# 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

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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}^...

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### 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 ...

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### 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 ...

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### 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 ...

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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 ...

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### 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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### 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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### “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,\ \...

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### 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 ...

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### 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 $\gamma$.
...

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### 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 ...

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### 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 $...

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### 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 ...

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### 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) ...

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### 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(\...

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### 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 ...

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### 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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### 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 ...

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### 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 ...

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### 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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### 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 ...

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### 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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### 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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### 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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### 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}(\...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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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### 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 ...

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### 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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### 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_{\...

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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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### 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 ...

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### 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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### 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 {\...

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### 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 ${\rm Area}(S) \leq {\rm Area}(P(S))$...

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### 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 "...

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### 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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### 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 $\...

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### 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 ...

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### 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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### 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 ...

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### 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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### 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 ...

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### 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 ...

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### 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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### 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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### 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.
...

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### 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....