The surfaces tag has no wiki summary.
0
votes
0answers
104 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
230 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 ...
1
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0answers
147 views
a good introduction to Laplace Beltrami operator over differential manifolds? [migrated]
I'd like to have a good reference to understand how the Laplacian operator get generalized over differential manifolds.
More concretely, I want to understand and prove the equation :
$$\Delta ...
4
votes
1answer
330 views
Triple bubble conjecture: Natural candidate?
Is there a standard natural candidate surface for
the shape that encloses three given volumes
in $\mathbb{R}^3$ and has minimal surface area?
I know the planar triple bubble conjecture was ...
0
votes
1answer
48 views
Is there a way to make MAGMA work with surfaces over weighted projective spaces?
Is there a way to use MAGMA to study surfaces defined over a weighted projective space (by "study" I mean computing e.g. invariants (e.g. $p_a$, $p_g$), singularities, etc)? For example, I was trying, ...
7
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0answers
193 views
Surfaces with many (but not solely) closed geodesics?
Let $S$ be a closed surface embedded in $\mathbb{R}^3$,
let's say of genus zero.
I seek examples of $S$ with the following property:
If one selects a random any point $p$ on $S$, and a random
...
3
votes
1answer
98 views
Locally trivial deformations of surfaces with quotient singularities
Let us consider the surface $\mathbb{A}^{2}/\mu_{6}$ where the action is given by
$$
\begin{array}{ccc}
\mu_{6}\times\mathbb{A}^{2} & \longrightarrow & \mathbb{A}^{2}\\
(\epsilon,x_{1},x_{2}) ...
10
votes
3answers
455 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
219 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
151 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
179 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
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0answers
130 views
Integral of Square of Mean Curvature
Let us assume $H$ is the mean curvature of a compact surface in $E^3$ and $g$ is its genus.
(1) When $g$ is arbitrary, we have $\int_{S^{2}}H^{2}dV=4\pi$ and $\int_{\Sigma}H^{2}dV\geq4\pi$.
(2)When ...
2
votes
1answer
220 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
218 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
149 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
438 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
317 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
242 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 ...
1
vote
2answers
209 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
150 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
156 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
180 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
355 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
313 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
271 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
899 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
361 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
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0answers
169 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
404 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
287 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
559 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:
...
15
votes
4answers
971 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 ...
9
votes
1answer
808 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 ...
4
votes
1answer
247 views
Representing groups with two generators as graph automorphisms
Suppose we have a group $G$ which can be generated by two elements $x$, $y$. Call $H$, $K$, $L$ the subgroups of $G$ generated by $x$, $y$ and $y^{-1}x^{-1}$, respectively.
With these data, we can ...
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2answers
957 views
Connected sum of surfaces
I'm looking for a detailed reference about connected sums. I'd like it to contain a proof that a connected sum of connected surfaces is independent - up to homeomorphism - of the various choices ...
2
votes
2answers
318 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
2answers
261 views
Abelian subgroups of ball quotient
Let $X$ be a compact complex surface of general type which a ball quotient. Is it true that $\pi_{1}(X)$ can not contain ${\mathbb{Z}}^{2}$ as a subgroup? What kind of infinite abelian groups can ...
2
votes
1answer
645 views
How to rigorously prove that simple closed curves on a surface are primitive closed curves ?
Let me first state the definitions :
A not-nullhomotopic closed curve / loop $c$ on an orientable surface $X,c:[0,1]\to X$ is called simple closed curve is $c|[0,1)$ is injective and [ $c(0)=c(1) ] ...
3
votes
2answers
434 views
Surface fitting with convexity requirement
Hi all,
Consider a cloud of points in 3D space (x,y,z). The data is well-behaved, once plotted the surface looks like some sort of spheroid. I assumed a form for the fitting function f(x,y,z) = c1 ...
3
votes
2answers
277 views
Moving a canonical divisor on a normal surface away from the singular locus
In a previous question Moving a Weil divisor on a normal surface away from a finite set of closed points I probably asked for too much. As J.C. Ottem pointed out, it is not always possible to move a ...
2
votes
3answers
351 views
Moving a Weil divisor on a normal surface away from a finite set of closed points
Let $Y$ be a normal surface and let $X$ be a closed subscheme of codimension 2, i.e., $X$ is a finite set of closed points.
Let $D$ be a Weil divisor on $Y$.
Question. Does there exist a Weil ...
1
vote
0answers
148 views
Is -(E,E) greater or equal to 2 for a minimal resolution
I'm quite confused by the terminology minimal resolution and minimal model.
Let $f:X\longrightarrow Y$ be a minimal resolution of singularities, where $Y$ is a normal surface.
Let $E$ be an ...
4
votes
1answer
697 views
intersection number
I vaguely recall the following fact that I'd like to use in my research. It should be easy to see that this holds (if it does) but I can't seem to prove it.
Let $p:X\longrightarrow S$ be a (regular) ...
2
votes
2answers
350 views
The exceptional locus of a minimal resolution of singularities
Let X be a surface. (A surface is an excellent integral normal separated 2-dimensional scheme.)
Let $\psi:Y\longrightarrow X$ be a minimal resolution of singularities and let $E$ be an irreducible ...
1
vote
1answer
545 views
Hirzebruch surfaces
I am sorry for too naive and stupid question,
How can I express the 2nd Hirzebruch surface, $F_{2}$ in terms of $SO(3)$. Can F_{2} be realizable as the total space of a bundle over $\mathbb{R}_{+}$ ...
1
vote
2answers
677 views
Names of certain surfaces
Are there any generally used names for the following algebraic and nonalgebraic surfaces? Any references to literature where the surfaces are studied are also appreciated.
Surface I. Implicit ...
1
vote
4answers
510 views
What is the quantity 2(handles)+crosscaps called?
It is well-known that up to homeomorphism, the complete set of orientable surfaces is $\lbrace S_g : g=0,1,\dots \rbrace$, where $S_g$ is the sphere with $g$ handles. The complete set of ...
1
vote
2answers
341 views
Can one obtain surfaces with interesting invariants as resolutions of singular surfaces?
(Perhaps a not very well defined question)
Let $(S_t)_t$ be a (flat) family of compact complex surfaces. Assume the generic member is smooth while $S_0$ has isolated singularities. As the simplest ...
15
votes
2answers
911 views
Simple curves on non-orientable surfaces.
Given an element in the (first) homology group of a surface, I would like to know if it can be represented as a simple closed curve. For orientable surfaces, this is well-known, but I wasn't able to ...
15
votes
5answers
867 views
Is a rhombus rigid on a sphere or torus? And generalizations.
If a rectangle is formed from rigid bars for edges and joints
at vertices, then it is flexible in the plane: it can flex
to a parallelogram.
On any smooth surface with a metric, one can define a ...
