# 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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### 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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### 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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### 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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### 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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### 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 $n$ distinct ...

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### 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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### 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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### 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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### 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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### 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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### 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 ...

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### How the hyperbolic metric changes when we add a puncture?

Suppose we have a surface of a finite genus, without boundary with a finite number of punctures. This surface admits a unique hyperbolic metric of curvature $-1.$
If I add a puncture somewhere, the ...

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### Why does this PDE have a solution?

Let $(M^3,g)$ be a compact $3$-manifold with boundary and let $\Sigma$ be a surface such that $\partial M \cap \Sigma = \partial \Sigma$ and the intersection is orthogonal ($\Sigma$ is a free-boundary ...

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### Order of separating Dehn twists in the image of Johnson homomorphisms

Let $S$ be a closed surface, $\Gamma=\pi_1 S$ and $\Gamma_i$ be the lower central series defined by $\Gamma_0=\Gamma$ and $\Gamma_i=[\Gamma,\Gamma_{i-1}].$
The Johnson filtration $\lbrace \mathcal{K}...

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### Lengths of edges of a triangulated surface

Consider a triangulated surface of genus $g,$ which is embedded in $\mathbb{R}^3$. A simple parameter counting shows that the lengths of edges of the surface satisfy $6g$ algebraic equations. Have ...

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169 views

### When are principal lines of curvature geodesics?

Let $S$ be a smooth surface embedded in $\mathbb{R}^3$.
When are (some of) the principal lines of curvature geodesics
on $S$? Perhaps on the ellipsoid below, the (blue) central
cycle, a max principal ...

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### open book decompositions of $\Sigma\times S^1$

Let $\Sigma$ be a closed orientable surface. Is there a standard open book decomposition on the $3$-manifold $M=\Sigma\times S^1$?
If the binding is allowed to be empty in the definition of an open ...

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### Transverse measures in pseudo-Anosov diffeomorphisms

I've recently begun doing research involving pseudo-Anosov diffeomorphisms, which are diffeomorphisms on surfaces $f : M \to M$ admitting two singular measured foliations $(\mathcal F^s, \nu^s)$ and $(...

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### Generators of the mapping class group for surfaces with punctures and boundaries

Let $\Gamma_{g,b}^m$ denote the mapping class group of a genus $g$ surface with $b$ non-permutable parametrised boundary curves and $m$ permutable punctures.
It is clear that in general, ...

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### What are sufficient conditions for “homotopy equivalence” of slice contours?

Apologies in advance for imprecision of the question. Thanks for improving it. Let M be a compact, connected, orientable surface in three dimensional Euclidean space without boundary and without self-...

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### Planar arc on a topologically embedded sphere or disk in $\mathbb{R}^3$

An arc is a set homeomorphic to the unit interval $[0,1]$; an arc in $\mathbb{R}^3$ is planar if it is contained in some plane.
The following questions are motivated by Anton Petrunin's Disc bounded ...

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### fundamental domains in H^2 containing large balls

I would like to construct a genus $g$ surface regularly tiled by triangles (for example by 238 triangles). Edmunds-Ewing-Kulkarni prove that the only obstruction to doing this is Euler characteristic ...

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### Hadamard theorem about embedding

The following theorem is commonly attributed to Jacques Hadamard.
Assume $\Sigma$ is a smooth locally convex immersed surface in the Euclidean space. Then $\Sigma$ is embedded and bounds a convex ...

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### Complex structures on topological surfaces

I am interested in the number of complex structures on a surface. More precisely, given a genus $g$ surface (topological manifold of real dimension 2) with $n$ punctures $X_{(g,n)}$, how many complex ...

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### Is this a 2-cyclic cocycle ? Does it have a nontrivial geometric interpretation?

Let $S$ be a surface in $\mathbb{R}^3$.
Inspired by the $2$-cyclic cocycle defined in page 20 of the book "Non-commutative geometry" by Alain Connes, we consider the following $3$-linear map on the ...

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### Thurston's preprint: “On the geometry and dynamics of diffeomorphisms of surfaces”

W. Veech on Teichmüller curves in moduli space, Eisenstein series and applications to triangular billiards says on the second paragraph of page 579:
"Thurston's original construction [8] corresponds ...

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### Invariant lifts of a closed curve on a surface of genus > 1

I am learning some things about surfaces of genus greater than $1$, and I am trying to answer this question :
Let $S$ be a compact and orientable surface of genus $g \geq 2$, and $c$ a closed curve ...

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### Does the Gauss-Bonnet theorem apply to non-orientable surfaces?

I hesitated for a long time to ask such an elementary-seeming question on Math Overflow, but when I asked and bountied it on Math SE, I found that a few experts seem to disagree on the answer, and I ...

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### Thrice intersecting closed geodesic on genus 2 orientable closed surface

Does there exist a closed geodesic on a closed genus 2 orientable surface (with hyperbolic metric) that self-intersects at only one point thrice?

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### Flat R-bundles on surfaces

I have questions about the definition of representation variety. In François Labourie's book "Lectures on representations of surface groups", Section 3.5, the author gives four models of the ...

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### The differential of the Gauss normal map from a Lie algebraic view point

Let $S\subset \mathbb{R}^3$ be a smooth surface with the Gauss normal map $N:S\to S^2$.
Then for every $x\in S$, the differential $(dN)_x:T_xS\to T_{N(x)}S^2$ can be considered as an ...

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### Is the conjecture true for n-sphere $(n>2)$? [closed]

This is higher dimension conjecture of Problem 3845 in Crux Mathematicorum and Theorem 2 in here:
PS: This figure is very nice, this is also generalization of Brianchon’s theorem, The Pascal theorem, ...

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### Projection of a ball in the ambient space to a manifold

Let $B_h (x)$ be the ball of radius $0<h \ll 1$ centered at $x\in \mathbb{R}^d$.
Let $I=[0,1]^{d-1}$ be the unit cube in $\mathbb{R}^{d-1}$, and let $f:I \to \mathbb{R}$ be a $C^2$ function. Then $...

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### Does this “algebraic” method for the application of the constructive proof of the classification of closed & compact surfaces have any use? [closed]

Disclaimer: This question is cross-posted in here. I have never asked a question in mathoverflow before, so if the level of this question is not appropriate for this site, please just vote close it.
...

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### How disconnected can a Seifert surface be?

Seifert surfaces
The standard definition of a Seifert surface for a link in $S^3$ is an oriented, compact surface embedded in $S^3$, bounding the link. Often, it is assumed to be connected, but given ...

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### Formula for Goldman Lie bracket of surface

Let $\Sigma_{g}$ be a closed oriented surface of genus $g$, Goldman defined a Lie algebra structure on the free module generated by the free homotopy
classes of loops on $\Sigma_{g}$. Roughly speaking,...

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### Isotopy of periodic homeomorphisms of a surface along periodic homeomorphisms

Let $\Sigma$ be an oriented compact surface with non-empty boundary that is not a disk or a cylinder (i.e. negative Euler characteristic). Let $\phi, \psi: \Sigma \to \Sigma$ be two orientation ...

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### Handel's Theorem for surfaces with boundary

Handel's Theorem(Entropy and semi-conjugacy in dimension two, 1987): let $M$ denote a closed surface. Let $\vartheta$ be a pseudo-Anosov (orientation-presrv.) homeomorphism of $M$ and $g$ be an (...

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### total mean curvature for singular surface

(a) The total mean curvature of a smooth oriented surface $S$ is defined as $$\iint_S H dA$$ where $H$ is the mean curvature (defined w.r.t. a choice of continuous unit normals on $S$.) Is there a ...

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### Examples of normal subgroups of a surface group with no embedded elements

As a consequence of the loop theorem, if $F$ is a closed surface in the boundary of a 3-manifold and if the kernel $N = \ker(\pi_1(F) \to \pi_1(M))$ is nonempty then there is a nontrivial element of $...

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### Lifts of geodesics on surfaces onto the universal cover [closed]

self-intersecting geodesic on hyperbolic surface of genus 2
Given a self intersecting geodesic on a hyperbolic surface of genus 2 as in the picture, how can I understand precisely what the lift to ...

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### Nice decomposition of surface diffeomorphisms

The question is very vague therefore any kind of suggestions, reference, ideas are welcome.
Suppose $S$ is an oriented surface with or without boundary. Let $m$ be an area form. Let $f$ be a ...

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### Generalizations of Dehn-Nielsen-Baer for topological branched cover?

For any manifolds $M$, a homotopy class of diffeomorphism gives rise to an automorphism of $\pi_1(M)$ (up to conjugacy since we are dealing with free homotopies). Moreover, in the specific case of ...

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### Normal generating set for the intersection of two normal subgroups of a surface group

Let $G = \left< a_1,b_1, ... , a_g, b_g | [a_1,b_1] \cdots [a_g,b_g] \right>$ be the fundamental group of a surface of genus $g$. Let $N_1$ and $N_2$ be two normal subgroups of $G$ that are ...

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### Relative Poincaré 2-complexes

I have a couple of questions about (relative) low-dimensional Poincaré-Duality spaces, also known as Poincaré complexes.
From Eckmann-Müller and Eckmann-Linnell we know that a CW complex is a ...

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### Is there a geometric interpretation of a Zariski dense surface subgroup?

Is there a geometric interpretation of a surface subgroup being Zariski dense? Or, conversely, given a $\Pi_1$ injective surface in a 3-manifold, is there a geometric or topological requirement on the ...

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### Equation for a geometrical half surface from folding a flat curved in one direction surface in half [closed]

I cannot not formulate the problem as i do not know how to model this idea. This is an open question not an mathematical exercise so if anybody has a good proposition i'm happy to use it.
Sorry in ...