**2**

votes

**1**answer

76 views

### Prescribing an induced metric

We know that, if we have a surface $z=f(x,y)$ with Euclidean space being ambient manifold, the induced metric is as follows (in matrix form):
$$g=\begin{bmatrix}
1+\left ( \frac{\partial ...

**2**

votes

**1**answer

301 views

### embeddings of product of spheres in Euclidean spaces [closed]

I notice that $T^2=S^1\times S^1$ can be embedded in $\mathbb{R}^3$ as a hypersurface (submnaifolds of codimension 1).
In general,
(1). could the product of spheres $S^{m_1}\times\cdots\times ...

**3**

votes

**0**answers

39 views

### Length and laplacian spectrum for quasi-fuchsian manifold

It is well known that, in the case of finite area hyperbolic surfaces, the length sprectrum (the collection of length of all closed geodesics) and the spectrum of the laplacian (acting on functions) ...

**3**

votes

**1**answer

69 views

### Homotopy classes of homeomorphisms of a multiple pointed space

Let $M$ be a multiple pointed space, i.e. $M$ is a topological space and there is a finite point set $M\supset P=\{p_1,...,p_k\}, k<\infty$. Such a $p_i$ is called a marked point. A map ...

**8**

votes

**2**answers

132 views

### Tangent fields spanning the distribution of principal directions on a surface

Suppose $S$ is an orientable regular surface in $\mathbb R^3$ without umbilical points (not necessarily compact, and with no boundary). There are two well-defined smooth $1$-dimensional tangent ...

**4**

votes

**1**answer

179 views

### Fair surfaces - general mathematical theory

Fairness measures for surfaces are, in general, functionals containing more complicated terms thatn the usual bending energy, and may depend not only on the mean curvature but also on principal ...

**6**

votes

**0**answers

110 views

### Harmonic map heat flow in positive curvature

Suppose I wish to relax/smooth a map $\phi:M\rightarrow N$ between two surfaces $M,N$ embedded in $\mathbb{R}^3$. I could try flowing the map using harmonic heat flow, which (as I understand it) is ...

**1**

vote

**1**answer

83 views

### Essential surfaces in the Exterior of Montesinos knots

Hatcher and Oertel computed the boundary slopes of essential surfaces of Montesinos knots in this paper. But they do not consider surfaces that do not intersect the boundary of the exterior. An ...

**5**

votes

**4**answers

410 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 ...

**1**

vote

**0**answers

136 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 ...

**12**

votes

**1**answer

1k views

### Naive definition of surface area doesn't work?

A first stab at a definition of surface area might go like this:
Let S be a surface. Select finitely many points from S and make a bunch of triangles having these points as vertexes. Add up the ...

**5**

votes

**1**answer

280 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

181 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

1k 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

254 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 ...

**8**

votes

**1**answer

575 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,
...

**5**

votes

**1**answer

139 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

315 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

172 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

172 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

63 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 ...

**0**

votes

**0**answers

255 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

**0**answers

312 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 ...

**2**

votes

**1**answer

82 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

129 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

151 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 ...

**5**

votes

**2**answers

449 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 ...

**6**

votes

**2**answers

226 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 ...

**3**

votes

**1**answer

93 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 ...

**5**

votes

**2**answers

1k views

### What is parameterization of the trefoil knot surface in R³?

What is a parameterization, say (x(u,v),y(u,v),z(u,v)), of the trefoil knot surface in R³ whose cross-section can be circular or, in general, elliptic?
Thanks!

**2**

votes

**1**answer

153 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.

**13**

votes

**0**answers

315 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

105 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 ...

**12**

votes

**3**answers

624 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 ...

**2**

votes

**0**answers

76 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

156 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

303 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

484 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

78 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, ...

**8**

votes

**0**answers

228 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

174 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}) ...

**2**

votes

**2**answers

362 views

### Contraction of curves on surfaces

Assume we have a surface $S$ (smooth if you want), and a map $f: S \to V$ that contracts a curve $C \subset S$. What condition would give a factoring of $f$ through a contraction $c: S \to V'$ ...

**1**

vote

**1**answer

373 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 ...

**3**

votes

**2**answers

341 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 ...

**6**

votes

**2**answers

358 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 ...

**4**

votes

**2**answers

226 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? ...

**2**

votes

**1**answer

246 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 ...

**0**

votes

**1**answer

199 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 ...

**9**

votes

**1**answer

720 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

334 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 ...