**1**

vote

**0**answers

90 views

### Spherical cap is the only compact constant mean curvature surface bounded by a circle

I would like to see that the only compact rotationally invariant constant mean curvature surfaces with boundary a planar circle, are either a planar disk or a spherical cap.
This is stated in the ...

**5**

votes

**1**answer

229 views

### Avoiding mean-curvature flow dumbbell neck-pinch by inflating a surface

It is well known that
Grayson's dumbbell neck-pinch1,2 separates
into disconnected pieces under
mean curvature flow:
Image ...

**2**

votes

**2**answers

170 views

### A Jordan Separation Theorem for Polyhedral Surfaces

Let me begin by defining what a polyhedral surface is.
A path-connected subset $ P $ of $ \mathbb{R}^{3} $ is called a polyhedral surface iff it is the union of a finite collection $ \mathcal{C} $ of ...

**21**

votes

**1**answer

955 views

### Homeomorphism historically: When did it reach its modern formulation?

Q. When did the notion of homeomorphism reach its
modern formulation as a bicontinuous bijection, i.e., a
continuous bijection
between topological spaces whose inverse is also continuous?
...

**3**

votes

**1**answer

233 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

**1**answer

112 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

**1**answer

279 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

**1**answer

159 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

**1**answer

148 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

**1**answer

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

**1**answer

73 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

**1**answer

124 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

**0**answers

138 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

**0**answers

289 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

**2**answers

191 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

**2**answers

416 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

**1**answer

86 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

**1**answer

498 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

**0**answers

239 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

**1**answer

99 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

**0**answers

73 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

**0**answers

142 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

**1**answer

300 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

**1**answer

436 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

**1**answer

65 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

**0**answers

218 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

**1**answer

152 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}) ...

**12**

votes

**3**answers

609 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

**2**answers

306 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

**1**answer

314 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

**2**answers

215 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

**0**answers

218 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

**1**answer

240 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

**2**answers

308 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

**1**answer

180 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

**1**answer

630 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

**2**answers

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

**1**answer

298 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

**1**answer

150 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

**2**answers

225 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

**0**answers

154 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

**2**answers

161 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

**1**answer

201 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

**2**answers

408 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

**1**answer

354 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

**2**answers

358 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

**3**answers

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

**2**answers

402 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

**0**answers

176 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

**1**answer

513 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 ...