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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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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12 votes
3 answers
891 views

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 ...
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1 vote
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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 ...
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1 vote
0 answers
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Are $f_1,f_2$ necessarily pseudo-Anosov?

Suppose $F: S_1\times S_2\to S_1\times S_2$ is an Anosov diffeomorphism (assume there is one), and $S_1,S_2$ are closed surfaces. If $F$ is homotopic to $f_1\times f_2$, a product of two surface ...
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1 vote
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108 views

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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5 votes
1 answer
197 views

"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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4 votes
1 answer
169 views

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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1 vote
1 answer
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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 ...
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8 votes
1 answer
175 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 $B_2$...
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2 votes
1 answer
90 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 ...
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4 votes
1 answer
290 views

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\...
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2 votes
0 answers
80 views

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 ...
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7 votes
1 answer
175 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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16 votes
3 answers
1k views

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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6 votes
0 answers
138 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 ...
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3 votes
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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1 vote
0 answers
230 views

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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8 votes
2 answers
189 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 ...
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6 votes
1 answer
241 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 ...
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9 votes
0 answers
153 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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0 votes
0 answers
116 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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1 vote
1 answer
183 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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1 vote
0 answers
177 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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2 votes
0 answers
153 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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5 votes
1 answer
116 views

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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2 votes
1 answer
161 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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  • 265
1 vote
0 answers
150 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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  • 11
7 votes
2 answers
289 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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1 vote
1 answer
285 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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2 votes
1 answer
119 views

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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8 votes
1 answer
698 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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0 votes
0 answers
62 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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2 votes
1 answer
145 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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3 votes
0 answers
177 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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1 vote
0 answers
35 views

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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0 votes
0 answers
90 views

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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9 votes
1 answer
138 views

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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9 votes
2 answers
266 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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5 votes
1 answer
198 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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  • 13.2k
2 votes
0 answers
111 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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  • 17k
3 votes
1 answer
140 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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  • 245
5 votes
1 answer
171 views

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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  • 103
2 votes
0 answers
40 views

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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6 votes
0 answers
335 views

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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0 votes
0 answers
131 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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6 votes
1 answer
279 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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1 vote
1 answer
120 views

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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6 votes
1 answer
383 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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1 vote
1 answer
142 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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6 votes
0 answers
63 views

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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