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
1answer
209 views
Reference for Arakelov's theorem: $K^2_f=0$ iff $f$ is locally trivial
Let $f:X\longrightarrow B$ be a family of curves, with $f$ relatively minimal, over a fixed curve $B$ ($B$ is projective, irreducible and smooth). The fibration $f$ is said locally trivial if all ...
5
votes
1answer
101 views
Translation surfaces & integer multiples of $\pi$
Richard Schwartz, in Mostly Surfaces (Vol. 60. American Mathematical Soc., 2011),
defines (on p.14) a translation surface as "a Euclidean cone surface, all of whose 'angle errors' are integer ...
6
votes
1answer
258 views
Which surfaces have only a finite number of connecting geodesics?
Q1. For a smooth, closed (compact) surface $S$ embedded in $\mathbb{R}^3$,
under which conditions is it true that, for every pair of points
$a,b \in S$, there are an infinite number of ...
1
vote
1answer
151 views
Compact surfaces with boundary of constant negative curvature
Consider a surface (with boundary) diffeomorphic to $S^1 \times [0, 1]$ and with constant negative curvature, sitting inside $\mathbb{R}^3$. All the examples I know of such surfaces are "part of" (or ...
7
votes
1answer
141 views
Foliation with leaves which are and are not dense
Do there exist a foliation on a closed surface (i.e. real dimension 2) which has a dense leaf and also a leaf which is not dense?
1
vote
1answer
58 views
Geodesic paths on a flat sphere
Let $S$ be a $2$-dimensional sphere endowed with a flat metric with $3$ conical singularities of positive curvature. Typically, $S$ is a metric space you get when you glue two copies of the same ...
2
votes
1answer
69 views
Triangulations of translation surfaces whose edges are shorter than the diameter
Let $S$ be a compact translation surface (i.e. a surface endowed with a singular flat metric such that singular points are locally isometric to a cone of angle an integer multiple of $2\pi$, and that ...
1
vote
1answer
122 views
Some questions about ruled surfaces defined over $\overline{\mathbb Q}$
definitions:
A non-singular complex projective surface $S$ is a ruled surface if it is birationally equivalent to $C\times_{\text{Spec} \mathbb C}\mathbb P^1_{\mathbb C}$ where $C$ is a non-singular ...
2
votes
0answers
130 views
Symplectic form on moduli space of connections
Let $M$ be the moduli space of flat $GL(n,\mathbb{C})$ connections on a compact oriented surface, and $\alpha$ the natural symplectic form on it.
Is there any known construction of a bundle with a ...
2
votes
0answers
276 views
Symmetry on a sphere
Let $u$ be a smooth function on the sphere $S^2$. Suppose there exists $C>0$ such that for all $R \in SO(3)$, the area of every nonempty connected component of $\{x\in S^2: u(x)> u(Rx)\}$ is at ...
6
votes
2answers
180 views
Geodesic flow on infinite surfaces
The geodesic flow on a compact hyperbolic surface (i.e. a surface with a riemannian metric of constant curvature $-1$) has been well-studied, in particular it has been known for a long time that it is ...
5
votes
2answers
397 views
Besides the tracioid are there other surfaces of revolution that have a constant negative curvature?
There is no surface in $ R^3 $ that can represent the complete hyperbolic plane (Hilberts theorem) so we always have to do with a surface that is not completely equivalent, has a cusp somewhere, but ...
3
votes
1answer
78 views
How to approximate volumes of m-dimensional manifolds with m-dimensional polyhedra
The areas of a sequence of polyhedra approaching a surface need not approach the area of the surface, but there are theorems guaranteeing that this be so. (T. Rado, On the Problem of Plateau, Chapter ...
7
votes
1answer
471 views
Do all combinatorially distinct fundamental polygons correspond to surfaces?
The topology of a closed surface can be constructed
by identifying edges of a fundamental polygon of an
even number $2n$ of edges.
Labeling the edges and using $\pm 1$ exponents to indicate
direction,
...
13
votes
0answers
191 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 ...
1
vote
1answer
95 views
Sutured Manifolds and minimal genus
Is there a result relating sutured manifolds and surfaces of minimal genus? perhaps someone has a very clever point of view of these two notions that can share.
In other matters, do we know how to ...
2
votes
0answers
70 views
Elliptic surfaces with different Kodaira symbols
Are there examples of surfaces $E$ of Kodaira dimension one that have two elliptic fibrations $p,q:E\to C$ over some curve $C$ such that $p$ has semi-stable fibres but $q$ has an additive fibre?
Can ...
0
votes
0answers
134 views
Third variation of area of a minimal surface
There is a formula for the third variation of area on page 96 of Nitsche's book,
Lectures on Minimal Surfaces, vol. 1 (English version). He says at the bottom of the
page it is good for normal ...
1
vote
1answer
294 views
Symmetry on a sphere
Let $u$ be a smooth function on the sphere $S^2$. Suppose there exists $C>0$ such that for all $R \in SO(3)$, the area of every connected component of $\{x\in S^2: u(x)> u(Rx)\}$ is at least ...
5
votes
1answer
413 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 ...
0
votes
1answer
61 views
Is there a way to make MAGMA work with surfaces over weighted projective spaces?
Is there a way to use MAGMA to study surfaces defined over a weighted projective space (by "study" I mean computing e.g. invariants (e.g. $p_a$, $p_g$), singularities, etc)? For example, I was trying, ...
7
votes
0answers
212 views
Surfaces with many (but not solely) closed geodesics?
Let $S$ be a closed surface embedded in $\mathbb{R}^3$,
let's say of genus zero.
I seek examples of $S$ with the following property:
If one selects a random any point $p$ on $S$, and a random
...
2
votes
1answer
148 views
Locally trivial deformations of surfaces with quotient singularities
Let us consider the surface $\mathbb{A}^{2}/\mu_{6}$ where the action is given by
$$
\begin{array}{ccc}
\mu_{6}\times\mathbb{A}^{2} & \longrightarrow & \mathbb{A}^{2}\\
(\epsilon,x_{1},x_{2}) ...
11
votes
3answers
577 views
Space of embeddings of circle in a surface
Let $S$ be a compact oriented surface of genus at least $2$ (possibly with boundary). Let $X$ be a connected component of the space of embeddings of $S^1$ into $S$.
Question : what is the ...
3
votes
2answers
273 views
Realizing homology classes on surfaces with boundary by simple closed curves
Let $\Sigma$ be a compact oriented surface with boundary. Assume that the genus of $\Sigma$ is positive. We say that an element $h \in H_1(\Sigma)$ can be realized by a simple closed curve if there ...
1
vote
1answer
281 views
Explicit computation of the action of a Dehn twist on the fundamental group of a surface
Let $S$ be a compact orientable surface of genus $g$. Now let $p\in S$ and $\gamma$ a closed simple curve on $S$ disjoint from $p$. It is not very difficult to compute the action of a Dehn twist along ...
4
votes
2answers
209 views
Some facts about cut-locus
Let $M$ be a 2-dimensional closed Riemmanian manifold diffeomorphic to $S^2$.
S.B.Myers says "the cut-locus of every point $x\in M$ is a finite tree."
How the set of point can be a tree? ...
0
votes
0answers
209 views
Integral of Square of Mean Curvature
Let us assume $\text{H}$ is the mean curvature of a compact surface in $\mathbb E^3$ and $g$ is its genus.
When $g$ is arbitrary, we have $\int_{\mathbb S^2}\text{H}^2dV=4\pi$ and ...
2
votes
1answer
237 views
Homotopy versus path-homotopy on punctured surface
I have some problems with homotopies.
The situation is this:
Let $X$ be a surface, which is homeomorphic to a 2-Sphere with a finite number (at least 3) of points removed (equivalently, an open ...
6
votes
2answers
291 views
How to compute the normals to Costa's minimal surface?
I am trying to draw Costa's minimal surface in high resolution using the PovRay raytracer. For this I need to compute points on the surface as well as the normals. It is relatively easy to compute the ...
0
votes
1answer
171 views
Find a simple closed curve in $S$ which represents a commutator in $\pi_1 S$
I am interested in the following problem : decide if a certain element of the fundamental group can be represented by a simple closed curve. The general case has already been asked and answered on MO ...
8
votes
1answer
575 views
Geodesics on the twisted pseudosphere (Dini's surface)
I wonder how difficult it is to compute geodesics on Dini's Surface,
a twisted pseudosphere?
Here is one parametrization, from
Alfred Gray's Modern Differential Geometry of Curves and Surfaces, ...
5
votes
2answers
329 views
Action of $\pi_1(S)$ on its commutator subgroup
Let $G$ be a group. It acts canonically on its derived subgroup by conjugation. Can on describe the orbits of this action when $G$ is the fundamental group of a compact orientable surface of genus $g ...
6
votes
1answer
290 views
Costa's minimal surface and the structure of lungs
Seeing this image of Costa's minimal surface
(MathWorld image)
made me wonder if the fine-grained structure
of the human lung is somehow composed of pieces of ...
2
votes
1answer
147 views
Looking for software that computes intersection numbers (Heegaard Diagrams)
As a part of my research I am working with intersection matrices of Heegaard diagrams. Is there some software that could help me compute such matrices for some examples?
Thanks.
2
votes
2answers
218 views
Dimension of the homology group with coefficients in $\mathbb{Z}/2\mathbb{Z}$
I asked this on math.stackexchange.com, but didn't get a single answer.
Charles Weibel writes in his survey of homological algebra
Riemann defined a surface $S$ to be $(n + 1)$-fold
connected ...
1
vote
0answers
153 views
Contractibility of union of smooth rational curves in famillies
Let $\pi:\mathcal{X} \to S$ be a family of smooth surfaces containing a subfamily $\mathcal{Y} \subset \mathcal{X}$ proper, flat over $S$ parametrizing contractible curves in $\mathcal{X}$ satisfying ...
3
votes
2answers
160 views
Random metrics on compact orientable surfaces
Hello everyone,
Let $S_g$ be a compact orientable surface of genus $g \geq 2$, and let $\mathcal{A}$ be the set of $\mathcal{C}^{\infty}$ Riemanniann metric on $S_g$ endowed with the topology of ...
5
votes
1answer
195 views
Fixing a proof of the systolic inequality for higher genus surfaces
I'm currently learning some stuffs about systolic inequalities. While reading the relevant sections (p329 to 340) in Berger's Panoramic View of Riemannian Geometry, I noticed a gap in one of the ...
4
votes
2answers
396 views
Surface Laplace-Beltrami without coordinates, exterior calculus?
Let $f: M \rightarrow \mathbb{R}^3$ be an immersion of a surface $M$. For pedagogical purposes (i.e., I'm teaching a class!) I am looking for an expression for the scalar Laplace-Beltrami operator ...
1
vote
1answer
342 views
Description of regular covering maps between surfaces.
This is an improved and hopefully a more precise version of the question Covering spaces of surfaces.
Question: Given a regular covering map $\pi:\Sigma_g\to\Sigma_h$, where $\Sigma_n$ denotes a ...
5
votes
2answers
332 views
What is the homotopy type of the space of simple closed curves isotopic to a given one?
For surfaces there are many statements along the lines of: if two simple closed curves are homotopic, they are isotopic. I'm interested in such questions for families of curves.
More precisely, let ...
3
votes
3answers
1k views
Covering spaces of surfaces
Let $\Sigma_g$ be a surface of genus $g\ge 2$, and let $\Sigma_k$ be an $m$-sheeted covering
space of $\Sigma_g$. It is known that $k=m(g-1)+1$.
An example of such a covering space is a regular ...
6
votes
2answers
388 views
Do the following set of Dehn twists generate the mapping class group?
If $S$ is the surface illustrated below, do the Dehn twists about the red curves generate the mapping class group $\operatorname{MCG}(S,\partial S)$?
1
vote
0answers
175 views
surfaces dominated by a product of curves
I would like to know for which projective smooth surfaces over a finite field there exists a dominant rational map from a product of curves to the surface
2
votes
1answer
486 views
Parallel translation on surfaces
Parallel translation of a vector along a geodesic in a surface is characterized by the following three properties:
The vector being transported moves continuously.
It has constant norm.
It maintains ...
3
votes
3answers
314 views
Lagrangian Kleinian bottles
I remember some talks some time ago about proofs of nonexistence of Lagrangian Kleinian bottles in C^2 for the standard symplectic structure, mentioning that this were the only compact surface for ...
5
votes
2answers
573 views
Pursuit-Evasion on a Manifold
I know pursuit-evasion has been studied in many contexts, including
on a manifold (e.g., Melikyan,
"Geometry of Pursuit-Evasion Games on Two-Dimensional Manifolds"),
but I have not seen this version:
...
16
votes
4answers
1k views
Morse theory in TOP and PL categories?
Apparently there are topological and piecewise linear versions of Morse theory. I would like to know of references that treat these topics.
How is a Morse function defined for compact manifolds (with ...
11
votes
1answer
1k views
Classification of surfaces and the TOP, DIFF and PL categories for manifolds
A surface is simply a 2-manifold. The classification theorem for compact connected surfaces (with boundary) is commonly regarded in the categories TOP, DIFF and PL. Well known proofs (e.g. via ...
