Questions tagged [curves-and-surfaces]
A surface is a generalization of a plane which needs not be flat, that is, the curvature is not necessarily zero. This is analogous to a curve generalizing a straight line
301
questions
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votes
0answers
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Nodes of rational plane sextic curves
According to the genus formula for plane curves, a nodal, irreducible rational plane curve $C\subset\mathbb{P}^2$ of degree six must have $10$ nodes.
Now, if we take $10$ general points in $\mathbb{P}^...
5
votes
2answers
126 views
Necessary and sufficient curvature condition for a regular planar curve to be simple and closed
Given a smooth, $2\pi$-periodic function $\kappa(s)$, the associated planar curve $\gamma(s)$ for which $\kappa(s)$ is the (signed) curvature, is uniquely determined up to Euclidean invariance: a ...
4
votes
0answers
113 views
Sections of fibrations of Kodaira dimension zero
Let $X$ be a projective variety with a morphism $f:X\rightarrow \mathbb{P}^1$, and let $F$ be a general fiber of $f$.
Assume that $F$ in turn has a fibration $g_{F}:F\rightarrow S$ with rational ...
5
votes
1answer
109 views
When does a spherical curve equal its tangent indicatrix?
Given a smooth regular curve $\gamma$ in $\mathbb{R}^{3}$, one defines the tangent indicatrix of $\gamma$ to be the spherical curve $\gamma'/\lVert \gamma'\rVert$. It is then natural to look for ...
3
votes
0answers
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Existence of developable ribbonization of a surface
Let $S$ be a smooth compact surface embedded in $\mathbb{R}^{3}$. It is well-known that there exists a triangulation of $S$. I am considering an alternative way of approximating $S$, where instead of ...
2
votes
1answer
264 views
When is the cut locus a tree?
Let $\gamma: \mathbf{S}^1 \to \mathbf{R}^2$ be a simple, closed curve. Assume additionally that $\gamma$ is regular, of class $C^2$ at least. The complement of $\gamma$ in $\mathbf{R}^2$ has two ...
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vote
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Understanding sheaves on normalisation of a curve: $v_* \mathcal{O}_{\tilde{C}} / \mathcal{O}_C$
Let $(C, \mathcal{O}_C)$ be a reduced irreducible curve and $(\tilde{C},\mathcal{O}_{\tilde{C}})$ its normalisation with $v : \tilde{C} \rightarrow C$. Then we have an imoprtant skyscraper sheaf $v_* \...
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votes
1answer
112 views
“Arc” length parametrization for surfaces
If we have a function $\phi:\Omega\subseteq\mathbb{R}^2\to\mathbb{R}$, $\phi\in C^2(\Omega)$ is there a way to find a function $d:\Omega\to\mathbb{R}, d\in C^2(\Omega)$ so that:
$|\nabla d(x,y)|=1,\ \...
2
votes
1answer
82 views
Hyperbolic length of curve that does not enter a collar
Let $\Sigma$ be a compact surface of genus at least $1$ with one boundary component, equipped with a hyperbolic metric so that the boundary is geodesic. There is a version of the collar lemma that ...
9
votes
3answers
567 views
Is there the longest geodesic?
Given a closed 2-surface $M$ together with a Riemannian metric $g$.
We pick a free homotopy class $\gamma \in \pi_1(M)$ and consider the set $C(\gamma)$ of all closed geodesics homotopic to $\gamma$.
...
2
votes
0answers
119 views
Degree $4$ curves on K3 double covers of Del-Pezzo surfaces
Let $S$ be a smooth del-Pezzo surface and $\pi : X \longrightarrow S$ be the double cover of $S$ ramified in a smooth section of $-2K_S$. Going through the classification of del-Pezzo surfaces, one ...
13
votes
1answer
274 views
Are any of these complex surfaces ever projective?
Let $C$ and $T$ be compact connected Riemann surfaces (or: smooth projective connected curves over $\mathbb{C}$) of genus at least two and let $X:=C\times T$. Let $(c,t)$ be a point of $X$, and let $...
5
votes
1answer
103 views
Representing relative homology classes orientable surfaces with boundary
Let $S$ be compact oriented surface without boundary. Then it is a classical result that a primitive class $\gamma \in H_1(S; \mathbb{Z})$ is always represented by a simple closed curve. It implies ...
2
votes
0answers
151 views
On cycloids and other roulettes
It is well known that the cycloid is the curve traced by a point on a circle as the circle rolls along a line without slipping.
Consider wheels with smooth convex shapes (not necessarily circular) ...
2
votes
0answers
73 views
intuition behind shape optimization using Hadamard's method
I'm trying to understand the intuition behind shape optimization using Hadamard's method. Please consider the following simple example:
Let $\lambda$ denote the Lebesgue measure on $\mathcal B(\...
3
votes
0answers
59 views
Shape derivative of boundary integrals and differentiability of the integrand on a tubular neighborhood
Let $d\in\mathbb N$, $U\subseteq\mathbb R^d$ be open, $$\mathcal A:=\{\Omega\subseteq\mathbb R^d:\Omega\text{ is bounded and open},\overline\Omega\subseteq U\text{ and }\partial\Omega\text{ is of ...
6
votes
2answers
238 views
An abstract characterization of line integrals
Let $M$ be a smooth manifold (endowed with a Riemann structure, if useful). If $\omega \in \Omega^1 (M)$ is a smooth $1$-form and $c : [0,1] \to M$ is a smooth curve, one defines the line integral of $...
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votes
0answers
121 views
Curvature without any reference to embedding or coordinates
On a 2 sphere, which is topologically just a compact simply connected 2 dim space, one can edow geometric structure by requiring each two geodesics have equidistant intersections.
Is this enough to ...
3
votes
0answers
142 views
The space of rearrangements of a plane curve
I learned of the paper "Closing curves by rearranging arcs" by L. Alese (arXiv link) by a post on Reddit and was curious about related questions and generalizations. Here a rearrangement of ...
1
vote
2answers
147 views
When is the inside of a Jordan curve open? [closed]
I'm working purely on intuition here. The Jordan curve theorem states that a Jordan curve separates the plane into a bounded component and an infinite component. For toy curves, it seems like this ...
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votes
5answers
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Surprising properties of closed planar curves
In https://arxiv.org/abs/2002.05422 I proved with elementary topological methods that a smooth planar curve with total turning number a non-zero integer multiple of $2\pi$ (the tangent fully turns a ...
1
vote
1answer
199 views
Non-isotrival fiber bundle over compact Riemann surface
In this paper, Kodaira constructed a fiber bundle $\Phi:M_{m,n}\to S$ from a compact complex surface $M_{m,n}$ to a compact Rieman surface $S$ of genus $>0$. In particular, (on p.212) for any point ...
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votes
0answers
81 views
What can we say about the complement of two homotopic simple closed curves on a compact orientable surface?
Do two homotopic simple closed curves separate a compact orientable surface?
If they are disjoint they do and bound a cylinder. But if they intersect? What can we say about the components of their ...
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vote
0answers
43 views
Difference of two functions with constant mean curvature
Define the set $\Omega := (-\epsilon,\epsilon) \times (-1,1)^{n-1}$, and define
$\Gamma := \{-\epsilon,\epsilon\} \times (-1,1)^{n-1} \subset \partial \Omega$.
Suppose I have two functions $u,v \in C^...
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votes
1answer
232 views
Geometric/Algebraic intersection numbers of curves on surfaces
I have the following problem, and struggling to find some references.
Suppose I start with a homology class of a curve on a closed genus $g$ surface $$h=(a_{1},b_{1},\dots,a_{g},b_{g})\in H_{1}(\...
2
votes
1answer
82 views
Coordinates for Laminations: geometric versus shear
Let $S$ be an orientable surface with a triangulation T.
A lamination $\ell$ is a simple closed curve on $S$, up to isotopy. We will assume that $\ell$ is drawn in such a way that it intersects the ...
3
votes
0answers
92 views
Chord of fixed length traveling around a Jordan curve
Let $C$ be a Jordan curve with nice enough properties whenever necessary (e.g. smooth, or just rectifiable, perhaps). I am interested in knowing how long can a chord be that "traverses" the ...
19
votes
0answers
754 views
I found a (probably new) family of real analytic closed Bezier-like curves; is it publishable?
Given $n$ distinct points $\mathbf{x} = (\mathbf{x}_1, \ldots, \mathbf{x}_n)$ in the plane $\mathbb{R}^2$, I associate a real analytic map:
$f_{\mathbf{x}}: S^1 \to \mathbb{R}^2$
with the following ...
6
votes
1answer
113 views
Stationary phase in spherical integral
I'm trying to find asymptotics for an oscillatory integral on $\mathbb{S}^{n-1}$, for which my advisor said I should use stationary phase arguments. The particular, he claims that:
If $\lambda\gg 1$...
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votes
0answers
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Controlling the intersection of two surfaces in $\mathbb{R}^3$
Let $F_1,F_2$ be two closed orientable surfaces embedded in $\mathbb{R}^3$ with genus $2g_1, 2g_2$, respectively (edit: with $g_1, g_2 \geq 1$). Is it possible to isotope around $F_1$ and $F_2$ so ...
7
votes
3answers
436 views
Examples of complicated parametric Jordan curves
For test purposes I need parametric Jordan curves that are complicated in the sense of having many inflection points and ideally no symmetries.
When doing online search I always land at complex ...
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votes
0answers
77 views
Is there a bijection between the set of simple closed curves and this space of functions?
A simple closed curve $\mathcal{C}$ in the plane is such that, going along the curve from a point $P$ thereon and getting back to it, the total angle has measure $2\pi$. So one can write $2\pi=\int_{\...
28
votes
4answers
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Conceptual proof of classification of surfaces?
Every compact surface is diffeomorphic to $S^2$, $\underbrace{T^2\#\ldots \#T^2}_n$, or $\underbrace{RP^2\#\ldots \#RP^2}_n$ for some $n\ge 1$.
Is there a conceptual proof of this classification ...
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vote
1answer
156 views
Area of a surface confined by a sphere II
[A followup on two related posts:
Area of a surface confined by a sphere
Area of a elliptic surface confined by a sphere
. Thanks to all the inputs so far.]
Let $S$ be a surface enclosed inside the ...
4
votes
1answer
151 views
Area of a elliptic surface confined by a sphere
Let $S$ be a surface enclosed inside the unit sphere in $R^3$. If every point of S is elliptic, then must $\operatorname{Area}(S)≤\operatorname{Area}(S^2)$?
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0answers
154 views
Area of a surface confined by a sphere
Let $S$ be a hypersurface enclosed inside the unit sphere in $R^n$. We may assume that every ray $\{t x: t \geq 0 \}$ intersects $S$ at most once.
Under what extra condition is ${\rm Area}(S) \leq {\...
3
votes
2answers
124 views
Is the radial projection map area increasing?
Let $S$ be a hypersurface enclosed inside the unit sphere in $R^n$. Assume that every ray $\{t x: t \geq 0 \}$ intersects $S$ at most once.
Is it always true that ${\rm Area}(S) \leq {\rm Area}(P(S))$...
4
votes
1answer
175 views
Essential simple closed curves on a punctured torus vs those in the torus
Let $T$ be a compact oriented torus, let $p\in T$ be a point, and let $T^*$ be $T - \{p\}$.
In Farb-Margalit's Primer on mapping class groups, in the discussion after Proposition 1.5 they say that "...
3
votes
0answers
124 views
Minimal extension of local systems
Let $M$ be a complex manifold of dimension $2$, $D \subset M$ be a connected, simple, normal crossings divisor and $L$ be a $\mathbb{C}$-local system defined over $M\backslash D$. Denote by $j: M\...
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votes
0answers
129 views
Given $\theta$, find $f$ such that $\int_{\mathbb{T}} \text{e}^{i\theta} \cos(h \cdot f) = 0,$ for all $h \in \mathbb{N}$
Let $\theta$ be a $C^{\infty}$ (resp. analytic) real-valued function on $\mathbb{T}=[0,2\pi]/\{0,2\pi\}$.
When can one find $f \neq 0$, $C^{\infty}$ (resp. analytic) real-valued function on $\...
4
votes
2answers
179 views
Closed simple curves in $\mathbb{R}\mathbb{P}^2$
EDIT: The well known Jordan curve theorem says: let $C\subset S^2$ be a closed simple curve on the 2-sphere. Then its complement $S^2\backslash C$ consists of two connected components, both ...
0
votes
1answer
218 views
A generalization of Jordan-Schoenflies theorem on simple plane curves
The well known Jordan-Schoenflies theorem says: let $C\subset \mathbb{R}^2$ be a closed simple curve. Then there exists a homeomorphism $f\colon\mathbb{R}^2\to \mathbb{R}^2$ such that $f(C)$ is the ...
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votes
0answers
90 views
Blaschke points
A Blaschke point of a metric space is a point so that every geodesic (i.e. locally shortest path) starting at that point and of length less than the diameter of the metric space is the unique shortest ...
4
votes
1answer
139 views
Minimal graph over convex domain is area-minimizing
I am looking for a reference stating that
If a graph $z=f(x,y)$ over a convex domain $D$ is minimal, then it is area-minimizing.
5.4.18 in Federer's "Geometric measure theory" and Lemma 1.1. in ...
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votes
0answers
166 views
Are Bernstein polynomials bounded by their coefficients?
I am representing a function by nth order Bernstein polynomial coefficients. I have bounded the coefficients between some $f_{min}$ and $f_{max}$. From what I can see experimentally, it appears that ...
8
votes
2answers
891 views
Text on old-fashioned differential geometry
I am seeking good books on the geometry of surfaces in Euclidean space, which would in particular discuss Darboux frames. Please explain for each suggestion why you like this book (classics are ...
3
votes
1answer
308 views
How to prove a developable surface must be ruled surface?
A developable surface is a smooth surface whose Gaussian curvature vanishes everywhere. A ruled surface is a surface where for each point there must be a line passing through the point lying on the ...
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vote
1answer
246 views
When is a graph morphism a regular embedding? [closed]
Let $f:X \to Y$ be a finite morphism, where $X$ is an affine, non-singular curve and $Y$ is an affine, non-singular surface over $\mathbb{C}$. Denote by $\Gamma_f \subset X \times Y$ the graph of the ...
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votes
2answers
184 views
Reference request: Reidemeister type moves for immersed curves on surfaces
Preliminaries
Let $\Gamma$ be a closed $1$-manifold (i.e. a union of finitely many circles) and let $\Sigma$ be a closed $2$-manifold (i.e. a surface). I'll adopt the following terminology.
...
6
votes
0answers
75 views
Rough classification of Peano Curves
By Peano curve I mean a continuous map from the unit interval that fills the unit square in $\mathbb R^2$.
In the paper:
Shchepin, E. V.; Bauman, K. E., Minimal Peano curve, Proc. Steklov Inst. Math....
