# 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.

**3**

votes

**0**answers

48 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

**1**answer

645 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

**1**answer

139 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

**0**answers

251 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

**1**answer

256 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

**1**answer

129 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 ...

**15**

votes

**0**answers

439 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 ...

**6**

votes

**2**answers

146 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

**2**answers

81 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

**1**answer

371 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 ...

**-2**

votes

**1**answer

446 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

**1**answer

174 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 $...

**1**

vote

**0**answers

58 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.
...

**0**

votes

**0**answers

32 views

### Edgebreaker algorithm over 2-manifolds

Suppose I triangulate a 2-manifold and make a dual pseudograph over it. If I do the Edgebreaker compression algorithm over this graph to generate a spanning tree, can exist an edge with three or more ...

**4**

votes

**1**answer

146 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

**1**answer

296 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

**1**answer

186 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

**2**answers

136 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

**1**answer

134 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

**1**answer

166 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

**0**answers

81 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

**1**answer

154 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

**1**answer

93 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

**1**answer

145 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 ...

**0**

votes

**0**answers

46 views

### Is there any simple expression for $(\dot{F}\phi_u)\times(\frac{\partial \dot{F}}{\partial u} \phi_v)$

$F$ is an isometry of two dimensional regular surfaces $A$ to $B$. $\phi$ and $F\circ\phi$ are the coordinate chart for $A$ and $B$ respectively.$\phi_u$ and $\phi_v$ are unit tangent vector of $A$ ...

**4**

votes

**0**answers

79 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

**1**answer

88 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

**1**answer

71 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 ...

**1**

vote

**1**answer

232 views

### Dehn twist generators for mapping class group of a genus zero surface with boundary

Can you help me find a reference or explain how to find explicit Dehn twist generators for $MCG(S_{0,n})$, the mapping class group of a genus $0$ surface with $n$ boundary components, fixing the ...

**3**

votes

**1**answer

267 views

### Symmetry of functions on $S^2$

Let $f$ be a continuous function on $S^2$ and suppose there exists a constant $C>0$ such that for every $\mathcal{R} \in SO(3)$ the area of every connected component of $\{f(x)\geq f(\mathcal{R}x)\}...

**4**

votes

**0**answers

155 views

### Right-veering periodic automorphisms of surfaces that are not composition of right-handed Dehn twists

Basically the title of the question. For the sake of completeness I state an introduction to the question.
In "Right-veering diffeomorphisms of compact surfaces with boundary I" and II, the authors ...

**5**

votes

**0**answers

110 views

### References on differential geometry and low-tech surveying

Apologies if this is a duplicate question, putting the word "surveying" into a search on this site is not very effective.
I'm interested in science education, and I was recently reminded of the old ...

**1**

vote

**2**answers

245 views

### Does there exist a surface over a finite field which contains three skew lines?

Does there exist an irreducible surface, other than Hermitian surface, in $\mathbb{P}^3 (\mathbb{F}_q)$ containing three skew lines?
I know that this is true for Hermitian surface. In fact, at every ...

**11**

votes

**2**answers

430 views

### Intrinsic vs Extrinsic geometry of convex surfaces

By Alexandrov's isometric embedding theorem, any locally convex metric prescribed on the sphere admits a realization as a convex surface in Euclidean 3-space, which, by Pogorelov's rigidity result, is ...

**2**

votes

**0**answers

92 views

### Conditions of a quasiconvex codimension 1 subgroup of a hyperbolic group that determine it to be a surface subgroup

A subgroup $H$ of a group $G$ is called quasiconvex if in the Cayley graph of $G$, $\Gamma (G,S)$, there exists $k>0$ such that all geodesics that join a pair of points in $H$ are subsets of a $k$-...

**4**

votes

**0**answers

95 views

### Positivity of the ramification divisor

Let $X$ be a non-normal surface such that $K_X$ is a pseudo-effective divisor
and ${\rm Bs}_{-}(K_X)$ (the diminished base locus of $K_X$) equals, at least set-theoretically, the non-normal locus of $...

**1**

vote

**0**answers

58 views

### Characterizing a surface in $R^3$ with a given metric [closed]

Let $g$ be a Riemannian metric in $\mathbb{R}^2$. How can I find a surface in $R^3$ such that their curvature are the same? The shape of the surface in $3$ space is important. I mean I dont like to ...

**2**

votes

**1**answer

93 views

### Constructing a Polyhedron given areas of its faces

I want to visualize a set of data as a polyhedron in 3d space. Imagine set A includes areas of such polyhedron's faces. I assume the first step is to check if there exist a polyehdron by making sure ...

**2**

votes

**1**answer

129 views

### Is there a pair of pants decomposition analogue for orbifolds?

The pair of pants decomposition is a useful tool is surface theory. Is there an analogous decomposition for orbifolds?
Thanks

**4**

votes

**0**answers

41 views

### Lower estimate on length of boundary of 2d Riemannian surface

Fix constants $\kappa\in \mathbb{R}, D>0,A>0$. Does there exist a constant $C>0$ depending on $\kappa, D, A$ only such that for any compact 2-dimensional Riemannian surface (or more generally ...

**7**

votes

**2**answers

314 views

### Handle decompositions using only 1-handles

Let $\Sigma$ be an oriented, compact, connected 2-manifold with boundary. Assume that its boundary is equipped with a disjoint union decomposition into two non-empty parts:
$$\partial\Sigma=\partial_{...

**8**

votes

**1**answer

288 views

### generalisation of umbilic surfaces

It is well known that if you have a complete surface in $\mathbb{R}^3$ with umbilic points, that is to say $k_1=k_2$ everywhere, where $k_1$ and $k_2$ are the principal curvatures, that is to say the ...

**5**

votes

**2**answers

221 views

### homogeneous surface in $\mathbb{R}^4$

It is well known that the only homogeneous surfaces in $\mathbb{R}^3$ are the spheres, cylinders or planes. My question is about other examples in dimension $4$. Such a surface should have "constant ...

**8**

votes

**3**answers

212 views

### Zone of negative curvature on surfaces embedded in $\mathbb{R}^3$

I consider the standard embedding of a compact oriented surface $\Sigma$ (say of genus 2) in the Euclidean space $\mathbb{R}^3$. I have coloured on the picture below the zone of this surface where the ...

**4**

votes

**1**answer

80 views

### Bishop-Gromov type inequality for Jordan curve on 2-sphere

Let $\Sigma$ be a 2-sphere with a (smooth) Riemannian metric $g$ of nonnegative curvature. Let $\mathcal{C}$ be a simple closed smooth curve on $\Sigma$. Then $\mathcal{C}$ splits $\Sigma$ into two ...

**3**

votes

**2**answers

137 views

### What is the homeomorphism from $\Gamma \backslash T_1 \mathbb{H}$ to $T_1(\Gamma \backslash \mathbb{H})$

Let $\mathbb{H}$ be hyperbolic plane, $\Gamma$ is a discrete subgroup of $PSL_2(\mathbb{R}$) so that $\Gamma \backslash \mathbb{H}$ is a compact hyperbolic surface. Maybe it will be very simple to you ...

**4**

votes

**1**answer

167 views

### The first eigenfunction of Dirac operator for surface

Let $M$ be a spherical oriented surface with Riemannian metric and with trivial spin structure. We know that the equation $$D \phi = \rho \phi$$, where $\rho: M\rightarrow \mathbb{R}$ is a real scalar ...

**1**

vote

**1**answer

102 views

### Surfaces of $\mathbb{R}^3$ invariant by an affine map

I have a rather elementary question.
I would like to know what are the surfaces of $\mathbb{R}^3$ which are globally preserved by the action of a linear or affine map in a non trivial way. This ...

**1**

vote

**1**answer

86 views

### Polar coordinates of a set with different radius and angle

Let $M$ be a $2$-dimensional Riemannian manifold and let $U\subset M$ be an open set. Suppose there exist polar coordinates $(r,\theta)$ with center $q\in M$ such that
$$U=\lbrace{ (r,\theta): 0<...

**17**

votes

**0**answers

180 views

### Modified Willmore energy and surfaces with infinitesimally narrow necks

Disclaimer: This is a copy of a question that I asked on the Mathematics Stack Exchange. It was suggested to me there that the question was worth asking over here.
There is an open problem in ...