that the Gaussian curvature is only a function ...
- 237
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297
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A local isometric immersion from $\mathbb H^{n}$ into $\mathbb R^{2n-1}$
I found this local isometric immersion from $\mathbb H^{n}$ into $\mathbb R^{2n-1}$, given by Schur (1886) in Über die Deformation der Räume constanten Riemannschen Krümmungsmaasses as follows, $(1\...
- 123
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votes
0
answers
103
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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 ...
- 587
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192
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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 ...
- 147k
6
votes
0
answers
156
views
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 ...
- 103
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votes
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140
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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 ...
- 2,001
1
vote
0
answers
291
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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 ...
- 123
8
votes
2
answers
224
views
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 ...
- 265
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votes
1
answer
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 ...
- 215
9
votes
0
answers
183
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 ...
- 5,299
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0
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133
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 ...
