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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1 vote
1 answer
219 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 ...
2 votes
1 answer
138 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 ...
1 vote
0 answers
54 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^...
6 votes
1 answer
272 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 $\lambda\gg 1$...
7 votes
3 answers
649 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 ...
1 vote
1 answer
242 views

Splines with bounded first derivative?

I have a set of points $(x_i,y_i)\in{\mathbb R}_+\times{\mathbb R}$, $i=1,...,n$, ($x_i$ are the independent variables and $y_i$ are the dependent variables or responses) that I want to fit using ...
5 votes
0 answers
122 views

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 ...
0 votes
0 answers
85 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_{\...
1 vote
1 answer
165 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 ...
5 votes
1 answer
174 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 $\operatorname{Area}(S)≤\operatorname{Area}(S^2)$?
1 vote
0 answers
163 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
2 answers
238 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
1 answer
532 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
2 answers
694 views

Does this surface contain all perfect cuboids?

Probably this is wildly known, but the closest I found was a surface with additional restrictive conditions. A perfect cuboid is a cuboid having integer side lengths, integer face diagonals and an ...
3 votes
0 answers
134 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\...
4 votes
0 answers
140 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 $\...
0 votes
1 answer
371 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 ...
4 votes
2 answers
294 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 ...
8 votes
0 answers
116 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
1 answer
260 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 ...
7 votes
0 answers
172 views

Which surfaces embedded in $\mathbb{R}^3$ have only axially-symmetric sections?

Dmitry Ryabogin and I considered the following question some time ago, but got nowehere: Let $M$ be a (smooth or algebraic) surface in $\mathbb{R}^3$. Suppose that for every section $S$ (an ...
0 votes
0 answers
250 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 ...
9 votes
2 answers
963 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
1 answer
915 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 ...
1 vote
1 answer
476 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 ...
8 votes
2 answers
289 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. ...
1 vote
0 answers
100 views

Reparametrization of a closed curve that balances sum of first derivatives

(Question in the yellow box below.) A few weeks ago I was wondering about the existence of a scalar function $f(s): S^1 \rightarrow \mathbb{R}$ and a turning angle $\phi(s):S^1 \rightarrow \mathbb{R}/...
7 votes
0 answers
96 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....
2 votes
0 answers
130 views

Hypersurfaces whose unit normal $N$ satisfies $[N,X] =0$ for every tangent vector field $X$

Let $M$ be a hypersurface of a Riemannian manifold, and assume that $M$ satisfies the following property: For each $p \in M$, given a unit normal vector field $N$ defined in a neighborhood $U$ of $...
8 votes
1 answer
385 views

To find a point in Teichmüller space or measured foliation, how many lengths of curves do you need?

To parametrize Teichmüller space, it suffices to measure the hyperbolic lengths of a finite number of curves. It is well-known that $9g-9$ curves suffice, by a standard pair-of-pants argument given in,...
2 votes
1 answer
139 views

The radius of an interval's image through a space-filling curve

Take $f:[0,1]\to [0,1]^n$ a continuous tour around $[0,1]^n,$ say, some iteration of a Hilbert curve. For $\varepsilon \in (0,1)$ what is the following thing called and are there any nontrivial upper ...
1 vote
1 answer
56 views

Two multi-curves in a surface with the same transverse measure

Let $(\cal F,\mu)$ be the stable measured foliation of a pseudo-Anosov on an oriented surface $S$. Can there be two non-isotopic multi-loops (collections of disjoint simple loops) $L_1,L_2\subset S$, ...
2 votes
0 answers
193 views

Orthogonality relation in $L^2$ implying periodicity

Let $\theta(t)$ and $\phi(t)$ be two real $C^1$ functions $[0,2\pi]\rightarrow \mathbb{R}$. Let us assume $\theta$ has the properties $$ \int_0^{2\pi} e^{i\theta(t)} dt=0. $$ Geometrically this means ...
7 votes
1 answer
348 views

Realizing Morse functions on $S^2$ as height functions

Let $h: \mathbb{R}^3 \to \mathbb{R}$ be the usual height function (i.e. $h(x,y,z) = z$). One way that Morse functions on $S^2$ are often described is by picking an embedding $i: S^2 \to \mathbb{R}^3$ ...
2 votes
1 answer
128 views

Closest points of curves on convex surfaces

Let $\Sigma$ be a convex surface with strictly positive curvature in $\mathbb{R}^3$ and $\Gamma \subset \Sigma$ be a closed simple space curve with parameterization $\gamma : \mathbb{S}^1 \rightarrow \...
1 vote
1 answer
565 views

When are principal lines of curvature geodesics?

Let $S$ be a smooth surface embedded in $\mathbb{R}^3$. When are (some of) the principal lines of curvature geodesics on $S$? Perhaps on the ellipsoid below, the (blue) central cycle, a max principal ...
1 vote
0 answers
148 views

Is the Frenet frame is independent of the choices of parameters?

I asked this question on StackExchange, but until now there is not any answer or hint. I hope I can get some help here. When I am reading ''A course in differential geometry'' of Klingenberg, I ...
2 votes
1 answer
169 views

Injectivity of the simple closed curves under geometric intersection number

Let $\Sigma$ be a closed surface of genus $g\geq 2$ and $\mathcal{C}$ be the set of all free homotopy classes of simple closed curves in $\Sigma$. Define $i:\mathcal{C}\rightarrow \mathbb{R}^{\...
5 votes
1 answer
395 views

Average distance to a curve of fixed length

Let $C$ be a continuous curve in the unit square having length $L$. Is there a lower bound on the average distance between the points in the unit square and $C$, as a function of $L$? Is there an ...
1 vote
0 answers
73 views

Classification of surface curves? [duplicate]

Having a genus $g$ closed orientable surface $\Sigma$ (connected sum of $g$ tori), how do I encode any embedded closed curve up to planar isotopy? For $g=1$ we can just state the coefficient $k \in \...
3 votes
0 answers
141 views

Reducing curves in surfaces by Dehn twists

Let $F$ be a compact, oriented surface. A Dehn move, $D_\beta$, on a simple closed curve (scc) $\alpha$ is a Dehn twist applied to $\alpha$ along a scc $\beta$ which intersects $\alpha$ once. Is ...
5 votes
1 answer
460 views

Explaining patterns in modular multiplication graphs

Let the multiplication graph $n/m$ be the graph with $m$ points distributed evenly on a circle and a line between two points $a$, $b$ when $an \equiv b\operatorname{mod} m$. These graphs often look ...
13 votes
0 answers
252 views

Planar arc on a topologically embedded sphere or disk in $\mathbb{R}^3$

An arc is a set homeomorphic to the unit interval $[0,1]$; an arc in $\mathbb{R}^3$ is planar if it is contained in some plane. The following questions are motivated by Anton Petrunin's Disc bounded ...
17 votes
3 answers
1k views

On closed simple curve with curvature at most 1

I am looking for the reference to the following theorem. I have to apply a similar statement, and it would be nice to trace the source. Please note, I know few proofs in fact it is Problem 3 in my ...
4 votes
1 answer
168 views

Length functions on Teichmuller space with constant difference

Let $S$ be a closed oriented surface of genus $g\geq 2$. Let $\mathcal{T}$ be the corresponding Teichmuller space. Given a free homotopy class of closed curve $[\gamma]$ we can define the length ...
12 votes
2 answers
456 views

Minimal area of Seifert surfaces

Let $K$ be a knot smooth knot in a 3-manifold $M$ and fix a metric on $M$. Let $F$ be a orientable surface of genus $g$ with one boundary component. Then we can consider the family of all maps $\...
14 votes
1 answer
884 views

Hadamard theorem about embedding

The following theorem is commonly attributed to Jacques Hadamard. Assume $\Sigma$ is a smooth locally convex immersed surface in the Euclidean space. Then $\Sigma$ is embedded and bounds a convex ...
10 votes
2 answers
1k 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
5 answers
4k views

Formulas for equidistant curves

I'm trying to draw on the computer a curve that keeps always the same distance(given as parameter) from a given curve. I know the formula for the given curve. I tried moving perpendicular to the first ...
5 votes
1 answer
119 views

Bernstein type theorems for CMC hypersurfaces in $\mathbb{R}^{n+1}$

Is there any Bernstein type theorems for CMC hypersurfaces in $\mathbb{R}^{n+1}$ in the literature? More precisely I would like to know if there is an answer to the following QUESTION: Let $f : \...

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