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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63 views

The area of the strip [closed]

Let $\gamma$ be a convex smooth plane curve of length $l$. I need to compute the area of the strip swept by the outer normal segments of length $r$ to $\gamma$ and the length of the outer boundary ...
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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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1answer
101 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}(\...
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1answer
56 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 ...
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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 Jordan ...
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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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1answer
79 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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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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3answers
350 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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74 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_{\...
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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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1answer
153 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 ...
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149 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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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 {\...
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2answers
114 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 ${\rm Area}(S) \leq {\rm Area}(P(S))$...
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Periodicity implied by the condition on level sets $\sum_{x \in \phi^{-1}(b)} \frac{\text{e}^{i \theta(x)}}{|\phi'(x)|}=0 $

Let $\theta$ and $\phi$ two real $C^h$ functions on $[0,2\pi]$, $h\geq1$, satisfying for all not critical values $b$ of $\phi$ $$ \sum_{x \in \phi^{-1}(b)} \frac{\text{e}^{i \theta(x)}}{|\phi'(x)|}...
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1answer
112 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 "...
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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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2answers
156 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 ...
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1answer
155 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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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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1answer
119 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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131 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 ...
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2answers
866 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 ...
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1answer
210 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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1answer
163 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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2answers
157 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. ...
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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....
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87 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}/...
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88 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 $...
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1answer
229 views

Properties of the affine curve-shortening flow

The affine curve-shortening flow (ACSF) seems to satisfy the following properties, even if the initial curve(s) is (are) self-intersecting (at least as long as they are "nice enough"): (1) The length ...
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1answer
305 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,...
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1answer
48 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) $L_1,L_2\subset S$, ...
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2answers
383 views

Does $\int_0^{2\pi} e^{i\theta(t)} (\phi(t))^n dt=0$ $\forall \; n\in\mathbb{N}_0$ imply $\phi$ periodic?

PROBLEM. Let $\theta(t)$ and $\phi(t)$ be two real analytic non-constant functions $[0,2\pi]\rightarrow \mathbb{R}$. I am trying to prove the following claim If the integral $$ \int_0^{2\pi} e^{...
6
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1answer
152 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 $i: S^2 \to \mathbb{R}^3$ ...
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0answers
175 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 ...
2
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1answer
98 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 \...
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1answer
172 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 ...
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69 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 ...
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1answer
118 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}^{\...
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69 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 \...
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0answers
132 views

Interesting geometric flow of space curves with non-vanishing torsion

Recently, while thinking about CMC surfaces, I came up with an interesting geometric flow for curves in $\mathbb{R}^3$ given by \begin{equation} \partial_t \gamma = \tau^{-\frac{1}{2}} n, \end{...
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0answers
132 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 ...
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0answers
219 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 ...
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2answers
248 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
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1answer
708 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 ...
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1answer
81 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 : \...
6
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1answer
184 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 ...
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2answers
874 views

A problem on Gauss--Bonnet formula

While teaching a course in differential geometry, I came up with the following problem, which I think is cool. Assume $\gamma$ is a closed geodesic on a sphere $\Sigma$ with positive Gauss curvature. ...

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