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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5 votes
0 answers
193 views

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 ...
9 votes
2 answers
1k views

Triple bubble conjecture: Natural candidate?

Is there a standard natural candidate surface for the shape that encloses three given volumes in $\mathbb{R}^3$ and has minimal surface area? I know the planar triple bubble conjecture was proved ...
21 votes
2 answers
891 views

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-...
21 votes
5 answers
1k views

Is a rhombus rigid on a sphere or torus? And generalizations

If a rectangle is formed from rigid bars for edges and joints at vertices, then it is flexible in the plane: it can flex to a parallelogram. On any smooth surface with a metric, one can define a ...
2 votes
1 answer
191 views

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 ...
3 votes
1 answer
185 views

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 ...
7 votes
2 answers
387 views

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 ...
2 votes
1 answer
129 views

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 \...
7 votes
4 answers
832 views

Lagrangian Kleinian bottles

I remember some talks some time ago about proofs of nonexistence of Lagrangian Kleinian bottles in ${\mathbb C}^2$ for the standard symplectic structure, mentioning that this were the only compact ...
12 votes
3 answers
996 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 ...
1 vote
0 answers
150 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 ...
5 votes
1 answer
308 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 ...
1 vote
0 answers
135 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 ...
16 votes
0 answers
627 views

Approximating homeomorphisms of 2-disk by diffeomorphisms

Any homeomorphism of a compact surface can be approximated by diffeomorphisms. Is there a parametrized version of this result, where the parameter space is an $n$-disk? In other words, if $S$ is a ...
4 votes
1 answer
184 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 $G$-...
8 votes
1 answer
206 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 $B_2$...
1 vote
1 answer
160 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 ...
2 votes
1 answer
114 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 ...
4 votes
1 answer
309 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\...
4 votes
1 answer
451 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 ...
4 votes
1 answer
260 views

Pseudo-Anosov diffeomorphisms vs reducible diffeomorphisms

I was wondering if anyone knew a 'simple' proof of the fact that a pseudo-Anosov diffeomorphism of a closed surface $\Sigma$ is not reducible, in the sense that it does not fix the free homotopy class ...
2 votes
0 answers
125 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 ...
7 votes
1 answer
218 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. ...
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 ...
3 votes
1 answer
155 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 ...
1 vote
0 answers
315 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 ...
8 votes
2 answers
245 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 ...
6 votes
1 answer
342 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 ...
9 votes
0 answers
202 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 ...
0 votes
0 answers
137 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 ...
1 vote
1 answer
232 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 ...
1 vote
0 answers
219 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 $...
2 votes
0 answers
158 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 $...
1 vote
0 answers
244 views

Why does this PDE have a solution?

Let $\Sigma$ be a compact surface and denote by $\nu$ the unit conormal for $\partial \Sigma$. Let $$E = \left\{ \phi \in C^{2,\alpha}(\Sigma) : \int_{\Sigma} \phi \, \mathrm{d}A = 0 \right\} $$ and $$...
7 votes
2 answers
351 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 ...
5 votes
1 answer
175 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 ...
26 votes
1 answer
820 views

Disc bounded by a plane curve

Let $\Sigma$ be a sphere topologically embedded into $\mathbb{R}^3$. Is it always possible to find a disc $\Delta\subset\Sigma$ which is bounded by a plane curve? It is easy to find an open disc ...
2 votes
1 answer
254 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 ...
1 vote
0 answers
218 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? ...
1 vote
1 answer
402 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 ...
6 votes
0 answers
385 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 ...
8 votes
1 answer
916 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 $\...
2 votes
1 answer
188 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 ...
0 votes
0 answers
81 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 ...
2 votes
1 answer
166 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 ...
4 votes
0 answers
229 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 ...
1 vote
0 answers
38 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 $(G,L,e)$ ...
6 votes
1 answer
318 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}^...
0 votes
0 answers
114 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 ...
9 votes
1 answer
164 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 $\Sigma_g$...