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.
184
questions
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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 ...
6
votes
1answer
181 views
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 ...
2
votes
3answers
184 views
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. ...
1
vote
1answer
142 views
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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0answers
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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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vote
0answers
53 views
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 $S$ ...
3
votes
1answer
264 views
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 ...
4
votes
2answers
156 views
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 ...
0
votes
1answer
155 views
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 ...
8
votes
1answer
341 views
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 $\...
6
votes
1answer
158 views
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^...
5
votes
1answer
130 views
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 ...
3
votes
1answer
192 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 ...
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0answers
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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 ...
8
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0answers
190 views
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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0answers
176 views
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 ...
3
votes
0answers
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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}...
3
votes
0answers
38 views
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 ...
1
vote
1answer
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 ...
6
votes
3answers
273 views
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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0answers
48 views
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 $(...
4
votes
1answer
212 views
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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0answers
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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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votes
0answers
219 views
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 ...
4
votes
1answer
278 views
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 ...
14
votes
1answer
708 views
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 ...
0
votes
1answer
173 views
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 ...
3
votes
0answers
291 views
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 ...
4
votes
1answer
309 views
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 ...
4
votes
1answer
161 views
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 ...
25
votes
1answer
808 views
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 ...
5
votes
2answers
182 views
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?
2
votes
2answers
95 views
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 ...
8
votes
1answer
383 views
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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votes
1answer
478 views
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, ...
2
votes
1answer
185 views
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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vote
0answers
62 views
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.
...
4
votes
1answer
203 views
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 ...
5
votes
1answer
366 views
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,...
8
votes
1answer
212 views
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 ...
6
votes
2answers
162 views
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 (...
3
votes
1answer
181 views
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 ...
6
votes
1answer
181 views
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 $...
2
votes
0answers
127 views
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 ...
0
votes
1answer
156 views
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 ...
3
votes
1answer
105 views
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 ...
3
votes
1answer
199 views
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 ...
4
votes
0answers
88 views
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 ...
3
votes
1answer
92 views
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 ...
1
vote
1answer
74 views
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 ...
